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N
N
+
β
kl
A
Ni
+
A
i
X
+
B
Nt
+
B
t
i
1
ρ
i
(ϑ(
t
))
1
ρ
t
(ϑ(
t
))
α
kl
X
i
=
t
=
+
B
Nt
+
B
t
i
T
+
β
lk
A
Ni
+
A
i
X
<
0
1
≤
k
,
l
m
(6.16)
≤
that is a necessary and sufficient condition for
-stability of parallel distributed
controllers (He and Duan
2006
), obtained as an extension of Chilali and Gahinet
(
1996
).
D
6.3 Fault Tolerant Control
6.3.1 System and Fault Modeling
Let us consider the faulty uncertain system represented by the following TS model:
IF
ϑ
1
(τ )
is
M
i
1
AND
...
AND
ϑ
p
(τ )
is M
ip
(6.17)
THEN
σ.
x
f
(τ )
=
(
A
Ni
+
A
i
)
x
f
(τ )
+
(
B
Ni
+
B
i
)
u
f
(τ )
i
=
1
,...,
N
Given a pair
x
f
(τ ),
u
f
(τ )
, the faulty state of the TS system can easily be inferred:
N
1
ρ
i
(ϑ(τ))
(
u
f
(τ )
σ.
x
f
(τ )
=
A
Ni
+
A
i
)
x
f
(τ )
+
(
B
Ni
+
B
i
)
(6.18)
i
=
n
x
where
N
is the number of subsystems,
x
f
(τ )
∈ R
is the faulty state vector,
u
f
(τ )
∈
n
u
is the faulty control input vector.
A
Ni
∈ R
n
x
×
n
x
,
B
Ni
∈ R
n
u
×
n
x
R
are the
i
th nominal
B
i
represent time varying parametric
uncertainties with appropriate dimensions corresponding to the
i
th subsystem.
The premise variables
A
i
and
state and input matrices, respectively.
are typically associated to changes in the operating
conditions and it is assumed that their values can be measured, computed, or estimated
in real-time using the available measurements. On the other hand, the time-varying
parameters
ϑ(τ)
B
i
associated to model uncertainties, are unknown, and cannot
be used to infer accordingly the controller, even though some knowledge
apriori
available can be exploited in the controller design phase.
The parametric faults
f
are assumed to be multiplicative and to belong to a set of
faults
A
i
and
F
that can be expressed as:
F = {
f
1
,
f
2
,...,
f
N
} =
[
F
1
]
×
[
F
2
]
×
...
×
[
F
N
]
(6.19)
=
F
i
,
F
i
,
∀
where [
F
i
]
i
=
1
,...,
N
.
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