Part 2 Stability Analysis of Linear Switched Systems

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

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 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

Variational Approach Pioneered by E. S. Pyatnitsky (1970 s). Basic idea: (1) relaxation: linear switched system → bilinear control system (2) characterize the “most destabilizing control” (3) the switched system is GUAS iff 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 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. 10

Positive Linear Systems with Theorem An example: 11

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

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

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

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

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. 18

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

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

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

Main Result: A Maximum Principle 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 1. Similar to the Pontryagin MP, but with one-point boundary conditions; 2. The unknown play an important role. 23

Comments on the Main Result 3. The switching function satisfies: 24

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

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

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

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

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 the switched system is GUAS. or 29

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. 30

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, … 31

More Information Margaliot. “Stability analysis of switched systems using variational principles: an introduction”, Automatica, 42: 2059 -2077, 2006. Fainshil & Margaliot. “Stability analysis of positive linear switched systems: a variational approach”, submitted. Available online: www. eng. tau. ac. il/~michaelm 32