1 Stability Analysis of Linear Switched Systems: An Optimal Control Approach Michael Margaliot School of Elec. Eng. Tel Aviv University, Israel Joint work with: Gideon Langholz (TAU), Daniel Liberzon (UIUC), Michael S. Branicky (CWRU), Joao Hespanha (UCSB). Part 1

2 Overview Switched systems Global asymptotic stability The edge of stability Stability analysis:  An optimal control approach  A geometric approach  An integrated approach Conclusions

3 Switched Systems Systems that can switch between several possible 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

Linear Systems Solution: 9 Theorem: Definition: The system is globally asymptotically stable if A is called a Hurwitz matrix. stability

10 Linear Switched Systems A system that can switch between them: Two (or more) linear systems: 10

11 Stability Linear switched system: Definition: Globally uniformly asymptotically stable (GUAS): AKA, “stability under arbitrary switching”. 11 for any In other words,

12 A Necessary Condition for GUAS The switching law yields Thus, a necessary condition for GUAS is that both are Hurwitz. Then instability can only arise due to repeated switching. 12

13 Why is the GUAS problem difficult? Answer 1: The number of possible switching laws is huge. 13

14 Why is the GUAS problem difficult? 14 Answer 2: Even if each linear subsystem is stable, the switched system may not be GUAS.

15 Why is the GUAS problem difficult? Answer 2: Even if each linear subsystem is stable, the switched system may not be GUAS. 15

16 Stability of Each Subsystem is Not Enough A multi-controller scheme plant controller 1 + switching logic controller 2 Even when each closed-loop is stable, the switched system may not be GUAS.

17 Easy Case #1 A trajectory of the switched system: Suppose that the matrices commute: 17 Then and since both matrices are Hurwitz, the switched system is GUAS.

18 Easy Case #2 Suppose that both matrices are upper triangular: Then 18 so Nowso This proves GUAS.

19 Optimal Control Approach Basic idea: (1) A relaxation: linear switched system → bilinear control system (2) characterize the “most destabilizing” control (3) the switched system is GUAS iff Pioneered by E. S. Pyatnitsky (1970s). 19

Optimal Control Approach Relaxation: the switched system: → a bilinear control system: where is the set of measurable functions taking values in [0,1]. 20

The bilinear control system (BCS) is globally asymptotically stable (GAS) if: Theorem The BCS is GAS if and only if the linear switched system is GUAS. 21 Optimal Control Approach

The most destabilizing control: Fix a final time. Let Optimal control problem: find a control that maximizes Intuition: maximize the distance to the origin. 22

Optimal Control Approach and Stability Theorem The BCS is GAS iff 23

Edge of Stability GAS 24 The BCS: Consider GAS original BCS

Edge of Stability GAS 25 The BCS: Consider GAS original BCS Definition: k* is the minimal value of k>0 such that GAS is lost.

Edge of Stability 26 The BCS: Consider Definition: k* is the minimal value of k>0 such that GAS is lost. The system is said to be on the edge of stability.

Edge of Stability 27 The BCS: Consider Definition: k* is the minimal value of k>0 such that GAS is lost. Proposition: our original BCS is GAS iff k*> k 0 1 k k*

Edge of Stability 28 The BCS: Consider Proposition: our original BCS is GAS iff k*>1. → we can always reduce the problem of analyzing GUAS to the problem of determining the edge of stability.

Edge of Stability When n=2 29 Consider The trajectory x* corresponding to u*: A closed periodic trajectory

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

31 Solving Optimal Control Problems 1. The HJB equation. Intuition: there exists a function and V can only decrease on any other trajectory of the system.

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

33 The HJB for a BCS: Hence, In general, finding is difficult. Note: u* depends on only.

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

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

36 Result #1 (Margaliot & Langholz, 2003) An explicit solution for the HJB equation, when n=2, and {A,B} is on the “edge of stability”. This yields an easily verifiable necessary and sufficient condition for stability of second-order switched linear systems.

37 Basic Idea The HJB eq. is: Thus, Let be a first integral of that is, Then is a concatenation of two first integrals and

38 Example: where and

39 Nonlinear Switched Systems with GAS. Problem: Find a sufficient condition guaranteeing GAS of (NLDI).

40 Lie-Algebraic Approach For the sake of simplicity, we present the approach for LDIs, that is, and

41 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.

42 Lie Brackets and Geometry Consider A calculation yields:

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

44 Nilpotency We saw that [A,B]=0 implies GAS. What if [A,[A,B]]=[B,[A,B]]=0? Definition: k’th order nilpotency - all Lie brackets involving k terms vanish. [A,B]=0 → 1st order nil. [A,[A,B]]=[B,[A,B]]=0 → 2nd order nil.

45 Nilpotency and Stability We saw that 1st order nilpotency Implies GAS. A natural question: Does k’th order nilpotency imply GAS?

46 Some Known Results Switched linear systems: k=2 implies GAS (Gurvits,1995). k 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)

47 A Partial Answer Result #2 (Margaliot & Liberzon, 2004) 3rd order nilpotency implies GAS. Proof: Consider the WCSL Define the switching function

48 Differentiating m(t) yields 2nd order nilpotency    no switching in the WCSL! Differentiating again, we get 3rd order nilpotency    up to a single switching in the WCSL.

49 Singular Arcs If m(t)  0, then the Maximum Principle provides no direct information. Singularity can be ruled out using the auxiliary system.

50 Summary Parking cars is an underpaid job. Stability analysis is difficult. A natural and useful idea is to consider the worst-case trajectory. Switched systems and differential inclusions are important in various scientific fields, and pose interesting theoretical questions.

Summary: Optimal Control Approach Advantages: reduction to a single control leads to necessary and sufficient conditions for GUAS allows the application of powerful tools (high-order MPs, HJB equation, Lie- algebraic ideas,….) applicable to nonlinear switched systems Disadvantages:  requires characterizing  explicit results for particular cases only 51

52 1. Margaliot. “Stability analysis of switched systems using variational principles: an introduction”, Automatica, 42: , Sharon & Margaliot. “Third-order nilpotency, nice reachability and asymptotic stability”, J. Diff. Eqns., 233: , Margaliot & Branicky. “Nice reachability for planar bilinear control systems with applications to planar linear switched systems”, IEEE Trans. Automatic Control, 54: , Available online: More Information