Passive Control of Sampling System - Intelligent Computing for Sustainable Energy and Environment - page 142

Proof. By passive controller, the closed-loop time-delay system

x ( k +1)= G 0 x ( k )+ G 1 x ( k h )+ H 0 Kx ( k )

=( G 0 + H 0 K ) x ( k )+ G 1 x ( k

−

− h ) (5)

Let P> 0, S> 0 are symmetric positive matrix, define the Lyapunov function
as

k − 1

V ( x ( k )) = x T ( k ) Px ( k )+ x T ( i ) Sx ( i )

t = k−h

The full differential form of ΔV ( x ( k )), along the trajectories is given by

ΔV ( x ( k )) = V ( x ( k +1)) V ( x ( k ))

= x T ( k +1) Px ( k +1)

−

x T ( k ) Px ( k ) −

k k− 1

x T ( i ) Sx ( i ) x T ( i ) Sx ( i ) + −

i = k +1 − h i = k − h

h ) Ψ x ( k )
x ( k

= x T ( k ) x T ( k

−

− h )

where

Ψ = ( G 0 + H 0 K 1 ) T P ( G 0 + H 0 K 1 )( G 0 + H 0 K 1 ) T PG 1
∗

G 1 PG 1

+ −

P + S 0
∗− S

Thereupon ΔV < 0 is equivalent with that
( G 0 + H 0 K 1 ) T P ( G 0 + H 0 K 1 )( G 0 + H 0 K 1 ) T PG 1
∗ G 1 PG 1

+ −

< 0
= ( G 0 + H 0 K 1 ) T

P + S 0
∗− S

P G 0 + H 0 K 1 G 1 + − < 0

P + S 0
∗−

G 1

S

Introducing passivity,

2 z T ( k ) ω ( k ) < 0

ΔV −

That is to say,
⎡

⎤ ⎡ ⎤

⎦ P 1 G 0 G 1 H 1 < 0

G 0
G 1
H 1

− P + S ∗ ∗

⎣ ⎦ + ⎣

0 − S ∗

H 2

− C 1 − C 2 − H 2 −