1 Stability Analysis of Continuous- Time Switched Systems: A Variational Approach Michael Margaliot School of EE-Systems Tel Aviv University, Israel Joint work with: Michael S. Branicky (CWRU) Daniel Liberzon (UIUC)

2 Overview Switched systems Stability Stability analysis:  A control-theoretic approach  A geometric approach  An integrated approach Conclusions

3 Switched Systems Systems that can switch between several modes of operation. Mode 1 Mode 2

4 Example 1 Switched power converter 100v 50v linear filter

5 Example 2 A multi-controller scheme plant controller 1 + switching logic controller 2 Switched controllers are “stronger” than regular controllers.

6 More Examples Air traffic control Biological switches Turbo-decoding …… For more details, see: - Introduction to hybrid systems, Branicky - Basic problems in stability and design of switched systems, Liberzon & Morse

7 Synthesis of Switched Systems Driving: use mode 1 (wheels) Braking: use mode 2 (legs) The advantage: no compromise

8 Gestalt Principle “Switched systems are more than the sum of their subsystems.“  theoretically interesting  practically promising

9 Differential Inclusions A solution is an absolutely continuous function satisfying (DI) for almost all t. Example:

10 Global Asymptotic Stability (GAS) Definition The differential inclusion is called GAS if for any solution (i) (ii)

11 The Challenge Why is stability analysis difficult? (i)A DI has an infinite number of solutions for each initial condition. (ii) The gestalt principle.

12 Absolute Stability [Lure, 1944]

13 Absolute Stability The closed-loop system: Absolute Stability Problem Find A is Hurwitz, so CL is asym. stable for any For CL is asym. stable for any

14 Absolute Stability and Switched Systems Absolute Stability Problem Find

15 Example

16 A Solution of the Switched System This implies that

17 Although both and are stable, is not stable. Instability requires repeated switching. Two Remarks

18 Optimal Control Approach Write as the bilinear control system: is the worst-case switching law (WCSL). Problem Find a control maximizing Analyze the corresponding trajectory Fix Define: 0.T 

19 Optimal Control Approach Consider as

20 Optimal Control Approach Theorem (Pyatnitsky) If then: (1) The function is finite, convex, positive, and homogeneous (i.e. ). (2) For every initial condition there exists a solution such that

21 Solving Optimal Control Problems is a functional: Two approaches: 1. Hamilton-Jacobi-Bellman (HJB) equation. 2. Maximum Principle.

22 HJB Equation Find such that Integrating: or An upper bound for, obtained for the maximizing Eq. (HJB).

23 Margaliot & Langholz (2003) derived an explicit solution for when n=2. This yields an easily verifiable necessary and sufficient condition for stability of second-order switched linear systems. The Case n=2

24 The function is a first integral of if We know that so Basic Idea Thus, is a concatenation of two first integrals and

25 Example: where and

26 Thus, → an explicit expression for V (and an explicit solution of the HJB).

27 More on the Planar Case [Margaliot & Branicky, 2009] Corollary GAS of 2nd-order positive linear switched systems. Theorem For a planar bilinear control system

28 Nonlinear Switched Systems where are GAS. Problem Find a sufficient condition guaranteeing GAS of (NLDI).

29 Lie-Algebraic Approach For simplicity, consider the linear differential inclusion: so

30 Commutation Relations and GAS Suppose that A and B commute, i.e. AB=BA, then Definition The Lie bracket of Ax and Bx is [Ax,Bx]:=ABx-BAx. Hence, [Ax,Bx]=0 implies GAS.

31 Lie Brackets and Geometry Consider Then:

32 Geometry of Car Parking This is why we can park our car. The term is the reason this takes so long.

33 Nilpotency Definition k’th order nilpotency: all Lie brackets involving k+1 terms vanish. 1 st order nilpotency: [A,B]=0 2 nd order nilpotency: [A,[A,B]]=[B,[A,B]]=0 Q: Does k’th order nilpotency imply GAS?

34 Known Results Linear switched systems: k = 2 implies GAS ( Gurvits,1995 ). k’th order nilpotency implies GAS ( Liberzon, Hespanha, & Morse, 1999 ) ( Kutepov, 1982 ) Nonlinear switched systems: k = 1 implies GAS ( Mancilla-Aguilar, 2000 ). An open problem: higher orders of k? ( Liberzon, 2003 )

35 A Partial Answer Theorem (Margaliot & Liberzon, 2004) 2nd order nilpotency implies GAS. Proof By the PMP, the WCSL satisfies Let

36 Then 1st order nilpotency   Differentiating again yields: 2nd order nilpotency    up to a single switch in the WCSL.

37 Handling Singularity If m(t)  0, the Maximum Principle does not necessarily provide enough information to characterize the WCSL. Singularity can be ruled out using the notion of strong extremality (Sussmann, 1979).

38 3rd Order Nilpotency In this case: further differentiation cannot be carried out.

39 3rd Order Nilpotency Theorem ( Sharon & Margaliot, 2007 ) 3rd order nilpotency implies Proof (1) Hall-Sussmann canonical system; (2) A second-order MP (Agrachev&Gamkrelidze).

40 Conclusions Stability analysis is difficult. A natural and powerful idea is to consider the “most unstable” trajectory. Switched systems and differential inclusions are important in various scientific fields, and pose interesting theoretical questions.

41 More info on the variational approach: “Stability analysis of switched systems using variational principles: an introduction”, Automatica 42(12): , Available online: