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Fig. 4.1
Closed loop control
Control
signals, u
system
Outputs, y
Controller
Plant
State variables, x
b ji
n
m
n
x i
˙
=
a ki ˜
x k +
c kj ˜
x k
(4.24)
k
=
0
j
=
1
k
=
0
That is,
n
n
m
b ji c kj ˜
x k
x i
˙
=
a ki ˜
x k +
(4.25)
k
=
0
k
=
0
j
=
1
which can be simplified as:
n
m
a ki +
˜
x i
˙
=
b ji c kj
x k
(4.26)
k
=
0
j
=
1
The expression ( 4.26 ) represents the equivalent mathematical model of a multi-
variable closed loop fuzzy control system, which may be nonlinear. The design of the
structure of the fuzzy controller is considered completely independent of the process
of modeling the plant, so that the partition of the universe of discourse of the states of
the plant and the controller does not have why to coincide. Similarly, the dynamics
of the plant and the controller need not have the same complexity, so the number of
rules each of these fuzzy systems is considered independent.
4.3 Synthesis of Stable Fuzzy Controllers by Design
One of the primary objectives of this chapter is to establish a methodology as general
as possible to allow the synthesis of stable fuzzy controllers by design, i.e., that
stability is guaranteed (fulfillment of the sufficiency condition) during the design
process itself. In order to make the methodology as general as possible, it is assumed
hypothesis that the dynamics of the plant is totally unknown, or complex enough to
not be possible to obtain a mathematical model of it. Obviously, in a practical case
would be possible to incorporate the known data to the system to facilitate both the
modeling of the plant, as the fuzzy controller design.
Lyapunov theory can be used to analyze the stability of fuzzy systems. In fact, the
literature has addressed as stabilization problems, as tracking problems with TS type
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