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(τ ),
(τ )
)
Given a pair
(
x
u
, the state of the TS system can easily be inferred by:
i
=
1
w
i
(ϑ(τ))(
N
A
i
x
(τ )
+
B
i
u
(τ ))
σ.
x
(τ )
=
=
1
ρ
i
(ϑ(τ))(
A
i
x
(τ )
+
B
i
u
(τ ))
i
=
1
w
i
(ϑ(τ))
i
=
(6.2)
ϑ
1
(τ),...,ϑ
p
(τ )
is the vector containing the premise variables,
where
ϑ(τ)
=
and
w
i
(ϑ(τ))
and
ρ
i
(ϑ(τ))
are defined as follows:
p
M
ij
ϑ
j
(τ )
w
i
(ϑ(τ))
=
(6.3)
j
=
1
w
i
(ϑ(τ))
i
=
1
w
i
(ϑ(τ))
ρ
i
(ϑ(τ))
=
(6.4)
where
M
ij
ϑ
j
(τ )
is the grade of membership of
ϑ
j
(τ )
in
M
ij
and
ρ
i
(ϑ(τ))
is such
that:
⎧
⎨
i
=
1
ρ
i
(ϑ(τ))
=
N
1
(6.5)
⎩
ρ
i
(ϑ(τ))
≥
0
,
i
=
1
,...,
N
6.2.2 Robust TS Controller Design
In this chapter, a robust TS framework that is based on the combination of robust
polytopic and Takagi-Sugeno design is proposed. In this framework, the variation of
the state matrix is due to the vector of premise parameters
, whose measurement
or estimation is supposed to be available, together with some bounded uncertain-
ties. The nominal TS model is used to generate a polytope described by its ver-
tices (e.g.
A
1
,
ϑ
A
5
in Fig.
6.1
). Later, the model uncertainties are taken into
account generating more polytopes, one for each vertex of the nominal polytope
(e.g.
A
1
,
1
,
A
2
,...,
A
1
,
4
around the first nominal vertex
A
1
). The robust TS design
problem is to obtain a state-feedback controller inferred by
A
1
,
2
,
A
1
,
3
,
ϑ
as a combination of
vertex controllers (e.g.
K
1
,
K
5
in Fig.
6.1
). Each vertex controller is designed
to satisfy some linear matrix inequalities (LMI) conditions at all the vertices of the
vertex polytope. Under some assumptions, the final result will be a TS controller
inferred by
K
2
,...,
ϑ
that is robust against bounded uncertainties.
6.2.3 Design Using LMI-Based Pole Placement
Consider the TS system (
6.1
) and assume that each subsystem is described by uncer-
tain state-space matrices as follows:
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