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The extension of the pendulum control to the full circle
x
1
∈[−
ˀ, ˀ
]
work space,
would be achieved similarly, only by adding two more rules to the TS fuzzy system.
8.3.5 Piecewise Bilinear Model
Recently, Piecewise Bilinear (PB) model is being used for control purpose. The
PB model is a fully parametric model to represent Linear/Nonlinear systems. The
obtained model is built on piecewise rectangular regions, and each region is defined
by four vertices partitioning the state space. As the conventional nonlinear system
control based on TS fuzzymodel represents a connection of linear state-space models
by sector nonlinearity, the PB model represents a convex combination of the vertices
defining piecewise regions.
In this approach, bilinear functions are used to regionally approximate any given
function. A bilinear function is a nonlinear function of the form
y
=
a
+
bx
1
+
cx
2
+
dx
1
x
2
, where any four points in the three dimensional space are spanned with
a bi-affine plane.
PB model has a good general approximation capability and it has a continuous
crossing over the piecewise regions. Its interpretability, simplicity and visibility facil-
itates the realization of controllers in industrial applications. A local error does not
trigger a global error and its interpolation nature generates robust outputs. A draw-
back of the PB model is that the stability analysis based on Lyapunov is difficult as
bilinear matrix inequalities (BMI) must be solved.
If a general case of an affine two-dimensional nonlinear control system is consid-
ered,
⊧
⊨
x
1
=
˙
f
1
(
x
1
,
x
2
)
x
2
=
˙
f
2
(
x
1
,
x
2
)
+
g(
x
1
,
x
2
)
·
u
⊩
y
=
h
(
x
1
,
x
2
)
where
u
is the input. For the PB representation of a state-space equation, a coordinate
vector
d
(˃, ˄ )
of the state space and a rectangle
R
ij
must be defined as,
T
d
(
i
,
j
)
≡
(
d
1
(
i
),
d
2
(
j
))
R
ij
≡[
d
1
(
i
),
d
1
(
i
+
1
)
]×[
d
2
(
j
),
d
2
(
j
+
1
)
]
where
i
∈
(
1
,...,
n
1
)
and
j
∈
(
1
,...,
n
2
)
are integers, and
d
1
(
i
)<
d
1
(
i
+
1
)
,
d
2
(
n
2
), where
n
1
and
n
2
represent the number of partitions of dimension
x
1
and
x
2
respectively.
Each value in the matrix is referred to as a vertex in the PB model. The operational
region of the system is divided into (
n
1
−
j
)<
d
2
(
j
+
1
)
. The PB models are formed by matrices of size (
n
1
×
1
×
n
2
−
1) piecewise regions that are
analyzed independently.
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