1. Introduction
Consider the switched system
(1)
where
,
. In Equation (1), the matrix
switches among
matrices
.
Switching signal
is piecewise continuous from the right function
and the switching times are arbitrary. For the switched system (1) with initial condition
and with switching signal
denotes the solution by
.
Definition 1. The origin is uniformly asymptotically stable (UAS) for the system (1) if for every
there exists
such that for every signal
and initial state
with
, the inequality
is satisfied for all
and uniformly on ![]()
![]()
If all systems in (1) share a common quadratic Lyapunov function (CQLF)
then the switched
system is UAS (T denotes the transpose).
In this case there exists a common
such that
(2)
and
is called a common solution to the set of Lyapunov matrix inequalities (2).
The problem of existence of common positive definite solution
of (2) has been studied in a lot of works (see [1] -[9] and references therein). Numerical solution for common
via nondifferentiable convex optimization 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
be a compact, for
the matrix
is a real
matrix. Is there a Hurwitz stable member (all eigenvalues lie in the open left half plane) in the family
![]()
or equivalently is there
such that
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.
2. Common Quadratic Lyapunov Function
For the switched system
![]()
consider the problem of determination of CQLF
where
. 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
such that
. (3)
Let
be
and
be an
symmetric matrix defined as
![]()
Define
(4)
If there exists
such that
and
then the matrix
is required solution. This problem can be reduced to the minimization of a convex function under convex constraints.
Consider the following convex minimization problem
(5)
Let
be a convex set and
be convex function. The vector
is said to be a subgradient of
at
if for all ![]()
.
The set of all subgradients of
at
is denoted by
. If
is an interior point of
then the set
is nonempty and convex. The following proposition follows from nondifferentiable optimization theory.
Proposition 1. Let
be defined as
(6)
where
is compact,
is continuous and differentiable in
. Then
![]()
where
is the set of all maximizing elements
in (6), i.e.
.
If for a given
the maximizing element is unique, i.e.
then
is differentiable at
and its gradient is
![]()
In the case of the Function (4)
![]()
If for the given
the maximizing
is unique and
is a simple eigenvalues, the differentiability of
at the point
is guaranteed [13] .
We investigate problem (5) by Kelley’s cutting-plane method.
This method converts the problem (5) to the problem
(7)
where
,
,
,
.
Let
be a starting point and
be
distinct points.
At the
th iteration, the cutting-plane algorithm solves the following LP problem
(8)
where
denotes a subgradient of
at
.
Let
be the minimizer of the problem (8).
If
satisfies the inequality
, where
is a tolerance then
is an approx-
imate solution of the problem (7).
Otherwise define
as the index for the most negative
, update the constraints in (8) by including the linear constraint
![]()
and repeat the procedure.
Recall that our aim is to find
such that
and
, but not the solution of the minimization problem (5), (7).
Theorem 2. If there exists
such that
![]()
where
is the minimizer of the problem (8), then the matrix
is a common solution to (3).
Proof:
![]()
![]()
and by (5),
is a common solution to (3).
For the problem (5), (7) Kelley’s method gives the following
Algorithm 1.
Step 1. Take an initial point
. Compute
and
. If
and
stop; otherwise continue.
Step 2. Determine
by solving LP problem in (8). If
and
then stop; otherwise continue. Set
, update the constraints in (8) and repeat the procedure.
Example 1. Consider the switched system
![]()
where
![]()
are Hurwitz stable matrices.
Choose the initial point
, then
![]()
,
and![]()
We obtain
by solving LP problem in (8). Calculations give the following Table 1, and
![]()
Since
and
,
![]()
Table 1. Kelley’s algorithm for Example 1.
![]()
is a common positive definite solution for
![]()
3. Stable Member in a Polytope
This part is devoted to the following question: Given a matrix family
where
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
. In this algorithm the uncertainty vector
varies in the whole space
. On the other hand we consider the case where
varies in a box
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
symmetric matrices and
![]()
![]()
where
,
.
Consider the problem
![]()
Theorem 3. There is a stable matrix in the family
if and only if ![]()
Proof:
![]()
![]()
![]()
By Lyapunov theorem, the matrix
is stable.
Example 2. Consider the family of matrices
![]()
where
![]()
For
,
is unstable. We apply the gradient algorithm to find a stable member in the family.
Let
and
. So
![]()
Then
![]()
Maximum eigenvalue of this matrix and its corresponding unit eigenvector are
![]()
respectively. Gradient of the function
at
is
![]()
The first tencomponent of the vector
should be on the ten dimensional unit sphere. Therefore
and
![]()
After 4 steps, we get
![]()
and
. Therefore
is stable.
4. 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.
NOTES
*Corresponding author.