IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROI, INSTRUMENTATION, VOL. IECI-22, NO. 1, FE,,BRUARY 1975

T(q,u) - <a- aq cos a,qL cos a,sL - a. sinlaL sin cxuLT(q,u)=

ZmQ(x) Jm(x) + RmMNm(x)Xm(X) Jm(x) + RmENm(X)

and the renmaining integrals must be done numerically.

ACKNOWLEDGMENTThe authors wish to thank Dr. S. J. Towaij for his

assistance in the computation of the results and Miss J.Gourlay for typing the manuscript.

REFERENCES[11 L. Solymar, "Step transducer between over-moded circular

waveguides," ProC. IEE Convention on Long Distance Trans-missio:n by Waveguide, Part B Supplement, pp. 129-131, Jan.1959.

[21 L. R. Yavich, "The input impedanlce of stepped converters,"Rladiotechnika (USS'R), vol. 17, pp. 22-2.5, March 1962.

[3] L. Youing, "'St-epped-impedance tranisformers and filter proto-types," IRE Trans. Microwave T7heory Tech., vol. MTT-10,pp. 339-359, Sept. 1962.

[4] D.. C. Thorn and A. W. Straiton, "Design of open-ended micro-wave resonanit cavities," IRE Trans. Microwave Theory Tech.,vol. MTT-7 pp. 389-390, July 1959.

[5] N. C. Wenger And J. Smetana, "Hydrogen density measure-ments using open-ended microw.ave cavity," IEEE Trans.Instrum. Meas., vol, IM-21, pp. 105-i44, May 1972.

[61 G. S. P. Castle and J. Roberts, "A microwave instrument forcontinuous monitoring of the water- content of crude oil, Proc.IEEE, Vol. 62, pp. 103-108, Jan. 1974.

[7] J. A. Stratton, Electroma netic 7heory. New York: McGraw-'Hiill Pp. 205-215, 1944.

[8] C. J. F. Bottcher, Theory of Electrical Polarization. Amsterdam, T:he Netherlands: Elsevier, pp. 417419, 1952.

59] L. Lewin, "'The electrical constants of a material Ioaded withspherical partiples," Proc. IEE, vol. 94, pt. III, p. 6.5, 1947.

[10] D. A. G. Bruggeman, Ann. Phys., ser. 5, vol. 24, p. 636, 193.5.0[11] R. F. Harrington, -Time Harmonic Electromagnetic Fields. New

York: McGraw-Hill, 1961, pp. 219-222.[121 J. Van Bladel, Elect-romagnetic Fields. New York: McGraw-

Hill, 1964, p. 448.[13] R. F. Harrinigton, Time Harmonic Electromagnetic Fields. New

York: McGraw-Hill, 1961, pp. 73-76.[14] S. 5. Stulchly, M. A. Rzepecka a id M. A. K. Hamid, "A micro-

wave open-enided cavity as a void fraction monitor for organiiccoolants," IEEE Trans. Ind. Electron. Contr. Instrum., vol.IECI 21, pp. 78-80, May 1974.

[15] P. Bhartia an'd M. A. K. Hainid, "Field distribution in acentrally loaded circeular cavity, IEEE Trans. on NuclearScience, vol. NS-16' pp. 27-34, April 1969.

[161 A. A. Oliner, "A new class of reactive-wall waveguides for low-loss applications," URSI General Assembly, Ottawa, Canada,Aug. 18-28, 1969.

[17] S. J. Towaij and M. A. K. Hamid, "RAdiation by a dielectric-loaded cylindrical antenna withi a wall air gap, Proc. IEE, Vol.119, no. 1, pp. 48-54, Jan. 1972.

A Predictive Controller for Automatic Voltage Regulation

oF DC Generators -A ybrid Simulation Study

M. H. NEHRIR, MEMBER, IEEE

Abstract-A hybrid predictive controller to maintain the plaintoutput at the desired level is proposed. The controller consists of afast-time linearized analog model of the controlled plant and acontrol logic, which is programmed on the digital section of thehybrid computer.The controller is applied to a linearized second order model of a

de generating system and the desired control is successfullyachieved.

NOMENCLATUREA Input signal determined by the predictive con-

troller.Amax Mlaximurn input.Amin Minimum input.C Scaled output of the system.CD System output rate (C (t)).

Manuscript received July 9, 1974.The author iIs with the Department of Electrical Engineering,

Pahlavi University, Shiiaz, Iran.

CMI Output of fast model.CM1D Output rate of fast model.e Error in the system output.-E Threshold limit of error used for the linear region.F Threshold limit of error-rate used for the liWear

region.REF Reference value at which the system output is

desired to remain.Te Exciter timne constant.T0 Generator time con1stant.X Coefficient of e(t) in equation (1).Y Coefficient of e (t) in equation (1).

INTRODUCTIONRECENT literature [1]; [2] -indicates that a sigrnificant

industrial interest exists in the use of fast timesiinulation in control systems. It has also been recognized[3] [4] that fast-time simulation can provide the responseof a system to certain inputs before the actual response is

43:

44 IEEI E TRANSACTIONS ON INDUSTRIAL ELECTRONICS ANTD CONTROI, INSTRUMI!NTATION, FEBRUARY 1975

completed. This is called a predictive control technique, model of the system make repetitive J)redictions to seeThe first major application of this technique was presented where the system response will end up if the inut isby Chestnut et at. In his work the predictive control tech- switched at the present tine. The fast model of the systemn,nique was used to control the landing of an aircraft via a in this case, operates 10 -times faster than the actual sys-digital-computer, which was used in an on-line fashion. tem. Note that the switching point may only occur in theIrn nearly all the applications of predictive control pre- second or fourth quadrant of the phase plane where thesented so far a digital comiputer has been used. error and error-rate have different polarities. When the

In this paper control of a linearized second order model system trajectory is in the first or third quadrant, theof a dc generating system is achieved by a hybrid predictive syste i response is going away from the desired referencecontroller. The predictive controller consists of a fast-time (REF ) and the ilnput is chosen to drive the trajectory toscale linear analog model of the simulated system and a the fourth or second quadrant, respectively. Thus, repeti-control logic. The fast mrodel is simulated on the analog tive predictions made by the model are only necessaryportion and the control logic is programmed on the digital when -the system trajectory is in the second or fourthpart of the hybrid system. The fast model is used to predict quadrant.the time response of the controlled system, and the control In the prediction region the fast model is repeatedlylogic is responsible for choosing the necessary input to the given the same state as the system (em = ei,ei, eicontrolled system via the actuator which in this case is but forced in the opposite direction (Fig. 3) until thea digital t'o analog convertor. A general block diagram of model error-rate em reaches zero. At this time the statethe control system is given in Fig. 1. of the model (e,,O) is compared with the state of the

system (eije ). If the point where em is equal to zero, eP'REDICTIVE CJONTROL STRATEGY has the same polarity as ei, point S has not yet been

Assume that the system to be con'trolled is of second reached. Therefore, A keeps the same value, Amxax andorder. The state of the system is then described by the again the model is given the same state as the systemoutput and its derivative. lIet the systenm input be A. (em - ei, em ei+±), and the procedure is repeated.As in all physical systems the input is bounded such that At some prediction by the model, say the rth, er and emAmin < A . Amax. when em is equal to zero will have different signs. This

Consider the error, error-rate phase plane of the system means point S was reached between the (r 1)th andas showvn in Fig. 2, where system error (e) is equal to sys- the rth prediction. Therefore, A is switched from Amaxtem output (C), less desired reference (REF). Pontry- to Amin. The switching will always occur late, after theagin's inaximuin principle [7] says that the fastest way system trajectory has passed point S. This could beto get from any point B in the phase plane to the origin prevented by making the time constants of the modelis to let the system input be Amax or Amin as required, and slightly larger than their corresponding systenm timeit is only necessary to switch onrce (at point S in Fig. 2) constants [4]. The amiount by which the time constantbetween the txwo levels of input. This allows the state of should be increased depends upon the system parametersthe systenm to move alonig r, the only trajectory with and the prediction rate.Amax or Amin that goes through the origin. In practice, since a relay is used for input switching, theThe switching point S is found by letting the fast-time system never comes to rest because the system trajectory

never passes exactly through the origin. It ends up ina limit cycle. To avoid this, a small rectangular linearregion is chosen around the origin, anid when the systemtrajectory enters this region the system input will bedetermined as follows:

A -Xe + Ye% ()J roto le J - output:input Syste_ output where X and Y are constants and best chosen by trial

Fig. 1. Conltrol system bl)ck diagram. and error, as they depend directly on the particular system

e(t) e(t)11 1.Prediction

No Prediction (erir eS~~~~~~~~~~~~~~~~~~r1V ~ S Linear region

A-A ~~~~~~~~~~~~~~~~~~~~~(e 1,61)max 2

A=A. l 2F Pt

e(t)=c(t)-gREF

X, . ~~~~~~~~~~~~~~~~~~~~~~~2ELNo Prediction

Fig. 3. System aild model trajectories (model trajectoiies are shownFig. 2. Error, error-rate phase plaie of the system. by dotted lines).

NEERIR: PREDICTIVE CONTROLLER

at hand. The rectangular linear region is shown in Fig. 3,and it is understood that the constraint Amin < A < Am ax

must be satisfied in the linear region, too. Assuming thecontrolled system transfer function be G(S), the controlsystem may be represented, in the linear region, by theblock diagram of Fig. 4. Since the reference is a stepfunction, the steady state error beconmes

REF1 + lim G(s).* X + sY]'

REF(s) C(s)0- >,X+sy G(s)

Fig. 4. Control system ill the linear region.

(2)

Assuming that G (S) has no poles or zeros on the imaginaryaxis, and that the gain of the generating system is GAIN,it follows that

REF

1 + GAI:N*X (3)

Equation (3) indicates that the steady state error of thesystem depends on the value of X.

THE CONTROL SYSTEM SIMULATIONAssuming a second order linear model for the de

generator-exciter system at hand, the transfer function forthe controlled system becomes

K(1 + s7e) (I + sT,) (4)

Fig. 5. Computer flow chart for the control logic.where K is the system gain, Te and Tg are exciter andgenerator time constants, respectively. A real-time analogmodel of this system is used in conjunction with a fafsttimee model which is operating 10 times faster than thereal-time model. Also, the time constants of the fast modelare chosen 1.8 times higher than the actual system timeconstants. This was done to prevent the slight time delayin the switching of the input when both the error anderror-rate of the model reach zero [6]. The factor 1.S waschosen by trial and error, as it seemed to give the correctswitching time for a satisfactory control.The analog simulation of the linearized, real-time model

and the fast-time model of the system is a straightforwardtask. The flow chart for the control logic part of thecontrol system which is programmed on the digital partof the hybrid computer is given in Fig. 5.The overall system was disturbed by positive and nega-

tive step signals, which correspond, respectively, to suddendecrease and sudden increase in the load of the de generat-ing system, and it was desired that the scaled outputof the simulated plant remain at a constant 5-volt level.The results of the experiment appear in Fig. 6 and Fig. 7.Fig. 6 shows the system response when a positive distur-bance is applied to the system, and Fig. 7 shows theresponse for application of a negative disturbance. In bothcases the output (C) is controlled to the desired level aftera short transient. In both figures the response of thepredictive controller is clear from the output and output-rate response of the fast model. It is also clear that whena positive disturbance is applied the error becomespsietive, thus the controller chooses A = Amin for the

2

Vertical Scalesare in Volts

3

Fig. 6. Rlesponse of the coistrol system simulation wheni positivedistuirbanice is applied.

45

I46EEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, FEBRUARY 1975

10

8

6

4

0

10

8

1

2

Vertical Scalesare in Volts

4

5

Fig. 7. Response of the control system simulation when negativedisturbanCe is applied.

systeimn until it is time for the input to switch. Similarlywhen a negative disturbance is applied to the system asin Fig. 7, the input A A ax is chosen for the systemuntil the time of switching. However, in both cases theinput does not change its level from Amax to Amin or viceversa, because at the time of switching the system, trajec-ftory has already entered the linear region and the inputis chosen by the linear control, which was previouslyexplained.

CONCLUSIONS:A hybrid predictive controller, which makes use of a

special purpose hybrid computer for on-line operation,is: designed. The proposed controller is specially advanta-geous for its fast time response and its independence of theplant parameters. Indeed the control performance can be.made quite independent of-the plant parameters, such assystem time constant and system gain, which are mostlikely to change during operation. In cases where thechange in system parameters is not negligible the controller

may be made adaptive to compute the systemr parametersinstantaneously. In this case the controller is dependenton the parameters of the plant.The data presented here show the applicability of thet

controller at worst cases (sudden disturbance) and illus-trate that approximations are acceptable for the systemparameters to be used in the fast model.

-REFERENCES[1] C. R. Kelley, "Closing the loop with predictive controller,"

Control Engineering, May 1968.[2] A. T. Schooley and P. G. Adams, "Analog computer provides

adaptive-predictive control of batch reactor," Control Engi-neering, July 1969.

[3] J. Billingsley and J. F. Coales, "Simple predictive con-troller forhigher order systems," Proc. IEE, October 1968.

[4] F. Fallside and N. Thedchanamoorthy, "Predictive cIontrol usingan adaptive fast model," Proc. IEE, November 1967.

[5] H. Chestnut, W. E. Sollecito, and P. H. Troutman, "Predictivecontrol system application." AIEE Trans. (Application andIndustry), vol. 55, July 1961.

[6] W. G. Rae, "Fast model search control systems InternationalJournal of Control, October 1967.

17] S. L. Chang Sheldon, Synthesis of Optimal Control Systems. NewYork: McGraw-Hill, 1961, p. 381.

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