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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 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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