Parameter Estimation Of Im At Standstill With Magnetic Flux Monitoring

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Parameter Estimation of Induction Motor at Standstill With Magnetic Flux Monitoring Paolo Castaldi and Andrea Tilli

Abstract—The paper presents a new method for the estimation of the electric parameters of induction motors (IMs). During the identification process the rotor flux is also estimated. The procedure relies on standstill tests performed with a standard drive architecture, hence, it is suitable for self-commissioning drives. The identification scheme is based on the model reference adaptive system (MRAS) approach. A novel parallel adaptive observer (PAO) has been designed, starting from the series-parallel Kreisselmeier observer. The most interesting features of the proposed method are the following: 1) rapidity and accuracy of the identification process; 2) low-computational burden; 3) excellent noise rejection, thanks to the adopted parallel structure; 4) avoidance of incorrect parameter estimation due to magnetic saturation phenomena, thanks to recursive rotor flux monitoring. The performances of the new scheme are shown by means of simulation and experimental tests. The estimation results are validated by comparison with a powerful batch nonlinear least square (NLS) method and by evaluating the steady-state mechanical curve of the IM used in the tests. Index Terms—Identification, induction motor (IM), magnetic saturation, parallel adaptive observer (PAO), self-commissioning drives.

I. INTRODUCTION

I

N RECENT years, the demand for high-performance electric drives based on induction motors (IMs) has been constantly growing. IMs are particularly attractive for industrial applications because of their low cost and high reliability. Moreover, power electronics and control electronics, essential to realize sophisticated 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 multivariable, nonlinear and strongly coupled. The concept of field-orientation, introduced in Blaschke’s 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 commercial IM drives for high-performance motion control. Nowadays, another kind of control strategy is becoming interesting for industrial IM drives: the direct torque control (DTC) technique 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 received in final form September 13, 2004. Recommended by Associate Editor A. Bazanella. The authors are with the Department of Electronics, Computer Science and Systems, University of Bologna, Bologna 40136, Italy (e-mail: pcastaldi@ deis.unibo.it). 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 solutions, 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 model is 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-established methods (in particular DFOC and IFOC). Although different approaches have been used [7], [8], only partial and quite poor results have been obtained in terms of performance robustness with respect to parameter uncertainties, particularly for sensorless control. Hence, at the state of the art, good knowledge 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 parameters 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 mechanical load. Recently, various parameter identification techniques 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

of approach is very interesting since it is possible “to track” the slow variation of the electric parameters during normal operation. 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 to identify 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 between electric and mechanical dynamics is exploited to obtain simultaneous speed and parameters estimation. In [13], the generalized total least-square (GTLS) technique is adopted. Filtered derivatives of the measured signals are still needed, but particular 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 identify 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 machine 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 operating 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 usually 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. Many offline identification methods have been proposed; some of

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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 parameters 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 also avoids 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 nonminimal state–space 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 framework. 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 observing that from nameplate data, usually quite rough, it is possible

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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 standstill, based on the two-phase equivalent representation, is reported 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 reported. 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 results are given; particular attention is paid to the discretization method 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 powerful 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 parameters are compared to validate the proposed method. In appendices, sketches of the proofs concerning nonminimal representation 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 balanced operating condition, the equivalent two-phase model of a squirrel-cage IM at standstill, represented in a stator reference , is [2], [44] frame

are stator/rotor resistances and inwhere is the mutual inductance ductances, respectively, while 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 in the expression of . factor From (1), the complete decoupling of the components “a” and “b” of the electrical variables at standstill can be noted. In addition, 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 condition 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 practical 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 repeated with different axis orientation. The resulting one-phase model is LTI but, as already mentioned 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 nominal load conditions and, generally, they are: the mechanical power, , the stator voltage, , the electric frequency, , the me, the stator current, , and the power factor, chanical speed, . The rated level of the magnetizing current can be deduced 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, by using typical nameplate inal stator flux rms value, at data and a simple dc measurement of the stator resistance is standstill. In fact, the expression of (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]:

(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: ,

(3) As will be shown in the following sections, the proposed identification 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 suitable to verify that no saturation phenomena occurs during the estimation process and, consequently, it guarantees that the estimated magnetic parameters are significant.

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According to Section II, consider now the a-phase LTI model of the IM (4) where

(a)

(b)

where and is the output . In the following, it will be shown how the previous secondorder model can be represented by the equivalent fourth-order model:

Fig. 1. (a) Series-parallel and (b) parallel adaptive observer schemes.

(5) III. NEW MRAS PARALLEL IDENTIFIER/OBSERVER FOR THE IM AT STANDSTILL During the last three decades, a considerable amount of work has 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 adaptive 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 identification 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 observer 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 excellent 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 accessible 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 robustness of the identification precess with respect to measurement noise, a PAO structure has been chosen for the proposed estimation 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 realization 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 -companion canonical forms. On the contrary, the proposed solution avoids the use of those canonical forms since they are numerically ill-conditioned [42].

where the couple is arbitrary, provided that it is completely reachable and is Hurwitz; while are related . to 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 equation 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 contribution 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 observer design in a noisy environment. To obtain the relation between the representations and Kreisselmeier’s result [41], holding for models in -companion canonical form, constitutes the starting point. Consider the IM model in K-companion form (6) where

In [41], it has been proved that system (6) can also be represented by the following nonminimal equivalent representation: (7) where is the th column of the identity matrix, companion form and

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is in

-

(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 . Obdepends on how the arbitrary and completely viously, matrix is chosen. reachable couple The relations between the models (4), (5) and the -companion 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 following relation between models (4) and (5) can be expressed, in order to impose the equivalence

their formal definition). In particular, if , then form,

is chosen in diagonal

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 dynamics (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 identifier/observer (PAO) is presented in this section. Referring to the IM nonminimal model (5), the structure of the adopted observer is the following: (12)

(10) Remark 2: As will be shown in the following, the final aim of the identification process is to calculate the IM physical parameters 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 and the product , starting the system parameters and , it is from and vectors. In order to determine 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 posand the “physical” state can be sible to obtain the matrix calculated by means of the following formula (the proof is in the Appendix): .. .

.. .

.. . (11)

where and ficient of

are matrices built with the polynomial coef, for (see the Appendix for

, and are, respectively, the estimate of the where 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 expois Hurwitz; nentially since is • in the case of the IM model (4) the initial state usually null. The parallel “nature” of the proposed scheme derives from the use of the estimated output in the output equation of (12) instead 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 must be added. the estimated parameters The proposed adaptation law is the following: (13) (14) and where defined as

are two filtered versions of the output error, (15)

while is an arbitrary positive scalar constant, are arbiis a positive–deftrary positive–definite gain matrices, and 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) and the guarantees asymptotic convergence of the states output to the actual ones , and . In addition, if persistency of excitation is guaranteed for the state variables, also

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TABLE I NAMEPLATE 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 is arbitrary, providing that it is controllable. the couple Actually, in order to implement a light and well-conditioned algorithm, 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 represent 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 not provide 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) depends on the physical parameters and . Hence, to obtain a recursive estimate of the flux during the the identification process, it is necessary to calculate an estimate of the previous parameters 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 discretized version of the adopted scheme, suitable for real implementation, 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 nonlinearity 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 “actual” parameters adopted during the simulations are reported in Table I. These parameters are related to the motor used in experimental 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. couple adopted in the proposed PAO is the folThe and In lowing: this way, the actual parameter values in the nonminimal realiza. These tion (5) are: are the values which have to be identified using the proposed scheme.

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 following linear matrix inequality (LMI) problem (see the Appendix): (16) where

(17) The set has been chosen in order to include the model matrix in canonical form [see (6)] for a wide range of possible of (16) has been obtained using the LMI IM. The solution 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 components 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 current at standstill is characterized by the slow (1–5 Hz) and fast (25–50 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 first set of figures (Figs. 3–5), the results of a simulation in ideal con-

ditions are reported. In Fig. 3, the temporal evolutions of the estimated parameters are reported. All of the estimations converge

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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. The convergence time is quite long; it can be reduced by increasing

the adaptation gains, but this will lead to larger oscillation in the transient. In Figs. 4 and 5, the current and flux estimates are

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TABLE II SIMULATION 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 parameters 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-monitoring capability of the proposed scheme. In Figs. 6–8, 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 identification-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 , where and error index defined as % are the vectors of the actual and estimated parameters, respectively: 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). Another index of the identification quality in a noisy environment is the

whiteness of the innovation. This characteristic has always been computed on the time interval 150–180 s, using a whiteness test variable, whose 99% based on an eight-degree-of-freedom confidence interval is 0–20.1 [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 identification and flux monitoring. This is essentially due to the parallel structure of the adaptive observer proposed. Moreover, a 25 mA-dead-zone has been inserted on the current estimation , in the adaptation law (13)-(14), according to stanerror, 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 accuracy will be(and the larger the convergence time). B. Discretization of the Proposed Scheme In order to obtain a really-implementable version of the proposed PAO it is necessary to develop a discrete-time version. Different discretization methods have been considered: forward differences, backward differences, Tustin, z-transformation with different input reconstructors. The sampling time that was s. expected to be used in real implementation is This a good a priori tradeoff between the dynamics of the observer (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 discretization 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:

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(18) (19)

CASTALDI AND TILLI: PARAMETER ESTIMATION OF INDUCTION MOTOR AT STANDSTILL

(20) (21) (22) (23) (24)

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Note that it is not convenient to stop the parameter identification 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 consciously increased without producing saturation of the magnetic core (Step 3). Hence, an optimization of the signal to noise ratio can be “safely” achieved.

where VI. EXPERIMENTAL RESULTS AND VALIDATION

and

and , and are the same gains used in the continuous version. The LTI dynamics (18), (19), and (21) have been discretized with an exact method in the hypothesis of constant inputs between two sampling times (z-transformation with zeroth-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 parameter 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 digital 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 correctly estimated when the IM model is LTI, but no results are available about the observation properties when magnetic saturation occurs. Consequently, the basic idea is to use the rotor flux level estimation to avoid the state of the IM exits from the linear region during the identification process. From the previous considerations, 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 convergence 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 test is satisfied.

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., nonrecursive) method based on NLS. The nameplate data of the adopted motor are reported in Table I. Its electrical parameters, roughly identified with traditional methods, have been used in the previous section to perform simulation tests. The stator resistance value, obtained with 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 imposed by a standard three-phase inverter with a 10 KHz symmetrical-PWM control. Simple techniques based on phase current 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, as previously 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 procedure shown in Section V to obtain good flux level, avoiding saturation. That means the flux level was kept under the maximum 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 sinusoidal components is 5 V. An additional equality constraint between 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 sinusoids 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. 9–11. It can be noted that the temporal evolution of both state and parameter estimate are very similar to the simulated ones. Only the final values of the estimated parameters are slightly different. Many other experimental tests were performed with different frequencies of the exciting sinusoids (always preserving linearity of the magnetic circuit by means of the procedure of Section V). The results are summarized in

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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 standard deviation, reported in the last two rows of Table III, are computed to evaluate the dispersion of different tests. In particular, the small standard deviation shows the good precision of the proposed method.

As underlined previously, the experimentally estimated parameters 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 proposed 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 “fminsearch” function of the optimization toolbox of Matlab [48];

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CASTALDI AND TILLI: PARAMETER ESTIMATION OF INDUCTION MOTOR AT STANDSTILL

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TABLE III INDUCTION MOTOR PARAMETERS EXPERIMENTALLY ESTIMATED WITH THE PROPOSED PAO (NO MAGNETIC SATURATION OCCURS)

TABLE IV INDUCTION 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 Nelder–Mead 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 estimated parameters and is the output simulated using these parameters. This method is very powerful so it represents a good touchstone. Obviously, it cannot be used directly in self-commissioning 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 corrupted 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 standard zeroing procedure before starting the identification algorithm. However, the robustness with respect to this kind of measurement and actuation trouble cannot checked by the comparison between the proposed scheme and the NLS method reported 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 mechanical 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 Hz–253 V sinusoidal three-phase voltage. The “traditionally-estimated” parameters are also considered. Very good matching is obtained between experimental data and the

Fig. 12. IM mechanical curve: “” from experiments; “3” simulated using the parameters estimated with the proposed scheme in test #2; “x” simulated using the “traditionally-estimated” parameters.

simulation results based on parameters estimated by the proposed solution, while a significant error can be noted when traditionally 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 comparison 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 validation methods is based on an “open-loop” experiment, therefore its results are not affected by feedback control algorithms which usually mitigate the effects of parameters mismatching.

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VII. CONCLUSION A new method for the estimation of the parameters of IMs at standstill has been presented. The proposed scheme is based on a PAO designed using a novel nonminimal representation derived from Kreisselmeier’s 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 rapidity of the approach. In particular, it has been shown that magnetic saturation can be avoided thanks to the good rotor flux estimation. 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 estimated 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 such that the matrix

,

is denoted as the matrix of the polynomial coefficients of . Proposition 1: Let • , and be the matrices and vectors defined in Section III-A , and be the matrices of the polyno• and mial coefficient of , respectively; hence, the following relation holds:

(26)

. where and Now, recalling the definition of , it is easy to verify

and the relation

(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]. is Proposition 3: The state of the IM model 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 , hence, the first part equation in (32), it results that of 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

Proof: Consider the problem (27)

(34)

The extension of (27) to (26) is straightforward. By means of and , it is easy to verify that relations (27) can be rewritten as

From (34) it can be deduced that, in this case, the convergence to to be Hurwitz. zero of error state requires the matrix and , it can be shown that Using (10) and the definition of . Then, in order to guarantee must be the global asymptotic stability of (34), the matrix Hurwitz.

(28)

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CASTALDI AND TILLI: PARAMETER ESTIMATION OF INDUCTION MOTOR AT STANDSTILL

Coming back to the general case reported in (32), the Hurwitz character of matrix is not strictly necessary in principle to design an adaptation law that guarantees asymptotic convergence. By the way, the matrix of the IM model (4) is certainly Hurwith, even if unknown. This characteristic has been exploited in the choice of the adaptation law (13), (14) as shown in the following convergence analysis. Now define the following Lyapunov-like function: (35) are arbitrary, and where is the solution of the following Lyapunov equation:

399

then, applying standard arguments related to persistency of excitation [32], it can be shown that an exponential convergence to zero of the parameter estimation error is obtained if the harmonic content of the input is large enough. Remark 8 The variable is similar to the augmented error typically used in adaptive systems [32]. In particular, it has been introduced 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

(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 practical point of view, some bounds can be defined on the IM parameters. belongs to a Hence, (36) can be translated in a LMI where certain set. Hence, a suitable can be found using the standard procedure for LMI solving [49]. is clearly positive defined on the error The function . The time derivative of along the state–space trajectories of (32) is

(37) has where the contribution of the initial state been neglected, since exponentially disappears with an arbitrary ratio. With simple computation (37) can be rearranged as follows:

(38) Considering the adaptation law reported in (13), (14), the definition (15) of the filtered error and recalling (36), the derivative of results as follows: (39) Hence, the error state is bounded and the Barbalat’s 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)

The authors would like to thank C. Morri and A. Casagrande for their valuable collaboration in testing the proposed procedure during their degree theses. The experimental tests were carried-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. REFERENCES [1] F. Blaschke, “The principle of field orientation applied to the new transvector closed-loop control system for rotating field machines,” Siemens-Rev., vol. 39, pp. 217–220, 1972. [2] W. Leonhard, Control of Electric Drives. Berlin, Germany: SpringerVerlag, 1995. [3] B. K. Bose, Power Electronics and Variable Speed Drives. Piscataway, NJ: IEEE Press, 1997. [4] P. Vas, Sensorless Vector and Direct Torque Control. Oxford, U.K.: Oxford Univ. Press, 1998. [5] G. Buja, D. Casadei, and G. Serra, “Direct stator flux and torque control of an induction motor: Theoretical analysis and experimental results (Tutorial),” in Proc. IEEE-IECON’98, Aachen, Germany, pp. T50–T64. [6] A. S. Bazanella and R. Reginatto, “Robustness margins for indirect fieldoriented control of induction motors,” IEEE Trans. Autom. Control, vol. 45, no. 6, pp. 1226–1231, Jun. 2000. [7] S. Peresada and A. Tonielli, “High performance robust speed-flux tracking controller for induction motor,” Int. J. Adapt. Control Signal Process., vol. 14, no. 2, pp. 177–200, 2000. [8] R. Marino, S. Peresada, and P. Tomei, “online stator and rotor resistence estimation for induction motors,” IEEE Trans. Contr. Syst. Technol., vol. 8, no. 3, pp. 570–579, May 2000. [9] N. R. Klaes, “Parameter identification of an induction machine with regard to dependencies on saturation,” IEEE Trans. Ind. Appl., vol. 29, no. 6, pp. 1135–1140, Nov.-Dec. 1993. [10] S. Sangwongwanich and S. Okuma, “A unified approach to speed and parameter identification of induction motor,” in Proc. IECON Int. Conf. Industrial Electronics, Control, and Instrumentation, 1991, pp. 712–715. [11] J. Stephan, M. Bodson, and J. Chiasson, “Real-time estimation of the parameters and fluxes of induction motors,” IEEE Trans. Ind. Appl., vol. 30, no. 3, pp. 746–759, May-Jun. 1994. [12] M. Velez-Reyes, K. Minami, and G. C. Verghese, “Recursive speed and parameter estimation for induction machines,” in Proc. IAS Conf. Rec., pp. 607–611. 19 889. [13] C. Moons and B. De Moor, “Parameter identification of induction motor drives,” Automatica, vol. 31, no. 8, pp. 1137–1147, 1995. [14] J. Holtz and T. Thimn, “Identification of the machine parameters in a vector-controlled induction motor drive,” IEEE Trans. Ind. Appl., vol. 27, no. 6, pp. 1111–1118, Nov.-Dec. 1991. [15] V. Pappano, S. E. Lyshevski, and B. Friedland, “Identification of induction motor parameters,” in Proc. 37th IEEE Conf. Decision and Control, Tampa, FL, Dec. 1998, pp. 989–994.

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Paolo Castaldi was born in Bologna, Italy. He received the Laurea degree in electronic engineering and the Ph.D. degree in system engineering from the University of Bologna, Italy, in 1990 and 1994, respectively. Since 1995, he has been a Research Associate in the Department of Electronics, Computer Science, and Systems (DEIS), University of Bologna. His research interests include adaptive filtering, system identification, fault diagnosis and their applications to mechanical and aerospace systems.

Andrea Tilli was born in Bologna, Italy, on April 4, 1971. He received the Laurea degree in electronic engineering and the Ph.D. degree in system engineering from the University of Bologna, Italy, in 1996 and 2000, respectively. Since 1997, he has been with the Department of Electronics, Computer Science, and Systems (DEIS), University of Bologna. Since 2001, he has been a Research Associate at DEIS. He is also a Member of the Center for Research on Complex Automated Systems “Giuseppe Evangelisti” (CASY), established within DEIS. His current research interests include applied nonlinear control techniques, adaptive observers, variable structure systems, electric drives, automotive systems, active power filters, and DSP-based control architectures.

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