Design and Modeling of Anti Wind Up PID Controllers - Complex System Modelling and Control Through Intelligent Soft Computations - page 34

where in LMI equivalent is given by

x a ð

T P

x a ð

k ÞU U x a ð k Þ k Þ Px a ð k Þ \ 0 ð 72 Þ

For the static output feedback problem the following LMI must be solved for
P and F (Mayer et al. 2013 )

P 1

U

0 ð 73 Þ

[

T

U P

with P − 1 > 0 and for the H ∞ controller synthesis the following LMI must be solved
considering

k T 2 x ð z Þ k 1 \ c ð 74 Þ

where is a robustness parameter that indicates the disturbance rejection of the

system and T 2 ˉ T2ω(z) is the discrete time transfer function of the closed loop system
(Kwan Ho et al. 2006 ). Then by solving for F and P in the following LMI the H ∞
static output feedback PID controller can be obtained.

ʳ

2
4

3
5 [

T PC cl

P 0 U

0 c I 0 0

0 ð 75 Þ

P U 0 P 0

C cl 00 c I

With these explanations the PID controller gains can be obtained by any of the
SOF controllers. In the next subsection an illustrative example is shown to evince
the numerical simulation of an anti windup controller for a DC motor in discrete
time.

6.2 Example 4

In this subsection a DC motor is stabilized with a PID anti windup controller in
MIMO form. The same DC motor model of example 3 is considered in this section,
so the following discretized state space model of the DC motor is obtained with a
sampling period T = 0.1 s

x

2 : 774 0 : 1749 0 : 1749 0 : 1609

x ð k þ 1 Þ ¼ ð k Þþ rð u ð k ÞÞ ð 76 Þ

1 : 993 0 : 9169 0 : 08306 2 : 153

x

10
01

y ð k Þ ¼ ð k Þ ð 77 Þ