2.TSK Fuzzy State Variable Model
When a TSK fuzzy model of a nonlinear dynamic
system is identified with input-output data.the
consequent of the fuzzy rule is a linear difference(or
differential)equation of input and output.This section
investigates the method transforming a TSK fuzzy
input-output model to a TSK fuzzy state variable
model.When a TSK fuzzy input-output model is
transformed to a TSK fuzzy state variable model.the
consistency of the two models has to be confirmed.
Generally.the linear equation of consequents of a
TSK fuzzy model has a constant term.so the state
equation also has a constant term.3.Pole Placement Design of TSK Fuzzy
Controller
A TSK fuzzy controller is designed from the
TSK fuzzy state variable model by using the
pole placement method which can be seen in
modem linear control systems.The control object
is z一0.The TSK fuzzy controller designed with
the pole placement method guarantees the stability of
the controlled system.As shown in Fig.1.the closed
loop system with the TSK fuzzy controller has the
same response as the linear system having the desked
State transition matrix
3.1.Discrete TSK fuzzy controller
A TSK fuzzy controller is designed from a
discrete TSK fuzzy state varable model having
the form of the TSK canonical forln(51.The i-th
rule of the controller corresponding to the i-th rule
Mi of model is
where is a state transition matrix of which
eigenvalues are the desired poles.The control input
u(k) is inferred by using the following consistency
condition[7].
ne next theorem sho•ws that the state transition
matrix of the closed loop system Call be arbitrarily
assigned by a TSK fuzzy controller.
Theorem:The behaviour of the fuzzy system(5)
controlled by the fuzzy controller(8)and the
consistency condition(11)is the same as the linear
system of which state transition matrix is the desired
one .
Proof
Remark:When me desired state transition matrix is
given by
,the Gi satisfying(9)cab ne obtained as follows as
follows:
Example l:A TSK fuzzy state controller for the next
system shown in【8】was designed.
The TSK fuzzy model of the
2.The i-th rule of the modle
system is shown in Fig
can be written as
Assume that the desired output is Yd and the desired
state is .The error state is
defmed as 。Then the rule(14)can be
expressed as
Where
The i-th rule of the TSK fuzzy controller designed
from the model(15)is as follows:
Fig.3 shows the responses of the plant(13)
controlled by the fuzzy controller when the desffed
pole
Fig.4 shows the responses of the linear system
and the plant(13)controlled by
diflferent fuzzy controller designed with different pole,