Robust Non-fragile Controllers Design for Uncertain Linear Switched Systems - Intelligent Computing for Sustainable Energy and Environment - page 253

Proof. The proof is very similar to the one for Theorem 1 with some modifica-
tions. For the sake of space, the detailed proof is omitted here.

In what follows, we will show that the design method of the robust non-fragile
controller based on the LMI framework [28].
Firstly, we give the sucient condition for the existence of non-fragile state
feedback controller with additive control gain perturbations of form (4).

Theorem 3. 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 , M ,ifthereexist X i > 0 ,W i , and the state feedback

gain matrix K i = W i X − 1

∀ i ∈

(if it exists), such that the LMIs condition

i

⎡
⎣

⎤
⎦

Γ i X i L i X i N i W i M i 0 E i X i N 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 (23)

∗∗ ∗ ∗− Ξ ( ε 4 i )0 0

∗∗ ∗ ∗ ∗−

ε 4 i I 0
∗∗ ∗ ∗ ∗ ∗− ε 4 i I

holds, then the closed-loop system (21) with additive control gain perturbations
(4) is exponentially stable with stability margin λ under arbitrary switching signal
in terms of average dwell-time satisfying τ D ≥ τ D .In(23), Γ i = He { A i X i +

+ ε 1 i D i D i + ε 2 i B i F i F i B i + ε 3 i E i E i +2 λ 0 X i , Ξ ( ε 4 i )= I

B i W i } − ε 4 i ( I +

M i F i F i M i ) .

Proof. Using Lemma 2, we can get

ε 1 i P i D i D i P i + ε − 1

1 i L i L i ,

He { P i ΔA i }≤

ε 2 i P i B i F i F i B i P i + ε − 1

2 i N i N i ,

He { P i B i ΔK i }≤

ε 3 i P i E i E i P i + ε − 1

3 i K i M i M i K i .

He { P i ΔB i K i }≤

Due to He { P i ΔB i ΔK i } = He { P i E i Δ B i ( t ) M i F i Δ K i ( t ) N i } , one has

( P i E i Δ B i ( t )+ N i Δ K i ( t ) F i M i ) He { P i ΔB i ΔK i }≤ ×

( P i E i Δ B i ( t )+ N i Δ K i ( t ) F i M i ) T .

Substituting the above inequalities into (22), it yields

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 + ε 1 i P i D i D i P i + ε − 1

1 i L i L i +

3 i K i M i M i K i +
( P i E i Δ B i ( t )+ N i Δ K i ( t ) F i M i )( P i E i Δ B i ( t )+ N i Δ K i ( t ) F i M i ) T < 0 .
(24)

ε 2 i P i B i F i F i B i P i + ε − 1

2 i N i N i + ε 3 i P i E i E i P i + ε − 1

Let us introduce the matrix X i = P − i and consider the change of variable
W i = K i X i . Then, pre- and post-multiplying (24) by P − i , and using Lemma 1,