Design and Modeling of Anti Wind Up PID Controllers - Complex System Modelling and Control Through Intelligent Soft Computations - page 33

where

2
4

3
5

A 00
CI0
C 00

A cl ¼

2
4

3
5

B
C
0

B cl ¼

2
4

3
5

ð 67 Þ

C 00
0 C cl ¼ I 0

C 0 I

2
4

3
5

0
0

D cl ¼

F ¼ ½ k p k I k D

The saturation nonlinearity can be modeled by the following de nition

(Li-Sheng et al. 2004 ; Zongli and Liang 2006 ; Shuping and Boukas 2009 ).

m n be given. For x n ,if Hy k 1

De nition 1 Let F ; H 2< 2< 1 then rð Fy a Þ2

n o where co{

co E j Fy a þ } denotes the convex hull and E j ¼

I E j where E j is the set of m × m diagonal matrices where all their elements are 1
or 0. With De

E j Hy a : j 2 ½ 1 ; 2 ·

nition 1 ( 65 ) can be transformed into:

x a ð k þ 1 Þ ¼ U x a ð k Þ ð 68 Þ

where:

B cl E j HC cl þ

U ¼ A cl þ B cl E j FC cl þ D cl FC cl ð 69 Þ

Considering de nition 1 and the following Lyapunov function:

x a ð V ð k Þ ¼ k Þ Px a ð k Þ ð 70 Þ

where P is a positive de nite function. The derivative of ( 70 ) is given by:

D V ð k Þ ¼ V ð k þ 1 Þ V ð k Þ

ð 71 Þ

D V ð k Þ ¼x a ð k ÞU

T P U x a ð k Þ x a ð k Þ Px a ð k Þ