µ
1
1
æ
1
1
ö
i
i
i
i
$
µ
em = L çw Ä i
i
i
m -
m +
Sαβ ÷
m =
k
w Ä m -
m +
k
dt
T
T Sαβ
m è
T
T
R
R
R
R
ø
$
$
q = i
Ä em
m
Sαβ
‘
µ
µ
w
q
D
k
w Kalman Filter
PI
m
Fig. 3. Reactive Power MRAS Speed Estimation
point (Aström & Wittenmark, 1997). In addition, this filter takes into account the signal
noise, which could be generated as pulse width modulation drivers. Assuming the definitions
⎡
⎤
⎡
⎤
1
T
− 1
x
⎣ − Bn
⎦
⎣
⎦
k =
ωk TL
, Am =
J
J
, Bm =
J
, Cm =
1
0
and yk = ω.
0
0
0
Then, the recursive equation for the discrete time Kalman Filter (De Campos et al., 2000) is
described by
− 1
K( k) = P( k)C Tm CmP( k)C Tm + R
(35)
where K( k) is the Kalman gain. The covariance matrix P( k) is given by
P( k + 1) = (I − Am ts) (P( k) − K( k)CmP( k)) (I − Am ts) T + (Bm ts) Q (Bm ts) T
(36)
Therefore, the estimated torque TL is one observed state of the Kalman filter
xk( k + 1) = (I − Am ts) xk( k) + Bm tsu( k) + (I − Am ts) K( k) ( ω − Cmxk( k)) (37)
T
giving ω ≈ ωk and xk( k) =
ωk TL
.
The matrices R and Q are defined according to noise elements of predicted state variables,
taking into account the measurement noise covariance R and the plant noise covariance Q.
7. State variable filter
The state variable filter (SVF) is used to mathematically evaluate differentiation signals. This
filter is necessary in the implementation of FLC and MRAS algorithms. The transfer function
of SVF is of second order as it is necessary to obtain the first order derivative.
ωsvf
Gsv f =
(38)
2
s + ωsv f
Feedback Linearization of Speed-Sensorless Induction Motor Control with Torque Compensation
105
where ωsv f is the filter bandwidth defined at around 5 to 10 times the input frequency signal
usv f .
The discretized transfer function, using the Euler method, can be performed in state-space as
xsvf ( k + 1) = Asvfxsvf ( k) + Bsvf usv f ( k)
(39)
where Asvf =
1
1
−ω 2
, B
and x
.
1 − 2 ω
svf =
0
svf =
x 1
sv f
sv f
ω 2
x
svf
2
The state variables x 1 and x 2 represent the input filtered signal and input derivative signal.
8. Experimental results
Sensorless control schemes were implemented in DSP based platform using TMS 320F2812.
Experimental results were carried out on a motor with specifications: 1.5cv, 380V, 2.56A, 60
Hz, Rs = 3.24Ω, Rr = 4.96Ω, Lr = 404.8 mH, Ls = 402.4 mH, Lm = 388.5 mH, N = 2 and nominal speed of 188 rad/s.
The experimental analyses are carried out with the following operational sequence:
1) The motor is excited (during 10 s to 12 s) using a smooth flux reference trajectory.
2) Starting from zero initial value, the rotor speed reference grows linearly until it reaches the
reference value. Thus, the reference rotor speed value is kept constant.
3) During stand-state, a step constant load torque is applied.
In order to generate load variation for torque disturbance analyses, the DC motor is connected
to an induction motor driving-shaft. Then, the load shaft varies in accordance with DC motor
field voltage and inserting a resistance on its armature. Fig. 4 and Fig. 5 depict performance
of both control schemes: FLC control and simplified FLC control with rotor speed reference of
18 rad/s. In these figures measure speed, estimated speed, stator (q-d) currents and estimated
torque are illustrated.
Fig.6 and Fig. 7 present experimental results with 36 rad/s rotor speed reference.
Fig. 8 and Fig. 9 show FLC control and Simplified FLC with 45 rad/s rotor speed reference.
The above figures present experimental results for low rotor speed range of FLC control and
Simplified FLC control applying load torque. In accordance with the figures above, both
control schemes present similar performances in steady state. It is verified that both schemes
respond to compensated torque variations. With respect to Simplified FLC, it is necessary to
carrefully select fixed gains in order to guarantee the alignment of the rotor flux on the d axis.
9. Conclusion
Two different sensorless IM control schemes were proposed and developed based on
nonlinear control - FLC Control and Simplified FLC Control. These control schemes are
composed of a flux-speed controller, which is derived from a fifth-order IM model. In the
implementation of feedback linearization control (FLC), the control algorithm presents a large
number of computational requirements. In the simplified FLC scheme, a substitution of FLC
currents controllers by two PI controllers is proposed to generate the stator drive voltage.
In order to provide the rotor speed for both control schemes, a MRAS algorithm based on
reactive power is applied.
To correctly evaluate whether this Simplified FLC does not affect control performance, a
comparative experimental analysis of a FLC control and a simplified FLC control is presented.
Experimental results in DSP TMS 320F2812 platform show the performance of both systems
106
Electric Machines and Drives
2
) 1.5
( A
I dt 1
rren 0.5
Cu
0
0
10
20
30
40
50
60
70
T ime( s)
(a) IM Stator Current Ids
2
) 1.5
( A
I q 1
tn 0.5
rreuC 0
−0.5
0
10
20
30
40
50
60
70
T ime( s)
(b) IM Stator Current Iqs
20
)
s/ 15
d
( ra 10
ω
eedp
5
S
ˆ ωk
0
otorR −5 0
10
20
30
40
50
60
70
T ime( s)
(c) Rotor Speed - Estimated and Encoder Measurement
2
) m 1
( N.e
0
orquTd −1
a
Lo
−2
0
10
20
30
40
50
60
70
T ime( s)
(d) Estimated Load Torque
Fig. 4. FLC control with 18 rad/s rotor speed reference
Feedback Linearization of Speed-Sensorless Induction Motor Control with Torque Compensation
107
2
) 1.5
( A
I d
1
tn
rre 0.5
uC
0
0
10
20
30
40
50
60
70
T ime( s)
(a) IM Stator Current Ids
1.5
) 1
( A
I qt 0.5
n
rreu 0
C
−0.5
0
10
20
30
40
50
60
70
T ime( s)
(b) IM Stator Current Iqs
20
)
s/d 15
( ra
10
ω
eedp
5
S
ˆ ωk
0
otorR
−5 0
10
20
30
40
50
60
70
T ime( s)
(c) Rotor Speed - Estimated and Encoder Measurement
1.5
) m 1
( N.e
0.5
orquTd 0
a
Lo
−0.5
0
10
20
30
40
50
60
70
T ime( s)
(d) Estimated Load Torque
Fig. 5. Simplified FLC control with 18 rad/s rotor speed reference
108
Electric Machines and Drives
1.5
)( A 1
I dtn 0.5
rreu 0
C
−0.5
0
10
20
30
40
50
60
70
T ime( s)
(a) IM Stator Current Ids
2
1.5
)( A 1
I qtn 0.5
rreu 0
C
−0.5
0
10
20
30
40
50
60
70
T ime( s)
(b) IM Stator Current Iqs
40
) s/ 30
d
( ra 20
ω
eedp
ˆ ω
10
k
S
otor
0
R
0
10
20
30
40
50
60
70
T ime( s)
(c) Rotor Speed - Estimated and Encoder Measurement
)
2
m
1
( N.e
0
orquT −1
da
Lo
−2
0
10
20
30
40
50
60
70
T ime( s)
(d) Estimated Load Torque
Fig. 6. FLC control with 36 rad/s rotor speed reference
Feedback Linearization of Speed-Sensorless Induction Motor Control with Torque Compensation
109
2.5
)
2
( A
I d 1.5
t
1
rren
Cu 0.5
0
0
10
20
30
40
50
60
70
T ime( s)
(a) IM Stator Current Ids
3
) 2
( A
I q 1
tn
rre
0
uC
−1
0
10
20
30
40
50
60
70
T ime( s)
(b) IM Stator Current Iqs
40
) s/ 30
d
( ra 20
ω
eedp
ˆ ω
10
k
S
otor
0
R
0
10
20
30
40
50
60
70
T ime( s)
(c) Rotor Speed - Estimated and Encoder Measurement
)
2
m
( N.
1
e
orquT 0
da
Lo −1
0
10
20
30
40
50
60
70
T ime( s)
(d) Estimated Load Torque
Fig. 7. Simplified FLC control with 36 rad/s rotor speed reference
110
Electric Machines and Drives
1.5
)
1
( A
I d 0.5
t
0
rren
Cu−0.5
−1
0
10
20
30
40
50
60
70
T ime( s)
(a) IM Stator Current Ids
3
) 2
( A
I q 1
tn
rre
0
uC
−1
0
10
20
30
40
50
60
70
T ime( s)
(b) IM Stator Current Iqs
50
) s 40
/d 30
( ra
ω
20
eedp
ˆ ω
10
S
k
0
otorR −10 0
10
20
30
40
50
60
70
T ime( s)
(c) Rotor Speed - Estimated and Encoder Measurement
)
2
m
1
( N.e
0
orquT −1
da
Lo
−2
0
10
20
30
40
50
60
70
T ime( s)
(d) Estimated Load Torque
Fig. 8. FLC control with 45 rad/s rotor speed reference
Feedback Linearization of Speed-Sensorless Induction Motor Control with Torque Compensation
111
2.5
)
2
( A
1.5
I dt
1
rren
0.5
Cu
0
0
10
20
30
40
50
60
70
T ime( s)
(a) IM Stator Current Ids
3
) 2
( A
I q 1
tn
rre
0
uC
−1
0
10
20
30
40
50
60
70
T ime( s)
(b) IM Stator Current Iqs
50
) s 40
/d
30
( ra
ω
20
eedp
ˆ
S
ω
10
k
0
otorR −10 0
10
20
30
40
50
60
70
T ime( s)
(c) Rotor Speed - Estimated and Encoder Measurement
)
2
m
( N.
1
e
orquT 0
da
Lo
−1
0
10
20
30
40
50
60
70
T ime( s)
(d) Estimated Load Torque
Fig. 9. Simplified FLC control with 45 rad/s rotor speed reference
112
Electric Machines and Drives
in the 18rad/s, 36 rad/s and 45 rad/s rotor speed range. Both c