Information Technology Reference
In-Depth Information
of fuzzy sets, fuzzy rules and a set of local linear models. The overall model of the
system is obtained by merging the local models through fuzzy membership functions.
In recent years, the interest in TS systems for FTC has grown due to the possibility
of using such a methodology to deal with nonlinear systems (Lopez-Toribio and
Patton
1999
). The TS theory is mainly used for designing controllers for non-faulty
systems, but recently it has also been used for active FTC (Dziekan et al.
2011
;Diao
and Passino
2001
; Ichtev et al.
2002
).
This chapter introduces the idea of the robust TS framework, that is obtained
as a combination of known results from the robust control area and the TS control
area. This framework can be used for fault tolerant control, with the advantage that,
depending on the information available about the fault, the proposed framework
can give rise to different FTC strategies: passive FTC, active FTC without controller
reconfiguration and active FTC with controller reconfiguration. Finally, the proposed
framework is illustrated by an application to a mobile robot.
The chapter is organized as follows. Section
6.2
presents the robust TS framework.
Section
6.3
shows how such a framework can be used to obtain different types of
FTC strategies. Section
6.4
describes the application example and some results are
presented in Sect.
6.5
. Finally, the main conclusions and the possible future work are
summarized in Sect.
6.6
.
6.2 The Robust TS Framework
6.2.1 TS Systems
TS systems, as proposed by Takagi and Sugeno (
1985
), are described by local models
merged together using fuzzy IF-THEN rules (Tanaka et al.
2001
), as follows:
IF
ϑ
1
(τ )
is
M
i
1
AND
...
AND
ϑ
p
(τ )
is M
ip
(6.1)
THEN
σ.
x
(τ )
=
A
i
x
(τ )
+
B
i
u
(τ )
i
=
1
,...,
N
where, following the notation used by Apkarian et al. (
1995
),
stands for the Laplace
variable
s
in the continuous-time case and for the
Z
-transform variable
z
in the
discrete-time case. Similarly,
σ
τ
will stand for the time
t
∈ R
in the continuous-time
∈ Z
+
in the discrete-time case. The notation
case and for the time samples
z
σ.
x
(τ )
stands for
for discrete-time signals.
Here,
M
ij
denote the fuzzy sets and
N
is the number of model rules;
x
x
˙
(
t
)
for continuous-time signals and for
x
(
k
+
1
)
n
x
(τ )
∈ R
n
u
is the input vector, while
A
i
and
B
i
are matrices
of appropriate dimensions. Finally,
is the state vector,
u
(τ )
∈ R
are premise variables that can
be functions of the state variables, external disturbances and/or time. Each linear
consequent equation represented by
A
i
x
ϑ
1
(τ ), . . . , ϑ
p
(τ )
(τ )
+
B
i
u
(τ )
is called a
subsystem
.
Search WWH ::
Custom Search