st 1 HYCON PhD School on Hybrid Systems www.ist-hycon.org www.unisi.it Stability and Stabilization of Hybrid Systems Mikael Johansson KTH Stockholm, Sweden mikaelj@s3.kth.se scimanyd suounitnoc enibmoc smetsys dirbyH lacipyt (snoitauqe ecnereffid ro laitnereffid) scimanyd etercsid dna stnalp lacisyhp fo fo lacipyt (snoitidnoc lacigol dna atamotua) fo senilpicsid gninibmoc yB .cigol lortnoc ,yroeht lortnoc dna smetsys dna ecneics retupmoc dilos a edivorp smetsys dirbyh no hcraeser ,sisylana eht rof sloot lanoitatupmoc dna yroeht fo ngised lortnoc dna ,noitacifirev ,noitalumis egral a ni desu era dna ,''smetsys deddebme`` ria ,smetsys evitomotua) snoitacilppa fo yteirav ssecorp ,smetsys lacigoloib ,tnemeganam ciffart .(srehto ynam dna ,seirtsudni HYSCOM IEEE CSS Technical Committee on Hybrid Systems 6 Siena, July 1 9-22, 2005 - Rectorate of the University of Siena
Goals and class structure Stability and stabilization of hybrid systems Goal: After these lectures, you should • Have an overview of some key results on stability and stabilization of hybrid systems • Be familiar with the computational methods for piecewise linear systems • Understand how the tools can be applied to (relatively) practical systems Mikael Johansson Three lectures: Department of Signals, Sensors and Systems KTH, Stockholm, Sweden 1. Stability theory 2. Computational tools for piecewise linear systems 3. Applications 1 2 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 A hybrid systems model Part I – Stability theory We consider hybrid systems on the form Outline: • A hybrid systems model and stability concepts • Lyapunov theory for smooth systems where • Lyapunov theory for stability and stabilization of hybrid systems The discrete state indexes vector fields while is the (discontinuous) transition function describing the evolution of the discrete state. Acknow ledgem ents: M. Heemels, ESI Unless stated otherwise, we will assume that is piecewise continuous (i.e., that there is only a finite number of mode changes per unit time interval). For now, disregard issues with sliding modes, zeno, … (precise statements in refs) 3 4 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Example: a switched linear system Stability concepts Focus: stability of equilibrium point (in the continuous state-space) Global asym ptotic stability (GAS): ensure that Global uniform asym ptotic stability (GUAS): ensure that (i.e., uniformly in ) (numerical values for the matrices A i (numerical values for the matrices A i can be found in the notes for Lecture 2) can be found in the notes for Lecture 2) 5 6 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005
Three fundamental problems Part I – Stability theory Problem P1 : Under what conditions is Outline: GAS for all (piecewise continuous) switching signals ? • A hybrid systems model and stability concepts • Lyapunov theory for smooth systems Problem P2 : Given vector fields , design switching strategy : • Lyapunov theory for stability and stabilization of hybrid systems Aim : establishing common grounds by reviewing fundamentals. is globally asymptotically stable. Problem P3 : determine if a given switched system is globally asymptotically stable. 7 8 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Lyapunov theory for smooth systems Converse theorem Under appropriate technical conditions (mainly smoothness of the vector fields) Consequence: worthwhile to search for Lyapunov functions (but how?) I nterpretation: Lyapunov function is an abstract measure of system energy System energy should decrease along all trajectories. 9 10 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Stability of linear systems Partial proof 11 12 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005
Stability of discrete-time systems Performance analysis Lyapunov-like techniques are also useful for estimating system performance. I nterpretation: System energy should decrease at every sampling instant (event) 13 14 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Switching between stable systems Part I – Stability theory Question: does switching between stable linear dynamics always create stable motions? Outline: Answ er: no, not necessarily. • A hybrid systems model and stability concepts • Lyapunov theory for smooth systems • Lyapunov theory for stability and stabilization of hybrid systems Both systems are stable, share the same eigenvalues, but stability depends on switching! Content: – Guaranteeing stability independent of switching strategy – Design a stabilizing switching strategy – Prove stability for a given switching strategy 15 16 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 P1: Stability for arbitrary switching signals The common Lyapunov function approach Problem : when is the switched system In fact, if the submodels are smooth, the following results hold. globally asymptotically stable for all (piecewise continuous) switching signals ? Claim : only if there is a radially unbounded Lyapunov function for each subsystem (can you explain why?) Hence, common Lyapunov functions necessary and sufficient. 17 18 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005
Switched linear systems Infeasibility test For switched linear systems It is also possible to prove that there is no common quadratic Lyapunov function: it is natural to look for a common quadratic Lyapunov function is a common Lyapunov function if Common quadratic Lyapunov function found by solving linear matrix inequalities (systems that admit quadratic Lyapunov function are sometimes called quadratically stable ) 19 20 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Example Example Question: Does GUAS of switched linear system imply existence of a common Sample trajectories of switched system (under two different switching strategies) quadratic Lyapunov function? 1 1 Answ er: No, the system given by 0 0 x 1 -1 x 1 -1 -2 -2 -3 -3 0 2 4 6 8 10 0 2 4 6 8 10 is GUAS, but does not admit any common quadratic Lyapunov function since Time Time 1 1 0.5 0.5 x 2 x 2 0 0 -0.5 -0.5 0 2 4 6 8 10 satisfy the infeasibility condition. 0 2 4 6 8 10 Time Time (there is, however, a common piecewise quadratic Lyapunov function) Even if solutions are very different, all possible motions are asymptotically stable 21 22 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 P2: Stabilization Stabilization of switched linear systems Problem form ulation: given matrices A i , find switching rule ν (x,i) such that is asymptotically stable. 23 24 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005 Mikael Johansson - mikaelj@s3.kth.se Hycon Summer School, Siena, July 2005
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