DYNAMIC FLUX OBSERVER FOR INDUCTION MOTOR SPEED CONTROL
S.Sathiakumar
Lecturer in School of Electrical and Information Engineering
University of Sydney, NSW2206, Australia
Abstract:
A dynamic flux observer suitable for speed estimation and control of induction motor is described
in this paper. The technique does not employ computationally extensive algorithm such as Model
Reference Adaptive System or Speed Adaptive Flux System, in which estimation accuracy is
influenced by chosen parameters of the adaptive mechanism. In the proposed algorithm, a stable
flux observer estimates the rotor flux based on the dynamic model of the machine and the speed of
the motor is then calculated using the estimated flux. The estimated speed is successfully used in
the speed control loop without employing any start up procedure. Extensive simulation and
experimental results suggest that the algorithm works with any initial conditions.
INTRODUCTION
The control of induction machine drive system
requires speed feedback, which is usually obtained
from mechanical transducers mounted on the machine
shaft. However, these transducers reduce system’s
robustness and reliability. In some cases it is difficult
to mount the speed sensor on the motor shaft.
Tachometer noise is also a problem for an accurate
control system. The new drive system, namely “speedsensorless control” overcomes the problems of
mechanical transducers. The concept of this technique
is that the rotor speed is estimated from easily
measurable voltage and current of the induction motor
and is used in the feedback loop of the speed
controller. Several speed estimation methods have
been proposed [1]-[9].
Almost all techniques proposed are for application in
the vector control [2],[3],[4],[5],[6],[8]and [9] which
are found to work well in field-orientation control
condition. While, the other techniques were only
suitable for the slip-controlled system application. In
the estimation technique [1], slip calculation is carried
out based on the steady state equivalent circuit and then
speed is calculated. So this technique can achieve good
performance only at steady state as detailed evaluations
given by [7]. In [2] the slip is calculated using the
dynamic variables, assuming a steady state conditions
for the flux. This work while presenting different
control methods does not discuss the effectiveness of
the speed estimation under changing flux conditions. In
the Model Reference Adaptive System (MRAS) [3],
[6], the speed estimation technique is based on two
models describing the motor dynamics. The error that
expresses rotor flux difference of the two models
drives an adaptation mechanism to generate estimated
speed. This technique could provide a good
performance in steady state and some transients except
in low speed range due to the presence of pure
integration in the algorithm causing drift in the
estimation. The drift problem is solved completely by
the speed estimation method proposed by H. Kubota
and K. Matsuse [4],[5]. In this algorithm, speed
adaptation is implemented in the full-order observer, so
high performance could be achieved even at very low
speed by choosing appropriate parameters of the
adaptive mechanism. The method presented in [8], [9]
seem to be the simplest due to direct calculation of
rotor speed from dynamic model of the machine.
However, good performance may not be achieved due
to the use of open loop integration in [9] or in the case
of rising and falling edges of step torque [8].
A dynamic flux observer is presented in this paper
which takes the applied stator voltage and the
measured stator current as inputs. The observer can be
shown to converge by properly choosing the pole
location of the observer. It has been found through
simulation as well as by experimentation that the
observer converges irrespective of any initial condition
chosen. The flux thus estimated is then used for speed
estimation which overcomes the drawbacks presented
above. This algorithm does not impose any specific
conditions on the control signals and does not include
pure integration. Through computer simulation, the
algorithm has been confirmed to be independent of the
drive system controlling the machine and performs well
even at very low speed under different dynamic
conditions of the motor including transient and steady
state operations.
DYNAMIC MODEL OF INDUCTION MOTOR:
The state equations of the induction motor are
expressed using space vector notation in the stationary
coordinate frame as follows:
r
r
dis
r
r
= a11 is + a12ψ r + B1v s
dt
(1a)
r
r
r
dψ r
= a 21is + a 22ψ r
dt
r
r
r
dFau
= (a 22 − La12 )ψˆ r + (a 21 − La11 )is
dt
r
− LB1 v s
(1b)
where :
D
;
Ls σ
L
L
1
K
a12 =
( m − jω r m ) =
− jω r K ;
Ls σ Lr Tr
Lr
Tr
L
1
a 21 = m ; a 22 = jω r − ;
Tr
Tr
a11 = −
1
B1 =
;
Ls σ
L
Tr = r .
Rr
Lm
K=
Lr L s σ
Rr L2m
; D = Rs +
;
L2r
r
r
r
ψ̂ r = Fau + Lis
r
is
r
vs
(3b)
a21 - La11
L
+
_
LB1
∫
r
Fau
+
r
ψˆr
+
+
r
The state variables of the motor are stator current is
r
r
and rotor flux linkage ψ r . The input variable is vs .
Rs , Rr are stator, rotor resistance, respectively and
Ls , Lr represent stator, rotor self-inductance. Lm is
L2m
mutual inductance and σ = 1 −
is motor
L s Lr
leakage coefficient. ω r is the rotor speed expressed in
electrical radians per seconds and Tr is rotor time
constant.
DYNAMIC FLUX OBSERVER
For the algorithm presented in this paper, it will be
shown that the rotor speed can be calculated directly
from the rotor flux. The flux required for speed
calculation will be estimated using a dynamic flux
observer as follows. In (1), two states of the machine
are represented by two differential equations involving
stator current and rotor flux. Stator current can be
measured using current sensor, so it is regarded as a
known state and rotor flux can be estimated by an
observer that is designed based on Gopinath’s reducedorder theory [9] and the algorithm proposed in [10].
The observer equation is presented in the following
form:
r
r
r
dψˆ r
= (a 22 − La12 )ψˆ r + (a 21 − La11 )is
dt
(2)
r
dis
r
+L
− LB1v s
dt
r
where ψˆ r is the estimated rotor flux vector and L is
feedback gain of the flux observer. In order to avoid
the derivative of stator current in the algorithm, the
new variable
(3a)
r
Fau is used and the flux observer
equation is given by (3).
a22 - La12
Figure 1. Configuration rotor flux observer
Figure 1 shows the configuration of the rotor flux
observer. The coefficients a12 and a 22 require the
rotor speed ωr and we use the estimated speed ω$r
instead. This is perfectly valid since the flux dynamics
is much faster than the mechanical speed, which is
calculated from the flux. In the case of no parameter
variations and for small speed estimation error,
dynamic equation of flux estimation error is presented
as follows:
r
r
r
r
de dψˆ r dψ r
=
−
= −( La12 − a 22 )e
dt
dt
dt
(4)
Observer poles are defined in the form of -αs ±jβs . The
complex constant L (L1 + j L2) can be so chosen to
place the poles of the observer in the stable region of
the complex plane. Solving the characteristic equation
of estimation error, we obtain:
αs =
^
^
KL1 + 1
+ KL2 ω r = α so + β so Tr ω r
Tr
^
β s = ( KL1 + 1) ω r −
^
KL2
= − β so + α so Tr ω r
Tr
(5)
where
α so =
KL1 + 1
KL2
and β so =
are absolute
Tr
Tr
values of pole components at zero speed. Desired poles
for condition of stability and convergence of the
observer is defined as:
p desired = −α d ± jβ d
(6)
Feedback gain L of the observer can be chosen to place
poles at the desired place by assigning
α so = α d and β so = β d
(7)
For higher positive speeds the poles move towards the
left half plane and the system becomes more stable
ROTOR SPEED CALCULATION
Equation (1a) representing the relations of stator
voltage, stator current, rotor flux and rotor speed can
be rewritten as equation (8). Assuming that voltage,
current and rotor flux vectors can be expressed in the
complex form, from (8) rotor angular speed is given by
equation (9).
________________________________________________________________________________________
r
L2m r
dis
L r
L
r
r
v s = ( R s + R r 2 ) is + Ls σ
− m ψ r + m jω rψ r
dt Lr Tr
Lr
Lr
^
^
di sβ
di sα
v
Di
L
ψ
(
−
−
σ
)
−
ψ
)
sβ
sβ
s
rα
rβ (v sα − Di sα − Ls σ
^
dt
dt
ωr =
r 2L
m
ψˆ
Lr
+
vsβ
D
-
isβ
ψˆ r α
*
(9)
+
Lsσ
(8)
ω̂ r
-
d dt
(x2+y2)
Lm/Lr
ψˆ rβ
vsα
+
D
isα
-
*
-
Lsσ
d dt
Figure 2. Speed Calculation
___________________________________________________________________________________________
where: sα and sβ - subscript refer to the stator
quantities in the stationary coordinate frame.
rα and rβ - subscript refer to the rotor quantities in the
stationary coordinate frame.
Figure 2 shows the block diagram of the speed
calculation algorithm. Thus the shaft angular speed is
determined only from components of terminal
quantities and rotor flux in the stationary rectangular
coordinate frame.
SIMULATION RESULTS
The proposed algorithm is verified for its applicability
in a slip controlled drive system as shown in Figure 3.
The speed feed-back required for the control system is
obtained using the algorithm described above. The
algorithm was verified for different speed ranges with
different initial conditions of the flux observer. The
performance of the rotor flux and the speed is studied
under the following test conditions. The speed
controller is given an initial set point of 314 rad/sec at
time 0 sec. and then the set point is changed to 157
rad/sec at time 2 sec. The load torque of 50 % of rated
value is applied at 3 sec. and removed at 4 sec.
a) Rotor Flux performance:
The stability and convergence of rotor flux observer
with different initial conditions at different speed of the
motor can be evaluated by error of rotor flux. Figure 4
shows the error of α-axis rotor flux with observer
initial conditions of 0.1 and 0.3 (Wb Turns) for the two
different set speeds. One can see that, flux error decays
fast from initial values of 0.1 or 0.3 and
Main
Source
ωr *
+ -
ω*sl
Speed
Controller
ω*s
+
Rectifier
Function
Generator
V*s
Static
Inverter
+
ω^r
Speed
Estimator
IM
Figure 3. The diagram of the sensorless slip-control drive system.
Figure 4. Error of rotor flux on α-axis with different initial condition of the observer
a) High speed test
(b) Low speed test
(Dark line is error at initial values of 0.1 and light line is at 0.3)
.
(a) Rotor flux
(b) Speed
Figure 5. Simulation results of the slip-controlled drive system at a speed of 314 rad/sec and 157 rad/sec.
Top plots are actual values ; Middle and bottom plots are estimated values at initial condition
0.1 Wb-Turns and 0.3 Wb-Turns, respectively.
remains zero for subsequent operations including
steady state, deceleration and load change. Decaying
time of rotor flux error depends mainly on speed and
not much on initial conditions. From figure 4a and 4b,
it can be seen that the convergence is faster at high
speed as expected from observer design. The
performance for error of β - axis rotor flux is similar.
b) Speed performance:
Figures 5a, 5b present the simulated time response of
the rotor flux and speed for the slip-control drive
Main Source
Rectifier
Computer
Limit
Speed Controller
V*
+
-
+
I/O
Board
DAS-1602
Vωr
Kωr
Control
Unit
+
Vfb
Vω$r
Vin Inverter
D/A
ω$r
Speed
Estimator
Kωr
Data
Acauisition
Control
and
Interface
VTa
VTb
VTc
A/D
PWM VSI
Va Vb Vc
ITa
ITb
ωr
Dynamometer
IM
Figure 6. The configuration of the experimental computer-based slip-controlled drive system
system. The simulation results show that the estimated
speed tracks the true speed well regardless of initial
conditions of the rotor flux observer in all transient and
steady state operations including high speed and low
speed. The only deviations are during the initial
transient of the flux observer.
EXPERIMENTAL VERIFICATIONS
Figure 6 shows the configuration of the experimental
set up for computer-based slip-controlled drive system.
The induction motor is driven from a three-phase
voltage-source inverter whose specification is listed in
appendix. The speed estimation and controller have
been implemented using a Pentium 586 computer.
Stator three-phase voltages were detected by three
voltage transducers VTa , VTb , VTc with the ratio of
1/100 and two-phase currents were measured by two
current/voltage isolators ITa , ITb with the ratio of
10V/5A. A 12 bit I/O board DAS-1602 is used for
A/D and D/A conversions. Interface between the
computer and outside circuits is carried out by the
Labvolts data acquisition interface and control board.
The control program is written in C++ and the
sampling period for one computing cycles used in
experiment is 350 microseconds.
Figure 7 presents the motor speed of the sensorless
system and the one with speed sensor during the
forward-reverse
transient
and
reverse-forward
transient. The experimental results reveal the fact that
the speed-sensorless control drive works well in both
a. Forward-reverse transient
b. Reverse -forward transient
Figure 7. Motor speed of drive system with sensor (top plots) and sensorless drive system (bottom plots)
transients and steady state under PWM condition of the
experimental inverter. Transients during starting from
rest and forward-reverse as well as reverse-forward are
smooth.
[3]
CONCLUSION
A dynamic flux observer that takes the applied stator
voltage and measured stator current as inputs is
presented in this paper. The observer poles are so
chosen to make it stable and it is shown to converge
irrespective of different initial conditions. An algorithm
for speed estimation is also presented which uses the
flux obtained from the observer. The proposed
algorithm has been verified by computer simulations
and found to perform well in both transients and steady
state including high speed and very low speed
regardless of initial conditions of the speed estimator.
When implementing the algorithm, suitable initial
conditions of the rotor flux observer can be chosen for
minimum deviations of estimated speed from actual
one during the starting from rest. The proposed
algorithm has been implemented successfully on a slip
controlled drive system without using any speed
sensor.
APPENDIX
Induction motor parameters :
175 W 1395 rpm 415 V 0.46 A ; Rs =47 Ω; Rr =
53.4 Ω;
Ls = Lr = 2.587 H; Lm = 2.399 H; Jm = 1/150 kgm2
[4]
[5]
[6]
[7]
[8]
[9]
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