Passive Control of Sampling System - Intelligent Computing for Sustainable Energy and Environment - page 140

2 Preliminaries and Problem Statement

Consider continuous control system corresponding to sampling system

⎧
⎨

x ( t )=( A 0 + ΔA 0 ) x ( t )+( A 1 + ΔA 1 ) x ( t − τ )+( B 0 + ΔB 0 ) u ( t )+

f ( x,u,t )+ B 1 ω ( t )
z ( t )= C 1 x ( t )+ C 2 x ( t

(1)

− τ )+ H 2 ω ( t )

⎩

x ( t )= x 0 ,t ∈ [ − τ, 0]

R n is control input vector, A 0 ,A 1
and B 0 are system matrix and control matrix with appropriate dimension respec-
tively, ΔA 0 ,ΔA 1 are uncertain system matrix with the appropriate dimension
respectively. Constant time-delay d> 0.
to discretization equation (1)

R n is the state vector, u ( t ) where x ( t ) ∈ ∈

⎧
⎨

x ( k +1)=( G 0 + ΔG 0 ) x ( k )+( G 1 + ΔG 1 ) x ( k−d )+( H 0 + ΔH 0 ) u ( k )+
f ( x,u,t )+ H 1 ω ( k )
z ( k )= C 1 x ( k )+ C 2 x ( k

(2)

− d )+ H 2 ω ( k )

⎩

x ( k )= x 0 ,k ∈ [ − d, 0]

where

G 0 = e A 0 h ,G 1 = h
0

e A 0 ( h−w ) dwA 1

H 0 = h
0

e A 0 ( h−w ) dwB 0 ,H 1 = h
0

e A 0 ( h−w ) dwB 1

ΔG 0 ,ΔG 1 ,ΔH 0 are uncertain matrix, and satisfy the following form
ΔG 0 ΔG 1 ΔH 0 = MF ( k ) E 0 E 1 E 2
Lemma 1. [9] For a given symmetric matrix S = S T = S 11 S 12
S 12 S 22

with S 11 ∈

R r×r , the following conditions are equivalent:

(1) S< 0
(2) S 11 < 0 , S 22 − S 12 S − 1

11 S 12 < 0

S 12 S − 1

22 S 12 < 0

(3) S 22 < 0 , S 11 −

Lemma 2. [10] For given matrices Q = Q T ,H,E with appropriate dimen-
sions,

Q + HF ( t ) E + E T F T ( t ) H T < 0
holds for all F ( k ) satisfying F T ( t ) F ( t )

≤ I if and only if there exists ε> 0 ,such

that

Q + ε − 1 HH T + εE T E< 0