an investigation of dual stator winding induction machines

ˆ ω
rm
The output error is expressed in terms of the speed information and states in (10.121).
i
qdsi
− iˆ
qdsi
= C ⋅ X − Cˆ
⋅ Xˆ
=
=
i
( Ci
− ∆Ci
) ⋅ ei
+ ∆Ci
⋅ X i
( C − ∆C
) ⋅ [ p ⋅ I − A + K
−1
⋅ C ] α ⋅ Xˆ
− Z ⋅ ( ω − ˆ ω )
i
i
i
i
i
The speed estimation using (10.117) and (10.121) becomes:
= G
+
*
*
⋅k{
Im[
ˆ λ ( − ˆ ) ] + [ ˆ ( − ˆ
qdr1
iqds1
iqds1
k1
Re λqdr1
iqds1
iqds1
) ] }
*
*
( 1−
k)
{ Im ˆ λ ( i −iˆ
) + k Re ˆ λ i −iˆ
}
+ G⋅
( 1−
k)
i
i
i
393
[ i i i ri ri ] + ∆Ci
⋅ X i
[ qdr2
qds2
qds2
] 2 [ qdr2
( qds2
qds2
) ]
ˆ *
−1
λ ⋅(
C − ∆C
) ⋅[
p⋅
I − A + K ⋅C
] [ α ⋅ Xˆ
− Z ⋅(
ω − ˆ ω ) ]
⎧ ⎡ qdr1
1 1
⎪ Im⎢
⎪
⎢
⎣+
ˆ *
λqdr1
⋅∆C
1 ⋅ X1
= G⋅
k⎨
*
⎪ ⎡ ˆ λqdr1
1 1
⎪+
k ⎢ 1 Re
*
⎪ ⎢
⎩ ⎣+
ˆ λqdr1
⋅∆C
1 ⋅ X
[ ]
−1
( C − ∆C
) ⋅[
p⋅
I − A + K ⋅C
] α ⋅ Xˆ
− Z ⋅(
ω − ˆ ω )
[ ]
−1
( C − ∆C
) ⋅[
p⋅
I − A + K ⋅C
] α ⋅ Xˆ
− Z ⋅(
ω − ˆ ω )
⎧ ⎡ ˆ *
λqdr2
⋅ 2 2
⎪ Im⎢
⎪
⎢
⎣+
ˆ *
λqdr2
⋅∆C
2 ⋅ X 2
⎨
*
⎪ ⎡ ˆ λqdr2
⋅ 2
⎪+
k ⎢ 2 Re
*
⎪ ⎢
⎩ ⎣+
ˆ λqdr2
⋅∆C
2 ⋅ X
1
⎤ ⎫
⎥ ⎪
⎥ ⎪
⎦ ⎪
⎬
⎤ 1 ⎪
⎥⎪
⎥
⎦⎪⎭
−1
( C − ∆C
) ⋅[
p⋅
I − A + K ⋅C
] α ⋅ Xˆ
− Z ⋅(
ω − ˆ ω )
2
2
1
1
1
2
1
1
2
1
2
1
2
2
1
1
2
1
2
1
(10.121)
⎤
⎥
⎥
⎦
[ ]
2
1
2
r1
r1
2
2
r1
2
r
r2
r2
r2
r2
⎫
⎪
⎪
⎬
⎤⎪
⎥⎪
⎥
⎦⎪⎭
(10.122)
To simplify the analysis that follows, the motor parameters are assumed to be
constant, which means C = 0 and α = 0 . Hence, the speed estimation is simplified as
in (10.123).
ˆ ω
rm
∆ i
⎪
⎧ Im
= −G
⋅ k⎨
⎪⎩ + k1
⎪
⎧ Im qdr
− G ⋅ ( 1 − k ) ⎨
⎪⎩ + k2
Re
i
[ ˆ *
−1
λ ⋅ ( ) ⋅ [ ⋅ − + ⋅ ] [ ⋅ ( − ˆ
qdr1
C1
p I A1
K1
C1
Z1
ωr1
ωr1
) ]
*
Re ˆ λ ( C ) ⋅ [ p ⋅ I − A + K
−1
⋅ C ] [ Z ⋅ ( ω − ˆ ω ) ]
⎬
[ qdr1
1
1 1 1 1 r1
r1
] ⎪⎭
[ ˆ *
−1
λ ⋅ ( C ) ⋅ [ p ⋅ I − A + K ⋅ C ] [ Z ⋅ ( ω − ˆ ω ) ]
2
2
2
2
⎬
[ ˆ *
−1
λ ⋅ ( ) ⋅ [ ⋅ − + ⋅ ] [ ⋅ ( − ˆ
qdr 2 C2
p I A2
K 2 C2
Z 2 ωr
2 ωr
2 ) ] ]⎪⎭ 2
2
r 2
⎪
⎫
r 2
⎪
⎫