Stability Analysis of Linear Switched Systems: An Optimal Control Approach 1 Michael Margaliot School of Elec. Eng. Tel Aviv University, Israel Joint work with Lior Fainshil Part 2

Outline 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 2

Linear Systems Solution: 3 Theorem: Definition: The system is stable if A is called a Hurwitz matrix. stability

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

Variational Approach Basic idea: (1) 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). 8

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” 9

Positive Linear Systems 10 Motivation: suppose that the state variables can never attain negative values. In a linear system this holds if Such a matrix is called a Metzler matrix. i.e., off-diagonal entries are non-negative. 10

Positive Linear Systems 11 with Theorem An example: 11

Positive Linear Systems 12 If A is Metzler then for any so transition matrix The solution ofis The transition matrix is a non-negative matrix. 12

Perron-Frobenius Theory 13 Definition Spectral radius of a matrix 13 Example Let The eigenvalues are so

Perron-Frobenius Theorem 14 The corresponding eigenvectors of, denoted, satisfy has a real eigenvalue such that: Theorem Suppose that 14

Some Perturbation Theory 15 Let be a smooth parameter-dependent non-negative matrix. Denote: dominant eigenvalue of dominant eigenvectors of Then, 15

Sketch of Proof 16 Differentiate with respect to 16

Positive Linear Switched Systems: A Variational Approach 17 Relaxation: “Most destabilizing control”: maximize the spectral radius of the transition matrix. 17

Positive Linear Switched Systems: A variational Approach 18 Theorem For any T>0, is called the transition matrix corresponding to u. where is the solution at time T of 18

Transition Matrix of a Positive System 19 If are Metzler, then eigenvaluesuch that: admit a real and The corresponding eigenvectors satisfy 19

Optimal Control Problem 20 Fix an arbitrary T>0. Problem: find a control that maximizes We refer to as the “most destabilizing” control. 20

Relation to Stability 21 Define: Theorem: the PBCS is GAS if and only if 21

Main Result: A Maximum Principle 22 Theorem Fix T>0. Consider Let be optimal. Let and let denote the factors of Define and let Then 22

Comments on the Main Result Similar to the Pontryagin MP, but with one-point boundary conditions; 2. The unknown play an important role. 23

Comments on the Main Result The switching function satisfies: 24

Comments on the Main Result 25 The number of switching points in a bang- bang control must be even. 25

Main Result: Sketch of Proof 26 Let be optimal. Introduce a needle variation with perturbation width Let denote the corresponding transition matrix. By optimality, 26

Sketch of Proof 27 Let Then We know that Since is optimal, so with 27

Sketch of Proof 28 We can obtain an expression for Since is optimal, so to first order in as is a needle variation. 28

29 Applications of Main Result Assumptions: are Metzler is Hurwitz Proposition 1 If there exist such that the switched system is GUAS. Proposition 2 If and either or the switched system is GUAS. 29

30 Applications of Main Result Assumptions:are Metzler is Hurwitz Proposition 3 If then any bang-bang control with more than one switch includes at least 4 switches. Conjecture If switched system is GUAS. then the 30

31 Conclusions We considered the stability of positive switched linear systems using a variational approach. 31 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,…

32 Margaliot. “Stability analysis of switched systems using variational principles: an introduction”, Automatica, 42: , Fainshil & Margaliot. “Stability analysis of positive linear switched systems: a variational approach”, submitted. Available online: More Information