Proceedings of the 2009 IEEEInternational Conference on Mechatronics and Automation

August 9 - 12, Changchun, China

A MRAS based Speed Identification Scheme for a PMSynchronous Motor Drive Using the Sliding Mode Technique

Weisheng Yan, Hai Lin, Hong Li, Huiping Li and Jian LuDepartment of Mechanical Engineering and Automatic Control,

College of Marine Engineering, Northwestern Polytechnical University,Xi'an 710072, China

Abstract- The paper focuses on a model reference adaptivesystem (MRAS) speed identification of permanent magnetsynchronous motor (PMSM) using the sliding mode approach.The new MRAS algorithm is based on the error between theoutputs of flux linkage-current model and flux linkage-voltagemodel. Then, the error is used to estimate rotor speed bya suitable adaptation mechanism. Moreover, the sliding modetechnique is used to MRAS based control of PMSM, which ismore robust to motor parameter uncertainly, model inaccuracyand external disturbance. Compared to the conventional MRASmethod, the new method has a much better dynamic, steadystate performance and robustness. Simulation results show thevalidity of the proposed method.

Index Terms-Permanent Magnet Synchronous Motor(PMSM), Model Reference Adaptive System, Sliding ModeControl, Speed Identification

I. INTRODUCTION

Permanent magnet synchronous motors (PMSM) has ap­plied to the many fields of numerical control machines,industrial robotics and flying vehicles etc. since its merits ofhigh power factor, high power density, high torque to currentratio, large power to weight ratio, and high efficiency etc..However, its drawbacks still exist in the practical application,for example, the current cannot be decoupled easily andthe control of the PMSM need the precise signal of therotor position and speed. Then, the research to solve theseproblems have attract more and more attentions. To decouplethe current of the motor, field oriented control (FOC [1]) wasintroduced to control method of the PMSM, which achievefast dynamic response and the flexibility normally obtainedin the control of separately excited de motors. In order toobtain precise signals of the rotor position and speed, manydetection devices were developed in recent years, such asHall effect sensor, photoelectric encoder and resolver etc.However, in some application field, such as pit gear, speedand position sensors cannot be used. In addition, speedand position sensors require the additional mounting spaceand increase the cost of a motor. Although signals of therotor position are easily obtained by the developed detection

*This work is supported National Natural Science Foundation of China(60875071) and program for New Century Excellent Talents in University(Ministry of Education of PRC[2005]290)

978-1-4244-2693-5/09/$25.00 ©2009 IEEE 3656

devices, they also lead to new problems, which effect thepractical application of the PMSM in some fields.

Consequently, for the purpose of obtaining signals of therotor position and speed effectively and accurately, severalsensorless control methods for PMSM have been developedand proposed in the last years. Extended Kalman filters (EKF)have found wide application in the estimation of rotor posi­tion and speed in PMSM drives [1]-[2]. The EKF approachdoes not require either the knowledge of the mechanicalparameters or the initial rotor position, and is robust toparameter variations of the motor. Although it has led toboth reduction of computational time and extreme reductionof sensors, limited to the current and de-link voltage trans­ducers, complex algorithm of EKF increase implementationdifficulty. More recently, the sliding mode technique forestimating or controlling nonlinear systems have been inves­tigated for a long time [3]-[5]. With the proposed a slidingmode observer, the back EMF components were estimated,which contain the information of rotor position and muchuseless disturbance. Then, the rotor position was extractedfrom the back EMF estimation via an extended Kalmanfilter. Nevertheless, this estimation method using the slidingmode with switched controls exists chattering phenomenaand torque perturbations. Specially, it has lower performanceat low speeds. In [6]-[8], estimation algorithms based onsaliency of the rotor structure have developed, which iseffective at zero and at low motor speed. To amplify the effectof saturation saliency of the stator core, high-frequency statorvoltage pulses were injected to the fundamental voltage.Then, the rotor position can be obtained by the amplitudedifference of three phase current responses.

Recently, Model Reference Adaptive System (MRAS)technique has been extensively applied to the control ofelectrical machines [9]-[12]. A MRAS using the state ob­server model with the current error feedback and the magnetflux model as two models was proposed to estimated theback-EMF [9]. According the errors between outputs oftwo models, the speed is generated by a speed adaptationmechanism. However, the precision of speed estimation isheavily depended on the exact parameter of the motor. Whenparameter uncertainties is taken into account, this methodmay not be applicable because it is based on the electrical

{~d: -R~d/L+ OJAq +Rl/J/L+Ud (5)

Aq - -RlqlL - roAd +uq

Then, the electromagnetic torque is

Where TL is the load torque, B is the friction coefficient ofthe motor, J is the moment of inertia of the motor.

For a uniform airgap surface-mounted PMSM motor, Ld ==Lq == L. According to (1) and (2), the state flux linkage ofa uniform airgap surface-mounted PMSM motor can also berewritten by

equations with the exact parameters of the motor. In thispaper, a new MRAS with flux-current model and flux-voltagemodel is investigated, which is robust to the parametervariation of the motor using the sliding mode technique. Thesimulation demonstrate that the proposed scheme has a goodspeed estimation performance and robustness compared withthe traditional MRAS.

II. MOTOR MODEL

The stator voltage equations of a PMSM can be describedin the synchronously rotating reference frame (d-q) by thefollowing equations [7]:

{Ud == Rid+pAd - ProAq

(1)uq == Ri..+pAq+ProAd

(7)

(9)

(11)

x == f(x) + g(x)u

x==Ax+Bu

_ [Ad] _ [Kl/J +Ud]x- A ,u- U 'q q

A == [- K ro] B == [1 0]-ro -K' 0 1 '

. ds dss== dXf(x)x+ dXg(x)u

Then, the equivalent control is derived by

where

When the system state trajectories are forced toward thesliding surface and stay on it, the dynamic performance of thesystem satisfy s(x) == 0 and s(x) == O. To describe the slidingsurface, the equivalent control approach is used to make thethe system state trajectories stay on the sliding surface, whichcan be obtained by s(x) == O. The dynamic of the slidingsurface is

and K == RIL.

ds _Idsueq==-(dxg(x)) dxf(x)x (10)

and the sliding mode dynamics can be obtained by substitut­ing ueq in (7).

B. Design of Sliding Mode MRAS Speed Identifier

The regulating model in (5) can be expressed in a compactmatrix form as

output error of two models and appropriate adaptive controllaw for adjusting the unknown parameters of the regulatingmodel. Considering that the speed is regarded as the unknownparameter, the reference model employs the equations in (2)and the regulating model is chosen to be (5).

A. Theory Of Sliding Mode

Sliding-mode control is an effective nonlinear robust con­trol approaches for its fast system dynamic responses and therobustness of the parametric uncertainties and disturbance.The construction of the system is continuously changed byswitching the controlled variable once the system dynamicsare controlled in the sliding mode. The design procedureof sliding-mode control is first to select a sliding surfacethat models the desired closed-loop performance in the statevariable space and then to design the control such that thesystem state trajectories are forced toward the sliding surfaceand stay on it [1].

For the affine system

where x E IRn, f(x), g(x) E IRnxm, u E IRm, the control variable(u) is varied logically on sliding surface (s(x) == 0) by

u == {u+,S(X) > 0 (8)u-,s(x) < 0

(2)

(6)

(3)

(4)

T; == 1.5P(l/Jiq - (Ld - Lq)idiq)

The mechanical dynamic equation is given by

Jpro == -Bto + T; - TL

where

{Ad = '": +l/JAq == Lqlq

and Ad, Aq, Ud, uq, id, iq, Ld, Lq are the stator flux linkage,stator voltages, stator currents, stator inductances in the frameof dq-axes respectively, co is the rotor speed in angularfrequency, l/J is the magnetic flux linkage, R is the statorper-phase resistance, P is the number of poles of the motor,and p is the is the differential operator (p == dIdt).

The electromagnetic torque (Te) expressed in terms of rotorflux linkage and d-q axis current is

t: == 1.5Pl/J iq

III. SLIDING MODE MRAS BASED SPEEDIDENTIFICATION

MRAS is a traditional method to estimate the rotor speedand position of the motor for its effective and physically clearconstruction. In the theory of MRAS, The reference modelwith the constant parameter and the regulating model with theunknown parameters to be estimated operate simultaneouslyand have the uniform outputs. Then, the output of the regu­lating model follow that of the reference model by using the

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where

(20)

(22)

(]o] ::;s)(]o] > -c)

ro == kosat(a)

coeq == co == (kosat(a) )eq

{a 1£

sat (a) == . ( )sign a

and the equivalent control is given by

Using (2), (12) and (20), a sliding mode MRAS represen­tation of the system is shown in Fig. 1, which has a robustperformance through combining the reference model and theregulating model.

If the system state trajectories are forced toward the slidingsurface, a == 0 and a == O. Considering (20), the actual rotorspeed is expressed as

co == kosat(a) + (xTx)-l (2KeTpx - eTPBu), (21)

is satisfied, the positive constant ko exist in all the domain ofsliding surface a. Thus, o!a < 0, V< 0 and the system statetrajectories are forced toward the sliding surface and stay onit in the finite time.

Theorem 2: The designed speed identification law in (20)is robust to the parametric uncertainties and disturbance.

Proof 2: Considering that the error existing in the speedidentification system affect the size of the designed sliding

Where i == 1,2, and £ is a positive constant.To reject the parametric uncertainties and disturbance

of the system, the speed identification law is designed asconstant switching control defined as

where ko ia a positive constant and sat(·) is the switchingfunction.

For avoiding the undesirable chatting in the sliding modetechniques, a improved switching function sat (.) is definedas

Remark: In practice, the signal of the estimated speedcontain much noise signal, which degrade the speed estima­tion performance in the system. To obtain the perfect speedestimation value, the estimated speed signal must be filteredby the low pass filter.

Theorem 1: With the designed speed identification law in(20), the estimated speed converge to the actual value quicklyand accurately. The reaching condition (ira < 0) is satisfiedand the control system will be stabilized.

Proof 1: Define the following Lyapunov function

aTaV== - (23)

2

Using (19) and (20), the derivative of the Lyapunovfunction (28) can be derived as

V == aT (coxTX- kosat(a)xTX- 2KeTpx + eTPBu) (24)

Considering that xT x > 0, and the inequality

ko >I co - (xTx)-l (2KeTpx - eTPBu) I (25)

(16)

(15)

(14)

(13)

(18)

(12)

(17)a == M

e==x-x

lime(t) == o.t----*O

e== Ax-Ax == Ae+~coPx

Thus, the general law of MRAS speed identification usingthe error between the estimated flux linkage obtained by thetwo models is derived by

where

Then, the dynamics of the sliding surface is expressed as

Using (12) and (14), the dynamics of the sliding surfaceis given by

where M == eTpx, kp and k, are positive constants.To improve the robustness of the general MRAS speed

identification, the sliding mode technique is introduced tothe system. According to the error dynamic (14), we designthe sliding surface as

According to (11) and (12), the dynamic of the definedstate error in (13) is expressed as

Mo = w- ro,P = [~1 ~].

According to the theorem of Popov hyperstability, if thefollowing condition

1) the forward path transfer matrix H(s) == D(sI - A)-l isstrictly positive real.

2) 11 (O,to) == J~OvTwdt 2:: -r6, where to 2:: 0, w == ~coPx,

v == e, and yJ is a finite positive constant, which isindependent of to.

are satisfied, the system of the MRAS speed identification isasymptotically stable in the large and

X= [~:] ,A= [=~ ~K]'where ~d and ~q are the estimated stator flux linkage in theframe of dq-axes, and ro is the estimated rotor speed.

The error vector between the state variable (x) and theestimated state variable (x) is

Then, define the estimation model with the tunable param­eter as

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(28)

Fig. 2. Block diagram of nonlinear control system

Value3000 rpm2.875 Q

8.5 mlf

8.5 mll0.175 Wb

40.089 kg ·m20.005 N im -s

ParametersBase speed COb

Armature resistance R,d-axi s inductance Ldq-axis inductance Lq

Magnet flux linkage t/JNumber of poles I'

Motor inertia JFriction coefficient B

TABLE I

PMSM MOTOR PARAM ETERS

I) The speed command (or") change from 300rpm to- 300rpm at t=0.7s .

2) The load torque (Td is increased to 5N· m at t=0.8sand decreased to - 5N .m at t= lAs.

3) The motor is started with the nominal stator resistance(RN), motor inertia (iN) and viscous coefficient (BN).Then, they are suddenly set to 3RN, 3JN and 3BN att=1.0s respectively.

Fig. 3-5 show the responses of the system under the con­dition of the speed command variation, load torque variationand motor parameter variation. It is easily seen in Fig. 3that the estimated speed is closed to the actual value andthe error between the estimated speed and the actual speedis nearly zero except for a bigger error in the stage of themotor starting. The current and torque responses have a gooddynamic performance, even if the speed command is abruptlyvaried from the positive value to the negative value. In Fig. 4,the transient speed, the speed error, the phase A current andthe torque responses also have a good dynamic performancewhen the load torque change. Especially, the estimated speedquickly approach to the actual value in case of two suddenlyload variation. The speed error is always kept at zero duringthe steady state. Fig. 5 shows that the proposed method worksvery well, and all the response (the speed, the speed error, thephase A current and the torque) appear the perfect robustnessto the parameter (the nominal stator resistance, motor inertiaand viscous coefficient) variation with the proposed speedidentifier. During the parameter variation, the estimated speedis kept at its actual value of 300rpm with an error of ± 3%.

surface, such as measure error, load disturbance and motorparameter variation etc., the dynamics of the sliding surfaceis rewritten as

Fig. 1. Sliding mode MRAS representation for identify ing the speed

cY = t.WXT X- 2ICeTPx +eTPBu + S (26)

where Sdenote the sum of various noise.Substituting (20) and (26) in (28), the derivative of the

Lyapunov function is

V= crT (wxTx - kosat(cr)xTx - 2ICeTPx +eTPBu + s) (27)

Then, the inequality is given by

ko >1w-(xTx)- 1(2 ICeT Px - eTPBu) + S 1

It is easily known that with the positive constant ko,V< 0, and the system state trajectories are still forced towardthe sliding surface and stay on it in the finite time. Thedesigned speed identification system have the robustness tothe parameter variation and external disturbance .

IV. SIMULATION RESULTS

The block diagram of the vector control system for aPMSM with the proposed scheme shown in Fig. 2. Basedon the traditional magnetic orientation control (FOe [I]),the based control method with two closed loop (the speedloop and current loop) is the control of the reference d axiscurrent equal to zero (id= 0). However, a robust sliding modeMRAS speed estimator is designed to obtain the rotor speedand position in the new system. The parameters of a PMSMused in Fig. 2 is shown in Tab. 1.

The parameter of the news system is chosen as: the speedPI regulator (Kp = 6, K, = 24), the d axis current PI regulator(Kp = 3, K, = 10), the q axis current PI regulator (Kp = 12,K, = 50), and the sliding mode MRAS identifier (ko = 14000,e = 0.01 and the bandwidth of low pass filter is chosen to be0.22Hz) . The simulation time of the system with the samplingperiod T, = 20J1s is set to 2s. The speed command changefrom a to 300rpm at t=Os. The system is started at no load.Then, the proposed scheme is verified by the test of threeoperating conditions defined by

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400..-------.-----,..----,..-------,

1.5Time (sec.)05o

Ii \(~j

# -: (tj, ,

;" :0 ,

50

350

150

'[ 200

300

t 150

e.- 100

1.5

(t)

Time (sec.)05

oooooo

• 0

,/ (t) : : '- - - - - - - - - -~-~- - - - - - - - - -·- - - - - - - - - - - - - -- - 1 - - - - - - - - - - - - - - - - -

of i L------f---------100

-4000

(a) The speed (a) The speed

: ~~ i : ; I§: -_00 05 1 1.5 1

ti mers)

: ~~ i i ; I§: -_00 05 1 1.5 1

tune rs)

(b) Speed error (b) Speed error

o 05 1tuners)

1.5~ JWf1 '~

o 05 1 1.5 1tuners)

(c) Phase A current (c) Phase A current

o 0 .5 1ti me ts)

1.5

':t31T, 1 1.,

" \..1 ]

-100 05 1 1.5 1

n me ts)

(d) The torque (d) The torque

Fig. 3. Dynamic response of the proposed system when the reference speedchange

Fig. 4. Dynamic response of the proposed system when the load torquechange

Fig. 6 illustrate the responses of the defined variable (M)using the conventional adaptation mechanism (16) and thepropose adaptation mechanism (20) under three conditionsmentioned on the above sector. The response of the de­fined variable (M) describe the performance of the speedestimation. The response of the defined variable (M) withthe proposed adaptation mechanism is controlled to be alittle range compared with that of the conventional adaptationmechanism. According to analysis of the simulation results,we can see that the proposed scheme is super to the traditionalscheme in the estimated accuracy, response speed and therobustness obviously, which achieve better dynamic andsteady performance.

is derived by the adaptation mechanism using the error ofthe estimated flux linkage estimated by the two models. Theproposed algorithm shows perfect dynamic speed responseunder the conditions of the speed command variation andthe load torque variation . Especially, it is robust under thevariation of the stator resistance, motor inertia and viscouscoefficient simultaneously. It is shown from the simulationresults that the proposed algorithm demonstrated very goodperformance on the estimated accuracy, stabilization androbustness, which keep the performance of the conventionalMRAS method . To verify the proposed method effectively,the experimental research will be further proceed in thefuture.

V. CONCLUSION

This paper has the investigation of a novel speed identifica­tion method of a permanent magnet synchronous motor. Thesliding mode technique is used in the general MRAS, whichcombined with the reference model and the regulating modelas two models for the flux linkage estimation. The rotor speed

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l.~

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! I

! I

1 1.5ti me t' )

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(b) Load torq ue variation

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Fig . 6. Dynamic response of the variable (M) under two controllaws(MRASand SM-MRAS)

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