1
Speed Sensorless Control of Induction Motor based on Indirect Field-Orientation
Marcello Montanaria, Sergei Peresadab , Andrea Tillia , Alberto Toniellia
a Dept. of Electronics, Computer
and System Sciences (DEIS)
University of Bologna
Viale Risorgimento 2
40136 Bologna, ITALY
Abstract - A novel speed sensorless vector control is presented for induction motor. Speed and rotor resistance estimator has been designed using Lyapunov-like technique.
Speed and flux controller has been realized using indirectfield orientation concept. With this choice, robustness with
respect to stator resistance uncertainties can be improved
by means of rotor flux reference feedback in the speedrotor resistance estimator. The performance of the solution proposed has been tested by simulations and by experiments.
I
I NTRODUCTION
In recent years, a large number of speed sensorless vector control systems for induction motor (IM) have been proposed.
Speed information is generally provided by a speed transducer
on the motor shaft; recently, low cost and high performance
digital signal processors (DSP) become available allowing to
obtain speed by means of digital estimators integrated with
motor control. This solution represents an advantage in terms
of costs, simplicity and mechanical reliability of the drive.
Several schemes of speed estimators have been proposed in
the literature; among them, the model reference adaptive system (MRAS) approach is very attractive and gives good performance [1] [2]. The classical MRAS method is based on the
adaptation of the rotor flux [3] [4]; with this scheme, some difficulties in terms of precise and robust speed estimation arise,
especially at low speed. The need of a pure integration in the
speed estimator represents a drawback in the low speed region,
due to drift and low frequency disturbances; moreover, parameter sensitivity (in particular to stator resistance) represents a
usual disadvantage for all model-based estimators [5]. To overcome these problems, alternative MRAS schemes based on
back-EMF or reactive power [6] have been presented, but it
seems that they don’t solve troubles at low speed. The common approach to increase dynamic performance and stability
of speed sensorless field oriented control systems is the on-line
parameter adaptation [7] [8] [9].
The main contribution of this paper is a novel speed sensorless vector control based on: a) a speed and rotor resistance es-
b Dept. of Electrical Engineering
Kiev Polytechnic Institute
Prospect Pobedy 37
Kiev 252056, UKRAYNE
timator, designed using Lyapunov-like technique, b) indirectfield oriented control (IFOC), c) suitable adjustments to improve robustness with respect to parameter variations, measurement errors and plant non-ideality. Rotor resistance estimation and other compensations adopted allow enhanced dynamic performance. Simulation results and experimental tests
illustrate the good performance of this solution, also in the low
speed region.
The paper is organized as follows: in section II induction
motor model and estimator strategy are given. Speed and rotor
resistance estimation analysis is presented by means of simulation and investigated by means of a physical approach. In
section III sensorless IFO control performance is analyzed in
presence of stator resistance uncertainties and some solutions
for increased robustness are presented and discussed. In section IV, digital implementation of the estimator algorithm is
considered and experimental results are given.
II
S PEED AND
ROTOR RESISTANCE ESTIMATION
Under the assumptions of linearity of the magnetic circuits and
equal mutual inductances, neglecting iron losses, the equivalent two phase model of the induction motor, represented in a
fixed stator frame (a; b) and expressed in state-space form, is
the fifth order model:
d a
= a ! b + Lm ia
dt
d b
= b + ! a + Lm ib
dt
dia
1
=
ia + a + ! b + ua
dt
dib
1
= ib + b ! a + ub
dt
d! 1 =
( a ib
(1)
b ia ) TL :
dt J
In (2), (ia ; ib ) are stator currents, ( a ; b ) are rotor fluxes,
(ua ; ub ) are stator voltages, ! is the rotor speed, TL is the load
torque; the other parameters depend on rotor and stator resistances and inductances Rs ; Rr ; Ls ; Lr , mutual inductance Lm
2
+ La^ z^b (1 + Lm )^ib
r
1
+ u + K ~i + ^
and moment of inertia J , as follows:
= Ls 1
L
= m
Lr
L
= m
Lr
L2m Ls Lr
R
= r
Lr
Rs
=
+ Lm
b
Introducing two auxiliary variables (za ; zb ) defined as za =
ia + a , zb = ib + b , IM model expressed in the statespace base (za ; zb ; ia ; ib ; ! ) is:
dza
dt
dzb
dt
dia
dt
dib
dt
d!
dt
= Rs ia + 1 ua
= Rs i + 1 u
b b
= ( + )ia !ib + za + !zb + 1 ua
= ( + )ib + !ia + zb !za + 1 ub
= J1 (za ib zb ia ) TL :
(2)
Note that (za ; zb ) variables are proportional to stator fluxes,
with coefficient 1= .
For the design of the rotor resistance estimation, let us define
the rotor resistance error a = Rr RrN , where RrN is the
nominal value for Rr . From now on, the subscript N on other
variables stands for the nominal value calculated on the basis
of RrN .
The design of the speed and rotor resistance estimator
is based on Lyapunov-like method. The following tenthorder dynamic system is designed, based on measured signal
ua ; ub ; ia; ib and the assumption of exactly known stator resistance and inductances:
dz^a
dt
dz^b
dt
dz^a1
dt
dz^b1
dt
d^ia
dt
=
=
=
=
Rs ^ 1
i + u + K1~ia
a a
Rs ^ 1
i + u + K1~ib
b b
Rs
1
ia + ua
Rs
1
i + u
b b
(3a)
(3b)
(3c)
(3d)
= (N + N )^ia !^^ib + N z^a + !^ z^b1
+ a^ z^ (1 + L )^i
m
Lr a
+ 1 ua + K2~ia ^b
a
d^ib
= (N + N )^ib + !^^ia + N z^b
dt
(3e)
!^ z^a1 +
2b
a
(3f)
da^
1 ~i z^ (1 + L )^i =
m a
dt 2 Lr a a
(3g)
+ ~ib z^b (1 + Lm )^ib
d!^ 1 ~
~ib z^a1 ^ia
=
i z^ ^i
(3h)
dt 3 a b1 b
d^a 1 ~
= i
(3i)
dt 4 b
d^b
1 ~i
=
(3j)
dt
4 a
^ and !^ are the estimated rotor resistance error and the
where a
estimated speed respectively, (~ia ; ~ib ) are current estimation errors defined as ~ia = ia ^ia , ~ib = ib ^ib . The stability proof
based on Lyapunov-like technique is reported in the appendix.
In the hypothesis of constant speed and rotor resistance, the
stator current estimation errors tend asymptotically to zero; if
the persistency of excitation (PE) condition is satisfied, also the
exponentially convergent estimate of ! and Rr is provided.
The adaptation laws (3g),(3h) are very similar to the ones
proposed in [8], but they are derived using a different approach.
As already stated in [8], with constant rotor flux amplitude it
is not possible to obtain correct simultaneous speed and rotor
resistance estimation, owing to lack of persistent excitation.
The estimator has been analyzed by means of a physical approach, to understand this phenomenon. It can be noted that:
if the flux modulus is constant and the motor torque is
zero, IM model doesn’t depends on Rr . Hence, rotor
resistance error doesn’t affect speed estimate and rotor
resistance estimate is not updated, because rotor currents
are zero,
if flux modulus is constant, stator current errors tend to
zero, but ! and Rr could be wrongly estimated. The relation between the estimation errors in steady-state is the
following:
! !^ =
Rr iq R^r iq
L
+ L Lm ;
Lr m
r
where iq is the torque current and
is the flux amplitude.
In order to overcome these problems and to obtain simultaneous ! and Rr estimation, a small AC frequency is added to the
magnetizing current command, with the purpose of obtaining a
variable flux and the PE condition satisfaction. Note that problems due to the lack of persistency of excitation don’t exist
if the speed estimation only is implemented. Therefore, disabling the rotor resistance adaptation, speed estimation can be
achieved imposing a constant reference flux amplitude.
3
Speed reference
Load torque
120
*
8
100
Speed
Control
IFOC
d,q->a,b
2/3
Uq*
Ub*
Uc*
PWM
inverter
IM
6
o^
80
(rad/s)
Ua*
Ud*
iq*
(Nm)
60
40
4
id
*
iq
2
a,b->d,q
3/2
ia
ib
20
0
0
−20
0
1
2
Time (s)
−2
3
0
1
2
Time (s)
3
Slip
s^
Rr^
Figure 1: Speed reference. Load torque.
Speed and estimated speed
^
ia
ib
va
vb
Figure 3: Block diagram of sensorless IFO control.
Rotor resist. est.
120
Speed
& Rr
Estimator
6
100
5.5
(rad/s)
80
(Ω)
60
40
20
motor, to test performance at high (100 rad/s), low (10 rad/s)
and zero speed ; at t = 0:9 s: a load torque equal to 6Nm
(100% rated torque) is applied. An initial 20% error on Rr is
assumed, Rr estimation is enabled at t = 0:9s.
5
4.5
0
−20
0
1
2
Time (s)
4
3
0
1
3
0.03
2
0.02
1
0.01
0
0
−1
−0.01
−2
−0.02
−3
0
1
2
Time (s)
3
Current a error
(A)
(rad/s)
Speed estimation error
2
Time (s)
3
−0.03
0
1
2
Time (s)
3
Figure 2: Speed and rotor resistance estimation with 20% initial error in Rr (RrN = Rr =1:2). Variable flux reference.
The estimator has been designed under the hypothesis of
constant speed and rotor resistance. Nevertheless, the speed
varies with a fast dynamic, while the rotor resistance is a slowvarying variable dependent on thermal variations. Thus, rotor
resistance estimation dynamic is imposed slower than speed
estimation dynamic; in this way, considering also the concept
of dynamic separation, the presented estimator achieves better
performance in terms of estimation errors.
Moreover, for the estimator stability proof, speed is supposed constant, hence rotor resistance estimation should be
disabled during speed transients, to avoid drift errors in the
estimate. Then, since rotor resistance adaptation law (3g) depends on estimated rotor currents (i.e. motor torque), rotor
resistance estimation is enabled only if rotor current amplitude
is sufficiently high, in order to avoid incorrect estimation due
to measurement noise and model errors.
The proposed estimator has been tested by means of simulation under the operating conditions shown in fig. 1, with
an IM whose parameters are reported in Table I. After an initial time interval [0; 0:4s:] needed to drive the motor flux to the
rated value, a speed reference profile is applied to the unloaded
Adding to the flux a 10 Hz sinusoidal signal whose amplitude is 5% of the rated flux value, ! and Rr estimation is
correctly achieved, as shown in fig. 2: ! and Rr errors tend to
0, thanks to flux modulation that allows PE satisfaction. Note
that in the ideal situation of correct IM model, good estimate is
obtained also at zero speed with an applied load torque. From
the speed estimation error in fig. 2 it can be noted that during
the speed transient starting at t = 0:4s, the rotor resistance error introduce an error in the estimated speed; from t = 0:9s,
thanks to correct Rr estimation, also the speed error is negligible.
III
S ENSORLESS IFO
CONTROL
A sensorless IFO control, based on the feedback of estimated
speed and rotor resistance, has been realized. The IFO control
scheme proposed in [10] is used. The structure of the IFO controller and of the estimator is reported in fig. 3. The use of
!^ and R^r in the vector controller allows correct synchronous
speed estimation. Performance of the sensorless vector control in the hypothesis of correct Rs value in the same operating
conditions of fig. 1 are reported in fig. 4. Good speed and
torque control is achieved also in the low speed region. Sensorless control with rotor resistance adaptation ensures good
performance in presence of load torque and during transients.
In presence of stator resistance uncertainties, errors in
mostly
in the low speed region. Moreover, since pure integration is
needed in equations (3a)-(3d), problems due to drift arise at
low speed.
(za ; zb ) estimation cause incorrect speed estimation,
In order to solve these problems and to increase robustness
to stator resistance uncertainties, a feedback based on the error of flux estimation with respect to flux reference has been
4
Speed and estimated speed
introduced in equations (3a)-(3d), that is:
40
20
−20
IV
E XPERIMENTAL RESULTS
Digital implementation of the estimator requires discretization
of equations (3), that may introduce estimation errors and instability. Hence, particular attention must be paid to the choice
of the sampling time and the discretization method [11].
The estimator (3) can be simplified removing the (3i),(3j)
equations without significant performance degradation.
Since a simple Euler discretization of the whole estimator
leads to performance degradation, a different approach is followed. During a sampling interval, the rotor speed, the rotor
resistance and the stator currents can be assumed constant if
the sampling time is small enough with respect to the system
dynamics. The stator voltages are structurally constant during a sampling interval since they are the outputs of a digital
controller. Hence, the discrete-time model of equations (3a)(3f) describing the electrical part of the observer can be computed by means of the exponential matrix function. Since the
exponential matrix function is dependent on estimated speed
0
1
2
Time (s)
4
3
0
Speed error
2
2
1
1
(rad/s)
3
0
−1
−2
−2
1
2
Time (s)
2
Time (s)
3
0
−1
0
1
Speed estimation error
3
−3
−3
3
0
1
2
Time (s)
3
Figure 4: Sensorless IFO control with 20% initial error in Rr
(RrN = Rr =1:2).
Speed and estimated speed
Rotor resist. est.
120
6
100
5.5
80
(rad/s)
Flux-reference-based error feedback can be applied in the hypothesis that the real flux (not measurable) is equal to the reference flux, thanks to IFO control. This solution seems to be
more effective than other adjustments proposed in the literature
to solve drift problems, such as the substitution of the pure integrator with a low pass filter [3]. In addition, this approach
increases the robustness of the estimator with respect to motor model inaccuracies and parameter variations, measurement
errors, actuator distortion.
In fig. 5 simulation results are reported. Even with a Rs
error, satisfactory tracking performance is achieved at low
speed, but a zero speed condition cannot be guaranteed with
load torque, owing to error in speed estimation (speed error is
present also in the hypothesis of correct rotor resistance value
in the estimator).
4.5
60
(Ω)
cos "0
sin "0:
5
0
40
20
5
4.5
0
−20
0
1
2
Time (s)
4
3
0
Speed error
(rad/s)
=
=
(Ω)
60
the IFO control, that is:
a
b
5.5
80
3
2
2
1
1
0
−1
−2
−2
0
1
2
Time (s)
2
Time (s)
3
0
−1
−3
1
Speed estimation error
3
(rad/s)
(4b)
(4c)
4 a
a1
a
a a
= Rs ib + 1 ub + K4 (ib + b z^b1 )
(4d)
where ( a ; b ) are the reference flux signals, expressed in a
fixed reference frame. ( a ; b ) are calculated by the reference
flux amplitude and by the orientation angle "0 computed in
6
100
(4a)
(rad/s)
= Rs ^ia + 1 ua + K1~ia + K4 (ia + a z^a)
= Rs ^ib + 1 ub + K1~ib + K4 (ib + b z^b )
= Rs i + 1 u + K (i + z^ )
(rad/s)
dz^a
dt
dz^b
dt
dz^a1
dt
dz^b1
dt
Rotor resist. est.
120
3
−3
0
1
2
Time (s)
3
Figure 5: Sensorless IFO control with 20% initial error in Rr
(RrN = Rr =1:2) and 10% error in stator resistance (RsN =
Rs =1:1).
and rotor resistance, the discrete-time model should be computed on-line. A second-order Taylor approximation of the
exponential matrix function is adopted in the digital implementation proposed. In this way, the discrete-time estimator
shows good performance and can be implemented on available
DSP with sample time equal to Ts = 200s. The adaptation
laws (3g),(3h) for speed and rotor resistance estimation are discretized with classical Euler method.
The experimental setup is based on a power inverter with
540V DC-link voltage. A 10kHz symmetrical PWM technique
5
5
−5
−100
0
0.5
Time (s)
1
1
ω−ω (rad/s)
^
0
0.5
Time (s)
0
−1
0
0.5
Time (s)
1
0
0.5
Time (s)
1
0
0.5
Time (s)
1
2
1
0
*
0
1
−2
1
2
ω −ω (rad/s)
ω*−ω^ (rad/s)
−1
1
0
−1
−5
0.5
Time (s)
1
0
−2
1
0.4
Current a error (A)
4
0.5
Time (s)
2
0
−2
0
0.5
Time (s)
1
0.5
Time (s)
−2
1
4
0.2
0
−0.2
−0.4
0
0
0.5
Time (s)
1
0.4
Current a error (A)
0
Current a (A)
ω*−ω (rad/s)
−5
Current a (A)
0.5
Time (s)
5
0
−4
−5
−10
0
5
0
^
−50
0
*
0
5
ω −ω (rad/s)
50
2
10
Speed (rad/s)
ω−ω^ (rad/s)
Speed (rad/s)
100
2
0
−2
−4
0
0.5
Time (s)
1
0.2
0
−0.2
−0.4
Figure 6: Four quadrant operation at high speed
Figure 7: Four quadrant operation at low speed
is adopted to drive the inverter switches. Stator currents are
measured by Hall-type sensors. Since voltage sensors are not
available in actual equipment, applied voltages in the estimator
equations are considered equal to reference voltages imposed
by the controller to drive inverter. Motor speed is measured
by an encoder with 5000 ppr. The control board adopted is
based on a DSP TMS320C32. The sampling time is set to
200s. The DSP performs data acquisition, generates speed
and flux profiles, implements estimator and IFO control algorithms and generates PWM inverter commands: the control
board is connected to a standard PC to program the DSP and
to display acquisition data. A 1.1kW induction motor (whose
data are reported in Table I) is used connected to a DC motor
that generate load torque. The estimator parameters are set at
K1 = 2000; K2 = 500; 3 = 1=500; K4 = 40.
The experiments are performed to test the speed sensorless
vector control proposed in different operating conditions: high
and low constant speed, speed transient, resistive and regenerative load torque.
In the first five tests, only the speed estimation is enabled,
while the rotor resistance is kept equal to its nominal value. In
^ represents the speed referthe figures, the symbols ! and !
ence and the estimated speed respectively.
Experimental results reported in fig. 6 and 7 are obtained
without applied torque, but the DC motor is connected to the
IM giving a large inertial load (J ' 0:015Kg=m2). In both
tests, a smooth speed profile is applied to the motor. Fig. 6
shows motor behavior during steady-state condition at high
speed (100 rad/s) and during transient with speed inversion.
Fig. 7 shows motor performance in the low speed region (maximum reference speed is 10 rad/s). From these tests, it can be
noted that during transient and with inertial load, sensorless
vector control shows good performance. In fact, a negligible
steady-state error is obtained either at high or low speed and
good tracking is achieved during transient. Current error do
not tend to null value owing to model errors.
In fig. 8 and 9 braking and regenerative load torque equal
to 6Nm (rated torque) is applied when a constant speed
reference of 100 rad/s is imposed. Good torque disturbance
rejection is achieved with an almost null steady-state error and
a speed transient of about 100ms. Note that, in presence of
active load, performance are worse; in fact, a longer and less
damped arises, but stability is still preserved.
In fig. 10 resistive load torque equal to 3Nm (50% rated
torque) is applied at t = 0:4s, with a constant speed reference
of 10 rad/s. The load torque rejection at low speed is inferior
with respect to high speed, owing to model errors and system
non-ideality. Anyway, the adjustments adopted to increase the
estimator robustness substantially improve the estimator performance.
Experimental results showing simultaneous speed and rotor
resistance estimation are reported in fig. 11; they are obtained
with a constant speed reference of 80 rad/s and an applied 3
Nm load torque. A 1:6Hz sinusoidal component is added to
the nominal flux to obtain the persistency of excitation. A non-
0.1
0.2
Time (s)
−10
0.3
ω −ω (rad/s)
10
*
−5
−10
0
0.1
0.2
Time (s)
0.3
10
10
5
5
−10
0
0.1
0.2
Time (s)
0
0
0.1
0.2
Time (s)
0.3
ω−ω (rad/s)
^
0
0.2
Time (s)
−5
−10
0.3
0
0.1
0.2
Time (s)
0
0.1
0.2
Time (s)
0.3
Figure 8: Dynamic response with resistive load
Although the presented results show good performance,
some problems still exist. As claimed in previous section,
sensorless control with rotor resistance adaptation should give
better performance with respect to the one without this kind
of adaptation. Anyway, with the actual experimental equipment, sensorless vector control based on simultaneous speed
and rotor resistance estimation has implementation problems
and does not produce significant improvements, due to measurement errors; moreover, at very low speed some degradation have been noted. It seems that this mismatch between
simulation and experimental results is due to measurements errors and model errors. In particular, the stator voltage used in
the estimator is obtained by the imposed voltage, that is different from the actual voltage applied to the motor, due to actuator
distortion, PWM dead-times and other non-ideality. Moreover,
stator resistance uncertainties and drift introduce errors in the
estimator, especially in the low speed region; although some
remarkable improvements have been obtained with the feedback action based on the error of flux estimation with respect
to flux reference, some problems still exist. Equipment improvement and solution for an increased robustness will be the
topic of future work.
0.2
Time (s)
0.3
0
0.1
0.2
Time (s)
0.3
0
0.1
0.2
Time (s)
0.3
5
0
−5
−10
0
0.1
0.2
Time (s)
0.3
0
−5
Figure 9: Dynamic response with regenerative load
V
sensorless vector control is used to drive the motor. An initial
rotor resistance error equal to 3
(55% of its nominal value)
is present; estimator ensures that speed and rotor resistance
estimation error goes to zero.
0.1
5
0
−5
0.3
0
10
5
0
−5
0.1
−5
−10
0.3
0
0
5
Torque current (A)
Current a (A)
90
0.3
−5
5
−5
0.2
Time (s)
0
^
0
0.1
95
Current a (A)
ω*−ω (rad/s)
5
0
100
^
0
−5
5
ω*−ω (rad/s)
95
90
0
^
100
105
*
5
10
ω −ω (rad/s)
105
110
Torque current (A)
10
Speed (rad/s)
110
ω−ω (rad/s)
Speed (rad/s)
6
C ONCLUSIONS
A sensorless IFO control has been realized. A new speed and
rotor resistance estimation MRAS scheme based on Lyapunovlike method has been used. Solutions for robustness improvement have been adopted. Simulation results show good performance of the sensorless vector control at high and low speed
and good robustness to parameter variations. Some problems
exist at low speed. Sensorless control with rotor resistance
adaptation gives better performance with respect to the one
without this adptation. Performance of the sensorless IFO control has been verified by means of experiments, which show
results comparable with the simulated ones. The realization
of the IFO control based on simultaneous speed and rotor resistance estimation has implementation problems, owing to
model errors and lack of precise information on the actual voltages applied to the motor. The vector control performs well at
high speed also with applied torque; problems exists at very
low speed with high applied load torque, owing to model errors and non-ideality; nevertheless, the adjustments adopted to
increase the estimator robustness allows to obtain performance
improvement.
A PPENDIX
Speed and rotor resistance estimator stability can be proved by
~ = ! !^ , a~ = a a^,
Lyapunov-like method. Let define errors !
~ia = ia ^ia , ~ib = ib ^ib , z~a = za z^a , z~b = zb z^b ,
z~a1 = za z^a1 , z~b1 = zb z^b1 , ~a = a ^a , ~b = b ^b ,
7
5
ω−ω^ (rad/s)
Speed (rad/s)
15
10
5
0
0
0.2
0.4
Time (s)
0
0
0.2
0.4
Time (s)
0.4
Time (s)
0.6
0
0.2
0.4
Time (s)
0.6
0
0.2
0.4
Time (s)
0.6
5
Torque current (A)
Current a (A)
0.2
0
−5
0.6
5
0
−5
0
5
ω*−ω^ (rad/s)
ω*−ω (rad/s)
5
−5
0
−5
0.6
Table I: Induction motor data
0
0.2
0.4
Time (s)
0
−5
0.6
Figure 10: Dynamic response with resistive load at low speed
4
Ref. flux current (A)
Speed (rad/s)
85
80
75
0
2
4
6
Time (s)
8
10
2
0
−2
−4
2
4
6
Time (s)
8
10
4
Rr error estimate (Ω)
Speed est. error (rad/s)
5
0
0
2
0
0
2
4
6
Time (s)
8
10
−4
0
2
4
6
Time (s)
8
10
Figure 11: Simultaneous ! and Rr estimation
where (a ; b ) are defined as a = ! z~a1 ; b = ! z~b1 and
can be proved to be constant. Assuming that ! and Rr are
constant, from the IM model (2) and the estimator equations
(3), the following error dynamics results:
dz~a
=
dt
dz~b
=
dt
Rs
+ K1 ~ia
Rs
+
K1 ~ib
number of pole pairs
speed
motor torque
power
stator resistance
rotor resistance
stator inductance
rotor inductance
mutual inductance
rotor inertia
2
1410rpm rated
6:0Nm rated
1:1Kw rated
6:7
5:5
0:475H
0:47H
0:45H
0:015Kgm2
dz~a1
=0
dt
dz~b1
=0
dt
d~ia
= (N + N + K2 )~ia !~ib + N z~a ~b
dt
+ La~ z^a (1 + Lm )^ia + !~ (^zb1 ^ib)
r
d~ib
= (N + N + K2 )~ib + !~ia + N z~b + ~a
dt
+ La~ z^b (1 + Lm )^ib !~ (^za1 ^ia)
r
1 ~i z^ (1 + L )^i da~
=
m a
dt
2 Lr a a
+ ~ib z^b (1 + Lm )^ib
d!~
1 ~i z^ ^i ~i z^ ^i =
b a1
a
dt
3 a b1 b
d~a
= 1 ~ib
dt
4
d~b 1 ~
= i:
dt 4 a
Let consider the function:
−2
−5
p
!
Tm
P
Rs
Rr
Ls
Lr
Lm
J
1
1
1
V = (~i2a + ~i2b ) + (~za21 + z~b21 ) + 1 (~za2 + z~b2)
2
2
2
1
1
1
+ 2 2 a~2 + 2 3 !~ 2 + 2 4 (~a2 + ~b2 )
where 1 > 0; 2 > 0; 3 > 0; 4 > 0 are tuning parameters.
and ont he basis of the model error
Choosing 1 = Rs +K
1
equations, it follows that the time derivative of function V is
V_ = ( + + K2 )(~i2a + ~i2b ) < 0:
It can be proved that stator current errors tend asimptotically
to zero. Moreover, if the persistency of excitation (PE) condition is satisfied, also the exponentially convergent estimate of
8
parameters ! and Rr is provided.
R EFERENCES
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[9] R. Marino, S. Peresada, and P. Tomei. Output feedback
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