Switched systems, common solution to the Lyapunov equation and semidefinite programming

Speaker: Helena Smigoc (UCD)

Time: 4:00PM
Date: Mon 8th October 2007

Location: Mathematical Sciences Seminar Room

Matrix A is called (Hurwitz) stable if all its eigenvalues lie in the open left half of the complex plane. If A is a stable matrix, then the linear time-invariant system for A is stable. A classical result of Lyapunov states that a matrix A is stable if and only if for arbitrary Hermitian positive definite Q, the Lyapunov equation AP+PA^*=-Q admits a positive definite
solution P.

By a switched system we mean a dynamical system consisting of a family of linear time-invariant systems and a rule that orchestrates the switching between them. To guarantee a stability of such switched systems under arbitrary switching signals, it is sufficient to show that there exists a common solution to the Lyapunov equations associated with all the linear time-invariant systems defining the switched system.

Determining the existence of a common solution to the Lyapunov equation for a finite set of linear time-invariant systems is very difficult. In this talk we will discuss some special cases for which nice easily checkable conditions can be found. In particular, we will present a solution for 2x2 matrices and for matrices whose difference has rank one. We will show how methods from semidefinite programming can be applied to this problem.

(This talk is part of the Algebra/Claude Shannon Institute series.)