Design and Modeling of Anti Wind Up PID Controllers - Complex System Modelling and Control Through Intelligent Soft Computations - page 22

Fig. 17 Saturation model

n ( e )

M

1

1

e ʱ

ʱ

M

Z

T

1
T

a 1 ¼ sin ðx t Þ n ð t Þ dt

T

ð 35 Þ

Z

T

1
T

b 1 ¼ cos ðx t Þ n ð t Þ dt

T

where is the angular frequency of the input signal e and in order to obtain the

Fourier series coef

ˉ

cients a sinusoidal input signal of amplitude E and period
T must be assumed as the input of the saturation. Considering that e is 2

ˀ

periodic

ˀ

or T =2 the following Fourier coef cients are obtained:

r

1 a 1 ¼ ð a

2E

þð E Þ ð E Þþ 4 Þþ

M
2

2 E Þ

2

sin 1 þ

p ð

ð 36 Þ

2 2

b 1 ¼ E
4

E E M

p

E

21 1

p

Considering the following PID controller:

F 2 u c ð

S þ s s 1

y c ð s Þ ¼ F 1 u c ð s Þþ F 3 u c ð s Þ ð 37 Þ

s

where F 1 , F 2 and F 3 are diagonal matrices of appropriate dimensions (He and Wang
2006 ) for the proportional, integral and derivative parts of the controller.
In order to obtain the anti wind up controller, consider the following linear time
invariant system G(c) given by: