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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 1, JANUARY/FEBRUARY 2001 51

Application of a Sliding-Mode Observer forPosition and Speed Estimation in Switched

Reluctance Motor DrivesRoy A. McCann, Mohammad S. Islam, Student Member, IEEE, and Iqbal Husain, Senior Member, IEEE

AbstractA sliding-mode observer is applied toward theoperation of a switched reluctance motor (SRM) drive. Thesliding-mode observer estimates rotor position and velocity tocontrol the conduction angles of the machine. Conventional onoffcontrol with hysteresis current control is included with thepositionestimation scheme. The particular case of an automotive brakesystem motor is considered in detail where the conduction anglesare modified with velocity feedback to provide optimum time re-sponse to brake system commands. Nonlinear modeling of a SRMis described and a computer simulation is developed based on datafrom an experimental SRM system. The sliding-mode observer

is implemented with fixed-point and floating-point digital signalprocessors (DSPs) and the discrete-time implementation is firstexamined under locked-rotor conditions. A comparison is alsomade between the implementation in two different types of DSPs.After confirming the accuracy of the computer simulation withexperimental data, the design considerations in selecting observercoefficients with regard to sampling time, convergence rate, andtransient stability are discussed. In conclusion, the effects of fluxestimation errors on the system time response during a startuptransient are examined.

Index TermsPosition and speed estimations, sliding-mode ob-server, switched reluctance motors.

I. INTRODUCTION

SWITCHED reluctance motors have received consider-able attention as an alternative to permanent-magnet dcmotors. 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 neodymiumironboron 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 00049, presented at the 1997 Industry Applications SocietyAnnual Meeting, New Orleans, LA, October 59, and approved for publica-tion in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the IndustrialDrives Committee of the IEEE Industry Applications Society. Manuscript sub-mitted for review November 1, 1999 and released for publication October 15,2000.This workwas supported inpart bythe National Science Foundation underGrant 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 Engi-neering, 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 estima-

tion 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 mod-

eled 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 observersusing 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 hy-

draulic 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

00939994/01$10.00 2001 IEEE
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52 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 1, JANUARY/FEBRUARY 2001

Fig. 1. Startup velocitytransient ofprototype systemwith andwithoutvelocityfeedback.

Fig. 2. Measured and simulated motor phase current during startup transient.

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 speedand 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.

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-cur-

rent-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)

Fig. 3. (a) Block diagram of model-based sensorless controller. (b)Sliding-mode-theory-based state observer.

Fig. 4. Experimental position estimates for locked-rotor condition.

For computer simulation purposes, the and char-

acteristics are stored in tabular form using experimentally mea-

sured data from the prototype motor. The state-space differential

equations of the SRM to be solved are

(3)

(4)

(5)


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MCCANN et al.: SLIDING-MODE OBSERVER FOR POSITION AND SPEED ESTIMATION IN SRMs 53

where , , and are the resistance, current, and voltage of

the th phase, respectively. It is assumed that each phase is mag-

netically decoupled from every other phase. The rotor angular

position and velocity are denoted by and , respectively. is

the hydraulic system load torque. A detailed computer simula-

tion was developed in MATLAB/SIMULINK using tabulated em-

pirical data with linear interpolation. The simulation also ac-

counts for discretization and processing delays due to the micro-processor 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%. Con-

sequently, the design and evaluation of the sliding-mode ob-

server under a variety of conditions may be carried out with

confidence based on simulation results.

IV. SLIDING-MODE OBSERVER

A. Theoretical Development

The sliding-mode observer incorporates a state-space modelof the SRM to estimate rotor position and velocity. The reader

is referred to [2][5] for the development of sliding-mode-ob-

server 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 onthe difference of the motor flux computed from the mathemat-

ical 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

using phase current and phase voltage measurements:

(6)

This method of estimating the phase flux will accumulate inte-

gration errors if the motor is operating at very low or zero speed.

It is, therefore, important to demonstrate the sensorless opera-

tion 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

(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)

Fig. 5. Experimental velocity estimates for locked rotor.

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

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 dy-

namics become

(9b)
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Fig. 7. Position estimate error during motor startup.

Thus, (9a) and (9b) describe the convergence properties of the

observer. Once the sliding surface is reached, the error

dynamics become

(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 mea-

surable machine variable. Assuming increases for movement

from the unaligned toward the aligned position, the error func-

tion is chosen as

with (11)

The error term includes a flux estimate based on the estimated

rotor position

(12)

The function is selected to ensure the error function (11)

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-timeformulation of the observer follows from (6), (7), and (10). The

flux observer is implemented as

(14)

where is the sampling time interval. The voltage is de-

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. 8. Velocity estimate error during motor start-up k = 512 and 2048.

TABLE IEFFECT OF OBSERVER GAINS ON MEAN-SQUARE ERROR

Fig. 9. Velocity response with initial position errors.

Fig. 10. Experimental position estimation at a speed of 32 rad/s.

An error term is calculated from (13)(15)

(16a)


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(16b)

(17)

where the estimated rotor position is expressed in electrical ra-

dians. 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 per-

forms a divide by two with each right shift operation. With eachshift operation the least significant bit is lost. Floating-point pro-

cessors 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 fil-

ters, 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 implementa-

tion lends itself to fixed-point arithmetic in the computation of

(18) where the sample interval is multiplied by an integer con-

stant. Thus, the sign of the error function is first computed, andthe integers and are either added or subtracted in com-

puting the position and velocity estimates in (18). The effects

of selecting , , and will be considered in the following

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 out-

puts. With a floating-point implementation, this error can be re-

duced significantly. Implementation wise, floating-point calcu-

lation 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.

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 imple-

mentation. The observer parameters in (14)(18) were assigned

nominal values of , , and . The

rotor was locked at 42 with an initial position estimate of 0

and an initial velocity estimate of zero. Experimental results

are shown in Fig. 4 for values of , , and. The position estimate converges after 1.25 ms in

each case. However, produces the largest variance in

the position estimate.

The corresponding velocity estimates are shown in Fig. 5.

The influence of the sample interval on the velocity error dy-

namics (21b) is evident once the sliding surface is reached. In

general, the numerical errors of the observer computations may

prevent the velocityestimate fromconverging. When ,

the steady-state velocity error is reduced although the position

variance is increased. For the position variance is re-

duced, however, the velocity estimate does not converge after

the sliding surface is reached. Thus, the selection of , ,

and should account for the computational limitations offixed-point arithmetic. It should be noted that, for floating-point

computations, the selection of , , and may be chosen

without these numerical constraints. It is also noteworthy that

the velocity error increases until the position estimate has con-

verged. 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 non-

linear magnetic characteristics and computational errors asso-

ciated with the real-time implementation. Use of the simula-tion 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 tran-

sient 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 ob-

server 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 measure-

ment. The corresponding observer position and velocity errors


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

Fig. 12. Experimental results at a speed of 56 rad/s. Actual and estimatedpositions (top trace). Actual and estimated speeds (bottom trace).

TABLE IIEFFECT OF INITIAL POSITION ERROR ON MEAN-SQUARE ERROR

TABLE IIIEFFECT OF FLUX OBSERVER ERROR ON MEAN-SQUARE ERROR

are shown in Figs. 7 and 8. The effect of varying the observer

parameters on motor performance is examined in the following

sections.

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 (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 re-

spect to time for the nominal values of and

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 vari-

ance for feedback control. Fig. 8 clearly shows the influence of

the slower velocity convergence rate with as ex-

pected from (21) once the position estimate reaches the slidingsurface. Optimum results are obtained when is first set no

greater than necessary to ensure rapid position convergence, and

then selecting to the maximum allowed by the limitations of

the sample interval as discussed in Section V.

B. Variation of Initial Position Error

The effect on motor performance of varying the initial posi-

tion error while the other observer values are held constant was

considered. Fig. 9 shows the velocity response with initial po-

sition 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 esti-

mation and its error are shown in Fig. 11 where (21b) is alsoverified as the error in position goes to zero in a finite time,

while the error in speed goes to zero only exponentially. De-

pending on the initial position error, the sliding surface could

be different and, consequently, be reflected in the position esti-

mation 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 mul-

tiple of and speed estimation converges exponentially

as before. This demonstrates the claim of the existence of mul-

tiple sliding surfaces. The position estimate converges after ap-

proximately 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

The effect on motor performance of introducing an error in

flux observer (12) while the other observer values are held con-

stant was considered. The error was described by a constant


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Fig. 13. Experimental results at a speed of 100 rad/s. Actual and estimatedpositions (top trace). Measured and estimated fluxes (bottom trace).

Fig. 14. Experimental results at a speed of 58 rads/s. Phase current, scale: 5A/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 estimatesare listed in Table III. An observer flux value of 0.90 indicates

that the observer estimate is 90% of the true motor flux. The ob-

server 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 demon-

strates its robustness toward parameter variations and nonlin-

earity 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

Fig. 15. Experimental results at a speed of 58 rads/s. Total torque (top), phasetorque (middle) scale: 0.1 N 1 m/div, and phase flux (bottom) scale 0.015 Wb/div.



by the hysteresis control method, which enforces the desired

torque level at low operating speeds. Motor net shaft torqueand 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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