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(
t d ) =
K P [
(
t d ) + μ
(
t d ) ] ,
K P =
k c (
T i
T s /
), μ =
2 T s /(
2 T i
)
u
e
e
2
Ts
(8.9)
The Two Inputs-Single Output fuzzy controller (TISO-FLC) block is character-
ized by the weighted average defuzzification method and by the SUM and PROD
operators in the inference engine assisted by the complete rule base
R 1
:
IF e
(
t d )
IS N AND
e
(
t d )
IS N THEN
u
(
t d ) = η
K P [
e
(
t d ) + μ
e
(
t d ) ] ,
R 2
:
IF e
(
t d )
IS N AND
e
(
t d )
IS ZE THEN
u
(
t d ) =
K P [
e
(
t d ) + μ
e
(
t d ) ] ,
R 3
:
(
t d )
(
t d )
(
t d ) =
K P [
(
t d ) + μ
(
t d ) ] ,
IF e
IS N AND
e
ISPTHEN
u
e
e
R 4
:
IF e
(
t d )
IS ZE AND
e
(
t d )
IS N THEN
u
(
t d ) =
K P [
e
(
t d ) + μ
e
(
t d ) ] ,
R 5
:
IF e
(
t d )
IS ZE AND
e
(
t d )
IS ZE THEN
u
(
t d ) =
K P [
e
(
t d ) + μ
e
(
t d ) ] ,
R 6
:
IF e
(
t d )
IS ZE AND
e
(
t d )
ISPTHEN
u
(
t d ) =
K P [
e
(
t d ) + μ
e
(
t d ) ] ,
R 7
:
IF e
(
t d )
IS P AND
e
(
t d )
IS N THEN
u
(
t d ) =
K P [
e
(
t d ) + μ
e
(
t d ) ] ,
R 8
:
IF e
(
t d )
IS P AND
e
(
t d )
IS ZE THEN
u
(
t d ) =
K P [
e
(
t d ) + μ
e
(
t d ) ] ,
R 9
t d ) ] ,
(8.10)
Equation ( 8.10 ) shows that the TS PI-FLC is a bumpless interpolator between
two separately designed digital PI controllers, one with the rule consequent in
Eq. ( 8.9 ) and another one with the rule consequent multiplied by the parameter
η
:
IF e
(
t d )
IS P AND
e
(
t d )
IS P THEN
u
(
t d ) = η
K P [
e
(
t d ) + μ
e
(
1, that alleviates the fuzzy control system overshoot. Therefore, the rule
base in Eq. ( 8.10 ) can be reduced to only two rules, these rules and Fig. 8.2 offer a
class of simple TS PI-FLCs. The TS PI-FLC design and tuning uses the following
tuning condition resulted from the modal equivalence principle:
,0
<η<
B
= μ
B e =
2 T s B e /(
2
β
T
T s ).
(8.11)
e
The application of the ESO method and of the modal equivalence principle yields
only three parameters for TS PI-FLCs included in the parameter vector
T
ρ =[ ρ 1 ρ 2 ρ 3 ]
1 = β, ρ 2 =
B e 3 = η,
(8.12)
where the subscript T indicates the matrix transposition. The design approach
dedicated to TS PI-FLCs consists of the following steps that result in the optimal
parameter vector
ρ .
Step 1 . Apply the ESOmethod to tune the parameters of continuous-time linear PI
controllers, set the sampling period, apply Tustin's method to compute the parameters
in Eq. ( 8.9 ), derive the sensitivity model with respect to k P or T and insert the
sensitivity model in the fuzzy control system structure involved in simulations and
experiments in order to evaluate the objective function.
Step 2 . Set the weighting parameter
in Eq. ( 8.5 ) to meet the performance speci-
fications of fuzzy control systems, set the final time moment t f to replace infinity in
σ
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