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where
τ
j
(
j
=0
,
1
,...,q
U
, while
I
=
χ
[
τ
j−
1
,τ
j
)
denotes the indicator function of the interval [
τ
j−
1
,τ
j
) which is defined as:
χ
I
(
t
)=
1
, t
−
1) is interval endpoints.
σ
q,j
∈
I
0
,
elsewhere
∈
(14)
Define
σ
q
=[(
σ
q,
1
)
T
,...,
(
σ
q,q
)
T
]
T
and
σ
q,j
=[
σ
q,
1
,...,σ
q,
r
]
T
,
j
=1
,...,q
.
Furthermore,
σ
q,j
rq
such that
σ
q
∈ Ω
.
Apparently,
u
q
depends on the control parameter
σ
q
. For each control variable
σ
q
∈ U
,
j
=1
,...,q
.Let
Ω
be the set in
R
q
, the control parameter vector
σ
q
Ω
has only a single satisfies (14). On
the contrary, the control parameter vector
σ
q
∈U
∈
q
has only a unique
σ
q
∈U
∈
Ω
.
So the
Problem1
can be approximated by the following: Minimizing
g
0
(
σ
q
)=
g
0
(
u
q
(
t
))
(15)
For
σ
q
Ω
.
To solve this problem, the following conclusion can be obtained:
For each
i
=0
,
1
,...,N
and each
j
=1
,...,q
, the gradient of the function
g
0
(
σ
q
) with respect to
σ
q,j
is:
∂g
0
(
σ
q
)
∂σ
q,j
∈
=
I
∂H
0
(
t,x,u
)
∂
u
dt
(16)
Thus, the above approximation problem can be transformed into the finite di-
mension optimal control problem.
4 Simulation Analysis and Verification
In order to verify the feasibility of optimal control parameters approach, we use
a Panasonic 5RS102ZAA01 compressor motor for simulations, which is run on a
1.5 air-conditioning compressor-driven experimental platform. The compressor
motor parameters are shown in Table 1.
Tabl e 1.
Panasonic 5RS102ZAA01compressor motor parameters
MOTOR PARAMTER Parameter values
Permanent Magnetic Flux 0.105Wb
Number of Pole Pairs 3
Stator Winding 0.49
Ω
Direct-axis Inductance 6.3mH
Quadrature axis Inductance 11.8mH
Moment of inertia
0.00063kg.
m
2
Differential equations (4) can be expressed as:
⎧
⎨
−
77
.
78
x
1
+5
.
62
x
2
x
3
+ 158
.
73
u
1
f
(t
,x,u
)=
−
41
.
53
x
2
−
1
.
6
x
1
x
3
−
26
.
69
x
3
+84
.
75
u
2
(17)
⎩
−
39
.
29
x
1
x
2
+ 750
x
2
−
1587
.
3
τ
L
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