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Asymptotic stability of linear systems is closely related to Hurwitz stability of the system matrices. For uncertain linear systems we consider stability problem through common quadratic Lyapunov functions (CQLF) and problem of stabilization by linear feedback.

Let linear uncertain system

be given where

where

If the system ( 2) has a common

The problem of existence of common positive definite solution P of (2) has been studied in a lot of works (see [

In the first part of the paper we treat the problem (2) as a nonconvex optimization problem (minimization of a convex function under nonconvex constraints) and apply a modified gradient method. The comparison with [

In the second part we consider the stabilization problem, i.e. the following question: for the affine family

where

According to [

Let

Then

The function

Proposition 1. Let

geneous

where

Proof: Since

Therefore the directional derivative of f at a in the direction of a is positive

On the other hand

and

Proposition 1 shows that under its assumption the minus gradient vector at the point a is directed into the unit ball (

Consider the following optimization problem

Since the matrix

The gradient vector of

where

Well-known gradient algorithm in combination with Proposition 1 gives the following.

Algorithm 1.

Step 1. Take an initial point

intersects the unit sphere

Step 2. Take

quired point. Otherwise find t such that the line

Example 1. Consider the switched system

where

are Hurwitz stable matrices. Let

For

Take the initial point

is positive definite. Eigenvalues of the matrix

are

Maximum eigenvalue 4.015 is simple and the corresponding unit eigenvector is

Gradient of the function

The vector

After 9 steps, we get

The same problem solved by the algorithm from [

As the comparison with the algorithm from [

On the other hand an obviously advantage of the method from [

In this section we consider a sufficient condition for a stable member which is obtained by using Bendixson’s theorem.

If a matrix is symmetric then it is stable if and only if it is negative definite. Therefore if a family consists of symmetric matrices then searching for stable element is equivalent to the searching for negative definite one.

On the other hand every real

where B is symmetric and C is skew-symmetric. Bendixson’s theorem gives important inequalities for the eigenvalues of A, B and C.

Theorem 1. ([

Bendixson’s theorem leads to the following.

Proposition 2. Let the family

1) If there exists

2) If there exists

Proposition 2 gives a sufficient condition for the existence of a stable element.

In the case of affine family

where

In the non-affine case of the family

Example 2. Consider affine family

LMI method applied to the matrix inequality problem

and

LMI method applied to the inequality

so the family

We have investigated Example 2 by the algorithm from [

Example 3. Consider non-affine family

Consider the function

We are looking for

For this example, gradient method gives solution after 7 steps:

(see

This example has been solved by the algorithm from [

multiplicity | ||||
---|---|---|---|---|

0 | 11.079 | 1 | ||

1 | 10.632 | 1 | ||

2 | 9.910 | 1 | ||

3 | 8.634 | 1 | ||

4 | 6.712 | 1 | ||

5 | 3.840 | 1 | ||

6 | 0.444 | 1 | ||

7 | −2.404 |

55 steps. We start with

The eigenvalues of

In the first part of the paper, we consider the stability problem of a matrix polytope through common quadratic Lyapunov functions. We suggest a modified gradient algorithm. In the second part by using Bendixson’s theorem a sufficient condition for a stable member is given.