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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
]
,ρ
1
=
β, ρ
2
=
B
e
,ρ
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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