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multiple Lyapunov functions. In [16-18], dwell- time have been used to study the
stability of the switched systems. In [19-21] have analyzed the stability of the
switched systems and designed the controllers by means of average dwell-time.
Robustness of control systems to uncertainties has always been the central
issue in feedback control and therefore for uncertain dynamical systems, a large
number of robust controller design methods have been presented [22] and [23].
Because controller implementation is subject to imprecision inherent in analog-
digital and digital-analog conversion, finite word length, and finite resolution
measuring instruments and roundoff errors in numerical computations and any
useful design procedure should generate a controller which also has sucient
room for readjustment of its coecients [24-26]. For linear continuous-time sys-
tems with structured uncertainties existing in the system matrix only, a design
method of a robust non-fragile state feedback controller has been suggested [12].
Also, a design method of a
H
1
controller for linear systems with additive con-
troller gain variations has been derived [27]. Oya, Hagino and Mukaidani [25] con-
sidered the problem of robust non-fragile controllers for linear continuous-time
systems. However, so far the design problem of robust non-fragile controllers
for linear switched systems with uncertainties which are included in both the
system matrix and the input one has not been discussed. From this viewpoint
on the basis of the existing result for quadratic stabilization, we present a de-
sign method of a robust non-fragile controller for linear switched systems with
structured uncertainties existing in both the system matrix and the input one.
In this paper, we show that sucient conditions for the existence of the robust
non-fragile controller are given in terms of linear matrix inequalities (LMIs).
In this paper, we use
P>
0(
0) to denote a positive definite (pos-
itive semi-defined, negative definite, negative semi-definite) matrix.
R
n
is n-
dimensional real space;
R
m×n
is set of all the
m
by
n
matrices. For any vector
or matrix
A
,
A
T
means the transpose of
A
;
He{A}
donates
A
+
A
T
.Inthesym-
metric matrix,
∗
means the symmetric part of the symmetric matrix. For the
real symmetric matrices
A
and
B
,
A<B
(
A ≤ B
)means
A−B
is negative def-
inite (semi-definite) matrix.
I
represents the identity matrix.
λ
min
(
P
)
,λ
max
(
P
)
denote minimal and maximal eigenvalues of matrix
P
.
≥
,<,
≤
·
denotes the usual
2-norm.
2 Preliminaries
Consider the following switched linear uncertain system
x
(
t
)=(
A
σ
+
ΔA
σ
)
x
(
t
)+(
B
σ
+
ΔB
σ
)
u
(
t
)
(1)
R
n
is the state,
u
(
t
)
R
m
is the control input, the right con-
where
x
(
t
)
∈
∈
tinuous function
σ
(
t
):[0
,
+
∞
)
→
M
=
{
1
,
2
,
···
,m
}
is the switching signal,
corresponding to it, the switching sequence
Σ
=
{
x
0
,
(
i
0
,t
0
)
,
(
i
1
,t
1
)
,
···
,
(
i
j
,t
j
)
,
···|
i
j
∈
M
}
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