On Two Problems for Matrix Polytopes

We consider two problems from stability theory of matrix polytopes: the existence of common quadratic Lyapunov functions and the existence of a stable member. We show the applicability of the gradient algorithm and give a new sufficient condition for the second problem. A number of examples are considered.


Introduction
Consider the switched system where ( ) n x t ∈   , 0 t ≥ .In Equation (1), the matrix A switches among N matrices

V x
x Px = then the switched system is UAS (T denotes the transpose).
In this case there exists a common 0 P > such that ( ) and P is called a common solution to the set of Lyapunov matrix inequalities (2).The problem of existence of common positive definite solution P of (2) has been studied in a lot of works (see [1]- [9] and references therein).Numerical solution for common P via nondifferentiable convex optimiza- tion has been discussed in [10].
In the first part of the paper, the problem of existence of CQLF is investigated by Kelley's method.This method is applied when CQLF problem is treated as a convex optimization problem.
Second part of the paper is devoted to the following question: Let l B ⊂  be a compact, for q B ∈ the matrix ( ) A q is a real n n × matrix.Is there a Hurwitz stable member (all eigenvalues lie in the open left half plane) in the family or equivalently is there * q B ∈ such that ( ) A q is stable?This problem is one of the hard and important problems in control theory (see [11]).Numerical solution of this problem is considered in [12].In this paper we reduce this problem to a non-convex optimization problem.

Common Quadratic Lyapunov Function
For the switched system x Px = where 0 P > .We are going to investigate it by Kelley's cutting-plane method.This method gives new sufficient condition (Theorem 2) and new algorithm (Algorithm 1) which is more effective in comparison with the algorithm from [10].
Consider the problem of existence of a common 0 P > such that ( ) , , , r x x x x =  and P be an n n × symmetric matrix defined as ( ) ( ) If there exists * x such that ( ) Consider the following convex minimization problem ( ) ( ) Let n X ⊂  be a convex set and : F X →  be convex function.The vector n g ∈  is said to be a sub- gradient of ( ) The set of all subgradients of ( ) is nonempty and convex.The following proposition follows from nondifferentiable optimization theory.

Proposition 1. Let ( )
x φ be defined as where Y is compact, ( ) where ( ) Y x is the set of all maximizing elements y in (6), i.e. ( If for a given x the maximizing element is unique, i.e. ( ) In the case of the Function ( : maximizes , is a corresponding unit eigenvector . If for the given x the maximizing i is unique and ( ) is a simple eigenvalues, the differentiability of φ at the point x is guaranteed [13].
We investigate problem (5) by Kelley's cutting-plane method.This method converts the problem (5) to the problem where At the ( ) where z be the minimizer of the problem (8).
, where ε is a tolerance then * k z is an approx- imate solution of the problem (7).
Otherwise define * j as the index for the most negative ( ) * k j c z , update the constraints in (8) by including the linear constraint ( ) ( )( ) and repeat the procedure.
Recall that our aim is to find * x such that ( ) * 0 P x > and ( ) * 0 x φ < , but not the solution of the minimiza- tion problem ( 5), (7).

Stable Member in a Polytope
This part is devoted to the following question: Given a matrix family ∈ where l B ⊂  is a compact, is there a stable matrix in this family?
In [12], a numerical algorithm has been proposed for a stable member in the affine matrix family ( )

{ }
: l A q q ∈  .In this algorithm the uncertainty vector q varies in the whole space l  .On the other hand we consider the case where q varies in a box l B R ⊂ and use the gradient algorithm for minimization of the nonconvex maximum eigenvalue function.By choosing appropriate step-size, we obtain the convergence.Let ( )  be a basis for the subspace of n n × symmetric matrices and ( , , , k q q q q =  .Consider the problem , minimize.min 0 There is a stable matrix in the family ( ) A q if and only if there exists , such that 0 , , , 1,1 A q A q A q A q A q q q = + + + ∈ − where is unstable.We apply the gradient algorithm to find a stable member in the family.

4
A q is stable.

Conclusion
Two important problems from control theory are considered: the existence of common quadratic Lyapunov functions for switched linear systems and the existence of a stable member in a matrix polytope.We obtain new conditions which give new effective computational algorithms.

Definition 1 .
arbitrary.For the switched system (1) with initial condition ( ) The origin is uniformly asymptotically stable (UAS) for the system (1) if for every 0 If all systems in (1) share a common quadratic Lyapunov function (CQLF) ( ) T

<
then the matrix ( ) * P x is required solution.This problem can be reduced to the minimization of a convex function under convex constraints.