Received 10 November 2014; revised 6 December 2014; accepted 25 December 2014

ABSTRACT

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.

Keywords:

Common Quadratic Lyapunov Functions, Uncertain System, Gradient Method, Bendixson Theorem

1. Introduction

Let linear uncertain system

(1)

be given where, are real matrices. Consider the following matrix inequalities

(2)

where and the symbol “” stands for positive definiteness. The matrix P is called a common solution to (2).

If the system ( 2) has a common solution, then this system is uniformly asymptotically stable [1] .

The problem of existence of common positive definite solution P of (2) has been studied in a lot of works (see [1] - [7] and references therein). Numerical solution for common P via nondifferentiable convex optimization has been discussed in [8] .

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 [8] shows that our approach gives better result in some cases.

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

where is a box, is there a stable member? We consider a sufficient condition which follows from the Bendixson theorem [9] .

2. Gradient Method

According to [2] , let be the set (subspace) of dimensional symmetric block-diagonal matrices of the form where is symmetric.

Let be a basis of, ,

(3)

Then has CQLF there exists such that. In this case the matrix is a common solution to (2) where

The function is positive homogenous. Therefore the vector can be restricted to the condition. The advantage of the restriction shows the following proposition.

Proposition 1. Let be the unit sphere, let the function be positive homo-

geneous and be differentiable at. Assume that. Then

where, denotes the gradient and denotes the scalar product.

Proof: Since is positive homogeneous, it increases in the direction of the vector a: for,

.

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 (Figure 1).

Consider the following optimization problem

Since the matrix is symmetric, the function (3) can be written as

The gradient vector of at a point a is:

(4)

where is the unit eigenvector of corresponding to the simple maximum eigenvalue [2] .

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

Algorithm 1.

Step 1. Take an initial point. Compute. If, find t such that the line

intersects the unit sphere (Figure 2).

Step 2. Take where satisfies the condition. If, is re-

quired point. Otherwise find t such that the line intersects the unit sphere and repeat the procedure.

Example 1. Consider the switched system

where

are Hurwitz stable matrices. Let

For

Take the initial point, then

is positive definite. Eigenvalues of the matrix

are

Maximum eigenvalue 4.015 is simple and the corresponding unit eigenvector is

Gradient of the function at is

The vector should be on the six dimensional unit sphere. Therefore and

After 9 steps, we get where

is a common positive definite solution for and.

The same problem solved by the algorithm from [8] gives answer only after 70 steps. We have solved a number of examples using the above gradient algorithm and by the algorithm from [8] . These examples show that this algorithm is faster than the algorithm from [8] in some cases.

As the comparison with the algorithm from [8] is concerned, the algorithm from [8] at each step uses the gradient only one maximum eigenvalue function, i.e. at 1 step it uses the gradient of, at 2 step the gradient of and so on. This procedure delays the convergence. In our algorithm we use the function and the corresponding gradient direction decreases the greates maximum eigenvalue.

On the other hand an obviously advantage of the method from [8] is the choose of the step size, which is given by an exact formula, whereas our step size is determined by the intersection of the corresponding rays with the unit sphere.

3. Sufficient Condition for a Stable Member

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 matrix A can be decomposed

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. ([9] , p. 40) If A is an matrix, and ,

are the eigenvalues of A, B then

Bendixson’s theorem leads to the following.

Proposition 2. Let the family be given and is the symmetric part of. Then

1) If there exists such that is Hurwitz stable then is also Hurwitz stable,

2) If there exists such that is positive stable (all eigenvalues lie in the open right half plane) then is also positive stable.

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

In the case of affine family

where, is a box or, the searching procedure for stable element in can be effectively solved by powerful tools of Linear Matrix Inequalities (Matlab’s LMI Toolbox).

In the non-affine case of the family the gradient algorithm for a stable element in is applicable.

Example 2. Consider affine family

. Then

LMI method applied to the matrix inequality problem gives the value within a few seconds

and, and consequently is stable.

LMI method applied to the inequality gives also

so the family contains positive stable matrix.

We have investigated Example 2 by the algorithm from [10] and positive answer is obtained after about 100 seconds.

Example 3. Consider non-affine family

. Here

Consider the function

We are looking for satisfying. If for some the maximal eigenvalue is simple then is differentiable at and its gradient can be easily calculated (by the analogy with (4)).

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

(see Table 1). The step size t is chosen from the decreasing condition of the function: t must be chosen such that

This example has been solved by the algorithm from [10] as well. Positive answer has been obtained only after

55 steps. We start with and the algorithm from [10] gives another stabilizing point

.

The eigenvalues of are,.

4. Conclusion

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.

References

NOTES

^*Corresponding author.