Design and Modeling of Anti Wind Up PID Controllers - Complex System Modelling and Control Through Intelligent Soft Computations - page 13

4 Internal Model Anti Windup Control of Discrete Time
SISO Systems

In this section the design of a discrete time anti windup controller for discrete time
SISO system is explained. In this case, an internal model PID controller compen-
sator is proposed to suppress the unwanted effects yielded by windup when the
integrator output is increased due to the actuator saturation. The nonlinearities
found in many control systems, specially saturation, deteriorates the system per-
formance similar as it occurs in the SISO continuous time counterpart. As explained
before, there are several anti windup control architectures for discrete time systems,
some of them are derived from the system frequency response as shown in (Chen
et al. 2003 ; Lambeck and Sawodny 2004 ) where the design of a frequency response
method in a cascade con guration, eliminates the effects yielded by windup. As

shown in (Wittenmark 1989 ) the incorporation of a back calculation compensator
improves the system performance and reduces the unwanted effects yielded by
saturation. This anti windup controller compensation is shown in (Baheti 1989 )
where a digital PID controller implementation is used to eliminate the unwanted
effects of windup when the input of the system is saturated.
The anti windup controller strategy shown in this section is based on the the-
oretical background shown in (Morales et al. 2009 ) where a standard IMC anti
windup compensator is implemented where the robustness of the control system is
analyzed and the stabilization of the system is done by an internal model controller.
The proposed strategy shown in this section is based on an IMC PID anti windup
compensator, where due to integral characteristic of the PID controller it is nec-
essary to cancel the windup effects yielded by saturation. The saturation nonlin-
earity model is obtained by a scalar function approach as explained in (Zhang et al.
2011 ), so the IMC PID controller can be derived in order to avoid the unwanted
effects yielded by windup.

4.1 IMC PID Anti Windup Controller for Discrete Time
SISO Systems

The anti windup internal model PID controller architecture is shown in Fig. 9 .
Where G c (z) is the digital internal model controller, R(z) is the anti windup
compensator filter, G p is is the continuous time transfer function discretized by a

sampler and p c ð is the equivalent discrete time transfer function implemented in

the internal model PID anti windup controller design. In Fig. 10 the equivalent
discrete time transfer function is shown, where this transfer function is obtained by
the implementation of the scalar sign function approach.
The resulting transfer function p c ð

z Þ

Þ

is implemented to design the anti windup
internal model controller with the robustness and internal stability requirements

z