Robust Non-fragile Controllers Design for Uncertain Linear Switched Systems - Intelligent Computing for Sustainable Energy and Environment - page 250

means that the i j th subsystem is active when t [ t j ,t j +1 ). A σ ,B σ are con-

stant matrices of appropriate dimensions. The uncertainties ΔA σ and ΔB σ are
assumed to satisfy the following assumption.

∈

Assumption 1. The uncertainties are in the form:

ΔA σ = D σ Δ A σ ( t ) L σ ,ΔB σ = E σ Δ B σ ( t ) M σ (2)

where D σ ,L σ ,E σ and M σ are the constant matrices with appropriate dimen-
sions, and ΔA σ and ΔB σ are the unknown, real and possible time-varying
matrices satisfying Δ A σ ( t ) Δ A σ ( t ) 0and Δ B σ ( t ) Δ B σ ( t ) ≤ I,t ≥ ≤ I,t ≥ 0,

respectively.

For the system (1), we design the state feedback controller as follows

u ( t )=( K σ + ΔK σ ) x ( t ) (3)

R m×n denotes the state feedback gain matrix and ΔK σ denotes
the control gain perturbation matrix. In this paper, we consider the following
two forms of the control gain perturbations.
(i) the additive form:

where K σ ∈

ΔK σ = F σ Δ K σ ( t ) N σ (4)

(ii) the multiplicative form:

ΔK σ = F σ Δ K σ ( t ) N σ K σ (5)

where F σ ,N σ ,F σ and N σ are the known constant matrices, Δ K σ ( t )and Δ K σ ( t )
denote time-varying uncertainties satisfying

I, Δ K σ ( t ) Δ K σ ( t )

Δ K σ ( t ) Δ K σ ( t ) ≤ ≤ I.

To conclude this section, we recall the following lemmas which will be used in
the proof of our main results.

Lemma 1 (Schur Complement). For any given constant real symmetric ma-

trix P = P 11 P 12
P 12 P 22

, the following three arguments are equivalent

( i ) P< 0;
( ii ) P 22 < 0 ,P 11 − P 12 P − 1

22 P 12 < 0;

P 12 P − 1 ( iii ) P 11 < 0 ,P 22 −

11 P 12 < 0 .

(6)

Lemma 2. Let U , V be real matrices of appropriate dimensions. Then, for any
matrix Q> 0 of appropriate dimension and any scalar ε> 0 , such that

UV + V T U T ε − 1 UQ − 1 U T + εV T QV .

≤ (7)