/
s
s
r m
σ s
L r
L )} I ,
23
A = 1 / (σ s
L ) I ,
I = ⎢
⎥ ,
J = ⎢
⎥ ,
0 1
⎣
⎦
1
0
⎣
⎦
T
T
T
ψ = ⎡
⎤
i = ⎡
⎤
v = ⎡
⎤
r
ψ
⎣ rα ψ rβ ⎦ is the rotor flux, s ⎣ si
s
i
α
β ⎦ is the stator current, s
v
⎣ s
v
α
sβ ⎦ is
the stator voltage and
2
σ = 1− L /( L L )
m
s r is the leakage coefficient.
Sensorless Vector Control of Induction Motor Drive - A Model Based Approach
79
Now, we introduce a new quantity into the motor model which when introduced will make
the right hand side of conventional motor model given by equations (1) and (2) independent
of the unknowns – the rotor flux and speed. Let’s define the new quantity as
Z = − 11
A ψ r (3)
A new motor model is obtained after introducing the new quantity as given below:
dψ
r = 12
A is + 14
A Z (4)
dt
di
s = 2
A 2 is + 23
A vs + 24
A Z (5)
dt
dZ
= 32
A is + 34
A Z (6)
dt
where
2
2
24
A = { L /
m
σ s
L r
L }
14
A = − I ,
(
) I , 32
A = ( L R / L ) I −ω( L R /
m r
r
m r
r
L ) J and 34
A = 11
A .
2.2 Observer structure and speed estimation
The proposed speed estimation algorithm is based on observing the newly defined quantity
which is a function of rotor flux and speed. Equation (5) and (6) are used for constructing a
Gopinath’s reduced order observer (Gopinath, 1971) for estimating the newly defined
quantity. The observer is as given below
ˆ
Z
⎛
ˆ
d
di
di ⎞
ˆ
s
s
= 32
A is + 34
A Z + G ⎜
−
⎟ (7)
dt
⎜ dt
dt ⎟
⎝
⎠
⎡ g
− g ⎤
ˆ
di
where
1
2
G = ⎢
⎥ is the observer gain. Using equation (5) for s the observer equation
g
⎣ 2
1
g ⎦
dt
becomes
ˆ
dZ
⎛ di
⎞
ˆ
s
ˆ
= 32
A is + 34
A Z + G
− 22
A is − 23
A
s
v −
⎜
24
A Z ⎟ (8)
dt
⎝ dt
⎠
The observer poles can be placed at the desired locations in the stable region of the
complex plane by properly choosing the values of the elements of the G matrix. In order to
avoid taking derivative of the stator current in the algorithm we introduce another new
quantity
ˆ
D = Z − Gis (9)
Finally, the observer is of the following form:
d
F = ( 32
A + 3
A 4 G − G 22
A − G 24
A G) is − G 23
A vs + ( 34
A − G 24
A ) D ` (10)
dt
80
Electric Machines and Drives
ˆ
Z = D + Gis (11)
The block diagram of the Z observer is shown in Fig. 1.
34
A − G 24
A
+
v
_
ˆ
s
+
Z
23
GA
D
+
+
is
32
A + 34
A G − G 22
A − G 2
A 4 G
G
Fig. 1. Block diagram of Z observer
Assuming no parameter variation and no speed error, the equation for error dynamics is
given by
d
d
Z =
( ˆ
Z − Z ) = (
34
A − 2
A 4 G) Z (12)
dt
dt
Eigenvalues of ( 34
A − 24
A G) are the observer poles which are as given below:
⎛ R
L
⎞
⎛
L
⎞
r
m
m
P 1,
obs 2 = −⎜
+
1
g ⎟ ± j ⎜ω −
g 2 ⎟
⎜
⎟
⎜
⎟ (13)
⎝ r
L
σ s
L r
L
σ
⎠
⎝
s
L r
L
⎠
The desired observer dynamics can be imposed by proper selection of observer gain G.
Next, let’s see how the rotor speed is computed. It can be seen that the observed quantity is
a function of rotor flux and speed. Performing matrix multiplication of
T
ψr J with equation
(3) we have
Z ψ
− Z
α
β
βψ α = ( 2
2
r
r
ψ rα +ψ rβ )ω (14)
This is a simple equation which does not involve derivative or integration. To use it directly
for speed computation we need to know the rotor flux; and as for Zα and Zβ we can use the estimated values. The required flux is obtained from the reference. Rearranging the above
equation we have the equation used for rotor speed computation as given by
*
*
ˆ
ˆ
Z ψ
− Z ψ
ˆ
α rβ
β rα
ω =
(15)
2
2
*
*
ψ rα +ψ rβ
The coefficient matrices A 32 and A 34 in the observer equation are updated with the estimated values of rotor speed.
Sensorless Vector Control of Induction Motor Drive - A Model Based Approach
81
It is to be noted here that the model of the motor used in implementing the observer
algorithm has been developed assuming that the derivative of the rotor speed is zero. It is
valid to make such an assumption since the dynamics of rotor speed is much slower than
that of electrical states. Moreover, such an assumption allows estimation without requiring
the knowledge of mechanical quantities of the drive such as load torque, inertia etc.
2.3 Simulation results
Simulation is carried out in order to validate the speed estimation algorithm presented. The
block diagram of the sensorless indirect vector controlled induction motor drive
incorporating the proposed speed estimator is shown in Fig. 2. The results of simulation are
shown in Fig. 3 - Fig. 5.
_
dc
V
*
*
*
+
*
i
ω +
sa +
em
T
s
i d
dq
_
_
*
2 h
i
INDIRECT
sq
*
i
+
ˆ
ω
FOC
sb
*
ψ
INVERTER
r
ρ
_
2 h
* sic
+
abc
_
2 h
*
ψr
v
i i
i
s
sa sb
sc
SPEED
ESTIMATOR
is
IM
Fig. 2. Sensorless indirect VC induction motor drive
Initially, the drive is run at no load. It is accelerated from rest to 150 rad/s at 0.15 sec. and
then, the speed is reversed at 2.5 sec. The speed is reversed again at 5.5 sec. The speed of the
motor (ω ) , estimated speed ( ˆ
ω ) and reference speed *
(ω ) are shown in Fig. 3 (a). Fig. 3 (b)
shows speed estimation error (ω − ˆ
ω ) . The newly defined quantity ( Z) and its estimated
value ˆ
( Z ) are shown in Fig. 3 (c) and its estimation error
ˆ
( Z − Z ) is shown in Fig. 3 (d).
The estimation algorithm and the drive response are then verified under loading and
unloading conditions. The unloaded drive is started at 0.15 sec and full load is applied at 1
sec; then load is completely removed at 2 s. Later, after speed reversal, full load is applied at
4 sec and the load is completely removed at 5 sec. Fig. 4 shows the speed estimation result
and response of the sensorless drive system.
Then, the sensorless induction motor drive is run under fully loaded condition at various
operating speeds. The drive is started at full load at 0.15 s to 150 rad/s and the speed is
reduced in steps in order to observe the response of the loaded drive at various speeds. Fig.
5 shows the estimation results and response of the loaded drive.
82
Electric Machines and Drives
200
15
Reference speed
Actual speed
10
100
ror
d/s ]
Estimated speed
n er
5
[ ra
0
atio
0
ad/s ]
stim
r [ -5
Speed -100
-10
Speed e
-200
-15
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Time [ s ]
Time [ s ]
( a )
( b )
120
3
100
2
80
Actual Z
1
Estimated Z
60
on error
Z
0
ati
40
msti -1
e
20
Z -2
0
-3
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Time [ s ]
Time [ s ]
( c )
( d )
Fig. 3. Acceleration and speed reversal at no load; (a) reference, actual and estimated speeds;
(b) speed estimation error; (c) actual Z and estimated Z and (d) Z estimation error 200
15
Reference speed
r
Actual speed
10
] 100
erro
Estimated speed
d/s
5
ation /s ]
0
d [ ra
0
tim [ rad -5
Spee -100
d es
-10
Spee
-200
-15
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Time [ s ]
Time [ s ]
( a )
( b )
120
3
100
2
r
80
1
60
Actual Z
on erro
Z
0
Estimated Z
ati
40
-1
20
estimZ -2
0
-3
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Time [ s ]
Time [ s ]
( c )
( d )
Fig. 4. Application and removal of load; (a) reference, actual and estimated speeds; (b) speed
estimation error; (c) actual Z and estimated Z and (d) Z estimation error
Sensorless Vector Control of Induction Motor Drive - A Model Based Approach
83
180
8
Reference speed
150
Actual speed
r
]
4
120
Estimated speed
erro
ad/s
90
ation
d [r
0
tim
rad/s]
60
Spee
es
[
30
-4
Speed
0
-8
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Time [ s ]
Time [ s ]
( a )
( b )
100
4
80
Actual Z
Estimated Z
or 2
60
err
Z
0
ation
40
tim
es
20
Z -2
0
-4
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Time [ s ]
Time [ s ]
( c )
( d )
Fig. 5. Operation at full load at various speeds; (a) reference, actual and estimated speeds;
(b) speed estimation error; (c) actual Z and estimated Z and (d) Z estimation error 2.4 Improvement in speed estimation
It is observed that the estimation algorithm presented above gives good estimation ac