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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.

Consider the switched system

where

Switching signal

Definition 1. The origin is uniformly asymptotically stable (UAS) for the system (1) if for every

If all systems in (1) share a common quadratic Lyapunov function (CQLF)

system is UAS (T denotes the transpose).

In this case there exists a common

and

The problem of existence of common positive definite solution

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

or equivalently is there

For the switched system

consider the problem of determination of CQLF

Consider the problem of existence of a common

Let

Define

If there exists

Consider the following convex minimization problem

Let

The set of all subgradients of

Proposition 1. Let

where

where

If for a given

In the case of the Function (4)

If for the given

We investigate problem (5) by Kelley’s cutting-plane method.

This method converts the problem (5) to the problem

where

Let

At the

where

Let

If

imate solution of the problem (7).

Otherwise define

and repeat the procedure.

Recall that our aim is to find

Theorem 2. If there exists

where

Proof:

and by (5),

For the problem (5), (7) Kelley’s method gives the following

Algorithm 1.

Step 1. Take an initial point

Step 2. Determine

Example 1. Consider the switched system

where

are Hurwitz stable matrices.

Choose the initial point

We obtain

Since

k | |||
---|---|---|---|

1 | ‒209.7383 | ||

2 | ‒127.1153 | ||

3 | ‒106.2473 | ||

4 | ‒63.4433 | ||

14 | 0.2694 | ||

15 | 0.2075 |

is a common positive definite solution for

This part is devoted to the following question: Given a matrix family

In [

Let

where

Consider the problem

Theorem 3. There is a stable matrix in the family

Proof:

By Lyapunov theorem, the matrix

Example 2. Consider the family of matrices

where

For

Let

Then

Maximum eigenvalue of this matrix and its corresponding unit eigenvector are

respectively. Gradient of the function

The first tencomponent of the vector

After 4 steps, we get

and

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.