Robust Non-fragile Controllers Design for Uncertain Linear Switched Systems - Intelligent Computing for Sustainable Energy and Environment - page 254

it can be deduced that
⎡

⎤

Γ i X i L i X i N i W i M i E i Δ B i ( t )+ X i N i Δ K i ( t ) F i M i
∗−

⎣ ⎦

ε 1 i I 0

0 0

∗∗− ε 2 i I 0 0 < 0 . (25)

∗∗ ∗− ε 3 i I 0

∗∗ ∗ ∗ − I

Furthermore, by simply algebraic manipulation to (25) gives
⎡
⎣

⎨
⎩

⎬
⎭

⎡
⎣

⎤
⎦

⎤
⎦

Γ i X i L i X i N i W i M i E i X i N i Δ K i ( t )

0

0

0000 Δ B i ( t )
0000 F i M i

∗−ε 1 i I 0 0 0

0

∗∗− ε 2 i I 0 0 + He < 0 . (26)

0 0

∗∗ ∗ ε 3 i I 0

0 0

∗∗ ∗ ∗ − I

0 0

UsingLemma2to(26),wecanget
⎡
⎣

⎤
⎦

Γ i + ε − 1

4 i ( E i E i

+ X i N i N i X i ) X i L i X i N i W i M i 0

∗ − ε 1 i I 0 0 0

∗ ∗ − ε 2 i I 0 0 < 0 . (27)

∗ ∗ ∗ − ε 3 i I 0

∗ ∗ ∗ ∗ − Ξ ( ε 4 i )

By applying Lemma 1, it is easy to verify that (27) is equivalent to (23).

Remark 1. In Theorem 3, there are many parameters to choose before solving the
LMIs condition (23). As we known, λ 0 is the stability margin of single subsystem,
which is determined prior. In (23), the inequality holds unless Ξ ( ε 4 i ) > 0, then
we will choose 0 <ε 4 i < 1. In general, the other parameters are always not too
large, such as we can choose them as 1.

Next, the following theorem will give the sucient condition for the existence of
non-fragile state feedback controller with multiplicative control gain perturba-
tions (5).
Theorem 4. Consider system (1), for given scalar ε 1 i > 0 ,ε 2 i > 0 ,ε 3 i >
0 ,ε 4 i > 0 ,λ 0 > 0 ,

∀ i ∈ M ,ifthereexist X i > 0 ,W i , and state feedback gain

matrix K i = W i X − 1 (if it exists), such that the LMIs condition

i

⎡ ⎤

Γ i X i L i W i N T i W i M i E i W i N T

0

i

⎣ ⎦

∗− ε 1 i I

0 0 0 0 0

∗∗− ε 2 i I

0 0 0 0

∗∗ ∗ − ε 3 i I 0 0 0 < 0 (28)

Ξ ( ε 4 i )0 0
∗∗ ∗ ∗ ∗ −ε 4 i I 0
∗∗ ∗ ∗ ∗ ∗−ε 4 i I

∗∗ ∗ ∗−

holds, then the closed-loop system (21) with multiplicative control gain perturba-
tions (5) is exponentially stable with stability margin λ under arbitrary