Application of a sliding-mode observer for position and speed ...

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 1, JANUARY/FEBRUARY 2001

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Application of a Sliding-Mode Observer for Position and Speed Estimation in Switched Reluctance Motor Drives Roy A. McCann, Mohammad S. Islam, Student Member, IEEE, and Iqbal Husain, Senior Member, IEEE

Abstract—A sliding-mode observer is applied toward the operation of a switched reluctance motor (SRM) drive. The sliding-mode observer estimates rotor position and velocity to control the conduction angles of the machine. Conventional on–off control with hysteresis current control is included with the position estimation scheme. The particular case of an automotive brake system motor is considered in detail where the conduction angles are modified with velocity feedback to provide optimum time response to brake system commands. Nonlinear modeling of a SRM is described and a computer simulation is developed based on data from an experimental SRM system. The sliding-mode observer is implemented with fixed-point and floating-point digital signal processors (DSPs) and the discrete-time implementation is first examined under locked-rotor conditions. A comparison is also made between the implementation in two different types of DSPs. After confirming the accuracy of the computer simulation with experimental data, the design considerations in selecting observer coefficients with regard to sampling time, convergence rate, and transient stability are discussed. In conclusion, the effects of flux estimation errors on the system time response during a startup transient are examined. Index Terms—Position and speed estimations, sliding-mode observer, switched reluctance motors.

I. INTRODUCTION

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WITCHED reluctance motors have received considerable attention as an alternative to permanent-magnet dc motors. For automotive applications, the switched reluctance motor (SRM) avoids the problems associated with magnet bonding, corrosion, and demagnetization. In addition, for systems requiring fast dynamic response it is often found that the torque to inertia ratio for the SRM is higher than permanent-magnet dc motors using ceramic ferrite or injection molded neodymium–iron–boron magnets. The SRM also holds promise for sensorless operation including position estimation at zero speed. Sensorless operation is important for automotive applications due to the need for minimum package size, high Paper IPCSD 00–049, presented at the 1997 Industry Applications Society Annual Meeting, New Orleans, LA, October 5–9, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Industrial Drives Committee of the IEEE Industry Applications Society. Manuscript submitted for review November 1, 1999 and released for publication October 15, 2000. This work was supported in part by the National Science Foundation under Grant ECS 9702370. R. A. McCann is with Delphi Automotive Systems, Saginaw, MI 48601 USA (e-mail: [email protected]). M. S. Islam and I. Husain are with the Department of Electrical Engineering, The University of Akron, Akron, OH 44325-3904 USA (e-mail: [email protected]). Publisher Item Identifier S 0093-9994(01)00903-3.

reliability, and low cost for electric motor driven actuators. In this paper, the sensorless operation of a SRM for an advanced brake system is considered. In this configuration, an electric motor drives a hydraulic actuator that controls the vehicle brake pressure. During antilock braking conditions, the electric motor modulates the brake system pressure. This application is considered because it combines the need for high reliability, small size, and fast dynamic response from the motor drive. Various methods of indirect (or sensorless) position estimation have been investigated for switched reluctance machines. Lumsdaine and Lang first proposed a model-based estimator [1] from which very good results were obtained. In observer-based state estimation schemes, the dynamics of the motor are modeled in state space while a mathematical model runs in parallel with the physical machine. The model has the same inputs as the physical machine and the difference between its outputs and the measured outputs of the real machine are used to force the estimated variables to converge to the actual values. In the case of the SRM, terminal measurements of the phase currents and voltages are sufficient to develop the observer. The computational simplicity and robust stability properties of sliding-mode controllers prompted the study of sliding-mode observers. Misawa et al. [2] first studied the design of observers using sliding-mode theory for nonlinear systems. Further study on sliding-mode observers was presented by Slotine et al. [3] and Pradeep et al. [4]. Husain et al. [5] first presented a sliding-mode-observer-based rotor position estimation scheme for SRMs. The results presented in that work were based on computer simulations of a linear magnetic model for the SRM. Blaabjerg et al. [6] and Y. J. Zhan et al. [7] demonstrated the operation of a sliding-mode observer with a floating-point digital signal processor (DSP). This paper extends the previous results by considering the discrete-time formulation of the observer, the advantages and limitations of fixed-point and floating-point computations, the effects of flux and current estimation errors, and the use of velocity feedback to modify the conduction angles of the motor during transient conditions. II. DRIVE SYSTEM OPERATION The experimental SRM drive system is connected to a hydraulic actuator that controls the brake pressure for a passenger vehicle. This application is examined because fast dynamic response is important as well as the need for small package size and high reliability. The system uses velocity feedback to modify the conduction angles of the motor in order to provide

0093–9994/01$10.00 © 2001 IEEE

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Fig. 1. Startup velocity transient of prototype system with and without velocity feedback.

Fig. 2. Measured and simulated motor phase current during startup transient. Fig. 3. (a) Block diagram of model-based sensorless controller. (b) Sliding-mode-theory-based state observer.

maximum torque to the shaft over a wide speed range. The prototype motor is a four-phase machine with eight stator poles and six rotor poles. Throughout this paper, the motor is considered energized from initial rest conditions (zero speed and current) with a torque command indicative of an emergency condition requiring maximum brake pressure. The response time is the time required for the motor to reach rated speed and torque. Controlling the conduction angles with respect to velocity has a significant impact on the response time. The per-unit motor time response of the prototype system is shown in Fig. 1 for the case where the motor is operated with and without velocity feedback. Without velocity feedback, the SRM only achieves 64% of rated speed. Thus, it is important that an observer-based sensorless control scheme provide not only position, but also velocity estimates in order to optimize the switched reluctance drive for fast dynamic response time. Fig. 4.

III. NONLINEAR MODELING OF THE SWITCHED RELUCTANCE MACHINE The magnetic nonlinearities of an SRM can be taken into account by appropriate modeling of the nonlinear flux-current-angle ( – – ) characteristics of the machine. The output electromagnetic torque of the machine can also be described by nonlinear torque-current-angle ( – – ) data. The machine model may be described by (1) (2)

Experimental position estimates for locked-rotor condition.

For computer simulation purposes, the and characteristics are stored in tabular form using experimentally measured data from the prototype motor. The state-space differential equations of the SRM to be solved are (3) (4) (5)

MCCANN et al.: SLIDING-MODE OBSERVER FOR POSITION AND SPEED ESTIMATION IN SRMs

where , , and are the resistance, current, and voltage of the th phase, respectively. It is assumed that each phase is magnetically decoupled from every other phase. The rotor angular is position and velocity are denoted by and , respectively. the hydraulic system load torque. A detailed computer simulation was developed in MATLAB/SIMULINK using tabulated empirical data with linear interpolation. The simulation also accounts for discretization and processing delays due to the microprocessor based motor controller. Simulated and experimental phase current measurements of the SRM drive are shown in Fig. 2. Because the simulation is calibrated with empirical data, the predicted phase currents, torque and speed response deviate from the experimental system with errors of less than 3%. Consequently, the design and evaluation of the sliding-mode observer under a variety of conditions may be carried out with confidence based on simulation results.

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Fig. 5. Experimental velocity estimates for locked rotor.

IV. SLIDING-MODE OBSERVER A. Theoretical Development The sliding-mode observer incorporates a state-space model of the SRM to estimate rotor position and velocity. The reader is referred to [2]–[5] for the development of sliding-mode-observer theory. Briefly, the sliding-mode observer estimates rotor position and velocity from phase current and terminal voltage measurements. An error correction term is computed based on the difference of the motor flux computed from the mathematical model and that derived from motor measurements. A block diagram of the observer-based motor drive is shown in Fig. 3(a). A sliding-mode-theory-based state observer implementation is shown in Fig. 3(b). The phase flux may be obtained from the following equation measurements: using phase current and phase voltage (6) This method of estimating the phase flux will accumulate integration errors if the motor is operating at very low or zero speed. It is, therefore, important to demonstrate the sensorless operation at very low speeds. However, operating at nonzero speed will limit accumulated errors because the current and, hence, the flux of each phase will periodically go to zero. An observer may be constructed to estimate the unknowns and in (4) and (5). Consider a second-order sliding-mode observer for the SRM of the form

Fig. 6.

Motor startup with sliding-mode observer.

Differentiating both sides of (8a) yields

Substituting (4) and (7a) produces

and using the velocity error definition (8b) yields the position error dynamics (9a) To derive the velocity error dynamics, begin with

(7a) (7b) where and are the estimated rotor position and velocity, respectively, and is an error function based on measured and estimated variables. To describe the observer error dynamics, we define the errors in the SRM observer as (8a) (8b)

Substituting (5) and (7b) yields

If it is assumed that may be selected to be large enough such that the first two terms may be neglected, the velocity error dynamics become (9b)

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Fig. 7.


Position estimate error during motor startup.

Fig. 8. Velocity estimate error during motor start-up k

Thus, (9a) and (9b) describe the convergence properties of the is reached, the error observer. Once the sliding surface dynamics become

=

512 and 2048.

TABLE I EFFECT OF OBSERVER GAINS ON MEAN-SQUARE ERROR

(10a) (10b) There are many possible error functions that will stabilize the error dynamics. In general, the error function must compare a variable dependent upon the estimated position and a measurable machine variable. Assuming increases for movement from the unaligned toward the aligned position, the error function is chosen as with

(11)

The error term includes a flux estimate based on the estimated rotor position (12) Fig. 9. Velocity response with initial position errors.

is selected to ensure the error function (11) The function forces the position estimate to converge to a motoring condition. This may be accomplished by defining the following: (13) where

is the number of rotor poles.

B. Digital Implementation The sliding-mode observer was implemented with fixed-point and floating-point DSPs. The discrete-time formulation of the observer follows from (6), (7), and (10). The flux observer is implemented as (14) is dewhere is the sampling time interval. The voltage termined from a measurement of the supply voltage and the switching state of the inverter power devices (MOSFETs). A flux or current estimate is computed from tabulated data using the estimated rotor position and the measured current (15)

Fig. 10.

Experimental position estimation at a speed of 32 rad/s.

An error term is calculated from (13)–(15) (16a)


(16b) (17) where the estimated rotor position is expressed in electrical radians. The sliding-mode observer is now written as (18a) (18b) The discrete-time error is defined in a manner similar to (8) (19a) (19b) The error dynamics become (20a) (20b) Once the sliding surface is reached, the error dynamics become (21a) (21b) Thus, the discrete-time velocity error decays exponentially on the sliding surface as in (10b). C. Fixed- and Floating-Point Processors For floating-point and fixed-point processors, the sampling time interval may be interpreted as a shift operator that performs a divide by two with each right shift operation. With each shift operation the least significant bit is lost. Floating-point processors have a higher cost level associated with larger silicon size when compared to fixed-point processors. The advantages of floating-point processors over fixed-point processors include accuracy and ease of implementation. A prototype system was implemented with both a fixed-point and a floating-point DSP. Luenberger observers and Kalman filters, in general, will suffer with fixed-point arithmetic due to limitations on sample rates and arithmetic precision. One of the advantages of the sliding-mode observer is that its implementation lends itself to fixed-point arithmetic in the computation of (18) where the sample interval is multiplied by an integer constant. Thus, the sign of the error function is first computed, and and are either added or subtracted in comthe integers puting the position and velocity estimates in (18). The effects will be considered in the following of selecting , , and sections. One of the concerns of fixed-point algorithms is poor numerical accuracy in computing the motor flux and current and, consequently, will tend to create errors in the observer outputs. With a floating-point implementation, this error can be reduced significantly. Implementation wise, floating-point calculation provides easier implementation with more flexibility and less error. However, floating-point calculations have a longer computation time compared to fixed-point computations given the same clock speed.

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V. LOCKED-ROTOR EXPERIMENTAL RESULTS The sliding-mode observer constants and and sampling time interval were initially investigated using the special case of a locked rotor with fixed-point implementation. This allows the flux observer (14) to be evaluated against a direct measurement of the flux or compared to results from a finite-element analysis. It also allows the observer constants to be designed for a desired convergence rate without the complication of rotor acceleration. In the sequel, the motor performance during a startup transient will be examined. The prototype system was developed with a four-phase SRM constructed with eight stator poles and six rotor poles. The rotor position is referenced to 0 at the aligned position of phase A. The electrical period is 60 mechanical degrees and is subdivided into 128 possible discrete rotor positions for the digital implementation. The observer parameters in (14)–(18) were assigned , , and . The nominal values of rotor was locked at 42 with an initial position estimate of 0 and an initial velocity estimate of zero. Experimental results , , and are shown in Fig. 4 for values of . The position estimate converges after 1.25 ms in produces the largest variance in each case. However, the position estimate. The corresponding velocity estimates are shown in Fig. 5. The influence of the sample interval on the velocity error dynamics (21b) is evident once the sliding surface is reached. In general, the numerical errors of the observer computations may , prevent the velocity estimate from converging. When the steady-state velocity error is reduced although the position the position variance is revariance is increased. For duced, however, the velocity estimate does not converge after , the sliding surface is reached. Thus, the selection of , should account for the computational limitations of and fixed-point arithmetic. It should be noted that, for floating-point , and may be chosen computations, the selection of , without these numerical constraints. It is also noteworthy that the velocity error increases until the position estimate has converged. This will be significant when velocity information is used to control the motor during a startup transient. VI. TRANSIENT PERFORMANCE The sliding-mode observer was further evaluated by using the knowledge from the simulation program which includes nonlinear magnetic characteristics and computational errors associated with the real-time implementation. Use of the simulation program has the benefit of considering a number of design variables including the effects of flux estimation error in (12) on the convergence properties of the observer. The startup transient considered in Section I is examined with the sliding-mode observer used to operate the SRM with velocity feedback to modify the conduction angles. Simulated operation with the sliding-mode observer is compared to the prototype system that uses direct position measurement in Fig. 6. In this case, the observer variables are the same as that used in Section V. As can be seen from Fig. 6, the response of the motor with the observer is almost identical to operation with direct position measurement. The corresponding observer position and velocity errors

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A. Variation of

and

The effect of varying and on motor performance was considered. The mean-square error of the position and velocity estimates during the startup transient is listed in Table I. The mean-square error of the position estimate was computed as MSE

Fig. 11. Experimental results at a speed of 32 rad/s. Actual and estimated speeds (top trace). Error in speeds (bottom trace).

(22)

where is total number of time steps during the 125 ms startup period for the particular sampling rate selected. The velocity mean-square error using per-unit values with a 3000-r/min base is computed as in (22). Fig. 8 shows the velocity error with reand spect to time for the nominal values of and with . From Table I, the nominal observer values provide a reasonable tradeoff between minimizing the position error while providing a velocity estimate with moderate variance for feedback control. Fig. 8 clearly shows the influence of as exthe slower velocity convergence rate with pected from (21) once the position estimate reaches the sliding is first set no surface. Optimum results are obtained when greater than necessary to ensure rapid position convergence, and to the maximum allowed by the limitations of then selecting the sample interval as discussed in Section V. B. Variation of Initial Position Error

Fig. 12. Experimental results at a speed of 56 rad/s. Actual and estimated positions (top trace). Actual and estimated speeds (bottom trace). TABLE II EFFECT OF INITIAL POSITION ERROR ON MEAN-SQUARE ERROR

TABLE III EFFECT OF FLUX OBSERVER ERROR ON MEAN-SQUARE ERROR

The effect on motor performance of varying the initial position error while the other observer values are held constant was considered. Fig. 9 shows the velocity response with initial position errors. In each case, the initial estimated position is set to 0 . Fig. 10 shows the rotor position response during the startup transient with the initial estimated position 0 . The speed estimation and its error are shown in Fig. 11 where (21b) is also verified as the error in position goes to zero in a finite time, while the error in speed goes to zero only exponentially. Depending on the initial position error, the sliding surface could be different and, consequently, be reflected in the position estimation as a lag or lead between actual and estimated position. The worst case shift is 180 electrical degrees. Fig. 12 shows a result where 120 phase shift exists, which is an integer muland speed estimation converges exponentially tiple of as before. This demonstrates the claim of the existence of multiple sliding surfaces. The position estimate converges after approximately 35 ms for the worst case condition when the initial error is 180 . The longer time is due to the initial position error causing the motor to rotate opposite the desired direction. In addition, there is little load torque at low speeds to oppose the motor acceleration. Thus, the position estimate takes longer to converge compared to the locked-rotor condition in Section V. The mean-square error of the position and velocity estimates for various initial position errors during the startup transient are listed in Table II. C. Variation of Flux Observer Error

are shown in Figs. 7 and 8. The effect of varying the observer parameters on motor performance is examined in the following sections.

The effect on motor performance of introducing an error in flux observer (12) while the other observer values are held constant was considered. The error was described by a constant


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Fig. 15. Experimental results at a speed of 58 rads/s. Total torque (top), phase torque (middle) scale: 0.1 N1m/div, and phase flux (bottom) scale 0.015 Wb/div. Fig. 13. Experimental results at a speed of 100 rad/s. Actual and estimated positions (top trace). Measured and estimated fluxes (bottom trace).

Fig. 14. Experimental results at a speed of 58 rads/s. Phase current, scale: 5 A/div (top trace). Phase voltage, scale: 20 V/div (bottom trace).

factor that resulted in an observer flux that was either greater than or less than the true motor flux. The mean-position error and the mean-square error of the position and velocity estimates are listed in Table III. An observer flux value of 0.90 indicates that the observer estimate is 90% of the true motor flux. The observer demonstrates insensitivity to flux estimation errors since the mean position and velocity estimates are not significantly degraded. Insensitivity to flux observer errors is a particularly strong characteristic of the sliding-mode observer that demonstrates its robustness toward parameter variations and nonlinearity in machine modeling. Fig. 13 shows the measured flux and the estimated flux from the observer along with the actual and estimated positions. Although there is a large error between the two fluxes, there is not a significant impact on the position estimate. VII. STEADY-STATE EXPERIMENTAL RESULTS This section presents the experimental results of steady-state motor operation and control using the estimated variables over a wide speed range. In the experiments, the motor was operated with a conventional hysteresis current regulator using the estimated rotor position. The motor was loaded with 0.22 N m load torque. Fig. 14 shows the phase current and voltage waveform during the position sensorless operation at a speed of 58 rads/s. This demonstrates that position estimation using a sliding-mode observer can be used for torque and velocity control of the motor. It is observed that the current is regulated



by the hysteresis control method, which enforces the desired torque level at low operating speeds. Motor net shaft torque and the contribution from the individual phase torques along with the associated phase flux waveforms are shown in Fig. 15. Ripple in the net shaft torque is relatively large because no efforts were taken to minimize torque ripple in the prototype system. Figs. 16 and 17 show the phase current and voltage at speeds of 152 rad/s and 337 rad/s, respectively. During high-speed operation, the motor operates in single-pulse mode where positive voltage is applied during the conduction period and negative voltage is applied during the demagnetization period. These experimental results demonstrate the capacity for variable-speed operation using a sliding-mode observer for position sensorless control.

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VIII. SUMMARY AND CONCLUSIONS A sliding-mode observer that provides position and velocity estimates to control the conduction angles of a SRM has been presented. The discrete-time formulation of the observer is derived and is implemented with fixed-point and floating-point DSPs. The implications of selecting the sample rate and observer parameters were examined. The convergence of the observer is considered when there is an initial error in the position estimate and when there is an error in the flux measurement. It is found that the sliding-mode observer is well suited to fixed-point arithmetic and that it has robust convergence properties under the conditions that arise in practical applications.

ACKNOWLEDGMENT The authors would like to thank R. Wainwright for his help and advice with software development for fixed-point DSP systems.

REFERENCES [1] A. Lumsdaine and J. H. Lang, “State observers for variable reluctance motors,” IEEE Trans. Ind. Electron., vol. 37, pp. 133–142, Apr. 1990. [2] E. A. Misawa and J. K. Hedrik, “Nonlinear observers: A state-of-the-art survey,” ASME J. Dynam. Syst. Meas., vol. 111, no. 3, pp. 344–352, 1979. [3] J. J. Slotine, J. K. Hedrik, and E. A. Misawa, “On sliding observers for nonlinear systems,” ASME J. Dynam. Syst. Meas., vol. 109, pp. 245–252, Sept. 1987. [4] A. K. Pradeep, J. P. Lyons, and S. R. MacMinn, “Application of sliding mode observers for state estimation in rotating machines,” in Proc. VARSCON’91, June 1991, pp. 57–64. [5] I. Husain, S. Sodhi, and M. Ehsani, “A sliding mode observer based controller for switched reluctance motor drives,” in Conf. Rec. IEEE-IAS Annu. Meeting, Denver, CO, 1994, pp. 635–643. [6] F. Blaabjerg, L. Christensen, S. Hansen, J. P. Kristofferson, and P. O. Rasmussen, “Sensorless control of a switched reluctance motor with variable-structure observer,” Electromotion, vol. 3, no. 3, pp. 141–152, July 1996. [7] Y. J. Zhan, C. C. Chan, and K. T. Chau, “A novel sliding-mode observer for indirect position sensing of switched reluctance motor drives,” IEEE Trans. Ind. Electron., vol. 46, pp. 390–