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WM2.1SLIDING MODE BRUSHLESS DC MOTOR CURRENT AND TORQUECONTROL ALGORITHMS

Wu-Chung Su, Sergey V. Drakunov, Umit OzgiinerDepartment of Electricd EngineeringThe Ohio State University2015 Neil AvenueColumbus, OH 43210

AbstractThe unique attributes of the brushless de motor makesliding mode control suitable and favorable to manycontrol designers. The essence of controller designfor brushless dc motors is in fact a set of switchinglogic for the three phase input voltages. Stemmingfrom the idea of sliding mode, three algorithms forcurrent and torque control are developed in this pa-per from different aspects of the same problem. Thefirst. algorithm takes an analytical approach tha t com-putes the equivalent controls for eacb phase voltageand converts them to suitable dc values. The sec-ond deals with physical phenomena and generates aproper switching sequence directly without the needof a conversion process. Finally, the third further assumes no knowledge about the system parameters.By using current feedback and rotor position infor-mation, a look up table is constructed to achieve thedesired performance. The proposed algorithms arewell suited for microprocessor control applications.

I. IntroductionA brushless dc motor has the same physical structureasa permanent-magnet ac synchronous machine. Thepermanent-magnet rotor provides a constant fieldagainst the armature winding on the stator, which iscommutated electronically by a six-step inverter th atswitches according to the shaf t position. Withproperswitching of the inverter, the need for slip rings in therotor assembly is then eliminated. The rotor positioninformation can be obtained from either Hall effectsensors or a shaft encoder mounted in the motor assembly. The Hall effect sensors divide each rotor rev-olution into six equal sectors (60 degreee each). Thiscan only reveal a coarse knowledge of the rotor pcrsition but is sometimes sufficient for switching logicdesign. The shaft encoder, on the other hand, canprovide a higher resolution for the rotor angle whichis especially useful for speed sensing and plays an im-portant role in (d - ) analysis for %phase motors.

CH3OOO-7/9!2/oooO-0910 1.0001992 IEEE- , -

With the above attributes, there are some inherentlimitations in the brushleaa dc motor as well. Dueto the dc nature of the driver and the physical wind-ing conntraint, the stator can only provide six dif-fuent magnetomotive forces (mmf), 60 degrees apar tfrom each other M show in Figure 1. The controlinputs (stator mmfs) being discrete values, a slidingmode control approach seems appropriate for mostmntrol purposes. As a matter of fact, there indeedexist eeveral theoretical and experimental results thatsuppor t t he idea of sliding mode control for brushlessdc motors[2]-[5].The dynamic equations in the orthogonal axes (d - q )connected with the rotor can be written in the form

where i d , i f ,d, c f are components of stator currentsand voltagea in the direct and quadrature rotor axes

U

Figure 1: Stator voltage vectors and rotor positionsectors

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respectively, xd,X , , R are active and passive resis-tances, F s the constant magnet flux coefficient, andn is the angular velocity of the rotor. The torqueproduced by the motor is given as(3)

Let 0 be the angular rotor position. For a tphaeemotor, we haveCd = 2 ( e A c c t s e + e B c o s (e -p )+c c o o ( e +p ) )4)e, =-+ +cBh e - =)+ec h e + 5r)) (5)

r = ((xd- X,)id +F ) i ,2 2

32 2

with the inverse transformation[6]e A = e d C O S @ - c q S h @ (6)

(7)(e)

2 22 2

3

c g = CdCOB (e - z ~ ) ,rin (e - 5 ~ )cc = C d ccts ( e+ X ) - , sin (e + - r )

where e A , e B , ec are phase voltages and can takeonly two discrete values, namely +V& and -V&.We consider current control M an important subjectin the study of brushless dc motors especially whenone is interested in torque output control, energy loss,circuit protection, and any other current related is-sues. To control the currents, a general approach isto utilize equations (1) and (2) and design an a ppropria te controller t o achieve the performance crite-rion. The control inpu ts e d , e , are functions of i d ,i,, and n together with those system parameters R,X d , X , , and F . The computed values will be trans-formed into the 3-phase voltages (e:, e>, e whichare then converted to t he discrete values (e,,, e g , ec)to be applied physically to the motor. Several pa-pers have successfully verified the above scenario byusing various control schemes[2]-[5]. In this paper,we have a different approach that takes into accountsome practical limitations of the syatem set up. First,the system parameters may not be known or the givenvalues may be far from accurate. This makes the com-putation of c d , e , cumbersome or even impossible.Secondly, the computation will take a long time andtherefore high-speed Digital Signal Processing (DSP)chips become inevitable for an implementation, whichincreases the cost of the controller. Or without a DSPchip, the performance is deteriorated due to the largesampling time required for he computation. Thirdly,one may be interested in designing a universal con-troller for all types of brushless dc motors withoutthe need of changing parameters for different plants.Here, we present simple, fast alg or it hm tha t can leadto cheap, robust , and universal current controllers forbrushless dc motors.

II. Controller Design

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The awence of designing controllers for brushleas demotors is in fact a set of ewitching logic for C A , cg ,eC . The switching logic depends on the states of themotor; namely i d , i ,, 8, and poeaibly the angular ve-locity n. For current control, the performance cri-teria would be to reach some desired point (i:, i ; )in the etate space. T his point can be determined byeither mathematical optimization or physical consid-erations. Particularly, in the torque control problem,one may desire to achieve some torque value and atthe rame time minimize the energy loss. Thus (iz, ; )can be solved by

e : + i l , = aatrch t h d ((xd- X,)id 4- F ) i , = 7 d (9)

where U = $.+iz is the instantaneous energy loss, andr d is the desired torque output. Or in many cases, ifwe have a round, symmetric rotor; ie xd=xq , hen itcan be essily taken M

(ii,; ) = ( o , ? ) (10)Algorithm1. ( i d , ip )omain-conventional approachThe sliding surfaces are chosen as

(11)(12)

The control inputs e d , eq are designed to satisfy thesliding conditions i& < 0, i= 1,2 (13)

(14)d = e* - Csgn(s1) 'c q = e - s g n ( s 2 ) (15)e = Rid - X, i ,n (16)

e = R i, + (Xdid +F ) n (17)The computed ( C d , e , ) will be transformed back to(e>, e$, e>) , the discretetime equivalent controls[7]for sliding eurfacea (11) and (12), by equations (6)-( 8 ) . Since (e:, e>, &) will generally not satisfy the dcconstraint, a modification will be mad& In [2], Utkinsuggested that e, be picked as

(18)

. ..l = d - d$2 = , - ;

We have

with h > 0 and

eJ = r g n ( e J ) Vde, j= A ,B,CThis method was indeed shown to be effective[5].Here we propose anothtr way of converting e;. to ejby building the following nonlineaT circuitry

ZJ = - V d , s g n ( A J ) + e ; , j= A , B , C (19)

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It is shown in [l] hat if V& > le;}mor, the ayatcmwill reach sliding mode in finite time and we have= (&sgn(ZJ))..r, j t.A, B ,C (20)

By lettingeJ = Vksgn(Z,), j = A , B ,C, (21)

we obtain the desired switchings for CA ,C B ,ec ,whichare the actual voltages applied to the motor.We can also attack the problem directly from thetorque-energy point of view and work in the (7, a)state space. Our aim then would be to provide thedesired torque value r d with minimum energy ~ o s o .A restriction is tha t during the tramient procem, thevalue of U is not to exceed amor. his is always a la-gitimate consideration to protect the circuit from anovercurrent.Algorithm 2. ( r p )domain - direct switching logicdesignThe s tate equations for r and U are

whereg- = (-(-Rid + Xqiqn)+

i*(-Riq - Xd:, + F)n) (24)go = - ( - R i d +X , i ,n ) +3 -Ri, - X & +F)n) (25)d

x d x,Let's substitute (4),(5) into (22),(23)hen we haveT = b r A e A + brBCB + bycec + gr= boAeA + b uB eB + b o c e c +go

(26)(27)where bi, = b i j ( i d , i , !@ ) , = r , u , = A , B , C arefunctions of currents td , i , , and rotor pasition 0.

Since there dre only eight possible values of thephase voltage vectors, e = (CA,CB,CC) E E =

{e1,e2,...,e*},t is more convenient to describe theowitching logic not analytically but a sequence of con-ditions. Let e* = e*(OIid,iq) denote the vector 85-signed a~ a control for the given values of 8 , d , i,.Switching Logic

1. If U > U,.,, consider the set El of all possiblee E E satisfying the condition U < 0 ( E1 # 0).If there e x i s t s such e E E1 that (T- Td)f < 0,we aesign e* = e, else, e* = e for any e E E l .2. If a < a we change the priority and con-sider the set E2 of all e E E that condition

(7 - d ) i < 0 s satisfied (El 0). If there ex-bts such e E2 that &(e) < 0, we assign e* = e ,else e* = e for any e E E2.This algorithm works directly on the switching logic( C A ,CB ,e= ) and hence does not require the conversionsteps in (19)-(21).Algorithm 3. Table Look-upThe above two algorithms assume a full knowledge ofthe system parameters. Now we will exploit a differ-ent method that enables current control without theneed for calculating ( e d , e , ) or ( e A , e B ,ec) but simplymake use of Hall effect sensor information to designour switching logic.In (l), (2), we observe that the incremental vector(Aid,Ai,) is a function of id , ,, n , ed , e,, and the sam-pling time T,.

T,A i d = - ( -Rad + x q l q n - k ed ) +o ( c ) ( 2 8 )x dAi, = TI(-& - x d l d + F)?Z+ e q ) +o ( c ) (29)x,

To drive the current state i d ( k ) , , ( k ) towardsi ; ( k ) , i ( k ) , equations (28) and (29) suggest that wechoose Cd , e, such that

( 3 0 )( 3 1 )

This is appropriate from the point of view of slidingmode control. The existence condition in equation(13) demands that

# g n ( e d ) = #On(;; - 9 ) = SdJgn(e,) = sgn(i; - q ) sp

('Rid +Xqlqn + e d ) ( i d - ; ) < 0 (32)( - R i q - X d i d +F)n + e q ) ( i q - ; ) < 0 (33)

Since we have no information about R, xd, d, andF , moat feasible choice of e d , ep would be to satisfy(30)-(31). Note tha t only the sign of e d and e, aredetermined here. The actual value can be calculatedfrom equations (4) and ( 5 ) . Knowing the desired signsfor ed, e,, the next step will be to determine which ofthe six stator mmf's Vn, n = 1,...,6, re to be applied.

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As depicted in Figure 1. 8, is the aesociated angle for8 , = (34)

8gn(ed) = Wn(C00 (8" - )) (35)sgn(eq)= S P n ( M (e, - )) (36)For each pair of desired rgn(ed, e f ) and rotor angle 8,there exists either one or two V,'s that satisfy (35)and (36). One can choose an appropriate V, andobtain the corresponding switching logic ( e A l B , = )by looking up Table 1.

Vn3With the rotor position 8 , it can be seen that

Hall sensor informationrector number

I tator m m f v ec t o r II ( e l . e a . e c ) I

( e l . e.e. e c )

Another way of finaing V, is by using the Hall sen-sor information. As seen in Figure 1, for each sec-tor( I,II,. ..lVI ) he rotor resides in, we can always findone V, that satisfies (35), (36). Table 2 s then con-structed as follows:

v* X ( - , - ) ( - ,+ ) x (+,+) (+.-)

Table 2: sgn(ed,eq) aa a function of V, and rotorpositionwhere X denoteds "don't care" since the correspond-ing sgn(ed,eq) is ambiguous. Finally, with Table 1and 2, we come up with a iingle table of switchinglogic for (eA,eB,eC)Table 3).

m. Numerical ExamplesThe system parameters of the motor ar e taken as:Phase Winding Resistance R = l.Onq-axis Inductance X, = O.llHd-axis Inductance xd = 0.34H

t V 11 ( 8 9 , -ad, ad) vdcVI 11 (ad, -Sd, - s q ) * vd cTable 3:Current control switching logic look-up tableTorque Constant F = l .ONmA-'

Aliorithm I2 , I

3 , I

iI

Figure 2: Time response of id, and i,

Alponlhm 1 . hue Plane

6 -4 2 0 2 4 0Id

Figure 3: Phase rajectory of id l and i,The samdin n time is 0.001sec and the desired toraue

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4 , b l lPq lhm 2 1

7-0 0.05 0 1 0 15 0.2 0.25 0 3 0.35 0 4 0.45 05t imetuc)

'0' 005 0'1 0 5 0'2 0 5 0'3 0.35 0 4 045 i 5rimetsec)

Figure 4: Time response of r , and U

4 I7 0-4 2 4 6 YiaFigure 5: Phase trajectory of id , and i ,

output is chosen as:Td = 3.0"

Newton's method is employed to search for the opti-mal values i; and ii in (9). It is found thatI; = 1.07Ai; = 2.41A

Thus, the minimal instantaneous energy loss isOmin = 6.94A'

and the maximal instantaneous energy is set to beOmoo = 12.0A2

2 Alnori lhrn 3.......................t

1v0 0.05 0.1 0.15 0.2 0.23 0.3 0.35 0.4 0.45 0.5

I10 005 01 015 0 2 0 2 5 0 3 0 3 5 0 4 0 4 q 0 5

Ilmc(YL)

Figure 6: Time response of i d , and i,

4 0 2 4 6 bId

Figure 7: Phase trajectory of i d , and i,

In Algorithm 1, the control goal is to drive ( i d , i , )to ( i : , i ; ) . The reaching time directly depend on themagnitude of t in (14) and (15). I t was assumed ini-tially to be 0.1. If one desires to speed up the processby increasing k, hen the sampling time T, needs tobe chosen smaller to maintain tollerable chattering.Figure 2 and Figure 3 show the simulation result.Algorithm 2 deals with 7 and U directly. As seenin Figure 4, it reaches the desired torque much morerapidly than Algorithm 1 and the amount of energyloss drops to the minimal value thereafter. Figure 5illustrates the corresponding id-iq phase portrait ofthe same result-sliding mode occurs on the constanttorque trajectory and the trajectory slides into the

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optimal point along the nonlinear surface.Figure 6 nd Figure 7 re the time response and phaseplot of Algorithm 3. Since it makes no use of thesystem parameters at all, the result is slower than thesecond algorithm. Nevertheless, the effectiveness andease for implementation make it worth considering.

IV. ConclusionsWe have completed three different algorithms forbrushless dc motor current and torque control. Thesame idea can be generalized to some other controlapplications, such as speed and position, where weneed to include one or two more equations t o fully de-scribe the system. Different approaches as they take,the major design philosophy stems from the theoryof sliding mode. One of the import ant concerns insliding mode control is the chattering frequency that.essentially dominates the system performance. To in-crease the chattering frequency as much as possible,it is desirable to reduce the switching logic computa-tion time. If the system parameters are available, anobserver for id and i , can save the time spent for thethree phase current sensing and d-q transformation.The sampling rate can then be speeded up signifi-cantly.Recently, discrete t ime sliding mode is receiving muchattent ion from the control society. The idea enablesus to separate the controller design from the continu-ous time driving circuitry. In the discretized system,the controller mainta in sliding mode a t each samplinginstant. During the sampling interval, the analogdriving circuitry will be able to provide a much higherchattering frequency than a microprocessor can. Webelieve this will be a promising direction for the fu-ture study of brushless de motor as well 88 any othersliding mode control applications.

References[l] Utkin, V.I . Sliding modes and fheir application invariable sfructure sysfems. Moscow, 1978[2]Utkinv, V.I. Sliding mode in optimization and con-irol. Moscow Nauka (in Russian), 1981.[3] Izosimov, D.B., Matic, B.,Utkin, V.I. and Sa-banovic, A., Using sliding modes in control of elec-trical motors, Dokl. AN SSSR , 241, No.4, pp. 769-772,1978.[4] Hashimoto, H., Yamomoto, H.,Yanagisava, S.,and Harashime, F. Brushleas servo motor controlusing variable structure approach, in Conference

Record of 1986 IEEE Zndusiry Applicafions SociefyAnnual Meafing, Pari I, pp. 72-79, 986[5] Low, T.S., Lee,T.H., Lock, K.S.,nd Tseng, K.J.,DSP-based instantaneous torque control in perma-nent magnet brushless d.c. drives, Mechatronics,Vol. 1,No. 2,pp. 203-229, 991.[SI Fitzgerald, A.E., Kingsley, C., and Umans, S.D.Eleciric Machinery . New York: McGrall-Hill, 1990.[7] Utkin V.I., Drakunov, S.V. On discrete-time slid-ing modes, preprints of ZFAC Conference on Nonlin-ear Control, Capri, Italy, 1989.

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