Variable Structure Position Control - IEEE Control Systems Society

Variable Structure Position Control N. N. Bengiamin Electrical Engineering Department, University of North Dakota, Grand Forks, North Dakota

B. Kauffmann American Microsystems Inc., 2 3 0 0 Buckskin Road, Pocatello, Idaho ABSTRACT: This paper describes the application of the variable structure control (VSC) concept for dc motor position control. The control scheme is derived, implemented, and tested in the laboratory where both digital and analog controllers have been used to controla representative servosystem. Thecontrol-systemschematicsaregivenandsampleresultsare shownforillustration.TheVSCexperiment should be of interest to educators and design engineers who wish to demonstrate and investigate sophisticated position-control methods and their applications. This experiment mayalso serve as a basis for further applications of VSC.

Introduction Position controlis a classical problem with wideindustrialapplications.Proportionalintegral-derivative (PID) feedback control is perhapsthemostcommoncontroldesign used by industry [ 13. Although the design of analogcontrollers is very well established, thedevelopmentsintransistortechnology have led to the introduction of small, inexpensive digital microcomputers with enough capabilitytoperformsophisticatedcontrol actions.Theflexibility ofmicrocomputer softwareprovidesthecontrolsystemsengineer with a more versatile meansof design andexperimentation.However,thelimitations of microcomputer speed, interface, and memory allocation should notbe overlooked when compared with analog controllers. The developmentofsteppermotorshasintroduced a new means for position control with readily available interface to digital systems andaredriven by simplecontrollogic. However, the slippage, the required holding torque,andthelowtorquecharacteristic Received January 10,1983;revisedMarch 16, 1984.Accepted in revised form by Technical Associate Editor, G . H. Hostetter.

where I+!I is the feedback gain, the value of of stepper motors have limited the range of which may switch as follows: their applications. Thispaperintroducesexperimentation if x , @ > 0 with positionVSC.Withinthe last twenty *= ( b (3) a if XI(+ < 0 sysyears,thetheoryofvariablestructure tems (VSS) has been developed in the USSR. The boundary D = 0, with C a constant, deBased on that theory, the VSC concept has finesaswitchingline in thephaseplane, been developed and applied to the control of that is, a wide range of processes. The reader is re0- = cx, + x2 (4) ferred to the survey paper by Utkin [2] and its subsequentpublications [3-51. Thebasic The feedback gainI+!Iwill be adjusted accordidea of VSC is for the system to change struc- ing to the location of the system state repture at certain instants so as to combine the resentedonthephaseplane asshown in useful properties of each of the structures. Fig. 1. This flexibility permits the use of good dyThe design problem to be considered is to namic properties of structures that cannot be determine values ofa,/3, and C such that the used over a long, continuous period of time. system state will be brought from any initial Operating such a system, in what is known position in the phase plane to the switching as the sliding mode, makes it insensitive to line D = 0. Then, the state trajectory is to parametervariationsanddisturbances.The slide along the switching line to the phasesliding mode has been analyzed using highplane origin. The phase trajectory along the gain methodologies and has proved to possessswitching line is known as the sliding mode properties of high-gain feedback systems161. of this controller. Whenever the system state This paper documents a VSC-based position- leavestheswitchingline,thecontroller control experiment that has been conducted changes structure to force the point back to in two parts. First, an analog controller was the switching line. Theoretically, the system tested,andtheswitchingofstructure is willchatteraroundtheswitchingline by illustrated for different modes of operation. changing structure at a very high speed such Second, the same control scheme was impleD = 0. that it stays in closeproximityto mentedusingamicroprocessor and digital Practically, the systemwill tend to overshoot systems interface. The results of analog and theswitchinglinesignificantly,onatime digital control are presented for comparison scaledependentonthedelayoftheconand further studies. trolled process.

Variable Structure Control Basics To illustrate the basicideaofVSC, we consider a second order system describedby the following canonical equations:

x,

=

x2

i z= -aZx2 - aIxI+ bu

(1)

where x I and x2 are state variables andthe control law is given by the equation u =

*x,

(2)

When approaching the origin of the phase plane, the overshoot tendsto become shorter in duration, resulting in a “chattering” in the steady state. There are several practicalsolutionstothisproblem,one of which is to switch structure to pure integral control when so thatthe neartothephase-planeorigin system will be driven smoothly after reaching a suitably small error level.

Position-Control Design Assuming much faster armature time con-

0272-1708/84/0800-0303$01.0001984IEEE

august 1984

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Forcing the system'sstate tomovetoward the switching linefromeachside requires satisfying the following condition [SI:

ffu< 0

(10)

that is.

Therefore, 0=O

where

T

e =bK

Fig. 1.

E

K

S(I

rS)

Phase plane.

.

x,

4)

=

-x.

E

a

= 0.038(6C2- 4C)

Since C determines the slope of the switching lineand is in the range of !CJ< 0.2, then IEj < 0.04. Therefore. condition (12) may be satisfied by

b

(5)

where x I = afl. x, = bw. and n and b are the gains of angleandspeedtransducers;see Fig. 2.

where r is the motor time constant andK is its gain in radians per volt second. 0 and E are the motor's angle and applied armature voltage. respectively. In smallmotors.such as used in this experiment. the nonlinear effect of brush friction on the motor time constant is substantial. However. by using a subsidiary tachogeneratorloop associated with an appropriate network. it was possible to reduce the brush friction effect and obtain a motortransferfunction similarto that of Eq. (5) 171. Therefore. the state equations of the motor are

Since ff =

C.TI

+ x..

(7)

where p is negative and a is positive and theirmagnitudesare _greater than 0.04. Figure 2 is a block diagram of the control system.

Implementation and Results

then

Butin the proximity of the switching line. = -C.Y!. Therefore.

x,

~~~

The s e n 0 system used in this experiment is the Feedback DC Modular MS 150 kit. The dcmotor is 24 V, 2 A. with series-wound transistor controlled split-field. It is coupled with a dc tachogenerator for speed sensing.

~

Fig. 2.

4

-

For thespecificsystemoftheexperiment, K = 8.2 rad/s * V, r = 0.25 s, a = 4.7 V/rad. and b = 0.78 V * s/rad; that is,

stant than the mechanical time constant for armature controlled dcmotors. the transfer function of the motor may be given by

-fl_ -

'nC' (,b

Block diagram of thecontrolsystem.

control systems magazine

Fig. 3a.

Schematic of thecontrolsystem.

-

I

!-*I*+

!

Multiplier

I sx1

a

.

-

Comparator

Fig. 3b. In additiontotheservomotor, this kit includesanoperationalamplifier,alternator, preamplifier, servo amplifier units, and other basic accessories. This experiment has two parts. The first part is the analog realization of the controller, and the second part is for theimplementation of thesamecontroller using digital equipment. The analog part is presented in thispaperforreference and comparison purposes. Any of the two parts may be promoted by itself without any lag in concept.

VSC schematic diagram.

The time responsefor a60” step change in position is shown in Fig. 4a, and the corresponding control signal and phase-plane trajectory are in Figs. 4b and 4 c , respectively. In the phase plane, x2 is amplified considerably to show the overshoot. As expected, overshootaroundtheswitchingline is dependent on the inertia of the system and the feedback gain. Severaltrials have been made to tune (Y and /3 to improve the system’s response. The actual plant behaves differently for positive and negative inputs because of motor-brushfrictioneffectandgearbacklash.Figure 5 showsthesystemresponse Analog Controller: Figure3showsa wheretheswitchingline is shiftedmore schematic of the control system. The analog In this case, the towardtheverticalaxis. computer we use is the EA1 TR-48 model. system changed structure a few times, then Otheranalogcomputersavailabletothe escaped from the sliding mode to the other reader may be used with few modifications, end of the switching line where it changed that is, the COMDYNA GP-6 has all the functions needed except the relay comparator structure again, then approached the phaseplane origin. that must be used externally. august 1984

Digital Controller: Themicrocomputer used is built around the Intel 8085A microprocessor. It containsadigital-to-analog (D/A) converterand ananalog-to-digital (A/D) converter.bothwith k.5 V analog ports. The angular position ( x , ) is read using the A/D converter. As theradianvelocity (x2)is also needed. it is calculated using the change in angular position and a fixed time base. The angular position is measured once each program cycle T. Therefore, x2 equals Axl multiplied by 1/T. The value of T was chosentogiveafourbinarydigitchange in x1 atthemaximumradianvelocityof 6 rad/s.Considerationinchoosing T was also given to simplify calculating x 2 , as the 8085A does not have a hardware multiply. The cycle timethen was chosen such that1 / T is a multiple of two, as “shift-left” instructionsareonlyrequired formultiplication. A cycletime of 62.5 mswaschosen

a little change may be noticed. The control system has been tested for a very wide range of the friction parameter, and it was evident that the response of the system is not sensitivetothechange in thatparameter. This substantiatesthesalientfeature of VSC systems when operating in the sliding mode. Fig. 4a. Time response.

Conclusions

~-!i-!!~T reverse

forward

Fig. 4b.

Control signal.

t

This paper presented the details of VSCa basedposition-controlexperiment. Various aspects of system'sperformancewerediscussedandillustrated by experimental results. Results of analog and digital control werefound to becomparable.Thesystem has been tested for a wide range of mechanical loading. The time response was found to be insensitive to change in load when operating in the sliding mode. This supports the theoryandproperties of high-gainfeedback systems. The simplicityandeffectiveness of the VSCconceptmakethisexperimentinteresting for educationalpurposes.Forpracticing engineers, the test results should help inpromotingfurtherinvestigation and exploration of the concepts.

Acknowledgment The authors would like to thank Mr. Gene H. Hostetter for his valuable comments and suggestions in preparing the text of this paper.

Fig. 4c.

Phase plane.

I

' I

(1/T= 16) forourexperiment.

Two other multiplications were needed to implement a VS controller. These are C * x, for u and @ * x1 for u. To prevent overflow in calculations,thedifferentvariableswere scaled and external amplifiers and attenuators were

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used at the terminals of the processor, Figure 6 is the digital system block diagram. The computer program flowchart andlisting. are in Fig. 7 and Table 1, respectively. The time response of the systemfor a 60" step change is shown in Fig. Sa, and the corresponding control signal and phase-plane trajectory are in 8b and 8c. To test the system's sensitivity to change in parameters, a high external friction was applied to the shaft of the motor. For a certain value of friction and twice as much, the 60" step response was recorded for comparison. Figure 9 shows thesystem's response where 6

Fig. Sa.

Control signal. 5b.Fig.

Time response.

Fig. plane. Phase 5c.

J

control systems magazine

TP

-

Pre Servo .amplifier-Ainplifier

- Motor

PP 8085A

D/A

9

Ipr eofseirtei on cne

Fig. 6.

Digitalsystem

block diagram.

Initialize Ports

7 reverse

1

Fig. Sa.

7 Error=,Pef.

(-x1).

R

Posi.

-X1,v+l

Time response

6 Call M u l t . S u b r o u t i n e

Fig. 8b. Control signal. Yes

@ = a

1 U.

= $

*

Error

C a l l X u l t .S u b r o u t i n e

r

ij; =

B

I

-

J

Output u D/A

Fig. 7.

august 1984

Program flowchart.

Fig. 8c. Phase plane.

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Table 1 Computer Program Listing ;VARIABLE STRUCTURE CONTROL PROGRAM

ORG 800H 8040 8000 ;INITIALIZATION LXI SP.87FOH 8043 800031F087 MVI A.06H 8045 8003 3E06 OUT 20H 8048 8005 0320 STA20FFH 8007 32FF20 8049 OLD :LOAD X1 804C 800A 3AE180 LOOP:LDA 80E1H 804F MOVC.A 8000 4F CONVERSION ;BEGIN 8050 MVI A,38H 800E 3E38 OUT 23H 8053 8010 D323 MVI A, 18H 8054 8012 3E18 8057 OUT 23H 8014 0323 AA: IN 23H 805A 8016 DB23 ANI 04H 8058 8018 E604 ;CONVERSION COMPLETE? 805E CDAOlO JNZ AA 801A C21680 IN 21H 8061 801D DE21 : ADJUST FOR A/D 8062 AD1 80H 801F C680 ;CMC FOR ROTATE RIGHT 8065 80 CMC 8021 3F X1H ;NEW 8066 STA 80E1 8022 32E180 8067 NOP 8025 00 MOV 8.A 806A 8026 47 RAR 8027 1F 8066 :x112 806E STA 80E2H 8028 32E280 8071 MOV. A.B 8028 78 :XlNEW-XlOLD 8074 SUB c 802C 91 RAL 8077 802D 17 8078 RAL 802E 17 RAL 8078 802F 17 807E NOP 8030 00 8080 NOP 8031 00 :x212 8082 STA 80E3H 8032 32E380 8085 28 NOP 8035 00 8086 NOP 8036 00 8088 84 NOP 8037 00 8089 NOP 8038 00 ;DESIRED POSITION (00) 808A LDA80EOH 8039 3AE08O ; SUBCURRENTXl 808D SUB B 803C 90 ;ERROR STA 80E8H 8030 32E880

Fig. 9.

Timeresponsefordifferent friction values (broken lines -- indicate doubled friction parameter).

References [ 11 P. B. Deshpande and R. H. Ash. "Elements

of Computer Process Control with Advanced Control.L\pplications." f n s r r . SOC. of ..ln~erica.198 I . [2] V.I. Utkin. "Variable Structure Systems with Slidlng Modes." I € € € Trans.. AC22. no. 2 . pp. 112-222. 1977. [?] K . D.Young."ControllerDesignfora Manipulator Using Theory of Variable Structure Systems." f E E € Trans.. MSC-8, no. 2. pp. 101-109. 1978.

3AE480 FElO CA5780 47 3AE280 CDAO8O 47 3AE380 80 C36680 3AE580 47 3AE380

BB:

47 3AE280 47 3AE280 A8 FA7480 3AE680 C37780 3AE780 47 3AE880 CDAOBO C680 D322 210018 3E00

85 C28580 C30A80

GG:

DD: EE:

zz;

LDA 80E4H CPI 10H JZ BB MOV B,A LDA 80E2H CALL MULT MOV B,A LDA 80E3H ADD B JMP GG LDA 80E5H MOV B.A LDA 80E3H CALL MULT MOVB,A LDA 80E2H ADD B MOV B.A LDA 80E2H XRA €3 JM OD LDA 80E6H JMP EE LDA 80E7H MOV B.A LDA 80E8H CALL MULT AD1 80H OUT 22H LXI H.1800H DCX H MVI A,OOH ORA H ORA L JNZ ZZ JMP LOOP END

;LOA0 C1 ;c1 = l ? ;x1/2 :Cl*X1/2 ;x212 ;a

+ x212

= Cl*X1/2

;LOAD C2 ;x212 ;C2*X2/2 ;x1/2 ; a=X1/2

+

C2'X2/2

;x112

:ERROR ;ADJUST FOR D/A ;OUTPUT

;TIMING LOOP

[4]K. D. Youngand H. G. Ktvatny. "Variable Structure Servomechanism Design and Applications to Overspeed Protection Control," Autornatica. vol. 18. no. 4, pp. 384-400, 1982. [5] N . N. Bengiamin and h ' . C. Chan, "Variable Structure Control of Electric Power Generation." IEEE Trans.. PAS-101,no. 2 , pp. 376-380. 1982.

[6] K . D. Young. P. V. Kokotovic,and V. I. Utkin, "A Singular Perturbation Analysis of High-Gain Feedback Systems."ZEEE Trans., AC-22. no. 6. pp. 931-938, 1977. [7] "Modular Servo System MS150," Book 3, FeedbackInstrumentsLd..MS150-3 EdA.0378, pp. 41-43. [8] C . Itkis, Control Systems of Variable Structure, John Wiley & Sons. 1976.

Nagp K . Bengiamin wasbornonNovember 26, 1948. in Egjpt. H er e c e i v e dt h e B.S.E.E.degreefrom AlexandriaUniversity in 1971.In1974. he went to Canada where he recelved his M . Engr. fromCarleton Universityand Ph.D:from CalgaryUniversity in 1976and 1979. respectively. Dr. Bengiamin is a member of IEEE and XSEE. His Interest is in the areas of digital and analog control systems. electric poner systems analysis and control. and robotics.

Brian R. Kauffmann received the B.S.E.E. L degree in 1981 and the M.S.E.E. degree in 1983 fromthe University ofNorthDakota, 1 Grand Forks, North Dakota. M r . KauffAmerican Inc. in a design shere he -ks in the design and development of MOS integrated circuits. Brian Kauffmannis a member of Tau Beta Pi, Eta Kappu Nu, and IEEE.

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