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

Outline Stability of linear switched systems Variational approach to stability analysis: -Relaxation: a bilinear control system -The “most destabilizing” control 2 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

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? 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 8 plant controller 1 + controller 2 switching logic 8

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). 9

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

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 16 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. 16

Positive Linear Systems 17 with Theorem An example: 17

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

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

Some Perturbation Theory 20 Let be a smooth parameter-dependent non-negative matrix. Denote: dominant eigenvalue of dominant eigenvectors of Then, 20

Sketch of Proof 21 Differentiate with respect to 21

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

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

Transition Matrix of a Positive System 24 If are Metzler, then eigenvaluesuch that: admit a positive The corresponding eigenvectors satisfy 24

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

An Example 26 T

Relation to Stability 27 Define: Theorem: the PBCS is GAS if and only if 27

Main Result: A Maximum Principle 28 Theorem Fix T>0. Consider Let be optimal. Let and let denote the factors of Define and let Then 28

Comments on the Main Result Similar to the Pontryagin MP, but with one-point boundary conditions; 2. The unknown play an important role. 29

Comments on the Main Result The switching function satisfies: 30

Comments on the Main Result 31 The number of switching points in a bang- bang control must be even. 31

An Example 32

Main Result: Sketch of Proof 33 Let be optimal. Introduce a needle variation with perturbation width Let denote the corresponding transition matrix. By optimality, 33

Sketch of Proof 34 Let Then We know that Since is optimal, so with 34

Sketch of Proof 35 We can obtain an expression for Since is optimal, so to first order in as is a needle variation. 35

36 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. 36

37 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 37

38 Conclusions We considered the GUAS of positive switched linear systems using a variational approach. 38 The main result is a new MP for the control maximizing the spectral radius of the transition matrix.

39 Margaliot. “Stability analysis of switched systems using variational principles: an introduction”, Automatica, Sharon & Margaliot. “Third-order nilpotency, nice reachability and asymptotic stability”, JDE, Margaliot & Branicky. “Nice reachability for planar bilinear control systems with applications to planar linear switched systems”, IEEE TAC, *Fainshil & Margaliot. “Stability analysis of positive linear switched systems: a variational approach”, SICON, *Hochma & Margaliot. “High-order maximum principles for the stability analysis of positive bilinear control systems“, OCAM, to appear, Available online: More Information