Stability Analysis of Positive Linear Switched Systems: A Variational Approach

1. Stability Analysis of Positive LinearSwitched Systems:A Variational Approach Michael Margaliot School of Elec. Eng. -Systems Tel Aviv University, Israel Joint work with Lior Fainshil

2. Outline • Stability of linear switched systems • Variational approach to stability analysis Relaxation: a bilinear control system The “most destabilizing” control • Positive linear switched systems • Variational approach Relaxation: a positive bilinear control system Maximizing the spectral radius of the transition matrix Main result: A maximum principle Applications

3. Linear Switched Systems Two (or more) linear systems: A system that can switch between them: Global Uniform Asymptotic Stability (GUAS): AKA, “stability under arbitrary switching”.

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

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

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

7. Switched Systems: An Example plant + controller1 controller2 switching logic 8

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

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

10. Variational Approach 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.

11. Variational Approach The most destabilizing control: Fix T>0. Let Optimal control problem: find a control that maximizes Intuition: maximize the distance to the origin.

12. Variational Approach and Stability Theorem The BCS is GAS iff

13. Variational 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

14. Part 2: Variational Approach for Positive Linear Switched Systems Basic idea: (1) positive linear switched system → positive bilinear control system (PBCS) (2) characterize the “most destabilizing control”

15. Positive Linear Systems Motivation: suppose that the state variables can never attain negative values. In a linear system this holds if i.e., off-diagonal entries are non-negative. Such a matrix is called a Metzler matrix. 16

16. Positive Linear Systems with Theorem An example: 17

17. Positive Linear Systems If A is Metzler then for any so The solution of is transition matrix The transition matrix is non-negative. 18

18. Perron-Frobenius Theorem Definition Spectral radius of a matrix Theorem Suppose that has a real eigenvalue such that: • • The corresponding eigenvectors of , denoted , satisfy • 19

19. Positive Linear Switched Systems: A Variational Approach Relaxation: “Most destabilizing control”: maximize the spectral radius of the transition matrix. 22

20. Positive Linear Switched Systems: A variational Approach Theorem For any T>0, where is the solution at time T of is called the transition matrix corresponding to u. 23

21. Transition Matrix of a Positive System If are Metzler, then admit a real and eigenvalue such that: The corresponding eigenvectors satisfy 24

22. Optimal Control Problem Fix an arbitrary T>0. Problem: find a control that maximizes We refer to as the “most destabilizing” control. 25

23. Relation to Stability Define: Theorem: the PBCS is GAS if and only if 26

24. Main Result: A Maximum Principle Theorem Fix T>0. Consider Let be optimal. Let and let denote the factors of Define and let Then 27

25. Comments on the Main Result 1. Similar to the Pontryagin MP, but with one-point boundary conditions; 2. The unknown play an important role. 28

26. Comments on the Main Result 3. The switching function satisfies: 29

27. Comments on the Main Result The number of switching points in a bang-bang control must be even. 30

28. Main Result: Sketch of Proof Let be optimal. Introduce a needle variation with perturbation width Let denote the corresponding transition matrix. By optimality, 31

29. Sketch of Proof Let Then We know that with Since is optimal, so 32

30. Sketch of Proof Since is optimal, so We can obtain an expression for to first order in as is a needle variation. 33

31. Applications of Main Result are Metzler Assumptions: is Hurwitz Proposition 1 If there exist such that the switched system is GUAS. Proposition 2 If and either the switched system is GUAS. or 34

32. Applications of Main Result Assumptions: are Metzler is Hurwitz then any Proposition 3 If bang-bang control with more than one switch includes at least 4 switches. Conjecture If then the switched system is GUAS. 35

33. Conclusions We considered the stability of positive switched linear systems using a variational approach. The main result is a new MP for the control maximizing the spectral radius of the transition matrix. Further research: numerical algorithms for calculating the optimal control; consensus problems; switched monotone control systems. 36

34. More Information • Margaliot. “Stability analysis of switched systems using variational principles: an introduction”, Automatica, 42: 2059-2077, 2006. • Sharon & Margaliot. “Third-order nilpotency, nice reachability and asymptotic stability”, J. Diff. Eqns., 233: 136-150, 2007. • Margaliot & Branicky. “Nice reachability for planar bilinear control systems with applications to planar linear switched systems”, IEEE Trans. Automatic Control, 54: 1430-1435, 2009. • Fainshil & Margaliot. “Stability analysis of positive linear switched systems: a variational approach”, submitted. • Available online: www.eng.tau.ac.il/~michaelm