parameter estimation of im at standstill with magnetic flux monitoring
TRANSCRIPT
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386 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 3, MAY 2005
Parameter Estimation of Induction Motor at StandstillWith Magnetic Flux Monitoring
Paolo Castaldi and Andrea Tilli
AbstractThe paper presents a new method for the estimationof the electric parameters of induction motors (IMs). During theidentification process the rotor flux is also estimated. The proce-dure relies on standstill tests performed with a standard drivearchitecture, hence, it is suitable for self-commissioning drives.The identification scheme is based on the model reference adaptivesystem (MRAS) approach. A novel parallel adaptive observer(PAO) has been designed, starting from the series-parallel Kreis-selmeier observer. The most interesting features of the proposedmethod are the following: 1) rapidity and accuracy of the identi-fication process; 2) low-computational burden; 3) excellent noiserejection, thanks to the adopted parallel structure; 4) avoidanceof incorrect parameter estimation due to magnetic saturation
phenomena, thanks to recursive rotor flux monitoring. The per-formances of the new scheme are shown by means of simulationand experimental tests. The estimation results are validated bycomparison with a powerful batch nonlinear least square (NLS)method and by evaluating the steady-state mechanical curve ofthe IM used in the tests.
Index TermsIdentification, induction motor (IM), magneticsaturation, parallel adaptive observer (PAO), self-commissioningdrives.
I. INTRODUCTION
I
N RECENTyears, the demandfor high-performance electric
drives based on induction motors (IMs) has been constantlygrowing. IMs are particularly attractive for industrial applica-
tions because of their low cost and high reliability. Moreover,
power electronics and control electronics, essential to realize so-
phisticated variable-speed drives, are becoming cheaper every
day. On the other hand, high-performance control of this kind
of electric machine is quite difficult. The IM model is multivari-
able, nonlinear and strongly coupled. The concept of field-orien-
tation, introduced in Blaschkes pioneering work [1], has led to
decoupling torque and flux control in induction machines. This
was the key point in developing direct and indirect field-oriented
control (DFOC and IFOC) algorithms [2], [3], adopted in com-
mercial IM drives for high-performance motion control. Nowa-
days, another kind of control strategy is becoming interestingfor industrial IM drives: the direct torque control (DTC) tech-
nique which directly takes into account the switching nature
of the inverter used to feed the motor [4], [5].
The basic versions of almost all the IM controllers rely on
rotor speed measurement and recently great deal of effort has
Manuscript received July 12, 2003; revised May 5, 2004. Manuscript re-ceived in final form September 13, 2004. Recommended by Associate EditorA. Bazanella.
The authors are with the Department of Electronics, Computer Science andSystems, University of Bologna, Bologna 40136, Italy (e-mail: [email protected]).
Digital Object Identifier 10.1109/TCST.2004.841643
been devoted to developing the so-called sensorless control for
IM, where no speed sensor is used [4]. A final solution for this
hard control task is still to be found. However, different solu-
tions, derived from standard field-oriented controllers and DTC,
are already available in commercial drives, in spite of the open
issues on both methodology and practice.
Unfortunately, field-oriented control and DTC techniques
both require accurate knowledge of electric parameters of the
machine continuous-time model in order to guarantee good
performance. In the case of classic IM control (i.e., with
a speed-sensor), it has been extensively proved [6] that the
control stability is quite robust with respect to variations of the
rotor time constant, which is the most critical IM parameter for
control commissioning. But, in terms of tracking fast variable
speed references, a significant reduction in performance can
be noted when the wrong parameters are adopted. In fact, the
wrong electrical parameters cause flux misalignment leading
to loss of efficiency and effectiveness in torque control. In
particular, besides the rotor time constant, the main inductance
plays an important role, since the wrong value leads to deflux
or to saturate the machine. The effects of errors in other IM
parameters are mitigated by current feedback control. In the
case of sensorless control, the effect of parametrization errors is
even more relevant; in fact, a partial or full IM electrical modelis usually adopted to estimate the rotor speed.
In nonlinear and adaptive control literature a great deal of
work has been devoted to developing other control algorithms
for IM or to improving the previously mentioned well-estab-
lished methods (in particular DFOC and IFOC). Although dif-
ferent approaches have been used [7], [8], only partial and quite
poor results have been obtained in terms of performance ro-
bustness with respect to parameter uncertainties, particularly for
sensorless control. Hence, at the state of the art, good knowl-
edge of the electric parameters of the model is a key point to
realize high-performance control of commercial IM drives. In
addition, also for the purpose of diagnosis the electric parame-ters of a healthy IM must be identified with high accuracy.
Traditionally, the IM electric parameters have been calculated
from the nameplate data and/or using the classical locked-rotor
and no-load tests. The resulting values are not usually enough
accurate to tune a high-performance drive and, moreover, the
no-load test requires the motor to be disconnected from any me-
chanical load. Recently, various parameter identification tech-
niques for IM have been proposed in the literature. These can
be divided into two main classes: the online techniques and
the offline techniques.
The online techniques perform the parameter identification
while the IM drive is operating in normal conditions. This kind
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CASTALDI AND TILLI: PARAMETER ESTIMATION OF INDUCTION MOTOR AT STANDSTILL 387
of approach is very interesting since it is possible to track the
slow variation of the electric parameters during normal opera-
tion. In fact, it is well-known that the values of the stator and
rotor resistances are strongly affected by the machine heating
and also the magnetic parameters considerably depend on the
level of the magnetic flux, particularly in the saturation zone
[9]. In [10], different theoretically rigorous, methods are used toidentify stator and rotor resistance during normal operation, but
filtered derivatives of the measurements are required. In [11],
an online method, based on the recursive least-square (RLS)
method, is presented to identify the electrical and mechanical
parameters of the system. A scaled version of the magnetic flux
is also estimated, but the derivatives of the measurements have
to be used and the computational load is quite heavy. A similar
approach is reported in [12], where the time-scale separation be-
tween electric and mechanical dynamics is exploited to obtain
simultaneous speed and parameters estimation. In [13], the gen-
eralized total least-square (GTLS) technique is adopted. Filtered
derivatives of the measured signals are still needed, but partic-
ular attention is devoted to the reduction of the noise effects.A constrained identification procedure is proposed to deal with
low signal-to-noise ratio conditions. In [14], the least-square
(LS) procedure has been applied in an original way to obtain
an estimate of the stator and rotor resistances and reactances.
No derivatives are required, but the proposed method is not
strictly recursive and the computational burden is quite heavy. In
[15][17], a theoretically elegant solution is presented to iden-
tify all the IM drive parameters, but knowledge of all of the state
variables and their derivatives is required. In [18] and [19], an
extended Kalman filter (EKF) has been used to identify the ma-
chine parameters; in [18] particular, attention has been paid to
the selection of noise covariance matrices and initial states. In[20], a sophisticated method, based on nonlinear programming,
is proposed. In [21], neuro-fuzzy technique is applied for online
identification of the rotor time-constant. In [8], a very interesting
technique to tune the stator and rotor resistances in normal op-
erating conditions is presented. The stability characteristics of
the proposed method are formally proved and experimentally
tested, and no derivative of the measurements is required.
The offline identification techniques perform the electric
parameter tuning while the IM drive is not operating normally.
From a philosophical point of view, it seems that the offline
techniques are useless since online techniques are available.
At present, from a control theory point of view, no online
identification method combined with an adaptive control has
been mathematically proved to be globally stable; only partial
simulative and experimental results are given. Moreover, even
if we set aside theoretical issues, the online techniques are usu-
ally characterized by a considerable computational burden, so
they are not suitable for cost-optimized industrial applications.
More important, online identification techniques are quite slow
so they cannot guarantee a safe starting of the drive if the
initial values of the estimated machine parameters are strongly
detuned. Hence, it results that offline methods are useful for
two reasons: 1) they can be used when no online method can
be supported; 2) they can provide a good initialization of the
machine parameters when online methods are adopted. Manyoffline identification methods have been proposed; some of
them require particular tests on the machine with free rotor shaft
and/or special measuring equipment [22][29]. The present
trend in drive technology is to perform the offline identification
at standstill, with the motor shaft connected to the mechanical
load and without any extra hardware. In this way, the set-up
of the control system can be automatically executed (and
repeated) after the drive installation (self-commissioning). In[30], a model reference adaptive system (MRAS) method [31],
[32] is used to perform parameter identification at standstill,
and a classical hyperstability approach is adopted to design
the adaptation law, but the motor torque-constant has to be
assumed known. In [34], the frequency response of the IM
at standstill is exploited, so this approach is suitable to avoid
the effects of inverter nonlinearities. In [35], the motor pa-
rameters are estimated by means of both time and frequency
responses of the stator current at standstill. In [36], the IM
is excited at standstill with a sinusoidal voltage in one of the
two equivalent phases, the equivalent impedance is identified
with RLS techniques and different frequencies are used to
identify the different magnetic parameters. This solution alsoavoids the effect of the inverter nonlinearities. In [37], a similar
approach has been implemented. The main difference is that
a simplified dynamical model replaces the typical steady-state
one. In [38], a standard linear LS technique is adopted to
estimate IM electric parameters similarly to the online methods
reported in [11][13], hence, filtered derivatives of the motor
voltages and currents are required. In [30], [34][38], a linear
model is assumed for the IM at standstill. While, in [9], offline
identification is carried out by relying on a deep knowledge of
the typical nonlinear behavior of the IM. In [39], a method is
proposed to identify the flux saturation curve at standstill. In
[40], the same purpose is pursued using EKF.In this paper, a novel offline identification method of the IM
electric model is proposed. This procedure relies on standstill
tests performed with a standard drive architecture, hence, it
is suitable for self-commissioning drives. Only one phase, in
the two-phases equivalent model, is excited to guarantee the
standstill condition without locked rotor. Under the hypothesis
of linear magnetic circuits, the IM model at standstill is linear
time invariant (LTI). A MRAS approach has been adopted.
The identification procedure is realized by means of a parallel
adaptive observer (PAO) [31], [32], which is based on a non-
minimal statespace representation of the the IM LTI-model,
derived from [41], and an original adaptation law involving
current measurements only (no measurement differentiation is
required). Unlike [30], none of the machine parameters has to
be assumed known. In accordance with the classical adaptive
observers theory, the theoretical analysis and design of the
proposed PAO has been carried out in a deterministic frame-
work. In fact, it is well-known that the parallel structure gives
the adaptive observer excellent noise rejection properties [31],
[43]. From a practical point of view, a key point for the correct
estimate of the IM LTI-model parameters is to avoid saturation
of the magnetic core. In fact, as is well-known [9], [23], the
magnetic parameters depend on the flux level and they can be
reasonably assumed to be constant only if the flux is not greater
than the rated one. On the other hand, it is worth observingthat from nameplate data, usually quite rough, it is possible
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388 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 3, MAY 2005
to deduce the nominal flux of the machine with acceptable
accuracy, but very poor information can be obtained about the
level of the magnetizing current [2]. Hence, in order to avoid
magnetic saturation during the identification process, the flux
should be monitored in some way. This requirement, often
neglected, is accomplished by the proposed scheme. In fact, the
adopted PAO gives a recursive estimate of the magnetic flux,during the identification process. Hence, this solution avoids
the incorrect estimation of the magnetic parameters due to
saturation phenomena.
The paper is organized as follows. The IM model at stand-
still, based on the two-phase equivalent representation, is re-
ported in Section II. In this section, the information that can
be deduced from standard nameplate data are discussed. In the
first part of Section III, the general structure of the PAOs is re-
ported. In Section III-A, the nonminimal representation of the
IM model, used in the proposed PAO, is shown. In Section III-B,
the original adaptation law together with the complete structure
of the adopted PAO is reported. In Section IV, simulation re-
sults are given; particular attention is paid to the discretizationmethod which has to be used in order to implement the proposed
algorithm on a real digital controller. Some simulation results
with noisy measurements are also presented. In Section V, it is
shown how the rotor flux estimate given by the proposed scheme
can be effectively used to avoid magnetic saturation during the
identification procedure. In Section VI, the experimental results
are reported. The estimation results of the proposed scheme
are compared with the estimates obtained by applying a pow-
erful batch nonlinear least square (NLS) method. The actual
steady-state mechanical curve of the IM under test and the one
obtained by simulation with the experimentally estimated pa-
rameters are compared to validate the proposed method. In ap-pendices, sketches of the proofs concerning nonminimal repre-
sentation and convergence properties of the proposed solution
are given.
II. INDUCTION MOTOR MODEL AT STANDSTILL AND
NAMEPLATE DATA ANALYSIS
Under the hypothesis of linear magnetic circuits and bal-
anced operating condition, the equivalent two-phase model of a
squirrel-cage IM at standstill, represented in a stator reference
frame , is [2], [44]
(1)
where are stator voltages, stator
currents, and rotor fluxes and is the magnetic torque
produced by the motor. Positive constants in model
(1), related to IM electrical parameters, are defined as:
,
where are stator/rotor resistances and in-
ductances, respectively, while is the mutual inductance
between stator and rotor windings. All the electric variables and
parameters are referred to stator. The transformation adopted
to map the three-phase variables into the two-phases reference
frame maintains the vectors amplitude, as indicated by the
factor in the expression of .From (1), the complete decoupling of the components a and
b of the electrical variables at standstill can be noted. In addi-
tion, the torque expression shows that if only one phase of the
equivalent model is excited then the produced magnetic torque
is null. Hence, if no external torque is applied, the standstill con-
dition is preserved. Therefore, in the following, only the first
two equations in (1) will be considered, while all the variables
of the -phase will be assumed to be equal to zero. From a prac-
tical point of view, this means that no voltage is applied in the
b-phase.
Remark 1: In order to mitigate the effects of the machine
asymmetries, the identification procedure described in the next
sections and based on the excitation of the a-phase, can be re-peated with different axis orientation.
The resulting one-phase model is LTI but, as already men-
tioned in the introduction, this condition is admissible only if no
significant magnetic saturation and thermal heating are present.
With respect to the magnetic effects, in general a linear behavior
can be assumed only if the level of the flux is lower than the
nominal value. Since this variable is not directly measurable,
it should be better to express this condition in terms of stator
currents, i.e., the magnetizing current has to be lower than the
rated one. Unfortunately, the nominal value of the magnetizing
current is not usually available from the IM nameplate data. In
fact, the data given by IM manufacturers are related to nominalload conditions and, generally, they are: the mechanical power,
, the stator voltage, , the electric frequency, , the me-
chanical speed, , the stator current, , and the power factor,
. The rated level of the magnetizing current can be de-
duced using a classical no-load test, but it is difficult to deduce
it with acceptable accuracy by means of a simple and fast test
at standstill. On the other hand, it is possible to obtain the nom-
inal stator flux rms value, , by using typical nameplate
data and a simple dc measurement of the stator resistance at
standstill. In fact, the expression of is
(2)
Therefore, it is reasonable to assume that the magnetic core is
not saturated, if the peak value of the rotor flux (referred to
stator) satisfies the following inequality [2], [9], [44]:
(3)
As will be shown in the following sections, the proposed iden-
tification procedure also produces a recursive estimation of the
rotor flux which asymptotically tracks the real one. Hence, this
solution, using the information derived from (2) and (3), is suit-
able to verify that no saturation phenomena occurs during the
estimation process and, consequently, it guarantees that the es-timated magnetic parameters are significant.
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CASTALDI AND TILLI: PARAMETER ESTIMATION OF INDUCTION MOTOR AT STANDSTILL 389
(a) (b)
Fig. 1. (a) Series-parallel and (b) parallel adaptive observer schemes.
III. NEW MRAS PARALLEL IDENTIFIER/OBSERVER FOR THE
IM AT STANDSTILL
During the last three decades, a considerable amount of workhas been done on the design of adaptive state observers with
MRAS configurations [33]. These schemes are suitable for both
state observation and parameter estimation owing to their adap-
tive nature. In the case of IM at standstill considered this kind of
approach can be used for estimating the machine parameters and
monitoring the nonmeasurable state variables during the identi-
fication process.
Fig. 1 shows the two possible classes of MRAS adaptive state
observers: the series-parallel adaptive observer (SPAO) which
uses the input and the output of the observed system in the ob-
server block and the PAO characterized by the absence of the
system output signal in the observer block.
It is well known [31] that the PAO is characterized by ex-
cellent noise-rejection properties, while the SPAO is preferable
only in the case of very high signal-to-noise ratio (SNR) because
of the larger amount of information carried by the output signal.
The solution proposed in the literature for both of the schemes
depends on the possibility of measuring the whole state vector.
For the SPAO several globally asymptotically stable solutions
have been developed both with accessible and not accessible
state [32]; while for the PAO only the solution in the case of ac-
cessible state is well-established. In the case of the IM the whole
state is not directly accessible and only noisy measurements of
the output current are available. In order to improve the robust-
ness of the identification precess with respect to measurementnoise, a PAO structure has been chosen for the proposed estima-
tion scheme. A new adaptation law has been developed to deal
with this case where the full state is not accessible.
A. Nonminimal Realization of the IM Model
The new PAO proposed is based on a nonminimal realiza-
tion of the IM model. This nonminimal form, whose order
is where is the order of the IM model, can be
considered as a generalization of the realization introduced
by Kreisselmeier [41], which is strictly based on the -com-
panion canonical forms. On the contrary, the proposed solution
avoids the use of those canonical forms since they are numer-ically ill-conditioned [42].
According to Section II, consider now the a-phase LTI model
of the IM
(4)
where
where and is the output .
In the following, it will be shown how the previous second-
order model can be represented by the equivalent fourth-order
model:
(5)
where the couple is arbitrary, provided that it is com-
pletely reachable and is Hurwitz; while are relatedto the original model (4) and the choice of .
The remarkable characteristic of (5) is that the relevant model
parameter vectors, and , appear linearly in the output equa-
tion only, while the state dynamics can be defined arbitrary.
Thereby, using this representation, the observation process can
be well separated from the adaptation process [32]. The matrix
is not very important in the model description since the con-
tribution vanishes, owing to the asymptotic
stability of matrix . In addition, note that filters both the
system input and output. This is a useful feature for a robust ob-
server design in a noisy environment.
To obtain the relation between the representations
and Kreisselmeiers result
[41], holding for models in -companion canonical form,
constitutes the starting point. Consider the IM model in K-com-
panion form
(6)
where
In [41], it has been proved that system (6) can also be repre-
sented by the following nonminimal equivalent representation:
(7)
where is the th column of the identity matrix, is in -
companion form and
(8)
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In the following, the conditions for the equivalence of models
(4) and (5) will be presented, using the relation (8), between the
canonical models (6) and (7). Consider the transformations
which sets the triple in the canonical form (6), and a
matrix satisfying relation . Ob-
viously, matrix depends on how the arbitrary and completely
reachable couple is chosen.
The relations between the models (4), (5) and the -com-
panion forms (6), (7) are the following:
(9)
Hence, the output equation of (5) can be rewritten as
Recalling the output expression in (7) and using (8), the fol-
lowing relation between models (4) and (5) can be expressed, in
order to impose the equivalence
(10)
Remark 2: As will be shown in the following, the final aim of
the identification process is to calculate the IM physical param-
eters from the estimation of vectors and . From the first two
equations in (10), it is straightforward to obtain the following
relations:
By solving the previous equations, it is possible to calculate
the system parameters and the product , starting
from and vectors. In order to determine and , it is
necessary to add the hypothesis of , which is usually
verified in practice, however in some types of induction machine
a different ratio is suggested [44], [45]. From that it follows that
then, since is known, it
is possible to determine and separately.
Remark 3: Given the IM physical parameters, it is also pos-
sible to obtain the matrix and the physical state can be
calculated by means of the following formula (the proof is in
the Appendix):
......
...
(11)
where and are matrices built with the polynomial coef-ficient of , for (see the Appendix for
their formal definition). In particular, if is chosen in diagonal
form, , then
and the following simplified expression for the matrixes results
:
B. New PAO for the IM at Standstill
In the previous remarks, it has been shown how the IM dy-
namics (4) can be described with a model of the form reported
in (5). In addition, in Remarks 2 and 3 it has been underlined
how it is possible to calculate the physical state
and the physical parameters , and from the state
and the parameters and of model (5).
On the basis of these results, a new MRAS parallel identi-fier/observer (PAO) is presented in this section. Referring to the
IM nonminimal model (5), the structure of the adopted observer
is the following:
(12)
where , and are, respectively, the estimate of the
output, the states, and the parameters of model (5). Note that in
the proposed observer structure no estimate of the initial state
is considered. The reason why is twofold:
the contribution of the initial state disappear expo-
nentially since is Hurwitz;
in the case of the IM model (4) the initial state is
usually null.
The parallel nature of the proposed scheme derives from the
use of the estimated output in the output equation of (12) in-
stead of the actual measurable output ; in this way the state
observation does not depend directly on the actual output.
In order to complete the proposed PAO an adaptation law for
the estimated parameters must be added.
The proposed adaptation law is the following:
(13)
(14)
where and are two filtered versions of the output error,
defined as
(15)
while is an arbitrary positive scalar constant, are arbi-
trary positivedefinite gain matrices, and is a positivedef-
inite matrix which has to satisfy some weak constraints (see the
Appendix).
In the Appendix, it is shown that the PAO given by (12)(15)
guarantees asymptotic convergence of the states and the
output to the actual ones , and . In addition, if per-sistency of excitation is guaranteed for the state variables, also
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CASTALDI AND TILLI: PARAMETER ESTIMATION OF INDUCTION MOTOR AT STANDSTILL 391
TABLE INAMEPLATE DATA AND TRADITIONALLY ESTIMATED PARAMETERS OF THE
ADOPTED INDUCTION MOTOR
the estimated parameters and converge to the actual values.
Some considerations about the choice of the adaptation law are
also reported.
Remark 4: From a theoretical point of view, the choice of
the couple is arbitrary, providing that it is controllable.
Actually, in order to implement a light and well-conditionedalgorithm, it is better to choose matrix in diagonal form.
Remark 5: The scalar and the matrices (usually in
diagonal form) define the adaptation gains. Their values repre-sent a compromise between the speed of convergence and the
noise rejection properties of the PAO.
Remark 6: The PAO scheme shown in (12) (15) does notprovide a direct estimate of the rotor flux. In order to calculate it,(11) has to be used, neglecting the initial state and replacing real
values with estimated ones. Note that the matrix in (11) de-
pends on the physical parameters and . Hence, to obtain a re-
cursive estimate of the flux during the the identification process,it is necessary to calculate an estimate of the previous parame-
ters following the procedure indicated in Remark 2.
IV. SIMULATION RESULTS AND DISCRETIZATION
The aim of this section is twofold: 1) to show, by means of
simulation results, the performances of the proposed PAO (both
ideal and noisy conditions are considered); 2) to introduce a dis-
cretized version of the adopted scheme, suitable for real imple-
mentation, and to show its behavior with respect to the original
continuous-time version.
In order to simulate the actual IM, the LTI model (4) has been
adopted. Hence, no magnetic saturation effect has been taken
into account at this stage. The issue related to the magnetic non-linearity will be discussed in next section. In this part, instead,
it is shown that the rotor flux is well-estimated whenever the
assumption of linear magnetic core is admissible. The IM ac-
tual parameters adopted during the simulations are reported in
Table I. These parameters are related to the motor used in exper-
imental tests. They have been identified by means of traditional
methods based on no-load and locked-rotor tests. The nameplate
data of the motor are also reported in Table I.
The couple adopted in the proposed PAO is the fol-
lowing: and In
this way, the actual parameter values in the nonminimal realiza-
tion (5) are: . These
are the values which have to be identified using the proposedscheme.
Fig. 2. Injected voltage waveform.
A. Simulations of the Continuous-Time Version of the PAO
The simulation tests reported in this part are related to the
PAO in continuous-time version, as introduced in Section III.B.
The adopted gains are the following:
and
In particular, the matrix has been chosen solving the fol-
lowing linear matrix inequality (LMI) problem (see the Ap-
pendix):
(16)
where
(17)
The set has been chosen in order to include the model ma-
trix in canonical form [see (6)] for a wide range of possible
IM. The solution of (16) has been obtained using the LMI
toolbox of Matlab [47].
In all of the tests performed the input voltage is given by the
sum of four sinusoids in order to guarantee the persistency of
excitation. The amplitude is set to 5 V for all the sinusoidal com-
ponents and the following Hz frequencies are adopted: 1, 3.18,
9, and 35 (see Fig. 2). The choice of these values is related to
some insights into the typical behavior of standard IM. In fact,
the transfer function between the stator voltage and stator cur-
rent at standstill is characterized by the slow (15 Hz) and fast(2550 Hz) poles. In addition, a zero is present near the slow
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Fig. 3. Continuous-time PAO, ideal case: estimation of the parameters p and q.
Fig. 4. Continuous-time PAO, ideal case: estimation of current and flux (beginning of the estimation process).
Fig. 5. Continuous-time PAO, ideal case: estimation of current and flux (end of the estimation process).
pole, but structurally on its left in the complex plane. In the firstset offigures (Figs. 35), the results of a simulation in ideal con-
ditions are reported. In Fig. 3, the temporal evolutions of the es-timated parameters are reported. All of the estimations converge
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CASTALDI AND TILLI: PARAMETER ESTIMATION OF INDUCTION MOTOR AT STANDSTILL 393
Fig. 6. Continuous-time PAO, noisy case: estimation of the parameters p and q.
Fig. 7. Continuous-time PAO, noisy case: estimation of current and flux (beginning of the estimation process).
Fig. 8. Continuous-time PAO, noisy case: estimation of current and flux (end of the estimation process).
to the real parameters, independently of their initial value. Theconvergence time is quite long; it can be reduced by increasing
the adaptation gains, but this will lead to larger oscillation inthe transient. In Figs. 4 and 5, the current and flux estimates are
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394 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 3, MAY 2005
TABLE IISIMULATION RESULTS FOR CONTINUOUS-TIME PAO IN NOISY ENVIRONMENT
compared with the real values. In Fig. 4, the beginning of the
simulation tests is considered, the estimates of the stator current
and the rotor flux are not very good; in fact, the estimated pa-
rameters are quite far from the real values. Instead, in Fig. 5(c)
and (d), where the end of the simulation is shown, the estimates
of both states are very good. This fact confirms the flux-moni-
toring capability of the proposed scheme.
In Figs. 68, the results of a simulation in noisy conditions
are reported. A white noise has been added on the output (the
stator current ). The adopted standard deviation is 10% of the
RMS value of the stator current in ideal conditions. In Fig. 6,
the temporal evolutions of the estimated parameters are shown.The adaptation process is very similar to the ideal case and the
convergence ratio is not influenced by the measurement noise.
In Figs. 7 and 8, the current and flux estimates are compared
with the actual values. In Fig. 7, the beginning of the simulation
test is considered, while in Fig. 8 the final part is shown. The
state estimate is still very good when the parameters are near
the correct values. Hence, also in a noisy environment the flux
monitoring can be performed. In particular, in Figs. 7(a) and
8(a), the current estimate is compared with the measured one
(impaired by noise). The difference between them represents
the so-called innovation or residual for the adopted identifica-
tion-observation scheme. Other tests have been performed with
different levels of noise, while other conditions are unchanged.
In Table II, the results are summarized. Only the product of
and is reported since these two parameters can be identified
separately only if some additional assumptions are considered
(see Remark 2). The quantity % represents an identification
error index defined as % , where and
are the vectors of the actual and estimated parameters, respec-
tively: and . In
particular, the estimated parameters in are the mean values
of the results given by the proposed PAO over a time interval
from 150 to 180 s, where the convergence transient is always
terminated. In Table II, the variance, over the same time interval,
of the estimated values is also indicated (in brackets). Anotherindex of the identification quality in a noisy environment is the
whiteness of the innovation. This characteristic has always been
computed on the time interval 150180 s, using a whiteness test
based on an eight-degree-of-freedom variable, whose 99%
confidence interval is 020.1 [51]. All the indexes considered
show good performances of the proposed PAO, even with large
noise. Some small differences can be noted among the results
of the simulation tests, owing to the white noise level. In fact,
as is well known, adaptation algorithms are usually biased in a
noisy environment [31]. However, the extensive simulation tests
confirm the robustness of the proposed solution for both identi-
fication and flux monitoring. This is essentially due to the par-
allel structure of the adaptive observer proposed. Moreover, a25 mA-dead-zone has been inserted on the current estimation
error, , in the adaptation law (13)-(14), according to stan-
dard practice of adaptive algorithms. Obviously, the gain
and also plays an important role in noise insensitivity: the
lower the gains, the greater the identification-observation accu-
racy will be(and the larger the convergence time).
B. Discretization of the Proposed Scheme
In order to obtain a really-implementable version of the pro-
posed PAO it is necessary to develop a discrete-time version.
Different discretization methods have been considered: forward
differences, backward differences, Tustin, z-transformationwith different input reconstructors. The sampling time that was
expected to be used in real implementation is s.
This a good a priori tradeoff between the dynamics of the ob-
server (similar to a typical IM) and the computation capability
of a standard DSP or microcontroller used in high-performance
drive. The criterion used to choose between the different dis-
cretization techniques was the following: a) to maximize the
likelihood between the continuous and the discrete version of
the PAO with the sampling time fixed above; b) to minimize
the computational complexity of the algorithm. The best results
were obtained with the following discretization:
(18)(19)
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CASTALDI AND TILLI: PARAMETER ESTIMATION OF INDUCTION MOTOR AT STANDSTILL 395
(20)
(21)
(22)
(23)
(24)
where
and
and , and are the same gains used in the continuousversion. The LTI dynamics (18), (19), and (21) have been dis-
cretized with an exact method in the hypothesis of constant in-
puts between two sampling times (z-transformation with ze-
roth-order reconstructor on the input). The nonlinear dynamic
and static equations (20), (22), (23), (24) have been discretized
using the Euler approximation.
The simulation tests performed for the continuous version
have been repeated for the discrete-time version proposed. The
results are very close to the continuous-time case, both for pa-
rameter estimation and flux observation, even with large noise
on the current measurement. Hence, the proposed discretization
method and the adopted sampling time are suitable for the dig-ital implementation of the original continuous version. (For the
sake of brevity, figures and tables related to the simulations of
the discrete version are not reported)
V. AVOIDANCE OF MAGNETIC SATURATION USING THE
PROPOSED ESTIMATOR
In this section, a procedure to avoid magnetic saturation,
based on the proposed estimator, is illustrated.
In previous paragraphs it was proved that the rotor flux is cor-
rectly estimated when the IM model is LTI, but no results are
available about the observation properties when magnetic satu-
ration occurs. Consequently, the basic idea is to use the rotor fluxlevel estimation to avoid the state of the IM exits from the linear
region during the identification process. From the previous con-
siderations, the following procedure can be defined as:
1) start the identification process with a low-voltage
signal (which guarantees very low flux, far from
saturation) satisfying the persistency of excitation
requirement;
2) wait for the flux and parameters estimation conver-
gence using a suitable innovation whiteness test;
3) slowly increase the voltage as long as a significant level
of the estimated rotor flux [obeying to (3)] is obtained;
4) stop the estimation algorithm when the whiteness testis satisfied.
Note that it is not convenient to stop the parameter identifica-
tion after the estimate convergence with low flux (Step 2 of the
procedure). In fact, in that condition the Signal to Noise Ratio
is very low and the nonideality of the power electronics device
used to feed the motor are relevant. By means of the proposed
procedure, based on flux estimation, the flux level can be con-
sciously increased without producing saturation of the magneticcore (Step 3). Hence, an optimization of the signal to noise ratio
can be safely achieved.
VI. EXPERIMENTAL RESULTS AND VALIDATION
In this section, the performances of the actual implementation
of the proposed identification scheme are shown. The obtained
results are compared with the output of a batch (i.e., nonrecur-
sive) method based on NLS.
The nameplate data of the adopted motor are reported in
Table I. Its electrical parameters, roughly identified with tra-
ditional methods, have been used in the previous section to
perform simulation tests. The stator resistance value, obtainedwith a simple dc test, is equal to 6.6 (as reported in Table I).
Using (2), it can be deduced that the nominal stator flux value is
Wb. Hence, recalling (3), no magnetic saturation
will arise if the rotor flux is maintained under 0.74 Wb.
During experimental tests, the stator currents were measured
using closed-loop Hall sensors. The stator voltages were im-
posed by a standard three-phase inverter with a 10 KHz sym-
metrical-PWM control. Simple techniques based on phase cur-
rent sign [52] were used to compensate for the effects of the
dead-time, set to 1.5 s. The proposed estimation scheme was
implemented on a control board equipped with a floating-point
DSP, TMS320C32. The adopted sampling time was 300 s, aspreviously indicated in Section IV-B. It is worth observing that
the motor shaft was connected to a mechanical load to avoid
rotor movements due to magnetic anisotropy. This solution is
typical for self-commissioning drives.
A set of experimental tests was performed using the proce-
dure shown in Section V to obtain good flux level, avoiding
saturation. That means the flux level was kept under the max-
imum value indicated previously. The voltage signal adopted is
formed by four sinusoids with the same frequencies reported in
Section IV and equal amplitudes of 2 V as starting values. After
stage 3 of the procedure, the amplitude for each of the sinu-
soidal components is 5 V. An additional equality constraint be-
tween the sinusoids amplitude was imposed to simplify the S/N
optimization procedure without impairing the overall estimation
performances. Note that in order to compare the simulations and
the experiments, an amplitude of 5 V was imposed to the sinu-
soids adopted in Section IV and the experimentally estimated
parameters are set to 0 at the end of stage 3 of the procedure
of Section V. The results of one of the experimental tests are
reported in Figs. 911. It can be noted that the temporal evolu-
tion of both state and parameter estimate are very similar to the
simulated ones. Only the final values of the estimated parame-
ters are slightly different. Many other experimental tests were
performed with different frequencies of the exciting sinusoids
(always preserving linearity of the magnetic circuit by meansof the procedure of Section V). The results are summarized in
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396 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 3, MAY 2005
Fig. 9. Experimental results: estimation of the parameters p and q.
Fig. 10. Experimental results: estimation of current and flux (beginning of the estimation process.
Fig. 11. Experimental results: estimation of current and flux (end of the estimation process.
Table III. For every different test, the estimated parameters are
the mean values on the time interval between 150 and 180 s.
The whiteness of the residual was checked by means of
the test used for simulations. The mean values and the stan-
dard deviation, reported in the last two rows of Table III, are
computed to evaluate the dispersion of different tests. In partic-
ular, the small standard deviation shows the good precision ofthe proposed method.
As underlined previously, the experimentally estimated pa-
rameters given by the proposed PAO are quite different from
the traditionally estimated data used in simulations as actual
values. In order to verify carefully the performances of the pro-
posed scheme, the experimental data have been processed with
a different identification algorithm, based on the nonrecursive
NLS method. This algorithm has been realized using the fmin-search function of the optimization toolbox of Matlab [48];
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CASTALDI AND TILLI: PARAMETER ESTIMATION OF INDUCTION MOTOR AT STANDSTILL 397
TABLE IIIINDUCTION MOTOR PARAMETERS EXPERIMENTALLY ESTIMATED WITH THE PROPOSED PAO (NO MAGNETIC SATURATION OCCURS)
TABLE IVINDUCTION MOTOR PARAMETERS EXPERIMENTALLY ESTIMATED WITH THE NLS METHOD (NO MAGNETIC SATURATION OCCURS)
fminsearch is a minimization procedure for a generic cost
function, based on the NelderMead method. The cost function,
, has been imposed equal to the difference, in the least square
sense, between the experimental data and the simulation with
the estimated motor parameters, that means
(25)
where is the experimental output, is the vector of the esti-mated parameters and is the output simulated using these pa-
rameters. This method is very powerful so it represents a good
touchstone. Obviously, it cannot be used directly in self-com-
missioning drives, since it has a heavy computational burden
and it can only be used in a batch way, without any recursive
monitoring of the rotor flux. The results obtained with the NLS
method for the experimental tests are reported in Table IV. The
parameters estimated with this approach are very similar to the
ones obtained with the proposed scheme.
The measurements collected during experiments can be cor-
rupted by typical sensor nonidealities such as current sensors
offset or typical actuation troubles such as unperfect dead-time
compensation. Hall sensors offset has been minimized by a stan-
dard zeroing procedure before starting the identification algo-
rithm. However, the robustness with respect to this kind of mea-
surement and actuation trouble cannot checked by the compar-
ison between the proposed scheme and the NLS method re-
ported since the potentially corrupted data are the same for
both algorithms. A practical method to check the correctness
of the estimated values is to compare the actual IM mechan-
ical curve (speed versus torque) with the one simulated using
the estimated parameters. This comparison is reported in Fig. 12
where the mechanical curves are derived by supplying the motor
with a 33.3 Hz253 V sinusoidal three-phase voltage. The
traditionally-estimated parameters are also considered. Verygood matching is obtained between experimental data and the
Fig. 12. IM mechanical curve: from experiments; 3 simulated using theparameters estimated with the proposed scheme in test #2; x simulated usingthe traditionally-estimated parameters.
simulation results based on parameters estimated by the pro-
posed solution, while a significant error can be noted when tradi-
tionally estimated parameters are considered. This result shows
that the robustness of the method presented combined with the
proposed measurements and actuation expedients guarantees a
very reliable IM parameter estimation.
Remark 7: The mechanical curve of an IM is very sensitive
to all of its electrical parameters [2], [44]. Hence, the compar-
ison between the actual speed-torque curve and the one obtained
simulating the IM model is a very effective method to validate
the estimated parameters used in the model. In addition, this val-
idation methods is based on an open-loop experiment, there-
fore its results are not affected by feedback control algorithmswhich usually mitigate the effects of parameters mismatching.
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398 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 3, MAY 2005
VII. CONCLUSION
A new method for the estimation of the parameters of IMs at
standstill has been presented. The proposed schemeis based on a
PAO designed using a novel nonminimal representation derived
from Kreisselmeiers canonical form and an original adaptation
law.
It has been proved, both theoretically and by implementation,that the proposed algorithm assures a simultaneous asymptotic
unbiased estimation of both the system parameters and the state
(i.e., stator current and rotor flux).
A discretized version, suitable for digital implementation, has
been developed, preserving the characteristics of the original
continuous-time procedure.
The simulation tests have shown the excellent noise rejection
properties of the proposed solution. This feature is related to
the parallel structure of the adopted adaptive observer and can
be tuned by varying the adaptation law gains.
Experimental results have proved the effectiveness and ra-
pidity of the approach. In particular, it has been shown that mag-netic saturation can be avoided thanks to the good rotor flux es-
timation. The identification results are strongly validated by two
methods: 1) the comparison with a powerful batch NLS method;
2) the comparison of the actual mechanical curve of the IM used
for the test with the one obtained by simulation using the esti-
mated parameters.
Finally, it has been shown that the algorithm is fast and simple
and may be easily implemented in self-commissioning drives.
APPENDIX
Definition 1: Given a generic second-order square matrix ,
the matrix such that
is denoted as the matrix of the polynomial coefficients of
.
Proposition 1: Let
, and be the matrices and vectors de-
fined in Section III-A
, and be the matrices of the polyno-
mial coefficient of and
, respectively;
hence, the following relation holds:
(26)
Proof: Consider the problem
(27)
The extension of (27) to (26) is straightforward. By means of
relations and , it is easy to verify that
(27) can be rewritten as
(28)
where .
Now, recalling the definition of and and the relation
, it is easy to verify
(29)By substituting (29) in (28) and noting that
, the proof is completed.
Proposition 2: [41] The state of the IM model
is linked to the state of the nonminimal representation
by the following relation:
(30)
Proof: [41].
Proposition 3: The state of the IM model is
linked state of the generalized nonminimal representation
by the following relation:
(31)
Proof: Straightforward by means of Propositions 1 and
2.
Now, the convergence properties of the proposed scheme
are discussed. The guidelines for the theoretical proof of
these characteristics are stated avoiding mathematical details.
Starting from the nonminimal parametrization of the IM with
locked rotor, given in (5), and the PAO expression, given in
(12), the following error model can be defined:
(32)
where
(33)
are the state, the estimation, and the output errors. From the first
equation in (32), it results that , hence, the first partof the state can be neglected in the convergence analysis.
Before considering the complete convergence analysis, it is
worth studying the case of perfect knowledge of the parameters
vectors and . In this condition, the error model is
(34)
From (34) it can be deduced that, in this case, the convergence to
zero of error state requires the matrix to be Hurwitz.
Using (10) and the definition of and , it can be shown that
. Then, in order to guarantee
the global asymptotic stability of (34), the matrix must beHurwitz.
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CASTALDI AND TILLI: PARAMETER ESTIMATION OF INDUCTION MOTOR AT STANDSTILL 399
Coming back to the general case reported in (32), the Hurwitz
character of matrix is not strictly necessary in principle to de-
sign an adaptation law that guarantees asymptotic convergence.
By the way, the matrix of the IM model (4) is certainly Hur-
with, even if unknown. This characteristic has been exploited in
the choice of the adaptation law (13), (14) as shown in the fol-
lowing convergence analysis.Now define the following Lyapunov-like function:
(35)
where are arbitrary, and
is the solution of the following Lyapunov equation:
(36)
with arbitrary . The solution of (36) exists,
since is Hurwitz. On the other hand, is unknown and it is
not possible to solve (36) directly. By the way, from a practicalpoint of view, some boundscan be defined on the IM parameters.
Hence, (36) can be translated in a LMI where belongs to a
certain set. Hence, a suitable can be found using the
standard procedure for LMI solving [49].
The function is clearly positive defined on the error
statespace . The time derivative of along the
trajectories of (32) is
(37)
where the contribution of the initial state hasbeen neglected, since exponentially disappears with an arbi-
trary ratio. With simple computation (37) can be rearranged as
follows:
(38)
Considering the adaptation law reported in (13), (14), the defini-tion (15) of the filtered error and recalling (36), the derivative
of results as follows:
(39)
Hence, the error state is bounded and the Barbalats
Lemma [50] can be applied. It results that
(40)
From the definition (15) and the expression of the output error
in the last of (32), it can be derived that
(41)
then, applying standard arguments related to persistency of ex-
citation [32], it can be shown that an exponential convergence
to zero of the parameter estimation error is obtained if the har-
monic content of the input is large enough.
Remark 8
The variable is similar to the augmented error typicallyused in adaptive systems [32]. In particular, it has been intro-
duced in order to have the parameter estimation errors in .
This allows persistency of excitation arguments to be applied
to achieve exponential convergence of the parameter estimates.
The remaining parts of the adaptation law are used to cancel
bad terms in .
ACKNOWLEDGMENT
The authors would like to thank C. Morri and A. Casagrande
for their valuable collaboration in testing the proposed proce-
dure during their degree theses. The experimental tests were car-
ried-out at the Laboratory of Automation and Robotics (LAR)of the University of Bologna. The authors would also like to
thank the anonymous reviewers for their valuable suggestions
about the experiments to test the proposed solution.
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Paolo Castaldiwas born in Bologna, Italy. He re-ceived the Laurea degree in electronic engineering
and the Ph.D. degree in system engineering from theUniversity of Bologna, Italy, in 1990 and 1994, re-spectively.
Since 1995, he has been a Research Associate inthe Department of Electronics, Computer Science,and Systems (DEIS), University of Bologna. Hisresearch interests include adaptive filtering, systemidentification, fault diagnosis and their applicationsto mechanical and aerospace systems.
Andrea Tilli was born in Bologna, Italy, on April 4,1971. He received the Laurea degree in electronic en-gineering and thePh.D. degreein system engineering
from the University of Bologna, Italy, in 1996 and2000, respectively.
Since 1997, he has been with the Department ofElectronics, Computer Science, and Systems (DEIS),University of Bologna. Since 2001, he has been a Re-search Associate at DEIS. He is also a Member of theCenterfor Research on ComplexAutomated Systems
Giuseppe Evangelisti (CASY), established withinDEIS. His current research interests include applied nonlinear control tech-niques, adaptive observers, variable structure systems, electric drives, automo-tive systems, active power filters, and DSP-based control architectures.