1 Stability Analysis of Switched Systems: A Variational Approach Michael Margaliot School of EE-Systems Tel Aviv University Joint work with 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 server

5 Example 2 Switched power converter 100v 50v linear filter

6 Example 3 A multi-controller scheme plant controller 1 + switching logic controller 2 Switched controllers are “stronger” than regular controllers.

7 More Examples Air traffic control Biological switches Turbo-decoding ……

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

9 Mathematical Modeling with Differential Inclusions easier ANALYSIS harder MODELING CAPABILITY weaker stronger

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

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

12 Stability The differential inclusion is called GAS if for any solution (i) (ii)

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

14 Absolute Stability

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

16 Absolute Stability and Switched Systems The Problem of Absolute Stability: Find

17 Example

18 Trajectory of the Switched System This implies that

19 Although both and are stable, is not stable. Instability requires repeated switching. This presents a serious problem in multi-controller schemes.

20 Optimal Control Approach Write as a control system: Fix Define Problem: Find the control that maximizes is the worst-case switching law (WCSL). Analyze the corresponding trajectory

21 Optimal Control Approach Consider as

22 Optimal Control Approach Thm. 1 (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

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

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

25 The HJB for a LDI: Hence, In general, finding is difficult.

26 The Maximum Principle Let Then, Differentiating we get A differential equation for with a boundary condition at

27 Summarizing, The WCSL is the maximizing that is, We can simulate the optimal solution backwards in time.

28 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

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

30 Example: where and

31 Thus, so we have an explicit expression for V (and an explicit solution of HJB).

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

33 Lie-Algebraic Approach For the sake of simplicity, consider the LDI so

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

35 Lie Brackets and Geometry Consider Then:

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

37 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?

38 Some Known Results Switched linear systems: k = 2 implies GAS (Gurvits,1995). k’th order nilpotency implies GAS (Liberzon, Hespanha, and Morse, 1999). (The proof is based on Lie’s Theorem) Switched nonlinear systems: k = 1 implies GAS. An open problem: higher orders of k? (Liberzon, 2003)

39 A Partial Answer Thm. 1 (Margaliot & Liberzon, 2004) 2nd order nilpotency implies GAS. Proof: Consider the WCSL Define the switching function

40 Differentiating m(t) yields 1st order nilpotency    no switching in the WCSL. Differentiating again, we get 2nd order nilpotency    up to a single switch in the WCSL.

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

42 3rd order Nilpotency In this case: further differentiation cannot be carried out.

43 3rd order Nilpotency Thm. 2 (Sharon & Margaliot, 2005) 3rd order nilpotency implies The proof is based on using: (1) the Hall- Sussmann canonical system; and (2) the second-order Agrachev-Gamkrelidze MP.

44 Hall-Sussmann System Consider the case [A,B]=0. Guess the solution: Then so (HS system) and

45 Hall-Sussmann System If two controls u, v yield the same values for then they yield the same value for measurable control can be replaced with a Since does not depend on u, we conclude that any bang-bang control with a single switch: and

46 3rd order Nilpotency In this case, The HS system:

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

48 For more information, see the survey paper: “Stability analysis of switched systems using variational principles: an introduction”, Automatica 42(12), , Available online: