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8.3.4 Parallel Distributed Compensation
The parallel distributed compensation (PDC) offers a procedure to design a fuzzy
controller from a given TS fuzzy model. To realize the PDC, a controlled nonlinear
system is first represented by a TS fuzzy model. Each control rule is designed from
the corresponding rule of a TS fuzzy model, and the designed fuzzy controller shares
the same fuzzy sets with the fuzzy model in the premise parts.
The fuzzy controller will have the following form,
Control Rule i:
IF
z
1
(
t
)
is
M
i
1
and
···
and
z
p
(
t
)
is
M
ip
,
THEN
u
(
t
)
=−
F
i
x
(
t
),
for
i
r
.
The fuzzy control rules have a linear controller (state feedback laws in this case)
in the consequent parts. The overall fuzzy controller is represented by
=
1
,
2
,...,
r
(
)
=−
h
i
(
(
))
(
)
u
t
z
t
F
i
x
t
.
i
=
1
The fuzzy controller design consists on determining the local feedback gains
F
i
in
the consequent parts. With PDC we have a simple and natural procedure to handle
nonlinear control systems. Other nonlinear control techniques require special and
rather involved knowledge.
The overall closed-loop system using the PDC controller method is,
r
r
x
˙
(
t
)
=
h
i
(
z
(
t
))
h
j
(
z
(
t
))
{
A
i
−
B
i
F
j
}
x
(
t
)
}
.
i
=
1
j
=
1
For this case, the following sufficient condition for stability can be obtained:
Theorem 8.3.4
The equilibrium of a fuzzy control system is globally asymptotically
stable if there exists a common positive definite matrix P such that
T
P
{
A
i
−
B
i
F
j
}
+
P
{
A
i
−
B
i
F
j
}
<
0
for h
i
(
z
(
t
))
·
h
j
(
z
(
t
))
=
0
,
∀
t, i
,
j
=
1
,
2
,...,
r.
A control problem in the framework of the TS fuzzy model and PDC design,
targets the design of a controller that ensures the stability of the closed-loop system.
Example 8.3.5
Considering the balancing of an inverted pendulum on a cart. The
motion equations of the pendulum are described by,
x
1
(
˙
)
=
x
2
(
)
t
t
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