Stability Analysis of Switched Systems A Variational Approach

Stability Analysis of Switched Systems: A Variational Approach Michael Margaliot School of EE-Systems Tel Aviv University Joint work with Daniel Liberzon (UIUC) 1

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

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

Example 1 server 4

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

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

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

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

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

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

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

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

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

Absolute Stability 14

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

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

Example 17

Trajectory of the Switched System This implies that 18

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

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

Optimal Control Approach Consider as 21

Optimal Control Approach Thm. 1 (Pyatnitsky) If (1) The function then: is finite, convex, positive, and homogeneous (i. e. , ). (2) For every initial condition there exists a solution such that 22

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

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

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

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

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

The Case n=2 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. 28

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

Example: where and 30

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

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

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

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. 34

Lie Brackets and Geometry Consider Then: 35

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

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, B]]=[B, [A, B]]=0 Q: Does k’th order nilpotency imply GAS? 37

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

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

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

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). 41

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

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

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

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

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

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

For more information, see the survey paper: “Stability analysis of switched systems using variational principles: an introduction”, Automatica 42(12), 2059 -2077, 2006. Available online: www. eng. tau. ac. il/~michaelm 48