228 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-27, NO. 3, AUGUST 1980

VI. CONCLUSIONSA complete microprocessor-controlled variable frequency dc

3 0 thyristor inverter system has been investigated.Microprocessor programming facilitates different modes of

motor load control and also allows a simple method to achievethe turn-off sequence, motor reversing, and any necessarystart-up sequence.The tendered directed bridge commutated inverter facilitates

programmable regenerative braking and a commutation cycleindependent of the load and requires no special starting se-squence. The resultant bridge is simple, versatile, of low cost,and light in weight.

ACKNOWLEDGMENTThanks are due to A.L. Davis of the University of Adelaide

for his helpful comments. Thanks are also due to the Schoolof Electrical Engineering of the South Australian Institute ofTechnology for making available experimental and workshopfacilities.

REFERENCES

[1] J. D. Edwards, "Three-phase digital P.W. M. invertor," Proc. IEE,vol. 122, no. 3, pp. 302-304, Mar. 1975.

[21 T. Mazda, "A digital logic P.W.M. speed control for single andpolyphase a.c. motors," in IEEE Conf. Rec. Annu. Meeting Indus-trial Appli. Soc., pp. 1-9, Oct. 1973.

(31 J. M. D. Murphy, Thyristor control of A.C. Motors. Elmsford,NY: Pergamon, 1975, ch. 3,5,6, and 7.

[41 B. W. Williams, "Impulse-commutated thyristor chopper," Proc.IEE, vol. 124, no. 9, pp. 793-795, Sept. 1977.

[51 B. D. Bedford and R. G. Hoft Principles ofInverter Circuits. NewYork: Wiley, ch. 7, pp. 165-230.

Microprocessor Control of Position or Speed

of an SCR DC Motor DriveJ. B. PLANT, SENIOR MEMBER, IEEE, S. J. JORNA, AND Y. T. CHAN

Abstract-In silicon-controlled rectifier (SCR) dc motor drive control,it is desirable for a microprocessor to perform both the control lawcomputations and the logical functions of SCR firing. However, becauseof their limited real-time capabilities, present-day microprocessorscannot accommodate excessively lengthy control algorithms. Thispaper presents a method, suitable for microprocessor implementation,of controlling the position or speed of an SCR dc motor drive whichuses a half-wave single-phase supply. The state-space equations of thesystem are first obtained. The derivation of a simple control law isgiven next, followed by the stability and error analysis of the controller.Finally, it is shown from simulation experiments that the scheme iseffective in controlling a 3-hp dc motor.

I. INTRODUCTIONT HE microprocessor has recently found increasing applica-

tions in silicon-controlled rectifier (SCR)' dc motor drives.Its role may be either to replace the logic circuits that controlthe SCR firing angle or in some instances, see [1 ], for example,it performs in addition the control law computations. Since a

control cannot be applied until it is computed, there is in-variably a delay between the beginning of computation andapplication of the control. In the case of SCR control, if thedelay is too long, the time corresponding to a particular firingtime may have already passed. The controller must then eitherfire at a different time in the present cycle or wait for the

Manuscript received August 20, 1978; revised October 23, 1979.The authors are with the Department of Electrical Engineenng,

Royal Military College of Canada, Kingston, Ont., Canada.

correct time in the next control cycle. However, a suboptimalcontrol will result in both cases. Thus in order to avoid longcomputational delays, and because the presently availablemicroprocessors are slow (as compared to minicomputers),the control law, if to be implementable by a microprocessor,must be simple. Ideally, determining the necessary controlsshould require only table lookup operations and any computa-tions should be kept to a minimum.This paper presents a method of controlling the position and

speed of an SCR dc motor drive using a half-wave single-phasesupply. A cost function of the errors (between desired andactual) in position and speed is first formulated. Next, theerrors are expressed as a function of ia, the average armaturecurrent supplied during the cycle. The controller determinesi*, the value which minimizes the cost function. The chosenia* is entered into a table lookup consisting of values of ia andtheir corresponding firing angles with motor back electromotiveforce (EMF) as a parameter. The desired firing angle is thenextracted from the Table. The control algorithm, because ofits simplicity, meets the real-time requirement, and its imple-mentation is well within the capability of the present-daymicroprocessor.Section II contains the derivation of the system's state-space

equations and the solution of the control problem. It isfollowed by the stability and error analyses of the controller inSection III; and the simulation results are in Section IV andthe conclusions in V.

PLANT et al.: POSITION OR SPEED OF SCR-DC MOTOR DRIRE

Fig. 1. SCR dc motor drive. Fig. 2. The control cycle.

II. SYSTEM MODEL AND CONTROL LAWFig. 1 depicts an SCR dc motor drive with a half-wave

'single-phase supply. The motor has a separate field excitationand the back-to-back connection of the SCR's allows for bothmotoring and braking actions. The symbols in Fig. 1, togetherwith the motor constants, are defined by

L A armature inductanceR A armature resistanceEb A back EMFJ A moment of inertia at motor shaft

Kt A motor torque constantKb A motor back EMF constantB A viscous friction coefficientCA coulomb friction coefficienti A armature currentN A output gear ratio

V sin cot A supply voltage of magnitude V and frequency co0 A angular position of load.

It is well known [2] that in the single-phase half-wave con-figuration, the current flow is discontinuous between cyclesduring motoring. However, in the case of braking, because thesupply voltage and back EMF are of opposite polarity, thecurrent can become continuous for a certain range of firingangles. The subsequent work is based on the assumption thatthe current is discontinuous; and, to ensure its validity, thecontroller is forbidden to select, during braking, firing anglesthat produce continuous current flow. Clearly, this is not asevere restriction. The motor simply takes a little longer tobrake in some instances.The control cycle (or interval), defined as one cycle of the

supply voltage, is divided into three regions as shown inFig. 2 where

ct A the firing time (also referred to as the firingangle) measured from the zero crossover ofthe supply voltage

,A the extinction time measured from zero cross-over

X A ca-fl, the time through which current flowsRegion I A region before firing and current flowRegion II A current flow regionRegion III A region between end of current flow and

beginning of the next control cycle.In Region II, the circuit and motor equations are given,

respectively, by

L d + Ri + Kb 0 = V sin ot

and

JO + BO + C sign (6) = Ktiwhere

1, 6>o

1, 6<O.The state-space equations are3 X 1 vector

XI(t) NO(t1X = x2(t) = 0(t)

x3(t)j i(t) j

Then (1) and (2) become

*(t) = Fx(t) + Gu(t)

y(t) = Hx(t)where

F=I

O N

B

0 -KbL

0

KtJ

RL

obtained by first defining a

(3)

0

C

J

ol

0

1

L

,H= [1 0 O]

and

u(t) -[sign (x2(t))]V sin cot

In Regions I and III of Fig. 2, where no current flows, the stateX3 is simply deleted from x to give

x(t) =Fx(t) + G u(t)

y=Hx(t) (4)where

t=[1 =[NO(t F N =

(1)

(2)

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230 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-27, NO. 3, AUGUST 1980

and Let

u(t) = sign (x2(t)).

Given a, and the initial condition of the state vector x atthe beginning of the control cycle, the solutions of (4) inRegion I, then (3) in Region II and then (4) again in RegionIII will give the value ofx at the end of the control cycle. Thetime when Region II ends is given by ,B, which, in a computersimulation and in practice, is the time when x3(t), the currentstate, goes to zero. Finding ,B in this manner requires theintegration of (3) through many small intervals in Region II.A less time consuming alternative uses a binary search techniqueto determine the a that makes X3 zero. Then (3) is integratedin one step from a to , to cover Region II. This latter approachwas used to obtain the table lookup below.We proceed next to the derivation of the control law. The

prime consideration here is that the computations be minimalso that the controller is realizable online by a microprocessor.To this end, the model given by (3) is reduced to a second-order one by treating the motor plus armature circuit as anonlinear system with the angular velocity and position asstates and the coulomb friction and armature current as inputs.Then (3) becomes

0 NI[x(t)1J2) 0 -rJ[X2(t)J

+ [j sign (X2(t)) + []i(t) (5)

B [x1(t)] _NO(t)where r = - and [ I ] The discrete solution [3]

Ux2(t).J.[(t) Jof (5) is given, assuming x2(t) does not change sign from kTto(k+ I)T,by

Xk+ 1 = IXk - h(Xk) + g(Xk, ak) (6)where Xk denotes the state vector at time kT, and ak is thefiring angle in the kth control cycle and where

[=Lo

-(1 - exp (-TT))

exp (-rT) J

C- Signi (X2k)

N-N(

h(xk)= N

and

N--{ - exp (-r(T- 6))}

g(xk, atk) = I L exp (-r(T- 6)) () (9)

The extinction time at the kth control cycle is denoted by

and clearly, it is a function of ak and of X2k, which determinesthe back EMF.

Pkexp (-(r(T - 6)) i(6) dS

k

rPki(6) dS

k

(10)

Since ak < 6 < 1k < T/2 < T, exp (-r(T - 6)) is close to aconstant over the interval Cl k .6 .< 1k so that (10) is approxi-mately given by

ek ; exp (-T(T- 6) t O,ka <6 <3k (11)where 0 < e < 1 is a constant. The validity of (11), as well asthe value of e, depend on r, the inverse of the motor's mechan1ical time constant. In general, the smaller XT is, the better is theapproximation (11) and the closer is e to unity. (But neverequals unity unless r = 0!) For example, let r = 0.025 s-.Then the two extreme values of Ek at 6 = ak = 0 and 6 = PkT/2 are, respectively, 0.99958 and 0.99979 for T = 6 s.Using the approximation ( 1) in (8) gives

-(1 -

g(xk, atk) = KtT

where

IJf klak=T i(S)dS

Tk

E)

i(12)

(13)

is the average armature current supplied during the kth con-trol cycle, and is a function of ak and X2k. Further, let p =

NIr (1 - e), a 2 X I vector, and define

ik= lak (14)

as the control function; then (12) becomes

(15)Consider the cost function

() Jk+ 1 = (Xk+ 1 - Xk+ 1 0+1X+1)(16)where xk+ 1 is the known desired state vector at (k + I)T and

'1 0(8) ;0 q w a positive semidefinite weight matrix, and q is

any number >0. Substitution of (15) into (6) then into (16)gives

Jk+ I 21 Xk+ 1 4Xk + h (Xk) -pk)TQ(Xk+ 1

- 'IXk + h (Xk) - Pik )-The control function ik that minimizes Jk+ I is obtained bysetting the derivate of Jk* 1 with respect to ik to zero. Theresult is

ikTQ

+ h(Xk)]1k = rxk+i1 - 1?Xk +hk).(18)

9(Xk, Ctk) . Pik -

(17)

PLANT et al.: POSITION OR SPEED OF SCR-DC MOTOR DRIVE

Fig. 3. Controller structure.

The strategy for the on-line controller is to calculate ik* from(18) and to enter ik and X2k into a table lookup and extractfrom it 1k, the desired firing angle that minimizes (16). Thetable lookup is constructed offline by evaluating (13) and (14)for a sufficient set of iak and ak with X2k as a parameter. Asnoted earlier, solution of (13) requires 1k which is found by abinary search technique.The selected ax from the table must satisfy the constraints

that i) the SCR can turn on only when forward biased and ii)in braking, current flow must cease before the start of thenext control cycle. Mathematically, constraints i) and ii) limita in the range

-sin- <.a - 1 --sin (9

TiKbx2 T 1 i1T Kbx 9)

for motoring and

a <T- X -T sin-1 ( bX2 (20)

for braking. The table lookup accommodates these constraintsby storing the maximum (minimum) allowed value when agreater (lesser) value of a is needed for certain ik.The structure of the controller is illustrated in Fig. 3. There

are two nonlinear operations in its implementation: one isrepresented by h(xk) which accounts for the stiction non-linearity of the motor; the other is represented by the tablelookup. The controller is suboptimal for two reasons. Thetable lookup, by necessity, stores only a quantized approxima-tion of the continuous function (13). Secondly, the use of(11) to give (15) is also an approximation. However, resultsin Section IV indicate that there are minimal differencesbetween the optimal and suboptimal responses.

III. CONTROLLER ANALYSISThis section examines the stability of the controller as well

as its performance in the steady-state under step and rampinputs. When the desired firing angle a exceeds the constraintsin either (19) or (20) and can therefore only take on the maxi-mum or minimum allowable values, the controller is said tobe in saturation (or in the large). Conversely, the controlleris not saturated (or in the small) if the desired a is within theconstraints (19) and (20). Stability in the large can be demon-strated if the element q in the weight matrix Q is zero. For ifthere is no velocity feedback (equivalent to q = 0), the systemin the large consists of a bang-bang controller driving a second-order system with unity feedback as shown in Fig. 4. Thissystem can easily be shown, by describing function analysis[31, to be stable. This follows because the Nyquist plot of the

Fig. 4. Saturation with only angular position feedback.

second-order system never intersects the negative real axis,where the locus of the inverse of the describing function of thenonlinearity lies. When q = 0, which is normally the case, theabove analysis does not hold. However, since velocity feedbackserves to increase the stability, one would expect the systemto remain stable. Indeed, this conjecture is borne out by thesimulation results in the next section.The stability in the small is determined by the eigenvalues of

the closed loop system. The closed loop state-space equationis obtained, by substituting (18) into (1 5) and using the resultin (6), as

Xk+ 1 =AIXk+ pTQp [Pk+ 1 - 4IXk + h(xk)] - h(xk).

Let

DD=PTQ I1 p2

pMTQp p2 +q pp

where

N(L1 -ep

pq1

qJ

(21)

(22)

(23)

Then (21) becomes

Xk+ 1 = (I - M)Xk +Mk+ 1 - (I- M)h(Xk).Let

A = exp (-rT).

Then the eigenvalues of the matrix (I - M)F are given by

zl =0 and Z2 = p2 +q{+P ()}.

(24)

(25)

(26)

The system is stable if zI and Z2 have magnitudes less than orequal to unity [3]. The design parameters e and q musttherefore be chosen to give Iz21 - 1. The weight q can beassigned any value from zero to infinity. However, recall from(11) and the subsequent discussion that e is a constant used toapproximate the range of values exp (-i-(T - 8)), 0 < 6 6 T/2.Thus the choice of e is limited. The design procedure is tofirst select a desired Z2 and e and solve for q from (26). As anaside, a negative eigenvalue (for example, Z2 = -0.9), althoughstable, was found to produce excessive "chatter" in thesteady-state velocity response and should therefore be avoided.We stress again that the "linear in the small" model for

which the eigenvalues in (26) were developed is an approxima-tion. Indeed, in reality and in the simulations, the "actual"e varies depending upon the duration of the current inputs.Hence the design point (or nominal) eigenvalues are onlyapproximations to the actual. To complete the stabilityanalysis, one should consider the sensitivity of Z2 with changesof e by evaluating

231

232 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-27, NO. 3, AUGUST 1980

dZ2 pN '2Z2- it (27)de re2(p2 +q) {2 1 _ e)

at the nominal Z2 and e. The upper and lower bounds on thevariations in e can be found by evaluating (10) at the extremevalues of a and ,B. These two bounds can then be used con-secutively in (27) to compute the corresponding bounds onthe z2 variations. For stability, the magnitudes of the Z2bounds must be less than or equal to unity. Notice that in(27), as q -e oo, dz2/de -* so that as q increases, the approxi-mate model responses will converge towards the actual modelresponses. This aspect is confirmed by simulation results.To obtain an expression for the error state vector, let this

vector be

ek =Xk - Xk ek+1 =Xk+ xk1 (28)etc., and subtract (24) from £k+ 1 to give

ek+1 = (I - M)4.ek + (1- M)[jk+ 1 - + h(Xk)I. (29)

For step inputs, Xk+ 1 = Xk for all k and 0k+ 1 - 4Fxk = 0 sincex2k, the desired velocity, is zero. Therefore, (29) becomes,in the steady state

ess = (I - M)I'e. + (I - M)h(Xk) (30)

where e. represents the step steady-state error vector and isgiven by

ess = [I - (I -M) I-' (I - M)h(Xk). (31)

Carrying out the matrix manipulations in (31), one obtains

-pq (32)SSpN(l )j ,2jhx)where xss represents the steady-state value of Xk. For a rampinput

Xk+ I=rXk (33)where

L NT(Dr =° 1

which implies that

Xk+ 1 - (Xk = (Dr (F)Xk. (34)Substituting (34) into (29) and by using (30) and (31), theramp steady-state error vector is

esr = ess + [I - (I - M)(D]-1 (I - M)Xss (35)where X^. is the desired steady-state vector. Again, carryingout the matrix operations in (35) gives

Fq ferT- (I )}I (1esr. es +36(1erT- (1 - i)} (36)ss

In (32), e. is zero if h(x,) is zero. This is to be expectedsince h(xSs) is given in (8) as a function of sign (x2) and ifthere is an eSs, the controller will continue to fire the SCR tocorrect for it. Thus a zero ess implies a stationary motor. For

the set of realistic motor parameters used in the simulationsbelow, both (32) and (36) predicted negligibly small errors andthis is confirmed by simulation. However, it is conceivablethat the errors can be large for some other situations.

IV. SIMULATION RESULTSThis section presents the simulation results to verify some of

the theoretical findings in Sections II and III. The programswere written in Fortran and executed on a PDP-11 computer.The pertinent simulation data are

AC Supply 120-V rms 60-Hz single phaseDC Motor 3 hp

-=52.44 92/HL

-t = 10.48 N * m/AJ

B = 0.025 s-1

- = 1 1.29 s-2J

Kb = 0.53 V s/rad

N=0.1

L = 0.031 H.The inductance L in the armature circuit contains an addedinductance which serves to limit a large instantaneous currentflow.In the first experiment, a unimodal golden section search

[4] was employed to find the optimal firing time a* thatminimizes the cost function in (16). The control is optimalbecause neither the approximation (11) nor a table lookupwas used. The optimal responses serve as a basis for comparisonwith those of the suboptimal controller. The weight qNquoted in the subsequent figures is equal to q X 1.6 X 106.The number 1.6 X 106 is a normalizing factor which ensuresthat position and velocity measurements, as seen by the costfunction (16), are of the same order of magnitude. Fig. 5contains the two optimal responses, for qN = 0 and qN = 6, toa unit step input. They confirm that i) the controller isstable and ii) the system becomes less oscillatory with increas-ing q. The nonlinear behavior of the system is evident in thechange in damped frequency of oscillation as time increases.Fig. 6 is a trace of the ramp response and shows that there isno steady-state ramp following error.The next experiment investigates the properties of the

suboptimal, but simpler controller. First, we found from (10)that ek, for the set of parameters used, varies from 0.9996 to0.9999. We chose e = 0.9998 and qN = 6. Then from (27),Z2 is found to vary from 0.0472 (at e = 0.9996) to 0.476(at e = 0.9999). The system is therefore stable. The tablelookup consists of two 25 X 15 matrices, one each for motoringand braking. The row index is given by ik in (14), in steps of0.1 rad/s2, from 0 to 2.5 rad/s2. The column index is given byX2, in steps of 4 rad/s, from 0 to 56 rad/s. Fig. (7) is the unit

PLANT et al.: POSITION OR SPEED OF SCR-DC MOTOR DRIVE

1. 4

1.05

.73

.35.

Secs

0.6 1.2 1.8

(a)

2.4 3.0 3.6 I

0.6 1.2 1.8 2.40

Fig. 7. Suboptimal step response qN = 6.

0.6 1.2 1.8

(b)

Secs

2.4 3.0 3.6

Fig. 5. Optimal step responses. (a) gN = 0. (b) qN = 6.

4.0-

2 . 0

step responsFig. 5. Figsteady-statethe optimal c

0 0.6 1.2 1.8 2.4 3.0 3.6

Fig. 8. Suboptimal ramp response qN = 6.

experiment the system was given a negative but stable eigen-value, i.e., Iz < 1. The response turned out to be excessively"chatter" and eigenvalues (all real in this problem) that arenegative, even though their magnitude is less than unity, shouldtherefore be avoided.

V. CONCLUSIONSA simple position or speed controller, suitable for half-wave

single-phase SCR dc motor control and realization by a micro-processor, has been presented. Briefly, the design steps are

1) Compute from (10) the range of possible ek'S.2) Select an e within the range. We found the average of

the lower and upper bounds on Ek is an acceptable choice.Secs 3) Select a stable eigenvalue Z2 and compute from (26) theI~~~~~~~~~Irqurdwegtq0.6 1.2 1.8 2.4 3.0 3.6 required weight q.

Fig. 6. Optimal ramp response qN~=6. 4) Check, using (27), that the eigenvalues remain stablewhen e is varied within the range. If not, simply increase

e and it is very similar to the optimal response inq and repeat this step. Of course, the nominal z2 is no

8 is the ramp response and except for a small longer the same as the one selected in Step 3.foffowing error, the response is again very close to 5) Construct offline the table lookup by using (13) and (14),follo winFig error, thepnssaaieyclsotogether with the constraints (19) and (20).ne Fig. 6.

Several other experiments were performed [51 but notreported here. In one experiment, an unstable eigenvalue waspurposely selected. The response was indeed unstable in thesa bu t sy e rm n st l i te ag I an hsmall but the system remained stable in the large. In another

Simulation experiments have demonstrated the effectivenessof the controller and confirmed some of the theoretical find-ings. Because of its simplicity (equation (18) requires fourmultiplications by a constant, five additions, and one logical

1.6 -

0.8

0.4

1.4 -

1.05-

ui)

..d

to

i- N

Secs

3.0 3.6

.70-

. 3 5-

n-

u)c

.,2a0;i

233

(11:et

*,i10f(W.

Secs

234 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-27, NO. 3, AUGUST 1980

decision), the control scheme is a promising candidate forimplementation in a microprocessor as a direct-digital real-timeon-line system.

REFERENCES[1] A. K. Lin and W. W. Koepsel, "A microprocessor speed control

system," IEEE Trans. Ind. Electron. Contr. Instrum., vol. IECI-24,pp. 241-247, Aug. 1977.

[21 A Kusko, Solid-State DC Motor Drives. Cambridge, MA: M.I.T.Press, 1969.

[3] K. Ogata, Modem Control Engineering. Englewood Cliffs, NJ:Prentice-Hall, 1970.

[4] D. A. Pierre, Optimization Theory with Applications. New York:Wiley, 1969.

[51 S. J. Jorna, "A direct digital position control algorithm for anSCR-dirven dc motor," M. Eng. thesis, Department of ElectricalEngineering, Royal Military College of Canada, Kingston, Ont.,Canada, 1978.

A Performance Study of Control Systems withDead Time

CHANG C. HANG, MEMBER, IEEE, C. H. TAN, AND W. P. CHAN

Abstract-This paper discusses the performance of the Smith DeadTime (SDT) controller for systems with appreciable dead time. Tuning,modeling accuracy, and the effects of parameter variations and measure-ment noise are all studied using a simulated second-order process withdead time. The corresponding performance of the conventional propor-tional plus integral (PI) controller is used as a reference for comparison.It is shown that the SDT controller yields improved results over the PIcontroller if the plant is fairly stationary and the noise level is moderate.Otherwise adaptive filtering and control will be needed to maintain thesuperiority of the SDT controller over the PI controller.

I. INTRODUCTIONI N THE PROCESS industries the occurrence of "dead time"

or "transportation lag" is quite common [11, [2]. Forsimple control loops like flow, pressure, and temperature, theamount of dead time is usually not significant when comparedto the system time constant. For more complicated controlloops like those for quality control, dead time is very signif-icant and may be even longer than the system time constant,due for instance to the down-stream location of the samplingpoint and the analyzer delay. Another class of example ischaracterized by a multitude of small lags such as a long bankof heat exchangers, giving rise to what is called "apparent"dead time [5].The difficulties caused by dead time in control systems have

been recognized for a long time. In a recent paper [3] it wasshown that for most practical control systems in which thedead time is not dominant and where speed is not critical,proportional plus integral (PI) control is very effective. Forsystems in which the dead time is appreciable and where highperformance is required, typically for quality control loops, PI

Manuscript received October 6, 1978.The authors are with the Department of Electrical Engineering, Uni-

versity of Singapore, Kent Ridge Campus, Singapore 0511, Singapore.

TABLE IMODEL OF A QUALITY CONTROL LooP

Parameter K r(min) 6(min)

Average 0.57 8.6 18.7

Maximum 0.93 11.0 24.0

Minimum 0.38 6.0 15.0

K exp(-s6)Transfer function of viscosity to top reflux =-e s

(1 + s)

control may not be good enough and other advanced control-lers would have to be considered.The increasing use ofprocess computers for improved control

performance arouses an interest in implementing adaptive con-trol. Previous attempts at using adaptive control with conven-tional controllers have suffered setbacks owing to insufficientincentives [41 for the increased system complexity. With pro-cess computers it is also easier to implement advanced controlschemes such as the Smith Dead Time (SDT) controller [1],[2]. Since more precise process knowledge is required fortheir implementation, these advanced control schemes wouldbe more sensitive to parameter variations and incentives foradaptive control may be higher than in the past.In this paper we shall center our study around a viscosity

control loop for a high vacuum distillation column. Themathematical model was obtained experimentally by a previousauthor [4] using pseudorandom binary sequences as probesignals. It was found that the transfer function between theviscosity and the reflux flow is adequately modeled by a deadtime plus two first-order lags and that the parameters wouldchange by 2 to 3 times as shown in Table I. In this study only

0018-9421/80/0800-0234$00.75 © 1980 IEEE