A Phase-frequency-locked Controller for Stepping Servo Control Systems

8/3/2019 A Phase-frequency-locked Controller for Stepping Servo Control Systems

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I12 IEEE TRANSACTIONS O N INDUSTRIAL ELECTRONICS, VOL. 39, NO. 2 , APRIL 1992

A Phase/Frequency-Locked Controller forStepping Servo Control Systems

Jung-Chien Li and Guan-Chyun Hsieh , Member, IEEE

Abstract-A phase-controlled oscillator (PCO), composed ofan adaptive digital-pumped controller (ADPC) and a voltage-controlled oscillator (VCO), is proposed as a novel steppingmotor driver. Therefore, a phase-locked stepping servomecha-nism (PLSS) is established and the PC O can provide an accurateand stable pulse train to adaptively drive the stepping motor.System modeling, analysis, stability investigation, design, andimplementation are all conducted. Computer simulation andexperiment result indicate that the performance of the PL SS isclose to the theoretical prediction. Some speed responses for40-1000 r/min are examined in the real PLSS. A good speedregulation of kO.15 r/min is achieved. An adaptive line densityselector can be used t o improve the system performance.I . INTRODUCTION

ECENT LY the stepping servomechanism plays an im-R ortant role due to the demand for speed and positioncontrol. It performs especially well in the low-speed range.The driving signal for the stepping motor is a train of pulses,which is easily combined with the digital system. The devel-opment of the microprocessor makes this servo system per-form better. However, most of the control strategies up todate usually use the digital code as the reference input. Inorder to achieve a better and more reliable performance, thecontroller design must be more complicated.This paper proposes a phase-controlled oscillator (PCO) asthe controller for the phase-locked stepping servomechanism(PLSS), based on the phase-locked technique with real-timeadaptive control capability. The PCO controller consists ofan adaptive digital-pumped controller (ADPC) [4] and avoltage-controlled oscillator (VCO) The PCO provides aphase-locked range of [ -2 n , 2n1, within which the associ-ated train of pulses will be used as the driving signal for thePLSS.Because the PCO controller employs the multirate sam-pling technique with the linear quantization property, thespeed of the servo system can be linearly and stably con-

trolled. Because the phase-locked technique can provide avery wide lock-in range, the VCO does not have to provide aconstant control profile. It can adaptively provide the opti-mum control signal. This phase-locked servo system provesto be adaptive, accurate, reliable, and practical.The param eters considered are multisampling rate N,um pvoltage P, ampling period T , and VCO gain K ,. Thispaper employs the Laplace and z transformations for PLSSsimulation and analy sis [ 5 ] . In stability analysis, the system iscomplicated since the stepping motor is at least of secondorder. Consequently, the computer-aided graphical methodcan be used for controller design. We propose a practicalexample to verify the simulation result. The speed range ofthe presented PLSS lies in 40-1000 r/min where a speedregulation of f .15 r/min is achieved. It is obvious that thissystem contributes to the low-speed range. It is also con-firmed that the digital phase-locked technique [l], [4] can besatisfactorily applied to motor speed control.

11. BASIC CHEMEF THE PC OFig. 1shows the PCO in the PLSS. The PCO consists ofan ADPC and a VCO so that the driving frequency for thestepping motor is proportional to the phase error. The de-tailed ADPC scheme can be seen in [4]. If the VCO operatesin the linear region, it can be viewed a s a constant gain K,.From [4], we have the mathematical model of the ADPCas

T = T I NManuscript received July 15, 1991; revised November 17, 1991. Thiswork was supported by National Science Council, Taipei, Taiwan, R.O.C.J.-C. Li is with the Department of Electronic Engineering, NationalProject no. NSC79-0404-E011-17.

8, is the phase error, and V, is the output voltage of theADPC.Taiwan Ocean U niversity, Keelung, Taiwan, R.O.C .Taiwan Institute of Technology, Taipei, Taiwan, R.O.C.G.-C. Hsieh is with the Department of Electronic Engineering, NationalIEEE Log Number 9106875.

111. MODELINGF THE STEPPING OTORFrom [lo], we have

0278-0046/92$03.00 0 992 IEEE


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LI AND HSIEH: A PHASEIFREQUENCY-LOCKED CONTROLLER 1 I3

wiroi I II * Adaptive 1,D i g i t a l - P u m p - - t VCOhase

I

Phase-Controlled Oscillator, PCO- 7- - - - - - - - - - - - - - - -

1em,Stepping -Motor

-ig. 1. PCO in the PLSSk , -+ 0 leads to With the z transformation, we obtain

L,z2 + L 2 z+ L ,( z - 1)(z2 2 D c o s ( E T ) z + D 2 )GH ( Z ) = A

where-- D

L , = 2bDEcos(ET) - 2b E + cETSince the real pole s = - / Lp lies in the negative U axisfar from the origin, its effect can be neglected. Thus, (4) canbe simplified as + 2 b2D in ( E T )- cD sin ( E T ) (9 )L 2 = 2bE - ~ c D E T c o s ( E T ) 2bD2E- 4b2D in ( E T )+ 2 CD in ( E T ) ( I O )( 5 )% ( S ) - 4,D

J- -4 s ) s2 + -s + L , = cD2ET+ 2 bD2E - 2 bDE COS ( E T )+ 2 b2D in ( E T ) - cD sin ( E T ) . (1 1)Equation (5) may be rewritten as The forward gain isK m-- TNPK,K, 1 - e p s TG(s )= . (12)(6)s ) -&(s) s2+ 2bs + c 2 a s(s2 + 2bs + C )

whereb =the parameter related to the rotational inertia andC =th e magnetic and electric parameter.Km/c =th e dc gain of the stepping motor, including the

With the z transformation, we obtainN J + N2G ( z )= - 2- 2 D c o s ( E T ) z+ D2viscous braking coefficient.

where27r factor.f; =the input frequency.U , =the output speed.

N, E - DE co s ( E T )- bD sin ( E T )N2 = D2 E- DE co s ( E T ) + bD sin ( E T ) . (14)(15)

Thu s, the closed -loop transfer fu nction will beIV . MATHEMATICALnalysis of the PLSSphase error, i.e ., in the phase-locked range. The linearequivalent model can then be derived as Fig. 2 [4], where nnumber of pulses produced by the optical encoder during onerevolution of the step ping motor.

G (4-uppose that the system is in the steady state with smallis the line density of the optical encoder, which denotes the

4 2)T ( z )=-, ( z ) - 1 + G H ( z )

A C ( Z 1) N , z + N2- n ( Z 3 + Qiz2 Q z Z e3From Fig. 2, we have the loop gain wherenTNPK,K, 1 - e p s T Q , = 2 AbDE co s ( E T )+ AcET + 2 Ab2 D in ( E T )

- 2AbE - AcDs in(ET) - 2 D c o s ( E T ) - 1 (17)H ( s ) = . ( 7 )2 n s2(s2+ 2bs + C )


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114 IEEE TRANSACTIONS ON INDUST RIAL ELECTRONICS, VOL. 39 , NO. 2 , APRIL 1992

Fig. 2. Linear equivalent model of the PLSS.

Q2 = 2 AbE - 2 AcDETcos ( E T )- 4Ab2D in ( E T )+D2 - 2 AbD2E+ 2 AcD sin ( E T )+ 2 D co s ( E T )(18)

Q3 2 AbD2E - D2 - 2 AbDE cos ( E T )+ AcD2ET+ 2 Ab2D in ( E T ) - Ac D sin ( E T ) (19)and

(20)( 2 1 )

1 / 2E = ( e - b 2 )D = e - b Tn TNPK K ,A = 2c2En '

Now consider the characteristic equation 1 + G H (z ) = 0 ,which leads toz3+ [2AbDEcos(ET)- 2AbE

+AcET + 2 Ab2D in ( E T )-Ac D sin ( E T ) - 2D co s ( E T ) - 11z 2+ [2AbE - 2 AbD2E+ D2- 2 AcDET co s ( E T )- 4 A b 2 D s i n ( E T ) + 2 D c o s ( E T )+2 AcD sin ( E T ) ]+ [ -D2 + 2 AbD2E - 2 AbDE cos ( E T )+AcD2ET+ 2 Ab2D in ( E T ) - AcD sin ( E T ) ]

= 0. (23)In order to consider the stability of the PLSS, we employ thebilinear transformation z = (1 + s)/(l - s) with T =2 n / n w and then apply the Routh-Hurwitz criterion. There-fore.

u0s3+ U,? + u 2 s+ U, = 0 (24)

whereU, = 2 - BcE/nw - [BcDEcos (E /n w) ]/ nw

-4BbD2E + 2D2-BcD2E/nw+ 4BbE + 4 0 cos (E l nu )- 8 B b 2 D s i n ( E / n o ) + 4BcDsin (E/nw)> 0 (25)

( B = NPK,K, /2c2Enw)U , = 4 - BcE/nw + 2BcDEcos (E /n w) /n w+ 8BbD2E

- 8BbDEcos (E ln w )+ 3BcD2E/nw- 40 '+8Bb2Dsin (E/nw)- 4BcDsin (E/nw)> 0

(26)U, = BcE/nw + [2BcDEcos (E /n o) ] / nw

- 3BcD2E/nw- 4BbE + 2 - 4BbD2E+ 2D2 - 4Dcos (E ln u)+8BbDEcos (E ln w) > 0 (27)U, = BcE/nw - [2BcDEcos (E/ nw )] /n w+ BcD2E/nw > 0 (28)

U,U? u,q. (29)For a given n , we can compute U,, U , , U,, U,, andU ,U 2- U,U, as functions of U . Using (25)-(29),we candetermine U,,, above which the system is stable. Fig. 3illustrates this computation for n = 200/32 = 6.25. It isclearly seen that U,," = 165 r/min for n = 6.25. Therefore,we can compute U,,, as a function of n fo r n = 6.25, 12.5,25 , 50, 100, and 200. Fig. 4 shows this result. It can beobserved that the larger the line density n , the lower theminimum speed U,,, that ensures the stable operation.

Now consider a phase ramp input O,(t)= Rtu,(t) for(16).From [ 5 ] , this is a type 1 system, and the steady-stateerror isRess= -K ,


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I ! / , / , /[60 -40

AND HSIEH: A PHASEJFREQUENCY-LOCKED CONTROLLER

3

0-1-2 -- 3 -

-

- 4

- 5 --

0 20 4 0 G O 00 100 120 140 1 G O I 8 0o ( r pn i )

Fig. 3 . Computation of U,,,," for n = 6.25

11s

nFig. 4 . U,,, versus n

where and circuit hardware. The software is written by the Turbo Clanguage. Its main functions include accept and display speed,read the speed, etc. As to the PC O controller, we may selectthe device and hardware design according to the specifica-tions such as speed resolution, maximum overshoot, steady-state error, rise time, and settling time.Speed resolution is determined by the jump voltage P . Th esensitivity of T (z ) with respect to P is

1T z-*1K , = - im [ ( z - l ) G H ( z)]

(3 1)Constant steady-state phase error means zero steadystatespeed error .

- A (& + L2 + L3)T[l + D 2 2 D c o s ( E T ) ] '

V . PLSS DESIGNThe objective of this system is to investigate the control ofthe motor speed. The stepping motor is a nonlinear devicewith high orders. This system contains computer software where T(z) = N ( z ) / D ( z ) c"X from (16). T%en

z3+ [ - 1 - 2 D c o s ( E T ) ] z 2 + [ D 2 + D c o s ( E T ) ] z - D2z3+ Q , z 2+ Q 2 z+ Q3s ,T'z) = (33)


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I I6 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 39, NO. 2, APRIL 1992

The speed variation is directly affected by P. P must beselected for the motor input frequency to vary within f Hzrange. The product NP determines the maximum outputvoltage of the ADPC, which is normally between 5 and30 V.In designing the PCO, the selection of N nd P is veryimportant. After the two parameters are determined accord-ing to the design procedure, we sub stitute them into thetransfer function to check if the system response satisfies thespecificatio ns. If not, we sea rch anoth er set of N and Pvalues until the specif ications are met.The PLSS design procedure can be stated as follows.

Step 1) Determine the maximum range of the PCOoutput frequency according to the steppingmotor parameters, system speed range, step-ping motor and driver gain K,, and VCOgain K , .Choose the possible line density n. Determinewmin according to Fig. 3. Compute Tma,2 n / n u m i nIn step 1, the maximum output voltage of theADPC is the product NP, here P is deter-mined by the permissible motor speed varia-tion during each pumping period TIN. heforward gain of the servo system can be de-fined as

Step 2)

Step 3)

NPK,K,2 n c (34)dc =

Therefore, the speed can be derived asw = Gdc8e (35)

where2x"N8 e = -.

For minimum 8e , we have N' = 1 and the jump voltage P.Thus, the speed jump is

which shows that A w is proportional to P. N can bedetermined by P and NP.The controller parameters can be derived fromstep 1 to step 3. With the derived steppingmotor and controller, design the phase compara-tor, ADPC, VCO, waveshaping circuit, and thehardware circuit of the com puter interface.The controller parameters computed from step 1to step 3 are used as the data for computersimulation. We can do the ex periment with step4, and find the optimum line density n .Step 6) The computer simulation of the PLSS can bestated as:Step 6-1)Use P, N, , K,, K,, b, c , and n tocompute the closed-loop transfer function from(16).

Step 4)

Step 5)

Step 6-2) We use the phase-ramp input to simulate thespeed response and compare with the systemrequirements, such as speed resolution, maxi-mum overshoot, settling time, and steady-stateresponse.Step 6-3) If the specifications can be met, the design isfinished. Otherwise, we return to step 1 andrepeat again.VI. D ESIG N xample, Simulation, and Experiment

Suppose that the parameters of the stepping motor areb = 30 rad/s and c = 2000 (rad/s)*. There are 200 steps foreach rotation. Now the system employed is half-step excita-tion and the driver and stepping motor dc gain is Km c =(2n /400) = 0.0157. Th at is, when a frequency of 400 Hz isused as the input, the motor will rotate at a speed of 1 r/s. Inthe hardware used, the optical encoder produces 200 pulsesduring one rotation. Thus the maximum line density is n =200. The PCO requires that when we have a phase-rampinput, the steady-state error should be less than 2%, themaximum overshoot be less than l o % , rise time be less than2 s, settling time be less than 3 s, permissible speed jump beless than 0.0471 rad /s, and speed range be 40-1000 r/min.The VCO has a gain of K, 454 Hz/V. Consequently,the maximum output voltage of the ADPC is NP = (1000r/min 2nrad)/(454 Hz/V * 0.0157 60) = 14.68 V. Wechoose NP o be 15 V. PK,Km/c 0.0471 rad/s. Thenchoose P = 6m V and N = 2500.Fig. 5 shows the block diagram of the PLSS. We have

Phase comparator: It consists of the MC4044 fre-quency-phase comparator IC and some digital logicgates. R denotes the reference phase and V denotes thevariable phase. If R leads V , U will produce pulseoutputs and D keeps high potential. On the other hand,if V leads R , will produce pulse outputs and Ukeeps high potential.Sampler: It consists of a NOT gate and a NAND gate,etc. N3 s just the multisampling rate N. t sam ples theD (o r U, epending on which keeps high potential)signal and provides this sampled signal as a clock pulseto the Up terminal of the counter if R leads V (or tothe D, terminal if V leads R) .ADPC: It is composed of a 74193 IC counter and a12-b D/A converter AD7541. In order to increasesystem resolution, the D/A converter used in this sys-tem is a 12-b IC. Then w e use three 4-b 74193 countersin cascade. If the clock pulse appears at the Up (or Dw)terminal, the counter will count up (or down). ThenAD7541 must be power supplied by the 7812 regulationIC, which provides a stable saturation voltage of V,,, =24.6 V. Therefore the D/A converter can provide2 '* = 4096 output voltage levels. Thus, the jump volt-age is P = 24.6V/4096 = 6 mV as desired.VCO and t NI:he VCO comprises LM331 VCO IC(with linear response) and pA741 (operational ampli-fier) so that better resolution can be achieved. Thecontrol voltage of the VCO is designed at 0.02V (for


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LI AN D HSIEH: A PHASE/FREQUENCY-LOCKED CONTROLLER 117

7 4 , 9 i I 7 4 7 6

I T + "1 S T E P P I XMOTOR(U. - -+ Ni + N I MICROPRGCESSOR DRIVER -1U. ____c < ( U ,P ~ I Ms 8n .x I 10 3 8932 OIEZP C / 4 T X2SS4 1 9 3 1 7 4 7 6~

F l L P - F O L PGSC

D I G I T A LMO D IFIER *

k M Hz 7 4 7 6

Fig. 5 . Block diagram of the PLSS where N , = 10, N 2 = 30-320, andN3 = 20-50.

171 Hz output frequency) - 16.03V (for 72.83 kHzoutput frequency). In order to operate with the steppingmotor, a frequency divider circuit ( +N , , N, 10) isincluded that consists of 74193 and 7476. Thus, K , =5) Driver: It is PMM-CS-803C-1. We need only providethe pulse signal at the input terminal. It provides half-step and two-phase excitations, clockwise (CW) andcountercloc kwise (CCW ) rotations, etc. The drivingsignal derived from the VCO and + N , circuit is ampli-fied and fee ds the driver inpu t terminal.6) Stepping motor: It is a 103-89 32-01E 2 hybrid PMstepping motor, with two-phase windings and a stepangle of 1.8"/step (200 steps constitute one rotation).

An optical encoder E680200C3J is mounted that in-cludes A and B terminals that generate 200 pulses with90" phase difference during one rotation, and a Cterminal that generates one pulse during one rotation.

7) Wa veshapin g circuit: Because the stepping motor ro-tates in a stepping manner, the vibration may occur.The pulse w aveform generated by the optical encoder israther complicated. The waveshaping circuit is used toimprove such a waveform. It uses optocoupler IC'sMCT2 for noise isolation, relays for CW/C CW isola-tion, and monostable multivibrators (one shots) withflipflops for waveshaping.8) Microprocessor and frequency generator: The fre-quency generator consists of an 8-MHz temperature-compensated crystal oscillator and a programmable di-

vider, whose divisors N2 an d N 3 come from themicroprocessor. This circuit generates the referencefrequency and the multisampling frequency. The micro-processor related circuit includes the PC/AT, multi-function interface cards, and a few buffers. As soon as

(72830 - 171)/(16.03 - 0.02)/10 = 454 Hz/V.

the speed command comes from the keyboard, theprogram computes the correct N2 and N3 values,which go to the programmable divider through ports Aand B. The optical encoder output waveforms go to themicroprocessor through port C. The program countsthe number of pulses during some fixed time intervaland obtains the output speed.This system becomes unstable when the PCO is not in thephase-locked range of [- a , + 2 a] , i .e. , when 1 0, I > 2 ain Fig. 2. From (34) and (35), 0, = 2 a l eads to w =NPK,Km/c, hich is the upper speed limit wH, eyondwhich the system is unstable. From the data of our example,we have wH = (PK,Km / c ) N = (0.0471 rad/s) * 2500 =

117.75 rad/s = 1124 r/min. From Fig. 4, the lower speedlimit is w L = 40 r/min. Thus, the speed range 40-1000r/min in our experiment is stable, and this is the reason whywe choose these comparatively modest operating speeds.The digital modifier provides the freque ncy-divide r func-tion for the output waveform of the optical encoder. Thus wehave n = 200, 100, 50, 25, 12.5 , and6.25. W euse n = 10 0to find the suitable values of b and c for different speedranges. Furthermore, we can find the appropriate line densityn for different speed ranges to obtain the optimum speedresponse. The re sult is show n in Table I.From Fig. 4, as far as the stability is concerned, n is smallfor high-speed rotation and large for low-speed rotation. Inorder to explain the effect of n, three families of speedresponses for IOW-,middle-, and high-speed rotations withadequate n's are examined and depicted in Figs. 6-8, respec-tively. It is clearly seen that the simulation and experimentresults are very close.Fig. 7 shows the speed response s for 60-120 r/min (withn = 25 and 50) and 60-180 r/min (with n = 25 and 50),


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IEEE TRANSACTIONSON INDUSTRIAL ELECTRONICS, VOL. 39, NO . 2, APRIL 1992

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for 0-60 rpmfor 0-90 rpm iSimulation : !xperiment : o o o for 0-60 rpmx for 0-90 rpm In 0.2 0. 4 0.6 0.8 1 1. 2 1.4 I . GTime (sec)

Fig. 6 . Speed responses for 0-60 r/min and 0-90 r/min (with n = 200) .130

12 0

1 10

100

90 Simulation : ~ n = 2 5..........

Experiment : o n = 25x x x n = 5 0

0 0.2 0.4 0.6 0.8 1 1 .2 1.4 1.GTime (sec)(a)

Simulation : ~ n = 25n = 50......~~~~Experiment : o o o n = 25

x x x n = 5 0

0 0.2 0.4 0.G 0.u I 1.2 I .4 1. G

Simulation : ~ n = 25n = 50.....~Experiment : o o o n = 25

x x x n = 5 0

0 0.2 0.4 0.G 0.u I 1.2 I .4 1. GTime ( s e c )

(b)(a) Speed responses for 60-120 r/min (with n = 25 and 50). (b ) Speed responses for 60-180 r/min (with n = 25 and 50) .


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LJ AND HSIEH: A PHASE/FREQUENCU-LOCKED CONTROLLER I19

5 0 0 , I- I45 0

400 I k IQ

a

20 0jnI ^10 0501/0

Ifor 60-360 rpmfor 60-480 rpmimulation : ___Experiment : o Q 0 for 60-360 rpm

Y x x for 60-480 rpm

5of0 0. 2 0.4 0. G 0. u 1 I .2 I .. I 1 .GT ime ( s e c )

Speed responses for 60-360 r /min and 60-480 r /min (withig. 8. n = 6.25) .

TABLE ISUITABLE, c , A N D n

Speed Range (r /min ) b , (rad/s) C , ((radls)*) n40-60 50 15000-9000 20070-90 40 4500-2500 200120-180 30 2400- 1200 50.25210-300 20 900-500 25, 12.5

420-480 9 180- 140 6.25330-390 10 425-200 12 A6 .2 5

respectively. The steady-state speed error is less than 0.15r/min. It is clearly indicated that in each case the rise timefor n = 25 is shorter than that for n = 50, but the overshootfor n = 25 is larger than that for n = 50. Remarkably, thesettling times are nearly the same. According to the threefamilies of speed responses mentioned above, it is concludedthat the determination of n is very important. It is found thatfrom Figs. 6 -8 , with an adequa te selection of n, horter risetime, smaller overshoot, and nearly the same settling timecan be obtained over the controlled speed range. The risetimes are all within 0.15-0.4 s, the overshoots are below5%, and the settling times are no more than 0.7 s. As aresult, if an adaptive n selector is carefully built according tothe speed and stability requirements, a better performancecan be obtained in this system. This adaptive n selector canbe easily formed by using an optical encoder followed by aprogrammable divider.

VII. CONCLUSIONSWe have proposed a PCO, which consists of an ADPC anda VCO. We build the mathematical model of the stepping

motor and PCO, from which N, P , T , stability investiga-tion, and design procedure can be determined. The steady-state speed error can be kept within kO.15 r/min over thecontrolled speed range. The larger the line density, the lower

the minimum stable speed. Under the steady rotation condi-tion, small line density corresponds to small rise time andlarge overshoot. We must find the appropriate line densityunder different speed ranges to obtain the optimum speedresponse. Therefore, an adaptive n selector can be used.ACKNOWLEDGMENT

The authors are very grateful to R . N . Jou for his help insimulation and experiment.REFERENCES

J . Tal, Speed control by phase-locked servo systems-new possibilityand limitation, IEEE Trans. Ind. Electron. Contr. Instrum., vol .IECI-24, pp. 118-125, Feb. 1977.DC M otors, Speed Co ntrols, Servo Systems, 4th ed., EngineeringHandbook, Electro Craft Corp., 1978.F. M . Gardner, Phaselock Techniques, 2nd ed. New York: Wiley,1979.G. C . Hsieh , Y. P. Wu, C. H. Lee, and C. H. Liu, An adaptivedigital pump controller for phase-locked servo systems, IEEE Trans.Ind. Electron, vol. IE-34, pp. 379-386, Aug. 1987.B. C. Kuo, Digital Control Systems. New York: Holt, Rinehart andWiston, 1980.I . J. Nagrath and M . Gopal, Confrol Systems Engineering. Ne wYork: Wiley, 1982.A. Hughes and P. J . Lawrenson, Electromagnetic damping instepping motor, IEE Proc . , vol. 122, no . 8, pp. 819-824, 1975.A. Hughes, Parameters governing the dynamic performance ofpermanent-magnet stepping motor, in Proc . Sixth Annual Symp o-sium of IMCSD, 1977, pp. 39-47.P . Lawrenson, A. Hughes, and P. P. Acarnley, Starting/stoppingrates of stepping motors, improvem ent and prediction, in Proc. Int.Conf. on Stepping Motors and Systems, 1976, pp. 54-60.Y . K. Hsu, Principles and Applications of Stepping Motors (inChinese). Taipei: Chuan-Hua Book Co., 1990.J.-C. Li, G.-C. Hsieh, and R.-N. Jou, A study on stepping servocontrol system by phase-locked technique, in Proc. IEEE I n t .Conf. on Industrial Electronics, Control, and Instrumentation,Kobe, Japan, Oct. 1991, pp. 366-370.

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A Phase-frequency-locked Controller for Stepping Servo Control Systems