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where τ j ( j =0 , 1 ,...,q
U , while I = χ [ τ j− 1 j )
denotes the indicator function of the interval [ τ j− 1 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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