1. Introduction
Let linear uncertain system
(1)
be given where
, ![](//html.scirp.org/file/8-7402584x9.png)
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
![](//html.scirp.org/file/8-7402584x15.png)
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
![]()
![]()
Figure 1. The direction
of the minus gradient.
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
![]()
Table 1. Gradient algorithm for example 3.
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.
NOTES
*Corresponding author.