Design and Modeling of Anti Wind Up PID Controllers - Complex System Modelling and Control Through Intelligent Soft Computations - page 3

where a discrete time controller is obtained by the design of a continuous time SISO
controller and then this controller is transformed to discrete time by one of the
several well known methods (Lambeck and Sawodny 2004 ).
In the case of MIMO continuous and discrete time systems several anti windup
controllers are synthesized usually by static output feedback (SOF) and then the
controller is found by the solution of the respective linear matrix inequalities
(LMI s to ensure the

stability of the system by traditional ways or by an H ∞ controller (Wu et al. 2005 ;
Henrion et al. 1999 ), allowing a

' s). The SOF control law can be found by solving the LMI '

fl flexible anti windup controller design when the

input of the system is saturated.
Based on the previous explanation of different kind of anti windup controller
architectures, this chapter is divided in the following sections so the first part of the

chapter is devoted to SISO continuous and discrete time systems and the second
part of this chapter is devoted to MIMO continuous and discrete time systems. In
Sect. 2 , the explanation of popular anti windup control techniques is explained to
introduce the proposed strategies shown in this article, where some continuous and
discrete time classical anti windup techniques found in literature are explained. It is
important to notice that in this chapter, the main objective is to design and obtain
stable PID controllers for the SISO and MIMO case, so in the following sections
this problem is considered for analysis. Based on the previous explanation, in
Sect. 3 the design of an internal model anti windup controller for continuous time
systems is explained, showing that is possible to obtain a desired anti windup PID
controller with an internal model controller (IMC) characteristics. In Sect. 4 an
internal model anti windup controller for discrete SISO system is shown where a
similar technique like the continuous counterpart is developed to eliminate the
unwanted effects produced by the system saturation by implementing a scalar sign
function approach (Zhang et al. 2011 ); an illustrative example is shown to compare
the performance of the system. In Sect. 5 the derivations of an anti windup PID
controller are done by SOF applying LMI

s that includes the saturation of the
system. The SOF control law is obtained by the stability characteristics of the
system and by a H ∞ design, so the controller and system performance can be
compared by the solution of these control problems. In Sect. 6 an anti windup PID
controller for MIMO discrete time systems is shown and similar to its continuous
counterpart, a SOF controller is implemented and then solving the LMI

'

s based on
the system stability or H ∞ the respective PID gains are found when the input of the
system is saturated; in this section an illustrative example is shown to compare the
systems performance. Finally, in Sects. 7 and 8 the discussion and conclusions of
this chapter are shown respectively so a complete analysis of all the proposed
schemes is done and then the conclusions are analyzed at the end of this chapter.

'