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University of Illinois at Urbana-Champaign Lecture 7: Stability Verification Sayan Mitra
University of Illinoisat Urbana-Champaign
Lecture7:StabilityVerification
SayanMitra
RecallStability
• Timeinvariantautonomoussystems(closedsystems,systemswithoutinputs)
• �̇� 𝑡 = 𝑓 𝑥 𝑡 , 𝑥[ ∈ ℝ�, 𝑡[ = 0 –(1)• 𝜉 𝑡 isthesolution• |𝜉 𝑡 | norm• 𝑥∗ ∈ ℝ� isanequilibriumpointif𝑓 𝑥∗ = 0.• Foranalysiswewillassume0tobeanequilibriumpointof(1)withoutlossofgenerality
Lyapunov stability
Lyapunov stability:Thesystem(1)issaidtobeLyapunov stable(attheorigin)ifforevery𝜀 >0thereexists𝛿ð > 0suchthatforeveryif𝜉 0 ≤ 𝛿ð thenforallt ≥ 0, 𝜉 𝑡 ≤ 𝜀.
𝛿ð
Asymptoticallystability
Thesystem(1)issaidtobeAsymptoticallystable(attheorigin)ifitisLyapunov stableandthereexists𝛿P > 0suchthatforeveryif 𝜉 0 ≤ 𝛿P thent → ∞, 𝜉 𝑡 → 𝟎.Ifthepropertyholdsforany𝛿P thenGloballyAsymptoticallyStable
Definingstabilityofhybridsystems
• Hybridautomaton:𝐀 = ⟨𝑉, 𝐴, 𝐷,Τ⟩– 𝑉 = 𝑋 ∪ {ℓ}
• Execution𝛼 = 𝜏[𝑎M𝜏M𝑎P …• Notation𝛼(𝑡):denotesthevaluation𝛽. 𝑙𝑠𝑡𝑎𝑡𝑒where𝛽
isthelongestprefixwith𝛽. ltime = 𝑡• |𝛼 𝑡 |: normofthecontinuousstate𝑋• AisLyapunov stable(attheorigin)ifforevery𝜀 >
0thereexists𝛿ð > 0suchthatforeveryif 𝛼 0 ≤ 𝛿ðthenforallt ≥ 0, 𝛼 𝑡 ≤ 𝜀.
• AsymptoticallystableifitisLyapunov stableandthereexists𝛿P > 0suchthatforeveryif 𝛼 0 ≤ 𝛿P thent →∞, 𝛼 𝑡 → 𝟎.
mode1𝑑 𝑥 = 𝑓M(𝑥)
mode2𝑑 𝑥 = 𝑓P(𝑥)
Pre𝐺MPEff𝑥 ≔ 𝑅MP(𝑥)
Question:Stability Verification• Ifeachmodeisasymptotically
stablethenisAalsoasymptoticallystable?
• No
ProofTechniques:Stability
CommonLyapunov Function• Ifthereexistspositivedefinitecontinuouslydifferentiablefunction𝑉:ℝ� → ℝ andapositivedefinitefunctionW: ℝ� → ℝ suchthatforeachmode𝑖, ßÿ
ß»𝑓Y 𝑥 < −𝑊(𝑥) forall𝑥 ≠ 0 thenV
iscalledacommonLyapunov functionforA.
• 𝑉 iscalledacommonLyapunov function
• Theorem.A isGUASifthereexistsacommonLyapunov function.
MultipleLyapunov Functions• Intheabsenceofacommonlyapunov functionthestability
verificationhastorelyofthediscretetransitions.• Thefollowingtheoremgivessuchastabilityintermsof
multipleLyapunov function.• Theorem [Branicky]Ifthereexistsafamilyofpositive
definitecontinuouslydifferentiableLyapunov functions𝑉Y: ℝ� → ℝ andapositivedefinitefunctionW\: ℝ� → ℝsuchthatforanyexecution𝛼andforanytime𝑡M𝑡P𝛼 𝑡M . ℓ = 𝛼 𝑡P . ℓ = 𝑖 andforalltime𝑡 ∈𝑡M, 𝑡P , 𝛼 𝑡 . ℓ ≠ 𝑖– 𝑉Y 𝛼 𝑡P . 𝑥 −𝑉Y 𝛼 𝑡M . 𝑥 ≤ −𝑊Y(𝛼 𝑡M . 𝑥)
time
)(tVi
V2 £ 𝜇 V1
• Average Dwell Time (ADT) characterizes rate of mode switches• Definition: H has ADT T if there exists a constant N0 such that for
every execution α, N(α) £ N0 + duration(α)/T.
N(α): number of mode switches in α
• Theorem [HM`99] H is asymptotically stable if its modes have aset of Lyapunov functions (𝜇, 𝜆[) and ADT(H) > log 𝜇/𝜆[ .
StabilityUnderSlowSwitching
mode1 mode2 mode2mode1
[HespanhaandMorse`99]
ProofTechniques:Stability
𝜕𝑉Y𝜕𝑥 ≤ −2𝜆[𝑉Y(𝑥)
RemarksaboutADTtheoremassumptions
1. If𝑓Y isgloballyasymptoticallystable,thenthereexistsaLyapunov function𝑉Y thatsatisfies
ßÿ�ßÈ
≤ −2𝜆Y𝑉Y 𝑥forappropriatelychosen 𝜆Y > 0
2. Ifthesetofmodesisfinite,choose𝜆[ independentof𝑖
3. TheotherassumptionrestrictsthemaximumincreaseinthevalueofthecurrentLyapunov functionsoveranymodeswitch,byafactorofμ.
4. Wewillalsoassumethatthereexiststrictlyincreasingfunctions𝛽Mand𝛽P suchthat𝛽M(|𝑥|) ≤𝑉Y 𝑥 ≤𝛽P(|𝑥|)
ProofsketchSuppose𝛼 isanyexecutionofA.Let𝑇 = 𝛼. 𝑙𝑡𝑖𝑚𝑒and𝑡M, … , 𝑡^ ú beinstantsofmodeswitchesin𝛼.Wewillfindanupper-boundonthevalueof𝑉ú ) .í 𝛼 𝑇 . 𝑥Define𝑊 𝑡 = 𝑒P_á»𝑉ú » .í(𝛼 𝑡 . 𝑥)
𝑊 isnon-increasingbetweenmodeswitches ßÿ�ßÈ
≤ −2𝜆[𝑉Y 𝑥
Thatis,𝑊 𝑡Y�M# ≤𝑊 𝑡Y𝑊 𝑡Y�M ≤ 𝜇𝑊 𝑡Y�M# ≤ 𝜇𝑊 𝑡YIteratingthis𝑁 𝛼 times:𝑊 𝑇 ≤ 𝜇^(ú)𝑊 0
𝑒P_á)𝑉ú ) .í 𝛼 𝑇 . 𝑥 ≤ 𝜇^ ú 𝑉ú [ .í(𝛼 0 . 𝑥)
𝑉ú ) .í 𝛼 𝑇 . 𝑥 ≤ 𝜇^ ú 𝑒#P_á)𝑉ú [ .í(𝛼 0 . 𝑥) = 𝑒#P_á)�^ ú abc d𝑉ú [ .í(𝛼 0 . 𝑥)
If𝛼 hasADT𝜏G then,recall,𝑁 𝛼 ≤ 𝑁[ + 𝑇/𝜏G and𝑉ú ) .í 𝛼 𝑇 . 𝑥 ≤𝑒#P_á)�(^á�)/Ñe) abc d𝑉ú [ .í(𝛼 0 . 𝑥) ≤ 𝐶𝑒)(#P_á�abcd/Ñe)
If𝜏G > log 𝜇 /2𝜆[ thensecondtermconvergesto0as𝑇 → ∞ thenfromassumption4itfollowsthat𝛼 convergesto0.
