Robust Non-fragile Controllers Design for Uncertain Linear Switched Systems - Intelligent Computing for Sustainable Energy and Environment - page 252

Assume that [ t k ,t k +1 ) is any switching interval, in which the i-th subsystem is
activated, then the time derivative of the V ( x ( t )) along the trajectory of system
(10) can be calculated as

V ( x ( t )) = x T ( A i P i + P i A i + K i B i P i + P i B i K i )
≤− 2 λ 0 x T P i x =

− 2 λ 0 V ( x ( t )) . (15)

From (15), it can be deduced that

e − 2 λ 0 ( t−t k ) V ( x ( t k )) , t

V ( x ( t )) ≤ ≥ t k . (16)

Let 0 = t 0 <t 1 < <t k = t N σ ( t, 0) denote the switching sequences in the

interval [0 ,t ). Substituting (13) into (16) yields

···

e − 2 λ 0 ( t−t k ) V ( x ( t k )) μ e − 2 λ 0 ( t−t k− 1 ) V ( x ( t k− 1 ))

V ( x ( t )) ≤ ≤ ≤···

μ N σ ( t, 0) e − 2 λ 0 t V ( x (0)) = e − 2 λ 0 t + N σ ( t, 0) ln μ V ( x (0)) .

≤ (17)

From (8) and definition of τ D , it can be concluded that

t
τ D

N σ ( t, 0)ln μ ≤ ( N 0 + )ln μ ≤ 2 α +2( λ 0 − λ ) t (18)

where α = N 0 ln μ
2

. Substituting (18) into (17), it yields

V ( x (0))e − 2 λt +2 α V ( x ( t )) ≤ (19)

From (14) and (19), we can obtain

≤ √ μ

e α−λt .

x ( t ) x (0)

(20)

Therefore, the system (10) is exponentially stable with stability margin λ .

Next, we consider the system (1) with the uncertainties (2), and the actual
controller (3) with perturbations (4)(or (5)). The closed-loop system can be
described as

x ( t )=[ A i + ΔA i +( B i + ΔB i )( K i + ΔK i )] x ( t ) , ∀ i ∈ M. (21)

Theorem 2. Given a scalar λ 0 > 0 , if there exist positive matrix P i and matrix
K i ,i ∈ M , such that

[ A i + ΔA i +( B i + ΔB i )( K i + ΔK i )] T P i + P i [ A i + ΔA i +
( B i + ΔB i )( K i + ΔK i )] + 2 λ 0 P i < 0

(22)

holds, then the closed-loop system (1) and (3) is globally exponentially stable
with stability margin λ ∈ (0 ,λ 0 ) for any switching signal with average dwell-time

τ D .

satisfying τ D ≥