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∈
≥
where
t
0 is the continuous time argument, the control signal
u
is a pulse
width modulated duty cycle,
d
is the disturbance input,
y
is the controlled output,
m
is the output of the saturation and dead zone static nonlinearity with the parameters
k
u
,
m
>
R
,
t
0 and 0
<
u
a
<
u
b
, and the state variables are
x
P
,
1
(
t
)
=
α(
t
),
x
P
,
2
(
t
)
=
ω(
t
),
(8.2)
where
is the (angular) speed. A simplified
process model used in the controller design and tuning is represented by the transfer
function of the linear subsystem in Eq. (
8.1
)
α(
t
)
is the (angular) position and
ω(
t
)
P
(
s
)
=
k
p
/
[
s
(
1
+
T
s
)
]
,
(8.3)
where
k
p
is the process gain,
k
p
=
is the small time constant.
Accepting that the inputs and are changing at the discrete sampling intervals the
discrete-time state-space model of the process Eq. (
8.1
)is
k
u
,
m
·
k
P
1
, and
T
⎧
⎨
0
,
if
|
u
(
t
d
)
|≤
u
a
,
m
(
t
d
)
=
k
u
,
m
[
u
(
t
d
)
−
u
a
sgn
(
u
(
t
d
))
]
,
if
u
a
<
|
u
(
t
d
)
|
<
u
b
,
⎩
k
u
,
m
(
u
b
−
u
a
)
sgn
(
u
(
t
d
)),
if
|
u
(
t
d
)
|≤
u
b
,
x
P
,
1
(
t
d
+
1
)
=
x
P
,
1
(
t
d
)
+
T
[
1
−
exp
(
−
T
s
/
T
)
]
x
P
,
2
(
t
d
)
+
(8.4)
+
k
P
[
T
s
+
T
exp
(
−
T
s
/
T
)
−
T
]
u
(
t
d
)
+
T
s
d
(
t
d
)
x
P
,
2
(
t
d
+
1
)
=[
exp
(
−
T
s
/
T
)
]
x
P
,
2
(
t
d
)
+
k
P
[
1
−
exp
(
−
T
s
/
T
)
]
u
(
t
d
),
y
(
t
d
)
=
x
P
,
1
(
t
d
),
where
t
d
∈
is the sampling period. The
model in Eq. (
8.3
) and its discretized form can be used as a benchmark as a simplified
model of complicated process models in many applications having in view the fact
that the parameters
k
P
and
T
Z
,
t
d
≥
0 is the discrete time argument, and
T
depend on the operating point. Therefore, the model
in Eq. (
8.3
) is viewed as a parameter varying model, and the sensitivity analysis with
respect to the variations of the process parameters
k
P
or
T
should be accounted for
in controller design and tuning.
The global performance specifications of control systems can be imposed by
means of the optimization problem
|
t
d
,ρ)
∞
ρ
∗
=
2
2
arg min
ρ
∈
D
ρ
I
(ρ),
I
(ρ)
=
e
(
t
d
,ρ)
|+
γ
σ
(
(8.5)
t
d
=
0
ρ
∗
is the optimal parameter
where
ρ
is the parameter vector of the fuzzy controllers,
vector,
D
is the
output sensitivity function obtained from the state sensitivity model with respect to
k
P
or
T
is the feasible domain of
ρ
,
e
(
t
d
,ρ)
is the control error,
σ(
t
d
,ρ)
ρ
,
γ
is the weighting parameter, and
I
(ρ)
is the objective function whose
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