Úvodní informace Matematické modelování · Literaturaii...

Uacutevodniacute informaceMatematickeacute modelovaacuteniacuteMatematickeacute metody pro ITS (11MAMY)

Jan Přikryl Bohumil Kovaacuteř

1 přednaacuteška 11MAMYčtvrtek 26 března 2020verze 2020-03-26 0147

Uacutestav aplikovaneacute matematikyČVUT v Praze Fakulta dopravniacute

1

Obsah přednaacutešky

O předmětu

Zaacutekladniacute organizačniacute informace

Seznam literatury

Hodnoceniacute předmětu

Vstupniacute znalosti

Vyacutestupniacute znalosti

2

Zaacutekladniacute informace

Přednaacutešejiacuteciacute

bull Dr Ing Jan Přikryl (prikrylfdcvutcz)přednaacutešky nepravidelně blokově uacutetndashčt 945ndash1445 on-line

Cvičiacuteciacute

bull Dr Ing Jan Přikryl (prikrylfdcvutcz)cvičeniacute nepravidelně blokově uacutetndashčt 945ndash1445 on-line

Garant předmětu

bull Dr Ing Jan Přikryl (prikrylfdcvutcz)

3

Zaacutekladniacute informace ndash Pokračovaacuteniacute

Domovskaacute straacutenka předmětu 11MAMYhttpzolotarevfdcvutczmamy

Cvičeniacute Pravidla jsou na straacutenkaacutech předmětu

Cvičeniacute pro druhyacute zaacutepis Zatiacutem naštěstiacute nemaacuteme

4

Literatura i

1 VELTEN Kai Mathematical modeling and simulation introduction for scientistsand engineers John Wiley amp Sons 2009

2 OGATA Katsuhiko Modern control engineering Prentice Hall PTR 2001

3 OPPENHEIM Alan V Alan S WILLSKY a Syed Hamid NAWAB Signals andSystems 2 vyd Upper Saddle River Prentice Hall 1997 957 s ISBN01-381-4757-4

4 JAMES Gareth et al An introduction to statistical learning New York Springer2013

5 DANGELMAYR Gerhard a KIRBY Michael Mathematical Modeling ndash AComprehensive Introduction Upper Saddle River Prentice Hall 2005

5

Literatura ii

6 HEATH Michael T Scientific computing ndash An Introductory Survey 2 vyd NewYork McGraw-Hill 2002

7 BERTSEKAS Dimitri P Dynamic programming and optimal control BelmontMA Athena Scientific 1995

8 KARBAN Pavel Vyacutepočty a simulace v programech Matlab a Simulink BrnoComputer Press 2006

9 Informace o prostřediacute MATLABhttpzolotarevfdcvutczmnihttpzolotarevfdcvutczmsp

httpwwwfdcvutczpersonalnagyivanPrpStatPrpMatIntropdf

10 Matematika-opakovaacuteniacutehttpeulerfdcvutczpredmetyml1

6

Zaacutepočet a zkouška

Celkovyacute počet bodů ktereacute lze ziacuteskat je 40 Počet bodů ktereacute studenti mohou ziacuteskat zesemestru je 20 Zkouška sestaacutevaacute z piacutesemneacuteho testu (20 bodů)

Zaacutepočet udělujeme od 10 bodů ze semestru vyacuteše za předpokladu že studentu uspěje vuacutevodniacutem testu

Body jsou rozděleny naacutesledovně

bull 10 bodů za implementaci jednoducheacuteho modelu z oblasti ITS

bull 10 bodů za aktivitu v semestru

Semestraacutelniacute projekt je skupinovyacute ndash skupiny po maximaacutelně třech studentech

7

Znalosti vstupniacute

Toto jsou znalosti u nichž předpoklaacutedaacuteme že je ovlaacutedaacutete Jejich neznalost seneomlouvaacute

1 Znalost zaacutekladniacutech pojmů a operaciacute s vektory a maticemi

2 Znalost praacutece s komplexniacutemi čiacutesly a zaacutekladů funkciacute komplexniacute proměnneacute

3 Znalost vlastnostiacute trigonometrickyacutech hyperbolickyacutech exponenciaacutelniacutech funkciacute

4 Znalost vyacutepočtu součtů nekonečneacute řady derivace a integraacutelů funkce jedneacuteproměnneacute

5 Znalost praacutece se zlomky algebraickyacutemi vyacuterazy a běžneacute středoškolskeacute matematiky

6 Zaacutekladniacute znalosti algoritmizace a prostřediacute SCILABMATLAB (v rozsahu 14ALG a11STAT)

8

Znalosti vyacutestupniacute

1 Znalost zaacutekladniacutech principů matematickeacuteho modelovaacuteniacute a matematickeacute teorie řiacutezeniacute

2 Zaacutekladniacute znalosti o typech modelů a jejich užitiacute

3 Zaacutekladniacute znalosti o měřeniacute a předzpracovaacuteniacute dat

4 Povědomiacute o lineaacuterniacute optimalizaci multikriteriaacutelniacute oprimalizaci a dynamickeacutemprogramovaacuteniacute

5 Znalost modelovaacuteniacute časovyacutech řad

6 Znalost prostřediacute MATLABSIMULINK pro modelovaacuteniacute dynamickyacutech systeacutemů ařešeniacute soustav nelineaacuterniacutech diferenciaacutelniacutech a diferenčniacutech rovnic

9

Matematickeacute modelovaacuteniacute systeacutemů

Modelovaacuteniacute je uměniacute kompromisu

Jakeacute ciacutele může modelovaacuteniacute dosaacutehnout

Klasifikace modelů

Faacuteze modelovaacuteniacute

Model systeacutemu

Vnějšiacute popis systeacutemů

Vnitřniacute popis systeacutemů

Iterace diferenčniacute rovnice

Uacutevod do teorie signaacutelů

Zaacutekladniacute spojiteacute signaacutely

Zaacutekladniacute diskreacutetniacute signaacutely

Odezva systeacutemu

10

Modely popisujiacute naše přesvědčeniacute o tom jak svět funguje

Provaacutezejiacute naacutes od nepaměti

bull obyčejnaacute mapa je dvourozměrnyacute model pohledu na krajinu

bull modely plaacutenovanyacutech budov ze saacutedry a dřeva

Většinou jde o zjednodušeniacute reality postihujiacute jen to co naacutes pro studium daneacutehoprobleacutemu opravdu zajiacutemaacute Pro naacutes nepodstatneacute detaily model zanedbaacutevaacute

11

V matematickeacutem modelovaacuteniacute naše přesvědčeniacute o fungovaacuteniacute světa překlaacutedaacuteme do jazykamatematiky

Maacute to řadu vyacutehod

1 Matematika je velmi přesnyacute jazyk

2 Matematika je vyacutestižnyacute jazyk s dobře definovanyacutemi pravidly pro manipulaci svyacuterazy

3 Všechny dřiacutevějšiacute vyacutesledky jsou naacutem k dispozici a můžeme je pro naacuteš model využiacutet

4 K provedeniacute numerickyacutech vyacutepočtů můžeme dnes použiacutet počiacutetače

12

Značnou čaacutest matematickeacuteho modelovaacuteniacute tvořiacute kompromisy

Většina reaacutelnyacutech systeacutemů je přiacuteliš složitaacute

Kompromis 1 Snaha identifikovat nejdůležitějšiacute čaacutestiacute systeacutemu ndash ty budou do modeluzahrnuty zbytek bude zanedbaacuten

Matematicky lze v naprosteacute obecnosti dokaacutezat mnoho ale použitelnost vyacutesledků zaacutevisiacutekriticky na formě použityacutech rovnic K jejich vyčiacutesleniacute použiacutevaacuteme počiacutetače a ty nejsounikdy zcela přesneacute

Kompromis 2 Použijeme-li k manipulaci s rovnicemi vytvaacuteřeneacuteho modelu počiacutetačnemusiacute to sice veacutest k elegantniacutem vyacutesledkům ale je to mnohem odolnějšiacute vůči změnaacutem

13

Matematickeacute modelovaacuteniacute

1 Rozvoj vědeckeacuteho poznaacuteniacute ndash prostřednictviacutem kvantitativniacuteho vyjaacutedřeniacute současnyacutechznalostiacute o systeacutemu (stejně jako znaacutezornit to co viacuteme můžeme takeacute ukaacutezat coneviacuteme)

2 Testovaacuteniacute vlivu změn v systeacutemu

3 Ziacuteskat informace pro podporu rozhodovaacuteniacute včetně31 taktickyacutech rozhodnutiacute manažerů32 strategickyacutech rozhodnutiacute plaacutenovačů

14

Modely

Deterministickeacute times stochastickeacute

Mechanistickeacute times empirickeacute

molekuly buňky orgaacuteny jedinec staacutedo

Niacutezkaacute Vysokaacute

15

Klasifikace modelů

Empirickyacute MechanistickyacuteDeterministickyacute Předpověď růstu dobytka z re-

gresniacute zaacutevislosti na konzumaci po-travy

Pohyb planet založenyacute na Newto-novskeacute mechanice popsaneacute dife-renciaacutelniacutemi rovnicemi

Stochastickyacute Analyacuteza rozptylu vyacutenosů odrůdpřes lokality a roky

Genetika malyacutech populaciacute za-loženaacute na Mendelovskeacute dědič-nosti popsaneacute pravděpodobnost-niacutemi rovnicemi

16

Čtyři hlavniacute faacuteze

Modelovaciacute projekty nepostupujiacute plynule od faacuteze tvorby modelu až po jeho použitiacute

Pokud dojde ke jakyacutemkoliv změnaacutem modelu pak se faacuteze studia a testovaacuteniacute musiacuteopakovat

Sestaveniacute Studium Testovaacuteniacute Použitiacute

17

Systeacutem

Definice (Systeacutem)Charakteristickeacute vlastnosti se kteryacutemi vystačiacuteme při modelovaacuteniacute

bull systeacutem považujeme za čaacutest prostřediacute kterou lze od jejiacuteho okoliacute oddělit fyzickounebo myšlenkovou hraniciacute

bull systeacutem se sklaacutedaacute z podsysteacutemů vzaacutejemně propojenyacutech součaacutestiacute

Je to čaacutest našeho světa kteraacute se svyacutem okoliacutem nějak interaguje napřiacutekladprostřednictviacutem vstupu a vyacutestupu

18

Co je modelovaacuteniacute

ModelZa model můžeme poklaacutedat naacutehradu nebo zjednodušeniacute skutečneacuteho objektureaacutelneacuteho světa z hlediska jeho vlastnostiacute a funkčnosti

Modelovaacuteniacute je možneacute pouze pokud zavedeme určityacute stupeň abstrakce a aproximace

19

Diskreacutetniacute a spojityacute model

Svstup vyacutestup

u(t) spojityacute systeacutem y(t)

u[n] diskreacutetniacute systeacutem y [n]

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Při analyacuteze navrženeacuteho modelu chceme učinit co možnaacute nejsilnějšiacute rozhodnutiacute nazaacutekladě maleacuteho množstviacute dat Spraacutevnost našeho naacutevrhu je nutneacute statisticky vyhodnotit

Probleacutemy

1 Vyacuteznamneacute diference ve sledovanyacutech parametrech mohou byacutet způsobeny špatnyacutemnaacutevrhem modelu přiacutepadně měřeniacutem dat

2 Je těžkeacute rozlišit zda diference v datech jsou skutečneacute nebo způsobeneacute bdquonaacutehodnyacutemvlivemldquo

22

Svět dopravy se neobejde bez měřeniacute

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

06Akustickyacute signaacutelminus naacutekladniacute auto

a teď jeho zvuk zvuk auta

23

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Other

5276747

Proč modelovaacuteniacute systeacutemů

Otaacutezky

bull Jak ověřiacuteme spraacutevnost vyacutepočtu rychlosti šiacuteřeniacute ptačiacute chřipky

bull Jak ověřiacuteme pevnost noveacuteho mostu

bull Jak ověřiacuteme bezpečnost softwaru zabezpečovaciacuteho zařiacutezeniacute

bull Jak předpoviacuteme dopravniacute zaacutecpu na daacutelnici

bull Jak zajistiacuteme spolehlivou funkci navigace přiacute vyacutepadku signaacutelu GPS

Pokud nemůžeme předem prokaacutezat určiteacute vlastnosti na samotneacuteho systeacutemu prokaacutežemehledaneacute vlastnosti na jeho modelu

24

Modely reaacutelneacuteho světa

25

26

27

Dynamickeacute systeacutemy

Dynamickyacute systeacutem maacute v každeacutem okamžiku stav danyacute množinou reaacutelnyacutech čiacutesel Tentostav lze reprezentovat jako bod ve stavoveacutem prostoru (tedy jako vektor často x)

Evolučniacute pravidlo (rovnice vyacutevoje stavu) popisuje přechody mezi jednotlivyacutemi stavydynamickeacuteho systeacutemu

bull většinou deterministickeacute

bull může byacutet stochastickeacute

Pozor V matematice a fyzice tak nazyacutevaacuteme systeacutemy citliveacute na počaacutetečniacute podmiacutenky(dvojiteacute kyvadlo Lorenzův atraktor)

28

θ +g

`sin θ = 0

29

partρ

partt+nabla middot (ρu) = 0

DuDt

= Fminus nablapρ

By MannyMax (original) - ImageBernoullisLawDerivationDiagrampng CC BY-SA 30httpscommonswikimediaorgwindexphpcurid=2870495

30

Exponenciaacutelniacute růst

N = rN

Logistickyacute růst

N = rN

(1minus N

K

)

31

By George Ioannidis - Own work CC BY 30httpscommonswikimediaorgwindexphpcurid=7920826

32

33

Vnějšiacute popis dynamickyacutech systeacutemů

Vnějšiacute popis vychaacuteziacute z popisu systeacutemu vektorem vstupu u a vektorem vyacutestupu y

u(t) S y(t)

u(t)

t

y(t)

t

Stavovyacute vektor systeacutemu x nepoužiacutevaacuteme Systeacutem chaacutepeme jako černou skřiacuteňku o jejiacutechžvlastnostech se dozviacuteme pouze tehdy jestliže budeme zkoumat jejiacute reakci na vnějšiacuteudaacutelosti (signaacutely data)

Vnějšiacute model popisujeme diferenciaacutelniacute rovniciacute pro systeacutemy se spojityacutem časem a diferenčniacuterovniciacute pro systeacutemy s diskreacutetniacutem časem Uvedenaacute rovnice je obecně vyššiacuteho řaacutedu než 1

34

Vnitřniacute popis synamickyacutech systeacutemů

Vnitřniacute tzv stavovyacute popis systeacutemu použiacutevaacute k popisu dynamiky systeacutemu vektorvnitřniacutech stavů x

Vektor vstupů u a vektor vyacutestupniacutech veličin y jsou druhotneacute veličiny vnitřniacuteho popisu

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

Stavoveacute modely popisujeme

bull soustavou diferenciaacutelniacutech rovnic prvniacuteho řaacutedu pro systeacutemy se spojityacutem časem abull soustavou diferenčniacutech rovnic prveacuteho řaacutedu pro systeacutemy s diskreacutetniacutem časem

35

Role matematiky

Modelovaacuteniacute neniacute samospasitelneacute

bull model je pouze přibliacuteženiacutem reality

bull vyacutestupy modelu je proto vždy třeba ověřovat

bull možneacute chyby jsou jak v modelu tak i v jeho vyacutepočtu

Rozlišujeme dva kroky ověřeniacute

Verifikace Počiacutetaacuteme spraacutevnyacute model

Validace Model počiacutetaacute spraacutevně

36

Přiacuteklady systeacutemů

R

Cu1(t) uC(t)

Napětiacute u1(t) na RC člaacutenku je součet napětiacute na rezistoru uR(t) a na kapacitoru uC(t)

u1(t) = uR(t) + uC(t)

37

Proud prochaacutezejiacuteciacute obvodem i(t) a časovyacute průběh napětiacute na rezistoru uR(t) je možnovyjaacutedřit jako

i(t) = Cddt

uC(t)

a proto

uR(t) = R middot i(t) = RCddt

uC(t)

38

Dosazeniacutem uR(t) ziacuteskaacuteme diferenciaacutelniacute rovnici prvniacuteho řaacutedu pro časovyacute průběh napětiacute nakapacitoru uC(t)

RCddt

uC(t) + uC(t) = u1(t)

Řešeniacute uvedeneacute rovnice maacute pro všechna t ge 0

α =1RC

u1(t) = U0

a pro počaacutetečniacute hodnotu uC(0) = 0 tvar

uC(t) = U0(1minus eminusαt)

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

Rovnice nabiacutedkyNabiacutedka dnes zaacutevisiacute na včerejšiacute ceně a to tak že nabiacutedka stoupaacute s rostouciacute cenou ProC gt 0 platiacute

n[k] = Cc[k minus 1] +Au[k]

Rovnice poptaacutevkyPoptaacutevka dnes zaacutevisiacute na dnešniacute ceně a to tak že poptaacutevka klesaacute s rostouciacute cenou ProD gt 0 platiacute

p[k] = minusDc[k] + Bu[k]

41

Rovnost nabiacutedky a poptaacutevkyn[k] = p[k]

pak vede na diferenčniacute rovnici prvniacuteho řaacutedu

c[k] +CD c[k minus 1] =

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

Rozdiacutel mezi lineaacuterniacutem a linearizovanyacutem

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace rovnice ceny

Odezva systeacutemu 45

Iterace

Iteraciacute rozumiacuteme obecně opakovaacuteniacute nějakeacuteho postupu v matematice napřiacutekladopakovaneacute vyhodnoceniacute funkce Vyacutesledky z jedneacute iterace se použijiacute jako vstup pro dalšiacuteiteračniacute krok

Iteračně napřiacuteklad

bull hledaacuteme nuloveacute body funkce f (xlowast) = 0

bull simulujeme vyacutevoj diskretizovaneacuteho dynamickeacuteho systeacutemu

46

Iterace

Diferenčniacute rovnici se vstupem u[n] a vyacutestupem y [n] můžeme přepsat do tvaru

y [n + 1] = f (y [n] y [n minus 1] u[n] u[n minus 1] )

stanovit tzv počaacutetečniacute podmiacutenky daneacute řaacutedem systeacutemu a potom napřiacuteklad pron = 2 3 s počaacutetečniacutemi podmiacutenkami y [0] y [1] y [2] iterativně počiacutetat vyacutestupydiskreacutetniacuteho systeacutemu y [3] y [4]

Podobnyacute postup se použiacutevaacute pro numerickou simulaci spojityacutech systeacutemů V tom přiacutepaděse systeacutemoveacute funkce vyhodnocujiacute v předem stanovenyacutech časovyacutech okamžiciacutech vyacutesledekneniacute tedy zcela přesnyacute

47

Diferenčniacute rovnicic[k] +

CD c[k minus 1] =

B minusAD x [k]

odvozenou na předešlyacutech slajdech přepiacutešeme do kanonickeacuteho tvaru

y [k] + γy [k minus 1] = βu[k]

a postupnyacutemi iteracemi nalezneme pro u[k] = 1[k] a počaacutetečniacute podmiacutenku y [minus1] = 0

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

y [0] = β minus γy [minus1] = β

Pro k = 1

y [1] + γy [0] = βu[1]

y [1] = β minus γy [0] = β minus βγ

49

Pro k = 2

y [2] + γy [1] = βu[2]

y [2] = β minus γy [1] = β minus βγ + βγ2

Pro obecneacute n

y [n] + γy [n minus 1] = βu[n]

y [n] = β minus γy [n minus 1] = β(1minus γ + γ2 + middot middot middot+ (minusγ)n

)

50

y [n] = β

nsum

m=0

(minusγ)m = β1minus (minusγ)n+1

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Jednotkovyacute skok

Exponenciaacutela

Periodickeacute a harmonickeacute funkce

Odezva systeacutemu

53

Diracův impuls

Tato funkce je definovaacutena na časoveacutem intervalu pro všechna t a jejiacute nenulovou hodnotupředpoklaacutedaacuteme pouze v okoliacute bodu t = 0 Plocha těchto funkciacute je rovna 1 pro každeacuteε gt 0

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

Funkci δ(t) definujeme jako δ(t) = limεrarr0 δε(t)

54

Diracův impuls

Funkce δ(t) se nazyacutevaacute Diracův impuls Diracova δ-funkce nebo jednotkovyacute impulsHodnota δ(t) pro t 6= 0 je δ(t) = 0 Jejiacute hodnota v t = 0 neniacute definovaacutena jako funkcepoužiacutevaacute se integraacutelniacute definice

int infin

minusinfinδ(t) dt =

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

pro každeacute ε gt 0

55

Funkce jednotkoveacuteho skoku byacutevaacute obvykle značena 1(t) a je definovaacutena jako

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

Uvažujme exponenciaacutelniacute funkcif (t) = eαt

kde α je reaacutelnaacute konstanta podle naacutesledujiacuteciacuteho obraacutezku

eαt

t

eαt

t

58

Exponenciaacutela

Exponenciaacutelniacute funkcef (t) = A eαt

kde α isin C je zajiacutemavaacute hlavně v přiacutepadě kdy α = iω

f (t) = A eiωt = A (cosωt + i sinωt)

59

Periodickaacute funkce

O spojiteacutem signaacutelu f (t) řiacutekaacuteme že je periodickyacute s periodou T jestliže

forallt f (t + T ) = f (t)

a tedy takeacute pro libovolneacute k isin Z

f (t) = f (t + T ) = f (t + 2T ) = middot middot middot = f (t + k middot T )

Nejmenšiacute možneacute T nazyacutevaacuteme fundamentaacutelniacute perioda značiacuteme T0

60

Sinusovaacute funkce

f (t) = A sin (ωt + Φ)

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

Konstanty A ω a Φ se nazyacutevajiacute amplituda uacutehlovaacute frekvence a faacutezovyacute posunSinusovka je periodickaacute se zaacutekladniacute periodou T = 2πω

61

Diskreacutetniacute jednotkovyacute impuls a skok

Diskreacutetniacute sinusovaacute posloupnost

Odezva systeacutemu

62

Vznik diskreacutetniacutech signaacutelů

Jak diskreacutetniacute signaacutely vznikajiacute

bull přirozeně (průměrneacute denniacute teploty denniacute kurzy počty studentů)

bull vzorkovaacuteniacutem spojityacutech signaacutelů (naměřeniacute teploty každou hodinu měřeniacutemprůtoku každyacutech 15 minut)

Diskreacutetniacute signaacutely jimiž se budeme v předmětu zabyacutevat jsou diskreacutetniacute v čase ale spojiteacuteve funkčniacute hodnotě

Digitaacutelniacute signaacutel je totiž často kvantovanyacute nabyacutevaacute tedy v každeacutem n pouze diskreacutetniacutemnožiny funkčniacutech hodnot napřiacuteklad 0 1 2 65535

63

Diskreacutetniacute jednotkovyacute impuls

Diskreacutetniacute jednotkovyacute impuls δ[n] je definovaacuten vztahem

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

Diskreacutetniacute jednotkovyacute skok

Diskreacutetniacute jednotkovyacute skok 1[n] je definovaacuten vztahem

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

Mějme sinusovyacute signaacutel f (t) = A sin(ωt + Φ) s periodou T = 2πωPokud tento signaacutel vzorkujeme s periodou Ts gt 0 ziacuteskaacuteme diskreacutetniacute sinusovyacute signaacutel

f [n] = f (nT ) = A sin(ωnTs + Φ) = A sin(ξn + Φ)

kde n = 0 plusmn1 plusmn2 a ξ = ωTs

A sin(ξn + Φ)

t

66

Periodickyacute signaacutel

Diskreacutetniacute signaacutel f [n] je periodickyacute jestliže existuje kladneacute celeacute čiacuteslo N takoveacute že platiacute

f [n] = f [n + N] = f [n + 2N] = middot middot middot = f [n + k middot N]

pro všechna n isin Z (z intervalu (minusinfin infin)) a pro libovolneacute k isin Z N se nazyacutevaacute periodadiskreacutetniacuteho signaacutelu

Nejmenšiacute možneacute N nazyacutevaacuteme fundamentaacutelniacute perioda a značiacuteme N0

67

Diskreacutetniacute sinusovyacute signaacutel nemusiacute byacutet nutně periodickyacute zaacuteležiacute na volbě vzorkovaciacuteperiody Ts Pro periodickyacute diskreacutetniacute sinusovyacute signaacutel s periodou N musiacute platit

N = m middot 2πTs

kde m isin N Maacuteme i N isin N proto 2πTs musiacute byacutet racionaacutelniacute čiacuteslo

Přiacuteklad (Neperiodickyacute sinusovyacute signaacutel)Signaacutel

y [n] = sin n

neniacute pro Ts = 01 s periodickyacute protože 2πTs neniacute racionaacutelniacute čiacuteslo

68

Odezva systeacutemu

Diskreacutetniacute systeacutem

Lineaacuterniacute a nelineaacuterniacute

Časově invariantniacute resp stacionaacuterniacute systeacutem

Kauzaacutelniacute přiacutečinnyacute systeacutem

Spojityacute systeacutem

Autonomniacute a neautonomniacute systeacutem

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

Definice (Impulsniacute odezva)

Odezvu systeacutemu na jednotkovyacute impuls δ[n] budeme nazyacutevat impulsniacute odezva aznačit h[n]

h[n] = Sδ[n]h[nm] = Sδ[n minusm] (1)

71

Definice (Přechodovaacute odezva)

Odezvu systeacutemu na jednotkovyacute skok 1[n] budeme nazyacutevat přechodovaacute odezva aznačit s[n]

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

Lineaacuterniacute systeacutem

Definice (Linearita)

V matematice označujeme funkci f (x) jako lineaacuterniacute v přiacutepadě že je

1 aditivniacute f (x1 + x2) = f (x1) + f (x2) a

2 homogenniacute f (αx) = αf (x)

Obdobně to platiacute i pro lineaacuterniacute systeacutemy

Definice (Lineaacuterniacute systeacutem)

Systeacutem je lineaacuterniacute pokud pro dva různeacute vstupniacute signaacutely u1[n] a u2[n] platiacute

Su1[n] + u2[n] = Su1[n]+ Su2[n] Sαu[n] = αSu[n]

73

Princip superpozice

Definice (Princip superpozice)

Pro dva různeacute vstupniacute signaacutely u1[n] a u2[n] platiacute

y1[n] = Su1[n]y2[n] = Su2[n]

a pro u[n] = αu1[n] + βu2[n] takeacute

αy1[n] + βy2[n] = y [n] = Su[n] = Sαu1[n] + βu2[n]

Obecně platiacute

u[n] =sum

i

aiui [n] rarr y [n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

Přiacuteklad (Lineaacuterniacute systeacutem)Uvažujme systeacutem

y [n] + a y [n minus 1] = u[n]

Je-li na vstupu lineaacuterniacute kombinace dvou různyacutech signaacutelů

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

y [n] = b1 (y1[n] + a y1[n minus 1]) + b2 (y2[n] + a y2[n minus 1])

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

Přiacuteklad (Nelineaacuterniacute systeacutem)Numerickyacute vyacutepočet druheacute odmocniny lze zapsat rekurentniacutem vztahem

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

Odmocnina z čiacutesla 10 je s přesnostiacute na 10 desetinnyacutech miacutest rovnaradic10 = 316227766017 Pro u[n] = u[0] = 10 dostaacutevaacuteme postupně

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

Pro obecnyacute vstupniacute signaacutel u[n] je pak odezva lineaacuterniacuteho systeacutemu

y [n] = Su[n] = S infinsum

m=minusinfinu[m] δ[n minusm]

=infinsum

m=minusinfinu[m]Sδ[n minusm] =

infinsum

m=minusinfinu[m] h[nm]

Vidiacuteme že chovaacuteniacute systeacutemu je zcela určeno jeho odezvami na různě posunuteacutejednotkoveacute pulsy h[nm]

77

Přechodovaacute odezva diskreacutetniacuteho lineaacuterniacuteho systeacutemu s[n] je daacutena prostyacutem součtemimpulsniacutech odezev pro 0 le m le n

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

Lze za nějakyacutech podmiacutenek zjednodušit h[nm]

78

Časově invariantniacute systeacutem

Systeacutem se nazyacutevaacute časově invariantniacute jestliže jsou všechny udaacutelosti zaacutevisleacute pouze načasoveacutem intervalu (rozdiacutelu časovyacutech udaacutelostiacute) n minusm a nikoliv na každeacutem časoveacutemokamžiku n a m samostatně

dnes y [n] = S [u[n]]

včera y [n minus 1] = S [u[n minus 1]]

Potom takeacute rovnice pro impulsniacute odezvu přejde z maticoveacuteho tvaru na prostyacute vektorovyacutezaacutepis

h[nm]rarr h[n minusm] = Sδ[n minusm]

79

h[n]

n

u[n] LTI y [n]

y [n] = u[0] middot h[n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

V důsledku časoveacute invariance dostaacutevaacuteme z rovnice konvolučniacute sumu

y [n] =infinsum

m=minusinfinh[n minusm] middot u[m] =

infinsum

k=minusinfinh[k] middot u[n minus k]

kterou pro uacutesporu miacutesta značiacuteme

y [n] = h[n] lowast u[n]

Pozor nejde o naacutesobeniacute

h[n] 6= y [n]

u[n]

81

Přiacuteklad

Přiacuteklad (Časově invariantniacute systeacutem)Uvažujme mikroekonomickyacute systeacutem variace ceny popsanyacute diferenčniacute rovnici

Protože jejiacute koeficienty nezaacutevisiacute na čase tj a je konstantniacute a neniacute funkciacute n zachovaacutevaacutetato rovnice tvar při zaacuteměně nrarr n minusm Impulsniacute odezva je potom

h[n] = (minusa)n1[n]

82

Přiacuteklad

Přiacuteklad (Časově proměnnyacute systeacutem)Uvažujme nyniacute obměněnou diferenčniacute rovnici

y [n] + n middot y [n minus 1] = u[n]

Koeficient u y [n minus 1] zaacutevisiacute na čase a tato rovnice nezachovaacutevaacute tvar při zaacuteměněnrarr n minusm Impulsniacute odezvu lze psaacutet ve tvaru

h[n] = (minus1)n n 1[n]

83

Kauzaacutelniacute systeacutem

Systeacutem je kauzaacutelniacute pokud jeho vyacutestup zaacutevisiacute pouze na současnyacutech a minulyacutechhodnotaacutech vstupů

Vyacutestupniacute signaacutel y [n] kauzaacutelniacuteho systeacutemu tedy zaacutevisiacute pouze nau[n] u[n minus 1] u[n minus 2] V konvolučniacute sumě proto

y [n] =infinsum

k=minusinfinh[k] u[n minus k]

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

musiacuteme položit všechny členy impulsniacute odezvy h[n] = 0 pro n lt 0

84

Konvolučniacute suma pro lineaacuterniacute časově invariantniacute a kauzaacutelniacute systeacutem maacute tvar

y [n] =infinsum

k=0

h[k] middot u[n minus k] =infinsum

k=0

u[k] middot h[n minus k]

Jestliže naviacutec budeme požadovat aby vstupniacute a vyacutestupniacute signaacutely měly dobře definovanyacutepočaacutetek tj aby foralln lt 0 u[n] = 0 y [n] = 0 (oba signaacutely mohou miacutet nenuloveacute členypouze pro n ge 0) potom platiacute

y [n] =nsum

k=0

h[k] middot u[n minus k] =nsum

k=0

h[n minus k] middot u[k]

85

Spojiteacute systeacutemy

Odezvu systeacutemu na Diracův impuls δ(t) budeme nazyacutevat impulsniacute odezva a značith(t)

h(t) = Sδ(t)h(t τ) = Sδ(t minus τ)

Odezvu systeacutemu na jednotkovyacute skok 1(t) budeme nazyacutevat přechodovaacute odezva aznačit s(t)

s(t) = S1(t) = Sint t

0δ(t minus τ) dt

86

V přiacutepadě spojiteacuteho casu postupujeme podobně a odvodiacuteme pro lineaacuterniacute časověinvariantniacute systeacutem konvolučniacute integraacutel

y(t) =

int infin

minusinfinh(τ)u(t minus τ) dτ =

int infin

minusinfinh(t minus τ) middot u(τ) dτ

Operaci často zapisujeme ve zjednodušeneacute formě jako

y(t) = h(t) lowast u(t)

Opět připomiacutenaacuteme že se v tomto zaacutepisu nejednaacute o naacutesobeniacute

87

88

Pro u(t) = δ(t) platiacute pro lineaacuterniacute a časoveacute invariantniacute systeacutem samozřejmě

y(t) = Su(t) =

int infin

minusinfinh(τ) middot δ(t minus τ) dτ = h(t)

89

Vyacutestupniacute signaacutel y(t) spojiteacuteho kauzaacutelniacuteho systeacutemu zaacutevisiacute pouze na hodnotaacutech vstupůpro předešleacute časoveacute okamžiky Z důvodu ktereacute klademe na kauzaacutelniacute chovaacuteniacute systeacutemupřejde konvolučniacute integraacutel na tvar

y(t) =

int infin

minusinfinh(τ) u(t minus τ) dτ

=

int 0

︸︷︷︸0

+

int infin

0h(τ) u(t minus τ) dτ

a hodnoty impulsniacute odezvy pro t lt 0 uvažujeme opět h(t) = 0

90

Konvolučniacute integraacutel pro lineaacuterniacute časově invariantniacute a kauzaacutelniacute systeacutem maacute tvar

y(t) =

int infin

0h(τ) middot u(t minus τ) dτ =

int infin

0u(τ) middot h(t minus τ) dτ

Jestliže naviacutec budeme požadovat aby vstupniacute a vyacutestupniacute signaacutely měly dobře definovanyacutepočaacutetek tj aby forallt lt 0 u(t) = 0 y(t) = 0 (oba signaacutely mohou byacutet nenuloveacute členypouze pro t ge 0) potom platiacute

y(t) =

int t

91

Charakteristiky systeacutemů

Definice (Autonomniacute systeacutem)Za autonomniacute systeacutem považujeme takovyacute kteryacute nemaacute vstup Diskreacutetniacute autonomniacutesysteacutem je popisuje tedy napřiacuteklad diferenčniacute rovnice vnějšiacuteho popisu

y [n + 1] + a y [n] = 0

Vyacutestup autonomniacuteho systeacutemu je odezvou na počaacutetečniacute podmiacutenky

V přiacutepadě že systeacutem maacute vstup u[n] tedy

systeacutem poklaacutedaacuteme za neautonomniacute

92

Definice stability

BIBO stabilita ndash bounded input bounded output

Odezva na omezenyacute vstupniacute signaacutel musiacute byacutet vždy omezenaacute ndash systeacutem je BIBO stabilniacute

Odezva systeacutemu je kombinaciacute

bull odezvy na vstupniacute signaacutel

bull odezvy na počaacutetečniacute podmiacutenky

93

Stabilita systeacutemů

Impulsniacute odezvu spojiteacuteho LTI systeacutemu lze vždy zapsat jako součet exponenciel ve tvaru

h(t) =nsum

micro=0

kmicroepmicrot

kde pmicro jsou tzv poacutely přenosoveacute funkce systeacutemu

Pro pmicro isin R je h(t) reaacutelnaacute exponenciela kteraacute pro t rarrinfin buď roste nade všechny meze(pmicro gt 0) nebo klesaacute k nule (pmicro lt 0)

Pro pmicro isin C je h(t) komplexniacute exponenciela kteraacute pro t rarrinfin kmitaacute a buď roste nadevšechny meze nebo klesaacute k nule zaacuteležiacute na ltpmicro

94

Z vyacuteše uvedeneacute uacutevahy vyplyacutevaacute jak z polohy poacutelů přenosoveacute funkce jednoduše odvodiacutemetvar impulsniacute odezvy a tedy stabilitu (limtrarrinfin h(t) = 0) respektive nestabilitu(limtrarrinfin h(t) =infin) systeacutemu

bull Pro stabilniacute systeacutem platiacute limtrarrinfin h(t) = 0

bull V impulsniacute odezvě se nachaacuteziacute minimaacutelně jedna rostouciacute exponenciela jež budehodnotě h(t) postupně dominovat a je tedy limtrarrinfin h(t) =infin

bull Systeacutem může byacutet ale nestabilniacute i v jinyacutech přiacutepadech kdy takeacute limtrarrinfin h(t) =infin

95

O predmetu
- Zaacutekladniacute organizacniacute informace
- Seznam literatury
- Hodnoceniacute predmetu
- Vstupniacute znalosti
- Vyacutestupniacute znalosti
- - Matematickeacute modelovaacuteniacute systeacutemu
    - - Modelovaacuteniacute je umeniacute kompromisu
      - Jakeacute ciacutele muže modelovaacuteniacute dosaacutehnout
      - Klasifikace modelu
      - Faacuteze modelovaacuteniacute
      - Model systeacutemu
      - Vnejšiacute popis systeacutemu
      - Vnitrniacute popis systeacutemu
      - Iterace diferencniacute rovnice
        
        
        Uacutevod do teorie signaacutelu
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        Casove invariantniacute resp stacionaacuterniacute systeacutem
        
        Kauzaacutelniacute priacutecinnyacute systeacutem
        
        
        
        Stabilita
        
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        0EndLeft
        
        anm0
        
        030
        
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        fdrm0

Page 2: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

O předmětu

Zaacutekladniacute organizačniacute informace

Seznam literatury

Hodnoceniacute předmětu

Vstupniacute znalosti

Vyacutestupniacute znalosti

2

Cvičiacuteciacute

Garant předmětu

3

4

Literatura i

5

Literatura ii

6

7

8

9

Klasifikace modelů

Model systeacutemu

Odezva systeacutemu

10

11

12

13

14

Modely

15

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
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        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 3: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Cvičiacuteciacute

Garant předmětu

3

4

Literatura i

5

Literatura ii

6

7

8

9

Klasifikace modelů

Model systeacutemu

Odezva systeacutemu

10

11

12

13

14

Modely

15

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

4

Literatura i

5

Literatura ii

6

7

8

9

Klasifikace modelů

Model systeacutemu

Odezva systeacutemu

10

11

12

13

14

Modely

15

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Literatura i

5

Literatura ii

6

7

8

9

Klasifikace modelů

Model systeacutemu

Odezva systeacutemu

10

11

12

13

14

Modely

15

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Literatura ii

6

7

8

9

Klasifikace modelů

Model systeacutemu

Odezva systeacutemu

10

11

12

13

14

Modely

15

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 7: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

7

8

9

Klasifikace modelů

Model systeacutemu

Odezva systeacutemu

10

11

12

13

14

Modely

15

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 8: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

8

9

Klasifikace modelů

Model systeacutemu

Odezva systeacutemu

10

11

12

13

14

Modely

15

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 9: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

9

Klasifikace modelů

Model systeacutemu

Odezva systeacutemu

10

11

12

13

14

Modely

15

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 10: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Klasifikace modelů

Model systeacutemu

Odezva systeacutemu

10

11

12

13

14

Modely

15

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 11: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

11

12

13

14

Modely

15

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 12: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

12

13

14

Modely

15

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 13: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

13

14

Modely

15

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 14: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

14

Modely

15

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 15: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Modely

15

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 16: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Klasifikace modelů

16

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 17: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

17

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 18: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Systeacutem

18

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 19: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

19

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 20: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Svstup vyacutestup

u(t) S y(t)

u(t)

t

y(t)

t

u[n] S y [n]

u[n]

n

y [n]

n

20

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 21: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Tvorba modelu

21

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 22: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Tvorba modelu

Probleacutemy

22

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 23: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

0 05 1 15 2 25 3 35 4 45 5minus08

minus06

minus04

minus02

0

02

04

23

Other

5276747

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 24: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Otaacutezky

24

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 25: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

25

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 26: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

26

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 27: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

27

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 28: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

28

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 29: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

θ +g

`sin θ = 0

29

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 30: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

partρ

DuDt

= Fminus nablapρ

30

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 31: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

N = rN

(1minus N

K

)

31

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
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        07
        
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        05
        
        04
        
        03
        
        02
        
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Page 32: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

32

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 33: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

33

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 34: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

u(t) S y(t)

u(t)

t

y(t)

t

34

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 35: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

u(t) x(t) =

x1(t)

x2(t)

xn(t)

y(t)u(t)

t

y(t)

t

35

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 36: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Role matematiky

36

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 37: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

R

Cu1(t) uC(t)

u1(t) = uR(t) + uC(t)

37

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 38: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

i(t) = Cddt

uC(t)

a proto

uC(t)

38

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 39: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

RCddt

uC(t) + uC(t) = u1(t)

α =1RC

u1(t) = U0

39

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 40: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

time [s]

u C [V

]

40

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 41: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

41

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 42: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

B minusAD u[k]

42

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 43: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

43

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 44: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

D1 D2 SP

QQ2Q1

P2

P1

44

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 45: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 46: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Iterace

46

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 47: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Iterace

47

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 48: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

CD c[k minus 1] =

B minusAD x [k]

48

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 49: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Pro k = 0

y [0] + γy [minus1] = βu[0]

Pro k = 1

y [1] + γy [0] = βu[1]

49

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 50: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Pro k = 2

y [2] + γy [1] = βu[2]

Pro obecneacute n

)

50

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 51: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

y [n] = β

nsum

m=0

1 + γ=

β

1 + γ+

βγ

1 + γ(minusγ)n

0 1 2 3 4 5 695

100

105

110

115

120

125

cena

nabiacute

dka

a po

ptaacutev

ka

51

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 52: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Odezva systeacutemu

52

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 53: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Diracův impuls

Exponenciaacutela

Odezva systeacutemu

53

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 54: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Diracův impuls

δε(t)

tminusε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

tminusε +ε

1ε

54

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 55: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Diracův impuls

int infin

int ε

minusεδ(t) dt =

int 0+

0minusδ(t) dt = 1

55

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 56: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

1(t) =

1 pro t ge 0

0 pro t lt 0

1(t)

t

1

56

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 57: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Platiacuteδ(t) =

ddt

1(t)

t

1

ε

57

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 58: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Exponenciaacutela

eαt

t

eαt

t

58

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 59: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Exponenciaacutela

59

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 60: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

60

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 61: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

A sin(ωt + Φ)

t

A

A sin(Φ)

T = 2πω

61

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
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        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
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        02
        
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        fdrm0

Page 62: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Odezva systeacutemu

62

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 63: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

63

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 64: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

δ[n] =

1 pro n = 0

0 pro n 6= 0

δ[n]

n

δ[n minus 2]

n

64

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 65: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

1[n] =

1 pro n ge 0

0 pro n lt 0

1[n]

n

1[n minus 1]

n

65

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 66: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

A sin(ξn + Φ)

t

66

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 67: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

67

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 68: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

N = m middot 2πTs

y [n] = sin n

68

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 69: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Odezva systeacutemu

Stabilita

69

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 70: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

u[n] LTI y [n]

u[n]

n

y [n]

n

70

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 71: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

71

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 72: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

72

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 73: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

73

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 74: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Princip superpozice

y1[n] = Su1[n]y2[n] = Su2[n]

Obecně platiacute

u[n] =sum

i

aiyi [n] =sum

i

aiSui [n]

74

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 75: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Přiacuteklad

u[n] = b1u1[n] + b2u2[n]

je na vyacutestupu

kde

y1[n] + a y1[n minus 1] = u1[n]

y2[n] + a y2[n minus 1] = u2[n] 75

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 76: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Přiacuteklad

y [n + 1] =12

(y [n] +

u[n]

y [n]

)

n y [n] y2[n]

1 3 92 3165 100172253 3162278 10000002149284 3162277660 9999999999568

76

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 77: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

=infinsum

infinsum

77

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 78: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

s[n] = S1[n] = S

nsum

m=0

δ[n minusm]

=nsum

m=0

Sδ[n minusm] =nsum

m=0

h[nm]

78

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 79: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

79

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 80: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

h[n]

n

u[n] LTI y [n]

u[n]

n

u[0] middot h[n]

n

y [n]

n80

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 81: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Konvoluce

y [n] =infinsum

infinsum

h[n] 6= y [n]

u[n]

81

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 82: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Přiacuteklad

82

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 83: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Přiacuteklad

83

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 84: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

y [n] =infinsum

=minus1sum

︸︷︷︸0

+infinsum

k=0

h[k] u[n minus k]

84

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 85: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

y [n] =infinsum

k=0

y [n] =nsum

k=0

85

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 86: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

0δ(t minus τ) dt

86

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 87: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

y(t) =

int infin

87

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 88: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

88

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 89: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

y(t) = Su(t) =

int infin

89

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 90: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

y(t) =

int infin

=

int 0

︸︷︷︸0

+

int infin

90

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 91: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

y(t) =

int infin

y(t) =

int t

91

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 92: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

y [n + 1] + a y [n] = 0

92

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 93: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

Definice stability

93

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 94: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

h(t) =nsum

micro=0

kmicroepmicrot

94

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

Page 95: Úvodní informace Matematické modelování · Literaturaii 6.HEATH,MichaelT.Scientiﬁccomputing–AnIntroductorySurvey.2.vyd.New York:McGraw-Hill,2002. 7.BERTSEKAS,DimitriP.Dynamicprogrammingandoptimalcontrol

95

O predmetu
- Seznam literatury
        
        
        
        
        
        Diracuv impuls
        
        
        Exponenciaacutela
        
        
        
        
        
        Odezva systeacutemu
        
        
        
        
        
        
        
        Stabilita
        
        0Plus
        
        0Reset
        
        0Minus
        
        0EndRight
        
        0StepRight
        
        0PlayPauseRight
        
        0PlayRight
        
        0PauseRight
        
        0PlayPauseLeft
        
        0PlayLeft
        
        0PauseLeft
        
        0StepLeft
        
        0EndLeft
        
        anm0
        
        030
        
        029
        
        028
        
        027
        
        026
        
        025
        
        024
        
        023
        
        022
        
        021
        
        020
        
        019
        
        018
        
        017
        
        016
        
        015
        
        014
        
        013
        
        012
        
        011
        
        010
        
        09
        
        08
        
        07
        
        06
        
        05
        
        04
        
        03
        
        02
        
        01
        
        00
        
        fdrm0

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