Design and Modeling of Anti Wind Up PID Controllers - Complex System Modelling and Control Through Intelligent Soft Computations - page 30

section it is shown that an ef cient control strategy is developed for the suppression of

the unwanted effects yielded by windup, and the controller synthesis is done by a
different saturation model.

6 Anti Windup Control of Discrete MIMO Systems
by Static Output Feedback (SOF)

In this section the derivation of an anti windup PID controller for discrete MIMO
system is proposed. The main idea behind this controller is to design an anti windup
controller/compensator that minimizes the windup effects when the input of the
system saturates producing an increasing of the integral action that deteriorates the
system performance. The controller design for this kind of systems consist in
deriving a static output feedback (SOF) control law, similar as the continuous time
counterpart (Bateman and Zongli 2002 ; Kwan Ho et al. 2006 ; Matsuda and Ohse
2006 ) and then the SOF gain is obtained by solving the LMI

' is as an optimization

problem.
In order to achieve suitable control gains for the PID controller, it is necessary to
implement a saturation model (Li-Sheng et al. 2004 ; Zongli and Liang 2006 ;
Shuping and Boukas 2009 ) where suf cient conditions are established in order to

solve the LMI s by a convex optimization problem (Shuping and Boukas 2009 ).

For the AWC design it is necessary to add a back calculation loop which consists in
the difference between the non saturated and saturated input signal, similar as the
continuous time counterpart, to reduce the effects of windup when the input system
saturates. Then using the saturation model (Li-Sheng et al. 2004 ) this nonlinearity
form is implemented to obtain the respective LMI

'

is solved by a convex optimi-
zation problem. Beside from the standard solution of static output feedback con-
trollers (SOF) a h ∞ SOF controller synthesis is obtained by solving the required
LMI

'

is (Lim and Lee 2008 ). In this section it is proved that a discrete time PID
controller can be obtained by a static output feedback control law, simplifying the
anti windup controller design and then the PID controller gains can be found by
solving the linear matrix inequalities for SOF and H ∞ SOF.
As occurs in the continuous time case, there are several numerical methods to
solve discrete time SOF problems by LMI

'

' s so with this method an optimal solution

of the LMI s and the satu-

ration nonlinearity model an optimal solution can be found by any of the algorithm
found in literature such as (Matsuda and Ohse 2006 ) for continuous time and
(Kwan Ho et al. 2006 ) for discrete time systems. The proposed anti windup PID
controller is designed taking into account the stability properties and characteristic
of the closed loop system and for the H ∞ SOF problem the robustness of the closed
loop system improves the system performance and reduces the deterioration of the
system operation when a reference signal needs to be tracked.

' s can be found. By implementing the appropriate LMI '