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**INTRODUCTION to SWITCHED SYSTEMS ; STABILITY under ARBITRARY SWITCHING**

Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign IAAC Workshop, Herzliya, Israel, June 1, 2009

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**SWITCHED and HYBRID SYSTEMS**

Hybrid systems combine continuous and discrete dynamics Which practical systems are hybrid? electric circuit thermostat cell division walking stick shift Picking the most relevant examples for Israel Walking: robots or humans On the last point, can mention a bouncing ball Which practical systems are not hybrid? More tractable models of continuous phenomena

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**MODELS of HYBRID SYSTEMS**

… [Van der Schaft–Schumacher ’00] [Proceedings of HSCC] Mention jumps The difference is in whether the emphasis is more on continuous, or on discrete, or equally on both Mention a more “balanced” hybrid model in terms of guards and location (see HYCON sheet) continuous discrete [Nešić–L ‘05]

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**SWITCHED vs. HYBRID SYSTEMS**

Switched system: is a family of systems is a switching signal Switching can be: State-dependent or time-dependent Autonomous or controlled Assuming common equilibrium Sigma is piecewise constant Hybrid systems give rise to classes of switching signals Measurable switching – related to control system (Filippov’s lemma), more on this later Details of discrete behavior are “abstracted away” Discrete dynamics classes of switching signals Properties of the continuous state : stability and beyond

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**STABILITY ISSUE Asymptotic stability of each subsystem is**

And vice versa: can stabilize, even if subsystems are unstable This is valid in dimensions 2 and higher unstable Asymptotic stability of each subsystem is not sufficient for stability

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**TWO BASIC PROBLEMS Stability for arbitrary switching**

Stability for constrained switching Second is more relevant for control First leads to second

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**TWO BASIC PROBLEMS Stability for arbitrary switching**

Stability for constrained switching In this talk we’ll mainly deal with the first problem, except for towards the end

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**GLOBAL UNIFORM ASYMPTOTIC STABILITY**

GUAS is: Lyapunov stability plus asymptotic convergence Reduces to standard GAS for non-switched systems Explain uniformity w.r.t. sigma GUES:

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**COMMON LYAPUNOV FUNCTION**

where is positive definite is GUAS if (and only if) s.t. quadratic is GUES

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**OUTLINE Stability criteria to be discussed:**

Commutation relations (Lie algebras) Feedback systems (absolute stability) Observability and LaSalle-like theorems Last bullet: WEAK Lyapunov function Common Lyapunov functions will play a central role

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**COMMUTING STABLE MATRICES => GUES**

(commuting Hurwitz matrices) “In order to elucidate the role of commutation relations…” For subsystems – similarly

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**COMMUTING STABLE MATRICES => GUES**

Alternative proof: quadratic common Lyapunov function [Narendra–Balakrishnan ’94] Alternative to LMI solvers for commuting case Last equation is a hyperlink . is a common Lyapunov function

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**NILPOTENT LIE ALGEBRA => GUES**

Lie bracket: Nilpotent means sufficiently high-order Lie brackets are 0 For example: (2nd-order nilpotent) Recall: in commuting case Gurvits proved it for second-order nilpotent In 2nd-order nilpotent case Hence still GUES [Gurvits ’95]

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**SOLVABLE LIE ALGEBRA => GUES**

Larger class containing all nilpotent Lie algebras Suff. high-order brackets with certain structure are 0 Lie’s Theorem: is solvable triangular form Example: This is a more general result exponentially fast exp fast quadratic common Lyap fcn diagonal [Kutepov ’82, L–Hespanha–Morse ’99]

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**SUMMARY: LINEAR CASE Lie algebra commuting subsystems:**

w.r.t. Assuming GES of all modes, GUES is guaranteed for: commuting subsystems: nilpotent Lie algebras (suff. high-order Lie brackets are 0) e.g. solvable Lie algebras (triangular up to coord. transf.) solvable + compact (purely imaginary eigenvalues) Quadratic common Lyapunov function exists in all these cases No further extension based on Lie algebra only [Agrachev–L ’01]

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**SWITCHED NONLINEAR SYSTEMS**

Lie bracket of nonlinear vector fields: Reduces to earlier notion for linear vector fields (modulo the sign)

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**SWITCHED NONLINEAR SYSTEMS**

Commuting systems GUAS Can prove by trajectory analysis [Mancilla-Aguilar ’00] or common Lyapunov function [Shim et al. ’98, Vu–L ’05] Linearization (Lyapunov’s indirect method) Lie bracket is defined for differentiable vector fields, but commutativity of the flows is enough Shim – for ES subsystems, Linh – for AS Comment on the iterative construction of common Lyapunov function “Common Lyapunov function” (actually, a transparent rectangle behind it) is a hyperlink Global results beyond commuting case – ? [Unsolved Problems in Math. Systems & Control Theory ’04]

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**SPECIAL CASE globally asymptotically stable Want to show: is GUAS**

Will show: differential inclusion is GAS Nilpotent of order 2 – covered by known results in linear case Some technical conditions are needed (see paper)

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**OPTIMAL CONTROL APPROACH**

Associated control system: where (original switched system ) Worst-case control law [Pyatnitskiy, Rapoport, Boscain, Margaliot]: fix and small enough Idea appeared before, but not pursued for nonlinear case. MM proposed to use it in this case, this is joint work with him Final time small for solutions to exist – but will show aposteriori that it could be arbitrary The optimal control problem is well defined (Filippov, Lin-Sontag-Wang) Explain how this will give us GUAS

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**MAXIMUM PRINCIPLE Optimal control: is linear in (unless )**

(along optimal trajectory) Optimal control: is linear in \lambda is costate, \varphi is switching function (defines optimal control) This is of course where Lie brackets come in Strong version of bang-bang theorem GAS is easy to prove from definition, and intuitively clear Note: commuting case covered in same way (worst-case law is just to follow one subsystem) To generalize, need to make sure terms multiplying u are 0 (unless ) at most 1 switch GAS

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**GENERAL CASE [Margaliot–L ’06, Sharon–Margaliot ’07]**

Want: polynomial of degree (proof – by induction on ) bang-bang with switches These \varphi define the optimal control Uniform bound on degree – guaranteed by suitable Lie bracket conditions (on next slide) Singularity is handled as before Explain difference with bang-bang theorems of Sussmann and Agrachev-Vakhrameev GAS [Margaliot–L ’06, Sharon–Margaliot ’07]

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**REMARKS on LIE-ALGEBRAIC CRITERIA**

Checkable conditions In terms of the original data Independent of representation Not robust to small perturbations Can use tools from Lie algebras (such as Killing form) to check No need to look for a specific basis (related to the previous point) They reveal the fact that what matters is the structure of the Lie algebra, and not its representation Strong negative emotions are well justified Theoretically appealing (but not very practical – unless we can measure “closeness”) In any neighborhood of any pair of matrices there exists a pair of matrices generating the entire Lie algebra [Agrachev–L ’01] How to measure closeness to a “nice” Lie algebra?

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**FEEDBACK SYSTEMS: ABSOLUTE STABILITY**

controllable Circle criterion: quadratic common Lyapunov function is strictly positive real (SPR): Common way in which a switched system arises Switched system is GUES Discuss passivity a little more, mention KYP (see notes) More on Popov later For this reduces to SPR (passivity) Popov criterion not suitable: depends on

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**FEEDBACK SYSTEMS: SMALL-GAIN THEOREM**

controllable Small-gain theorem: quadratic common Lyapunov function Again, switched system is GUES Can handle norms other than 1 (clear)

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**OBSERVABILITY and ASYMPTOTIC STABILITY**

Barbashin-Krasovskii-LaSalle theorem: is GAS if s.t. (weak Lyapunov function) is not identically zero along any nonzero solution (observability with respect to ) observable => GAS Example: Will spend some time on observability, but let me give motivation first V satisfies the usual properties of candidate Lyapunov function Remember to clarify the meaning of “weak” verbally The connection between LaSalle and observability is not always made explicitly, but has been discussed by several people (Artstein) C gives fictitious output

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**SWITCHED LINEAR SYSTEMS**

[Hespanha ’04] Theorem (common weak Lyapunov function): Switched linear system is GAS if s.t observable for each infinitely many switching intervals of length The condition on sigma: prevents sliding/chattering in the kernel of the output matrix (assume same output for subsystems). We’ll see the reason for this assumption later (we’ll need a piece of y of length tau to say anything about x). The counterexample is in Joao’s paper. Fast switching, state is bounded but doesn’t converge. This is not GUAS – have uniformity over p-dwell (but only in the linear case) Detectability is not enough (unless no switching in unobservable modes) – doesn’t decay enough in \tau time units Observability is the only one of the three conditions that is linear in nature (weak Lyap fcn – already had the nonlinear version, last one – on sigma only) To handle nonlinear switched systems and non-quadratic weak Lyapunov functions, need a suitable nonlinear observability notion

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**SWITCHED NONLINEAR SYSTEMS**

Theorem (common weak Lyapunov function): Switched system is GAS if s.t. infinitely many switching intervals of length Each system is norm-observable: V has the usual properties of a candidate Lyapunov function \tau used in the obs condition is the \tau from the last one (which we don’t know – but can have a slow switching version) It’s better to first list hypotheses and then give conclusion? Will come back to this result in the next lecture [Hespanha–L–Sontag–Angeli ’05]

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