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where:
A 0 þ
B 0 M
A cl ¼ ð
Þ
ð
51
Þ
I 0
00
C cl ¼
where I is an identity matrix of an appropriate dimension. Solving the LMI shown
in ( 48 ) and ( 50 ) for P and M the controller parameters can be extracted from
M obtaining all the PID controller gain matrices found by these optimization
techniques.
In this subsection is proved that an anti windup PID controller for MIMO
continuous time system can be implemented by solving a LMI based optimization
problem. In the following subsection, the control of a DC motor is done in order to
show by an illustrative example the application of these control strategies, it is
proved that
finding the respective matrix M the rest of the controller variables can
be obtained. The solution of these LMI can be obtained by several numerical
methods found in literature, such as shown in (He and Wang 2006 ) for example.
5.2 Example 3
In this section the stabilization and control of a DC motor by an anti windup PID
controller for MIMO systems is shown to illustrate the advantages of the proposed
technique.
Consider the following DC motor transfer function (Cockbum and Bailey 1991 ):
x L ð
s
Þ
k m
Þ ¼
ð
52
Þ
v a ð
ð
þ
Þ
s 2
þð
þ
J L B L Þ
þ
k m
s
J m L
J L L
J m R
s
ˉ L is the angular velocity of the model, v a is the applied armature voltage, J L
is the inertial load, J m is the motor inertia, L is the inductance, R is the resistance, B L
is the viscous friction constant and k m is the motor constant. Converting ( 52 ) to state
space the following equation is obtained:
where
"
# x 1
x 2
¼
J m R þ J L B L
J m þ J L L
1
_
x 1
x 2
K m
J m þ J L L
0
ð
53
Þ
"
# 0
v a
00
0
þ
K m
J m þ J L L
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