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3 Non-fragile Controller Design
In this section, we will show how to design state feedback gain
K
i
and switching
law
σ
(
t
) for switched linear uncertain system (1) to be exponentially stable.
Definition 1.
[19] For the switched signal
σ
and any
t
≥
τ
≥
0
,let
N
σ
(
t,τ
)
denote the system switching times in the open interval
(
τ,t
)
.If
N
0
+
t
−
τ
τ
D
N
σ
(
t,τ
)
≤
(8)
holds for
τ
D
>
0
and
N
0
≥
0
,then
τ
D
is called average dwell-time and
N
0
is
said to be the chatter bound.
Definition 2.
[15] The switched system (1) is exponentially stable if all the state
trajectories satisfy
k
1
e
−k
2
t
x
(
t
)
≤
x
(0)
(9)
for some
k
1
>
0
and
k
2
>
0
,
k
2
is called stability margin.
We first consider the nominal system of the switched system (1). That is
x
(
t
)=
A
σ
x
(
t
)+
B
σ
u
(
t
)
(10)
and the state feedback controller
u
(
t
)=
K
σ
x
(
t
)
.
(11)
Theorem 1.
Given a scalar
λ
0
>
0
, if there exist positive matrix
P
i
and matrix
K
i
, such that
A
i
P
i
+
P
i
A
i
+
K
i
B
i
P
i
+
P
i
B
i
K
i
+2
λ
0
P
i
<
0
,i∈ M
(12)
holds, then the closed-loop system (10) and (11) is globally exponentially stable
with stability margin
λ
∈
(0
,λ
0
)
for any switching signal with average dwell-time
satisfying
ln
μ
2(
λ
0
−
τ
D
=
τ
D
≥
λ
)
where
a
=sup
i∈M
λ
max
(
P
i
)
,b
=inf
i∈M
λ
min
(
P
i
)
and
μ
=
b
.
Proof.
Define the piecewise Lyapunov functional candidate
V
(
x
(
t
)) =
x
T
P
σ
x
,
which is positive definite since
P
σ
∈
R
n×n
is a positive definite matrix. When
the i-th subsystem is activated,
V
(
x
(
t
)) =
V
i
(
x
(
t
)) =
x
T
P
i
x
. From the definition
of
a,b,μ
,for
∀
i,j
∈
M
, we have the following two inequalities.
V
i
(
x
(
t
)) =
x
T
P
i
x
μx
T
P
j
x
=
μV
j
(
x
(
t
))
≤
(13)
2
x
T
P
i
x
2
b
x
≤
≤
a
x
(14)
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