Design and Modeling of Anti Wind Up PID Controllers - Complex System Modelling and Control Through Intelligent Soft Computations - page 21

5.1 PID Anti Windup Controller Design for MIMO
Continuous Time Systems

The PID anti windup controller design for MIMO continuous time systems, consist
of a back calculation PID controller by a loop which includes the saturated and non
saturated input signal of a linear time invariant MIMO system. In Fig. 16 the anti
windup controller is shown where v is the back calculation signal that is imple-
mented to avoid the windup effects which deteriorates the system performance. The
MIMO system is represented by G(s) and the anti windup PID controller and
compensator is represented by G c (s).
The anti windup controller takes the non saturated and saturated difference signal
v to suppress the increment of the integral action when the system saturates. As
occurs in the SISO case, the windup phenomena yields unwanted effects that
deteriorates the system performance; the settling time and overshoot generally are
damage when the input signal is saturated, so the compensator corrects the effects of
windup by the back calculation of the input signal added by an extra loop.
The approach explained in this section consist in obtaining a saturation repre-
sentation by a describing function approach (Taylor and O ' Donnell 1990 ), where

this method simpli es the PID controller synthesis and provides an accurate rep-

resentation of the equivalent control systems.
In Fig. 17 the saturation model is depicted in order to be represented by a
describing function that helps to obtain an equivalent anti windup controller.
The saturation model shown in Fig. 17 depicts the parts in which this model is
divided in order to obtain the Fourier series coef cients of the describing function.

n(e) is the saturation output, e is the saturation input, is the input limit and M is the

saturation output limit. The describing function is given by the following transfer
function:

ʱ

a 1 þ b 1 s

/ð s Þ ¼ ð 34 Þ

E

where a 1 and b 1 are the Fourier series coef cients when a sinusoidal input signal

with amplitude E is implemented. The coef cients of the Fourier series imple-

mented in this analysis are:

Fig. 16 Multivariable control
system

r

+
-

G ( s )

+

-

v

G c ( s )

Saturation