An Introduction to Fuzzy Control - Fuzzy Logic: An Introductory Course for Engineering Students - page 195

and

x 1 − d 1 (˃)

ʱ =

d 1 (˃ + 1 ) − d 1 (˃)

x 2 − d 2 (˄ )

ʲ =

d 2 (˄ + 1 ) − d 2 (˄ )

j
2 ∈[

i

in which case .

In every region of the PBmodels, i.e.: f 1 (

w

1 ,w

0 , 1 ]

, the values are computed through
bilinear interpolation of the corresponding four vertexes as shown in the figure.

x 1 , x 2 )

Note that the approximation is made by only using the values of a nonlinear
function at the vertexes of R ij 's.

8.3.6 Vertex Placement Principle

Generally speaking, if a plant model P and a desired (closed-loop) plant model DP
are given, we may have a strategy to design a Controller C by setting a relation
C

= −

P . This strategy works for linear systems but does not work in general for
nonlinear systems as we know. However if nonlinear systems are modeled with PB
systems, we can take this strategy by applying Vertex Placement Principle (VPP).
This is a general principle to design a LUT-controller to utilize the characteristics
of the PB model of an objective plant. Given a partition of the sate space into sub-
regions, the property of a PB model can be completely described by the values of a
PB model at the vertexes of regions. Therefore, to design a LUT-controller means to
assign the values of a table at the vertex positions. VPP guides us to assign the values
of a controller at the vertex positions based on the values of a PB model describing
the nonlinear plant. In this sense, VPP is similar to the idea of Pole Placement in
linear systems.

DP