Current Control For Induction Motor Drives Using ... - IEEE Xplore

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 45, NO. 5, OCTOBER 1998

Current Control for Induction Motor Drives Using Random PWM Cursino Brand˜ao Jacobina, Member, IEEE, Antonio Marcus Nogueira Lima, Member, IEEE, Edison Roberto Cabral da Silva, Senior Member, IEEE, and Andrzej M. Trzynadlowski, Senior Member, IEEE

Abstract— Current control in voltage-source inverters with random pulsewidth modulation (RPWM) is investigated. The random modulation is introduced to alleviate the undesirable acoustic, vibration, and EMI effects in inverter-fed ac drive systems. A novel RPWM digital technique with dithering of the switching frequency and compensation of the processing time is described. Design of the current control loop is discussed. Results of investigation of an experimental drive system are presented, proving the feasibility of the proposed solutions. Index Terms—Current control, induction motor, inverter, random PWM.

I. INTRODUCTION

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N INDUSTRIAL practice, adjustable-speed drive systems (ASD’s) with ac motors are becoming increasingly popular. Most ASD’s are comprised of a pulsewidth modulation (PWM) voltage-source inverter (ASD’s with current-source inverters are less common), an ac motor, and a control system that generates switching signals for the inverter. In lowperformance drives, feedforward voltage control, based on the regular-sampling or voltage space-vector PWM techniques, is employed [1]. In high-performance ASD’s, closed-loop (feedback) control of the motor is often used. With nonlinear current controllers, such as hysteresis, ramp-comparison, or predictive controllers, the current loop directly produces switching signals for the inverter [2]. However, certain manufacturers of drive systems prefer a scheme in which linear current controllers (usually of the proportional integral (PI) type) in the and axes generate a reference vector of the stator voltage to be implemented by pulse modulation [3]. Deterministic PWM (DPWM) results in concentration of the output power at discrete frequencies related to the fixed switching frequency. The voltage and current harmonics cause annoying tonal sound and increase the drive’s susceptibility to resonant vibration [4]. One approach to alleviate these undesirable phenomena is to increase the switching frequency beyond the audible range and move the harmonics to a band

in which there are no resonant vibration modes [5]. Another solution consists in employing random PWM (RPWM) [6]. It can easily be realized by parameter randomization of a selected DPWM technique. Dithering the switching frequency is particularly effective, but other randomization methods are also known [7]. Randomized switching patterns result in continuous voltage and current spectra, free of highorder harmonics. Beneficial effects of the RPWM on the acoustic and vibration characteristics of ASD’s, and even on the electromagnetic interference (EMI) conducted to the mains, have been documented in several publications, e.g., [8]–[10]. Most papers on RPWM, e.g., [11]–[13], have been devoted to control algorithms, inverter voltage spectra, and effects on the drive systems, but no studies on the impact of the random modulation on current control have been published. As an aside, it is worth noting that the direct, hysteresis current control scheme results in a sort of “spontaneous RPWM.” However, this paper deals with a different, “indirect” type of current control, the robustness of which may be compromised by the randomization or switching frequency. An ac drive was investigated, in which the system was designed to effectively generate random switching patterns in the inverter and accurately control the stator current in the motor. The major goal of the study was to evaluate the feasibility of the current control in an RPWM inverter and compare the control quality with that in a DPWM inverter. Realization of the randomization strategy in a digital control system of the inverter is presented first, followed by a discussion of the sampling rate selection in relation to the switching frequency. Based on the dynamic model of the induction motor in an arbitrary reference frame, the current control law is formulated next, and its implementation in a closed-loop system is illustrated. Experiments with a low-power ac drive system are described. Evaluation of the experimental results and final comments conclude the paper. II. RANDOMIZATION STRATEGY

Manuscript received April 30, 1997; revised February 3, 1998. Abstract published on the Internet July 3, 1998. C. B. Jacobina, A. M. N. Lima, and E. R. C. da Silva are with the Departamento de Engenharia El´etrica, Universidade Federal da Para´ıba, 58109-970 Campina Grande, PB, Brazil. A. M. Trzynadlowski is with the Department of Electrical Engineering, University of Nevada, Reno, NV 89557-0153 USA. Publisher Item Identifier S 0278-0046(98)07016-6.

Dithering of the switching (carrier) frequency is employed as the means of randomization of the regular-sampling PWM strategy used for control of the inverter voltage. A of randomly selected distinct values of the switchnumber to range, each ing frequency are contained in the The individual assigned certain probability

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JACOBINA et al.: CURRENT CONTROL FOR INDUCTION MOTOR DRIVES USING RPWM

probabilities add up to a unity, and the average switching can be calculated as frequency (1) To ensure uniform distribution of power in the spectrum of the inverter output voltage, all the probabilities are made Individual values equal, i.e., of the randomized switching frequency are determined as In the experimental system, the RPWM strategy was implemented in a microcomputer system with two timers. The first timer, TPMI, located on the microcomputer mother board, is and the sampling used to define switching intervals where denotes the sampling frequency. interval The second timer, TPMF, located on the multifunction board, provides timing of the switching signals for the inverter switches, i.e., it enforces the switching pattern. In the deterministic PWM strategy, the pulsewidth of the switching signal for phase 1 of the inverter is given by (2) denotes the clock frequency of TPMF, is the where which is the -axis component of dc-link voltage, and is the reference voltage the reference voltage vector The superscript in the machine variable and indicates the stator reference frame. Pulsewidths for the other two phases, 2 and 3, are determined similarly. in (2) originates from the three-to-two-phase Constant transformation used and corresponds to the case in which the power is assumed invariant [14]. In the random case, in (2) Consecutive values of is replaced with are supplied by a random number generator. is The local-average voltage of phase 1 (3) From a control point of view, a voltage-source inverter represents an actuator transforming the output of the voltage controller into the inverter output voltage. In the system under in the consideration, the output voltage of the actuator stator reference frame can be expressed as (4) is a constant gain, and is a time delay, depending where The voltage on the switching frequency, i.e., represents the disturbance term due to the pulsed vector modulation waveform of the inverter voltage. With the DPWM is a vector of deterministic high-frequency strategies, signals with distinct harmonic clusters. It becomes a randomsignal vector when an RPWM technique is employed and, in most cases, it is desirable to have it approaching the whitenoise quality.

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III. SAMPLING RATE Design of a digital controller begins with the choice of the sampling interval. In a microcomputer-based control drive system, the sampling clock determines timing of the realtime interrupts for implementation of the digital controller. is independent of the In general, the sampling frequency switching frequency , but it is required that However, it is not so in space-vector technique, in which the switching pattern for the first half of the switching interval is reversed in the second half of the interval [15]. The , between the switching and corresponding relation sampling frequencies, results in minimization of the number of switchings and maximization of the control bandwidth. Although feasible in the investigated system, such reversing scheme was not employed in this study. Otherwise, term would have to be replaced with in all pertinent equations. The length of the sampling interval is dictated by the dynamics of controlled systems or processing capabilities of the microcomputer, while the maximum switching frequency is mostly limited by switching times of the inverter switches or the acceptable level of switching losses. The following options can be considered. 1) In the first case, it is assumed that the maximum sampling frequency is of the order of the maximum switching frequency. Two approaches to the sampling are then possible. a) The control system operates with a variable sampling interval synchronized with the switching interval of the RPWM. Consequently, gains of the current controller must be modified at the beginning of every switching interval. b) The control system operates with a fixed sampling interval, equal to the maximum switching interval, A fixed-gain controller can be i.e., used, at the expense of a reduced flow rate of the feedback information. 2) In the second case, it is assumed that the maximum sampling frequency is much lower than the maximum switching frequency, that is, the converter switches faster than the microcomputer computes. The control system can operate with a fixed sampling interval, similarly to option b), above. In all cases, the equivalent DPWM strategy would be characterized by a constant switching frequency given by (1). The sampling frequency would also be constant, equaling the switching frequency in option 1), and that selected for RPWM strategy in option 2). Operation of the control system with constant and variable sampling interval is subsequently discussed in detail. A. Constant Sampling Interval Choice of a constant sampling interval allows avoiding gain adjustment in the current controller when the switching and using the frequency changes. Assuming that stator reference frame, the controller can be implemented in the

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Fig. 1.

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 45, NO. 5, OCTOBER 1998

Illustration of the processing time compensation.

following way. Pulsewidths – are computed within each one can compensate for the sampling interval. Since processing time of the microcomputer (to acquire new data, to execute the control algorithm, and to output new actuator values). If a sampling occurs at then the output reference is available only at Compensation of voltage the processing time is realized by calculating the reference voltage vector as for for

Fig. 2. Timing diagram of the RPWM strategy with a constant sampling frequency.

(5)

is the reference voltage vector to be synthesized by where the PWM inverter and (6) As a result, the local-averaged reference voltage determined by the control algorithm for a given sampling interval is effectively realized by the inverter. The corresponding timing diagram is given in Fig. 1. The timing diagram of the control system is shown in Fig. 2. It should be analyzed together with the flow graph of Fig. 3, which indicates the tasks to be executed every time an interruption request is generated by the TPMI. In Fig. 2, which must be less than represents the elapsed time between the TPMF and TPMI programming and pulsewidth determination, that is, the time necessary to execute the tasks related to blocks 3–7 in Fig. 3. The time interval required to execute the tasks related to blocks 1 and 2 is small enough to be neglected. The tasks 1–7 are executed in every sampling interval. B. Variable Sampling Period In this case, consecutive sampling intervals equal the randomized switching intervals, which simplifies the control algorithm. The timing diagram and task flow chart are shown in Figs. 4 and 5, respectively. For robustness of the current control, the gains of the controller should be adjusted in every sampling interval.

Fig. 3. Flow chart of the RPWM strategy with a constant sampling frequency.

JACOBINA et al.: CURRENT CONTROL FOR INDUCTION MOTOR DRIVES USING RPWM

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reference frame [16] is used: (7) (8) (9) (10) (11) Fig. 4. Timing diagram of the RPWM strategy with a variable sampling frequency.

, and represent the stator voltage, current, and flux, respectively. Similarly, by replacing subscript with respective vectors of the rotor variables, referred to the stator windings, are expressed. The other symbols in (7)–(11) denote the following: angular velocity of arbitrary the reference frame with respect to the stator; angular velocity of the rotor; stator resistance; rotor resistance referred to the stator; stator inductance ; rotor inductance referred to the stator ; stator leakage inductance; rotor leakage inductance referred to the stator; magnetizing inductance; electromagnetic torque developed in the motor; number of pole of the motor.

where vectors

V. DIGITAL CONTROLLER

Fig. 5. Flow chart of the RPWM strategy with a variable sampling frequency.

IV. MOTOR MODEL In the subsequent considerations, the continuous-time dynamic model of a three-phase induction motor in an arbitrary

In general, the control system of an adjustable-speed drive consists of at least three cascaded control loops: the current loop, the flux loop, and the speed loop (the fourth loop, for position control, may be employed in positioning systems). Many schemes have been proposed for current control, e.g., in [2], [3], and [17]–[19]. The current control loop is the innermost loop in high-performance ac drives. The first step in designing a digital current controller involves a dynamic model of the induction machine, required for expression of the current–voltage relationship. Equations (7)–(10) can be used for that purpose. The two following classes of current control systems can be distinguished. Class 1: The design of the control loop is based on a firstorder single-input/single-output (SISO) model of the motor. The cross-coupling impacts of flux and current components are treated as disturbances to be compensated at the output of the controller. Class 2: The current control loop is designed using the second-order multiple-input/multiple-output (MIMO) model [20]. The continuous-time current–voltage relationship (Class 1) in the stator reference frame, indicated by the superscript

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 45, NO. 5, OCTOBER 1998

is obtained from (7)–(10) as (12)

where

and differentiation operator ; total leakage factor of the motor ; stator time constant ; rotor time constant . This MIMO representation of the current–voltage dynamics was derived in [20] and used with good results in [21] for parameter estimation of the induction motor. In the described system, conversion from continuous time to discrete time is performed using the forward shift operator operator) [22]. It is assumed that the stator voltage and the rotor speed do not change during the sampling interval. It means that the source of the stator voltage can be modeled as a zero-order hold and that the mechanical modes are slower than the electrical ones. Equation (12) can be rewritten, in general terms, as (13) is the output vector, which in this case represents where is the input vector, representing the the stator currents, and are matrices of polynomials stator voltages, and operator. The input vector is synthesized by the of the inverter. As already mentioned, an RPWM inverter makes the stator voltage be mixed signals composed of deterministic fundamentals and a noise component due to randomness of the modulation technique. This consideration allows us to rewrite (13) as (14) is the random term due to [see (4)]. where Equation (14) indicates that, in the random-modulation case, the design of the current control loop constitutes a stochastic control problem, the solution of which requires the probaOnce this characterization bilistic characterization of is achieved, it is possible to employ stochastic control laws, like that of the minimum variance strategy [22]. However, disregarding the random nature of the RPWM would certainly offer a simpler solution. The question is whether such an approach would still guarantee an acceptable level of control quality. The answer to that question depends on the current algorithm employed. For simplicity, the SISO (Class 1) motor representation is considered. In this study, the synchronous reference frame, indicated by superscript was chosen. Then, where denotes the frequency of stator current. In this frame, the originally sinusoidal quantities are transformed into dc quantities, convenient for the control system design.

Fig. 6. Block diagram of the current control system.

In the synchronous reference frame, (7)–(10) can be rearranged to (15) (16) where (17) (18) denote components of the EMF vector of the motor. Assuming PI-type controllers in the and axes, the control law in the discrete-time domain can be reformulated as (19) where (20) and are the gains of the controller. Compensation and of the EMF terms is not considered here, although it could be included in the control algorithm at the expense of an increase in the processing time. The gains of the controller were determined using the optiand integral, mum damping criteria [23]. The proportional, gains of the continuous controller were computed first, and the discrete gains are obtained by the Tustin approximation [22] as (21) (22) The block diagram of the current control system is shown performs signal transformation from the in Fig. 6. Block synchronous frame to the stator frame described by (23) denotes the where, depending on the control strategy used, position of the rotor flux vector in the stator reference frame (vector-controlled motor) or just the time integral of the stator (scalar-controlled motor). The current vectors frequency

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Fig. 7. The experimental drive system. TABLE I INDUCTION MOTOR PARAMETERS AND CONTROLLER GAINS

in the synchronous reference frame, and are obtained from the vector current in the stator reference frame, and using the inverse of the matrix given in (23). Block allows the reference three-phase voltages to be obtained from the voltages. Block PWM VSI IM represents the pulsewidth are the PI-type current modulator, inverter, and motor, and controllers. VI. EXPERIMENTAL RESULTS The current control strategy presented in the preceding section was implemented in a small-scale laboratory ac drive system, as shown in Fig. 7. This drive system was used for experiments with both the DPWM and RPWM techniques for the inverter. Motor parameters and controller gains are listed in Table I. Several tests were conducted with the DPWM used as a benchmark. Clock frequencies of the TPMI and TPMF were

1.19318 and 10 MHz, respectively. The length of sampling intervals was restricted to the range of 120–600 s. These values limited the switching frequency employed in RPWM to the 1.667–8.333-kHz range. With the frequency resolution 9 b, up to 572 distinct frequency values could be generated. The resolution for the pulsewidth was 12 b, thus, over 4000 distinct values were feasible. The RPWM strategy was inor distinct values of the vestigated with switching frequency. Also, randomization of the switching was carried out in two ways: 1) the frequency was changed with every sampling interval and 2) the frequency was maintained constant within a one-sixth subcycle of the inverter output voltage. The following six experimental cases are described in this paper. kHz and 1) DPWM with s—This is a benchmark for cases 2) and 3). — 2) RPWM with The distinct values of the switching frequency were 1.667, 2.083, 2.778, 4.166, and 8.333 kHz. Note that equals in case the average switching frequency 1). 3) RPWM with —This case is similar to case 2), but with much as in case 2). higher , and the same kHz and s—This is a 4) DPWM with benchmark for cases 5) and 6). 5) RPWM with five distinct values, 1.667, 3.333, 5kHz, 6.667, and 8.333 kHz, of the switching frequency, and with the same fixed sampling interval of 600 s as in Case 4)—The switching frequency was randomly varied from one sampling interval to another. The average equals in Case 4). switching frequency

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 45, NO. 5, OCTOBER 1998

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Fig. 8. Oscillograms of the reference stator current, actual stator current, and current error. (a) Case 1), reference current. (b) Case 1), actual current. (c) Case 1), current error. (d) Case 2), reference current. (e) Case 2), actual current. (f) Case 2), current error. (g) Case 3), reference current. (h) Case 3), actual current. (i) Case 3), current error.

6) RPWM with and as in case 5)—However, the switching frequency randomly changed every one-sixth of the cycle. In all experiments, the same 10-Hz reference current waveform was employed, starting with a peak value of 1 A and reducing it to 0.5 A. The instant of the amplitude change was not fixed in the time because of the varying sampling intervals. Oscillograms of the reference stator current, actual stator current, and current error for cases 1), 2), and 3) are shown in Fig. 8. It can be seen that the current control is effective, and the average absolute current error is small in all cases. However, with random modulation, the current error tends to have distinctly higher variance, easily explained by the switching frequency dithering. The same variable for cases 4), 5), and 6) are illustrated in Fig. 9, with similar results, i.e., the current control effectiveness and the higher variance of the

current error when the DPWM is replaced with RPWM. The performance of the current loop in case 6) is higher than that in case 5) and comparable with that in the deterministic case 4).

VII. CONCLUSION A simple digital technique of RPWM in voltage-source inverters with linear current control has been described. Two general-purpose programmable interval timers that define the switching and sampling frequencies and pulsewidths of switching signals are employed. Randomization of the switching frequency and the voltage control algorithm are implemented in software. An effective method of compensation of voltage control errors due to the finite processing time has been proposed to be applicable when the switching frequency is greater than the sampling rate.

JACOBINA et al.: CURRENT CONTROL FOR INDUCTION MOTOR DRIVES USING RPWM

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Fig. 9. Oscillograms of the reference stator current, actual stator current, and current error. (a) Case 4), reference current. (b) Case 4), actual current. (c) Case 4), current error. (d) Case 5), reference current. (e) Case 5), actual current. (f) Case 5), current error. (g) Case 6), reference current. (h) Case 6), actual current. (i) Case 6), current error.

The implementation of current control in an RPWM inverter-fed ac drive system has been discussed. Several choices exist, depending on the relationship between the switching and sampling frequencies. A low-power drive system was experimentally investigated, in which a first-order current control law was employed using PI controllers in and axes. In spite of the utmost simplicity of this scheme, good practical results have been obtained. The experiments involved four variants of the RPWM strategy. Two DPWM strategies were used as benchmarks, by making the fixed switching frequency of the deterministic modulation equal to the average switching frequency of the random modulation. Comparative analysis of the results has shown effective current control in all the investigated cases, but a higher variance of control error when RPWM was employed. Random varying of the switching frequency from one subcycle of the inverter voltage to another has resulted in somewhat better performance of the

current control loop than that when the switching frequency was varied in consecutive sampling intervals. This pilot study has demonstrated the feasibility of RPWM in inverters with feedback current control. Further research is necessary to improve the quality of the control, investigating more advanced concepts, such as stochastic control laws. The topic of current control in RPWM inverters is quite important, as ac drive systems with such inverters could combine performance advantages of both the closed-loop current control and random modulation of the inverter voltage. REFERENCES [1] A. M. Trzynadlowski, “An overview of modern pwn techniques for three-phase, voltage-controlled, voltage-source inverters,” in Conf. Rec. ISIE’96, 1996, pp. 25–39. [2] D. M. Brod and D. W. Novotny, “Current control of vsi-pwm inverters,” IEEE Trans. Ind. Applicat., vol. 21, pp. 526–570, May/June 1985.

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[3] T. M. Rowan and R. J. Kerkman, “A new synchronous current regulator and an analysis of a current-regulated pwm inverter,” IEEE Trans. Ind. Applicat., vol. 22, pp. 678–690, July/Aug. 1986. [4] R. J. L. Bemans, L. D’Hondt, A. J. Vandenput, and W. Geysen, “Analysis of the audible noise of three-phase squirrel-cage induction motors supplied by inverters,” IEEE Trans. Ind. Applicat., vol. 23, pp. 842–847, 1987. [5] W. L. Erdman, R. Hudson, J. Yang, and R. G. Hoft, “A 7.5-kw ultrasonic inverter motor drive employing mos-controlled thyristors,” IEEE Trans. Ind. Applicat., vol. 26, pp. 756–768, 1990. [6] A. M. Trzynadlowski, S. Legowski, and R. L. Kirlin, “Random pulse width modulation technique for voltage-controlled power inverters,” in Conf. Rec. IEEE-IAS Annu. Meeting, 1987, pp. 863–868. [7] A. M. Yrzynadlowski, F. Blaabjerg, J. K. Pedersen, R. L. Kirlin, and S. Legowski, “Random pulse width modulation techniques for converterfed drive systems—A review,” IEEE Trans. Ind. Applicat., vol. 30, pp. 1166–1176, Sept./Oct. 1994. [8] T. G. Habetler and D. M. Divan, “Acoustic noise reduction in sinusoidal pwm drives using a randomly modulated carrier,” IEEE Trans. Power Electron., vol. 6, pp. 356–363, May 1991. [9] H. Kragh, F. Blaabjerg, and J. K. Pedersen, “Reduction of the acoustic noise effects from pwm-vsi inverter controlled ac drives by music and random modulation,” in Conf. Rec. PCC, 1993, pp. 85–92. [10] S. Bolognani, R. Conton, and M. Zigliotto, “Experimental analysis of the EMI reduction in PWM inverters using random space vector modulation,” in Conf. Rec. ISIE’96, 1996, pp. 482–487. [11] J. T. Boys and P. G. Handley, “Spread spectrum switching: Low noise modulcation technique for pwm inverter drives,” Proc. Inst. Elect. Eng., vol. 139, pt. B, pp. 252–260, Mar. 1992. [12] A. M. Stankovic, G. C. Verghese, and D. J. Perrault, “Analysis and synthesis of randomized modulation schemes for power converters,” IEEE Trans. Power Electron., vol. 10, pp. 680–693, Nov. 1995. [13] J. K. Pedersen, P. Blaabjerg, and P. S. Fredrikson, “Reduction of acoustical noise emission in ac machines by intelligent distributed random modulation,” in Conf. Rec. EPE, 1993, pp. 369–375. [14] D. C. White and H. H. Woodson, Electromechanical Energy Conversion. New York: Wiley, 1959. [15] H. W. Van der Broeck, H. C. Skudelny, and G. V. Stanke, “Analysis and realization of a pulsewidth modulator based on voltage space vectors,” IEEE Trans. Ind. Applicat., vol. 24, pp. 142–150, Jan./Feb. 1988. [16] P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electric Machinery. Piscataway, NJ: IEEE Press, 1995. [17] B. K. Bose, “An adaptive hysterese-band current control technique of a voltage-fed pwm inverter for machine drive system,” IEEE Trans. Ind. Electron., vol. 37, pp. 402–408, Oct. 1980. [18] M. P. Kazmierkowski, M. A. Dzieniakowski, and W. Sulkowski, “Novel space vector based current controllers for pwm-inverters,” IEEE Trans. Power Electron., vol. 6, pp. 158–166, Jan. 1994. [19] L. Malesani, P. Mattavelli, and P. Tomasin, “Improved contantfrequency hysteresis current control of vsi inverters with simple feed-forward bandwidth prediction,” in Conf. Rec. IEEE-IAS Annu. Meeting, 1995, pp. 2633–2639. [20] C. B. Jacobina, E. B. Souza Fl., A. M. N. Lima, and J. D. P. Rolim, “Current control for induction motor drives based on input-output dynamic discrete-model,” in Conf. Rec. IECON’92, Nov. 1992, pp. 133–137. [21] L. A. de S. Ribeiro, C. B. Jacobina, and A. M. N. Lima, “Dynamic estimation of the induction machine parameters and speed,” in Conf. Rec. PESC’95, June 1995, pp. 1281–1287. [22] K. J. Astrom and B. Wittenmark, Computer-Controlled Systems: Theory and Design. Englewood Cliffs, NJ: Prentice Hall, 1990. [23] H. Buhler, Reglages echantillonnes, vol. 1, 1st ed. Lausanne, Switzerland: Presses Polytechnique Romandes, 1983.

Cursino Brand˜ao Jacobina (S’78–M’78) was born in Correntes, Pernambuco, Brazil, in 1955. He received the Bachelor’s degree in electrical engineering from Federal University of Para´ıba, Campina Grande, Para´ıba, Brazil, in 1978 and the Diplˆome d’Etudes Approfondies (DEA) and the Doctoral degree from Institut National Polytechnique de Toulouse, Toulouse, France, in 1980 and 1983, respectively. Since 1978, he has been with the Electrical Engineering Department, Federal University of Para´ıba, where he is currently a Professor. His research interests include electrical drives, power electronics, control systems, and system identification.

Antonio Marcus Nogueira Lima (S’77–M’89) was born in Recife, Pernambuco, Brazil, in 1958. He received the Bachelor’s and Master’s degrees in electrical engineering from Federal University of Para´ıba, Campina Grande, Para´ıba, Brazil, in 1982 and 1985, respectively, and the Doctoral degree from Institut National Polytechnique de Toulouse, Toulouse, France, in 1989. He was with the Escola T´ecnica Redentorista, Campina Grande, Para´ıba, Brazil, from 1977 to 1982 and was a Project Engineer with Sul-Am´erica Philips, Recife, Pernambuco, Brazil, from 1982 to 1983. Since September 1983, he has been with the Electrical Engineering Department, Federal University of Para´ıba, where he is currently a Professor. His research interests are in the fields of electrical machines and drives, electronic instrumentation, control systems, and system identification.

Edison Roberto Cabral da Silva (M’92–SM’95) was born in Pelotas, Brazil, in 1942. He received the B.C.E.E. degree from the Polytechnic School of Pernambuco, Recife, Pernambuco, Brazil, the M.S.E.E. degree from the University of Rio de Janeiro, Rio de Janeiro, Brazil, and the D.Eng. degree from the University Paul Sabatier, Toulouse, France, in 1965, 1968, and 1972, respectively. In 1967, he joined the Electrical Engineering Department, Federal University of Paraiba, Campina Grande, Para´ıba, Brazil, where he is currently a Professor and Director of the Research Laboratory on Industrial Electronics and Machine Drives. In 1990, he was with COPPE, Federal University of Rio de Janeiro, and from 1990 to 1991, he was with WEMPEC, University of Wisconsin, Madison, as a Visiting Professor. He is the author or coauthor of more than 60 publications in the general area of solid-state power conversion and its applications. His current research work is in the areas of power electronics and motor drives. He was the General Chairman of the 1984 Joint Brazilian and Latin-American Conference on Automatic Control, sponsored by the Brazilian Automatic Control Society. Dr. Da Silva is currently a Member-at-Large of the Executive Board of the IEEE Industriy Applications Society.

Andrzej M. Trzynadlowski (M’83–SM’86) received the M.S. degree in electrical engineering, the M.S. degree in electronics, and the Ph.D. degree in electrical engineering from the Technical University of Wroclaw, Wroclaw, Poland, in 1964, 1969, and 1974, respectively. From 1966 to 1979, he was a Faculty Member at the Technical University of Wroclaw. In the following years, he worked at the University of Salahuddin in Iraq, University of Texas, Arlington, and University of Wyoming, Laramie. Since 1987, he has been with the University of Nevada, Reno, where he is currently a Professor of Electrical Engineering. During the second half of 1997, he was at Aalborg University, Aalborg, Denmark, as the Danfoss Visiting Professor. He has authored or coauthored over 90 publications in the areas of power electronics and electric drive systems and has been granted 11 patents. He is the author of The Field Orientation Principle in Control of Induction Motors (Amsterdam, The Netherlands: Kluwer, 1994). Dr. Trzynadlowski is an Associate Editor of the IEEE TRANSACTIONS ON POWER ELECTRONICS and IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS and a member of the Industrial Drives and Industrial Power Converter Committees of the IEEE Industry Applications Society (IAS). He was the recipient of the 1992 IEEE-IAS Myron Zucker Student–Faculty Grant.