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⎧
⎨
L
d
di
d
dt
=
−
R
s
i
d
+
n
p
ωL
q
i
q
+
u
d
L
q
di
q
(1)
dt
=
−
R
s
i
q
−
n
p
ωL
d
i
d
−
n
p
ωφ
+
u
q
⎩
J
d
dt
=
2
n
p
[(
L
d
−
L
q
)
i
d
i
q
+
φi
q
]
−
τ
L
where
i
d
and
i
q
are direct axis and quadrature axis current, respectively.
J
is the
Moment of Inertia,
τ
L
is the load moment,
φ
is the permanent magnetic flux,
R
s
is the Stator Resistance,
n
p
is the role pair number of motor,
u
d
and
u
q
are
direct axis and quadrature axis voltage, L
d
and L
q
, respectively, d-q axis stator
inductance.
x
and
u
are defined as follows.
⎛
⎞
⎛
⎞
⎠
,u
=
u
1
u
2
=
u
d
u
q
.
x
1
x
2
x
3
i
d
i
q
ω
⎝
⎠
=
⎝
x
=
(2)
(1) can be rewritten as:
d
x
(
t
)
dt
=
f
(
t,x
(
t
)
,u
(
t
))
(3)
,x
n
]
T
n
,and
u
=[
u
1
,
,u
r
]
T
r
are the state and
where
x
=[
x
1
,
···
∈
R
···
∈
R
,f
n
]
T
n
is as follows.
control vectors, respectively and
f
=[
f
1
,
···
∈
R
⎡
⎤
L
d
x
1
+
n
p
L
q
R
s
1
−
L
d
x
2
x
3
+
L
d
u
1
⎣
⎦
n
p
L
d
n
p
φ
R
s
1
f
(
t,x
(
t
)
,u
(
t
)) =
−
L
q
x
2
−
L
q
x
1
x
3
−
L
q
x
3
+
L
q
u
2
.
(4)
3
n
p
(
L
d
−L
q
)
2
J
x
1
x
2
+
3
n
p
φ
1
−
2
J
x
2
−
J
τ
L
here
τ
L
is a function with a constant period, the fitting curve in a cycle shown
in Fig. 1.
In one period,
τ
L
can be expressed by:
τ
L
=2
.
29
−
1
.
75 cos(131
.
26
t
)
−
3
.
38 sin(131
.
26
t
)
−
0
.
53 cos(262
.
51
t
)
(5)
+0
.
58 sin(262
.
51
t
)
Theinitialvalueofthestatevariables
x
for differential equations (3) is denoted
by:
x
(0)=
x
0
(6)
We also define
U
1
and
U
2
as follows
U
1
=
υ
=[
υ
1
···
n
:(
E
i
)
T
υ
υ
γ
]
T
∈
R
≤
b
i
,i
=1
,
···
,q
}
(7
a
)
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