~--.-.... --. -.' --- ...- - - -.'-.....l, -~ I

D. E. BourneO. FigueiredoM. E. Charles 289James L. White 294H. A l-ShahristaniD. G. Andrews 299A. A. NicolJ. T. Mc I.ean 304J. D. FordR. W. Missen 309W. KozickiC. TiuA. R. K. Rao 313D. A. CyganP. D. Richardson 321P. BourgeoisP. Grenier 325K. V. S. ReddyR. J. FlemingJ. W. Smith 329R. D. VoyerA. T. Miller 335H. D. GoodfellowW. F. Graydon 342W. F. PetryschukA. 1. Johnson 34·81. H. S. HendersonS. G. Ladan. 3551. H. S. HendersonS. G. Ladan 361Theodore P. LabuzaMax Rutman 364

B. R. DickeyL. D. Durbin 369V. G. ChantR. Luus 376Pierre R. Latour 382 .>M. A. LusisG. A. Ratcliff 385

S. L. HaganD. A. Ratkowsky 387

The Canadian Journalof

Chemical EngineeringCONTENTS

Laminar and Turbulent Flow In Annuli of Unit Eccentricity

Motion of Continuous Surfaces Through Stagnant ViscousNon- ewtonian Fluids

Heat Transfer from Teflon-Treated Surfaces under FlowConditions

Boiling Heat Transfer from a Rotating Horizontal Cylinder

On the Conditions for Stability of Falling Films Subject to SurfaceTension Disturbances; the Condensation of Binary Vapors

Filtration of Non-Newtonian Fluids

A Transcendental Approximation for Natural Convection at SmallPrandtl Numbers

The Ratio of Terminal Velocity to Minimum Fluidizing Velocityfor Spherical Particles

Maximum Spoutable Bed Depths of Mixed Particle-Size Beds

Improved Gas-liquid Contacting in Co-current Flow

Dependence of Electrostatic Charging Currents on FluidProperties

The Mathematical Representation of a Light Hydrocarbon RefiningNetwork

The Preparation and Structure of Electrodeposited SpongeCadmium Electrodes

The Electrolytic "Sintering" of Nickel Powder

The Effect of Surface Active Agents on Sorption Isotherms of aModel Food System

Continuous and Discrete Time Response Analysis of the BackftowCell Model for Linear Interphase Mass Transfer on a Distil-lation Plate

Time Sub-Optimal Control of the Gas Absorber

On the Relation Between State and Adjoint Variable InitialConditions in Optimum Control Theory

Diffusion in Binary Liquid Mixtures at Infinite Dilution

Ole to the EditorLaminar Flow in Cylindrical Ducts Having Regular Polygonal

Shaped Cores

On the Relation Between State and Adjoint VariableInitial Conditions In Optimum Control Theory

PIERRE R. LATOUR'

School of Chemical Engineering, Purdue University, W. Lafayette, Indiana, U.S.A.

The algebraic relation between state and adjoint vari-able initial conditions for time-optimum control of a par-ticular second order linear system is reported. The relationbetween adjoint initial conditions and switching time sug-gests that difficulties might arise when boundary-valuesearch methods arc employed to solve two-point boundary-value problems. These analytical results support the con-clusions in a recent paper by Paynter and Barrkoff.

Manyof the difficulties involved in thc application of moderncontrol theory for the optimum dcsign and control of

chemical processe~ arc of a computational narurc"-". RecentlyPaynter and Bankoff!" emphasized that much more computingexperience with realistic problems will have to be accumulatedbefore these methods become routine tools of the chemicalengineer. The computational di fficulties arc associated withtwo-point boundary-value problems which result from a calculusof variations or Pontryagin's maximum principle formulationof the necessary conditions for the optimum function!".

The most natural method of attack, boundary-conditioniteration, involves guessing some initial conditions, 'integratingthe process and adjoint differential equations using a controlwhich satisfies the maximum principle to determine terminalconditions, and adjusting the initial conditions based upon thedeviation between the calculated and specified terminal con-ditions't-:". For the problem of optimum cooling jacket designon a nonlinear turbular reactor with recycle, Paynter and Bankofffound boundary-condition iteration to be notably unsuccessfulbecause of rhe inherent instability of the adjoint equations andthe multimodal nature of the terminal condition error contoursurface. The optimum forcing function was of bang-bang type.

In supporr of their results, this communication reports thealgebraic relation between state and adjoint variable initialconditions for time-optimum control of a particular second-order linear system. The optimum forcing function is also ofbang-bang type. The relation between adjoint initial conditionsand switching time suggests that difficulrics might arise ifnumerical search methods employing boundary-condition itera-tion are used to solve the two-point boundary-value problem inagreement with the results of Paynter and Bankoff.

Statement of the ProblemConsider ,1 process which has ovcrdamped second-order

dynamics for small excursions from steady-state(5,6,;) representedbv the differential equation for rhe process error

b eel) + (I + b) e(t) + eel) = r ~ ",.(t) ... (I )

-Present address: U.S. Army. Manned Spacecraft Center, NASA, Houston,Texas, U.S.A.

On presente la relation algebr-ique entre Ie regrme etIes conditions initiales, variables et accessoir es, pour Iemeilleur reglage du temps dans Ie cas d'un systeme lineaireparticuIier de second ordre, La relation existant entre lesconditions initiales et accessoires et Ie temps de declenche-ment indique qu'il peut surgir des diff'icrrltes Ioi-squtonutilise des methodes de recherche de la valeur-limite pourresoudr e des pr-oblernes comportant des valeurs-limites itdenx points; ces resultats analytiques corroborent les con-clusions exprrmees dans un travail recent fait par Paynteret Bankoff.

with given initial conditionsc(O) eo

e(O) eor - Cn ~ ()

wheree(f) - l' - e(t)

r = set pOlOt, piecewise constant value of the desiredprocess output, c(t); 0 < T < 1

met) = manipulated variable, the process Iflpur to be de-termined; 0 ;;; 7ll(t) ;;; I

b = dimensionless minor process time constant; () <b < I

and the dot denotes differentiation with respect to time, t.Dimensionless time, t, the ratio of real time to the major processtime constant, is used throughout. The optimum controlfunction, m*(t), which drives the output from Co to r (or efrom eo to zero) in minimum time is bang-bang at the extremevalues 0 or 1 allowed for m with at most one switch at time t,during the rransicntwv. From a design point of view, a secondswitch to the new steady-state 1IZ = r is needed when the finalstate (e = e = 0) is reached at the final time, (5, to hold theprocess at the new set point thereafter. U sing state-variablenotation in terms of the error state vector e' = (e,e)t, Equarion(1) becomes

~(t) ( 0 1) e..etl + (0) (I'1 1 + b 1--- -

=b =b b

~'(O) = (eo,Iio), ~'(t.) = (0,0)

, ,(2)-- met») ...

where the transition tune, t" IS to be minimized.svstcm IS

The adjoint

Yt) = (0 ~) ,,-(L)1 + b -

-1--b

'/I.' = ('/1.1,'/1.,)

.. (3)

382 'I'he CUI/adian [ournai of Chemical Engineering: Vol. 46, October, 1968

+The prime (,) denotes transpose; the bar (_) denotes column vector.

with initial and final conditions as yet unspecified. Maximizinga Hamiltonian

H = },,(t) e(t) + Alt) (- e(t) - (1 + b) e(t) + r - m(t)j&(.lj

leads to! 1, A2(t) < 0

m*(I) = 1l0, A,(t) > 0'

according to Pontryagin's maximum principle!", and A,(t,) = O.It can be shown that A2(t) has, at most, one zero-".

A(ljoint Boundary Conditions

Although there is as yet no explicit general relation for rhcadjoint boundary conditions A(O) or A(t.) in the Ponrryaginformulation of problems with specifi~d initial and terminalsYStem states (the fixed right-end problem), for this parricu larsystem using the known switching time and switching functionequations, a relation can be written between arbitrary :'0 andA(0) = Ao= (AIO,A20)', Search procedures have been suggestedby Knud~en(8), Gilbert(9), and Lewallen'v to satisfy all boundaryconditions for more complex optimization problems. T rom-betta(IO) reports that the solution for met) is often very sensitivero the assumed Au, and convergence is not assured (4,l0). Thissensitivity will b~ evident in the results obtained here, and forsome Ao,'m(t) never switches.

The solution for the homogeneous adjoint system is

(I - b) Al(t)/AIO = (I - Ano)exp(t) - (b - '\nO)expel/b) .. (6)

(1 - b) A2(t)/A10 = b(l - Ano)exp(t) - (b - Ano) exp(t/b) (7)

where.. (8)

An interesting conclusion from Equations (5) and (7) is thatboth values of AIOand A20are not needed to precisely determinem*(t) in a mathematical sense, only their ratio and the sign ofone is needed. There is only one free parameter, A"o, cor-responding ro the single switching time, t.. From the definitionof switching time A2(t,) = 0 with Equation (7):

(1 - Ano/b)exp( -1,(1 - b)/h) = A"

(1 - nO)

This one-co-one correspondence between Anoand t, is illustrated inFigure 1. To insure that t, ~ 0, the constraint 0 ~ AnO< b mustbe satisfied, a severe restriction on the allowable Ao. It followsthat AlOand A20have the same sign. The sensitivity of t, > 0to Ano is evident in Figure 1, particularly for large switchingtime. (The function of Equation (9) is graphed for t, < 0 forgeneral illustration and later use.) Notice when b < AnO< 1,t, has the form of a natural logarithm of a negative number,which has no real value. Rearranging Equation (9)

, _ exp( -t,) - exp( -t,/b) _".0 - -/(1,).b=' exp( -- I,) - exp( ... I,/b)

... (10)

su~gests that Ano is almost independent of t, as b approachesHlllry.

For arbitrary initial conditions, Co, the switching time canbe calculated directly!" from the implicit equation

(1 - r)(l-b) (eo + eo - r + exp(t,/b»b =

b e« + eo - r + exp(t,). .(11)

provided 111*(0+) = O. This is the equation for the locus ofpoints eo to the left of the phase plane switching curve with thesame time t, before arrival at the switching curve along optimumtrajectories. The equation was obtained directly from theswitching function for a second-order system with dead time(6. 7).

The switching function gives the locus of points which are adead time away from the final path to the phase plane origin.

(5)

IIIIIII______ ..J _

: bII

Fi~llre I-Relation between adjoint initial condition ratioand switching tirne,

A companion equation when m*(O+) = 1 to the right of theswitching curve is obtained by replacing r by 1 - r, e by -e,and e by - e. Substituting t, from Equation (9) gives the directrelation from eo co Ano:

(9 )

.. (12)

Since this relation is implicit, the difficulty of formulating ageneral explicit relation between Ao and eo is evident. If Anofrom this equation satisfies the constraint as-well, then m*(O+) =o and AIQ,A20> O. If the constraint is not satisfied, the companionequation can be solved for Ano, implying that 117*(0+) = 1 andAIO,A20 < o.

The ratio of the adjoint final conditions

.. (13)

provides a second parameter which can be associated witheach point :"0through a one-to-one relarion with the time on thefinal portion of thc trajectory

If == /.5 - /.,. (1-+ )

When evaluated at f = t. = t, + ff' the ratio of final conditions,A2(t)/Al(t) from Equations (6) and (7), contains t.; fj, andA"o. By substituting for A"o from Equation (9), on I)' t, and t,remain. However, t, divides om and the result is

, . = exp(lf) - exp(ldb)"n, = f( -If)· . (15)b=' exp(lf) - exp(ldb)

which is the same function as A"o,but with a negativc argument.The similarity between Equations (15) and (10) suggests thatthe relation between A"5 and If of interest for tf > 0 is shown inFigure I to the left of the origin, (that is, replace t, by - If).

The> Canadian Journal of Chemical Engineering, Vol. 46, October, 1968 383

Now a value of An 5 > 1 or < 0 can be assigned to each initialportion of optimum trajectories since all points on such a curvehave a common If. These initial trajcctory curvesv" are given by

[eo + eo - I' Jb b eo + eo - l'

-1 + (1 - 1') expCtdb) = -1 + (1 - r) exp(lf)' .(16)

Eliminating tf between Equations (15) and (16) gives

b eo + eo - r( 17)

-1 + (1

Companion equations exist when m*(O+) = I. Equations (12)and (17) constitute the mapping from eo to A"o, Ans. From adesign point of view, knowledge of tf- (or t5) is needed as astopping criterion for m*(t) to switch to the final steady-state711 = r. Hence, both Ano and Ans are needed for completeness.Also, it is possible to write the inverse mapping from Ano, A".\to eo explicitly.

-The singularity in An 5 at tf = (b - 1) -1 b In b occurs atthe intersection in the phase plane of the optimum final path(or switching line) and the slow eigenvector to an extremalnodc'" . For most initial conditions, including rest :'0 = (eo, 0),optimum trajectories have tf less than this value. In the limitas leol becomes arbitrarily large, initial trajectories from restapproach the slow eigenvector and hence the intersection singu-larity. These topological characteristics illustrate the nature ofthe computational difficulties that might be encountered.

For an ntb order linear process the n-parameter search forA(O) can be reduced to an n-l paramcter search plus deter-mination of the polarity of the nth parameter, provided that thefinal time is not of interest. This results because

m*(l) = (SGN A,)(SGN(A.(t)/A.)),. (18)

and the adjoint system is homogeneoust". This is in agreementwith Gilbert's statement'?' for ntb order linear systems with asingle manipulated variable, that m*(t) is independent of themagnitude of the zz-vector, ~o.

For time-optimum contro I of

x=Ax+bm- --(19)

Knudsen=' has shown that the general mapping ~o = i(~0,t5)'obtained from the general solution of Equation (19), is

t6.\'0 = - fexp(-At)bSGN(b,Aoexp(-A't))dt. .(20)- 0 - - -

wherem

(x,y) = 1: x, y,-- i<=<l

~(-AI)" .exp( -A t) = 1: ---, all nXn matrrx

n=O n

and A' is the transpose of matrix A. Knudsen gives someproperties of this mapping and shows that it is impossible ingeneral to evaluate the inverse mapping 0.0, Is = [(:::0) analyti-cally.

Conclusions

The relation (it is implicit) between the adjoint and stateboundary conditions for time-optimum control of a particularsecond-order system was obtained analytically. The twoadjoint initial conditions always have the same sign for positiveswitching time, and their ratio is directly related to switchingtime. The sign of these initial conditions determines the startingpolarity of the manipulated variable, and only the value of theirratio is needed to specify the optimum control function and solvethe problem in a mathematical sense. The final value of theirratio is needed to determine the final time and completelyspecify the optimum manipulated variable in a design sense.Since only one switch is required for the system under study,only one search for a single parameter in a restricted range isneeded. The relation between this ratio and switching time hasthe form of a decaying exponential over the positive real line,hence the sensitivity of the forcing function to the adjointinitial conditions as observed by Trombetta. In view of Figure1, a general search for adjoint initial conditions could often leadto difficulty.

The ratio of adjoint final conditions is directly related to theremaining time; but often, in the vicinity of the singularity,large changes in the final ratio would have little effect upon theremaining time. Selection of the adjoint boundary conditionsin certain ranges, b < Ano, An 5 < 1, leads to nonreal functionsfor switching and final times, and no switching in met) wouldbe expected. This behavior was exhibited in the numericalresults of Paynter and Bankoff.

AcknowledgmentsThe autbor expresses appreciation to L. B. Koppel for guidance and

direction during tbis work. The helpful suggestions of D. R. Cougbanowrare acknowledged.

Financial support from tbe National Science Foundation in the formof Graduate and Co- operative Fellowships and from Purdue Universityis acknowledged with thanks.

Nomenclaturebc(t)eel)met)

dimensionless minor time constant; 0 < b < 1process outputerror, deviation of output from set point; r - c(t)manipulated variable, the process input to be deter-mined; 0 ::S: m(t) ::S: 1set poin t, piecewise constan t desired val ue of c; 0 < r < 1dimensionless time, real time/major process timeconstantvalue of I at which switch in m*(t) occurstime on final trajectory from I, until e(l) = (0,0)';Equation (14) ,..,total duration of optimum transient; e(ls) = (0,0)'adjoint variable vector; (Al(I), A2(t)),-

Equation (8)Equation (13)

rI

References(I) Paynter, J. D. and Bankoff, S. G., Can. J. Cbem. Eng., 44, 340

(1966) and 45, 226 (1967).(2) Fan, L. T., "The Continuous Maximum Principle", Jobn Wiley,

New York, 1966.(3) Pontryagin, L. S., et a!., "Mathematical Theory of Optimal Pro-

cesses", John Wiley, New York, 1962.(4) Lewallen, J. M., "An Analysis and Comparison of Several Tra-

jectory Optimization Methods", Ph.D. Thesis, Univ. of Texas, 1966.(5) Koppel, L. B. and Latour, P. R., "Time Optimum Control of

Second-Order Overdamped Systems with Transportation Lag", Ind.Eng. Chern. Fund., 4, 463 (1965).

(6) Latour, P. R., Koppel, L. B. and Coughanowr, D. R., "Time-Optimum Control of Chemical Processes for Set-Point Changes",Ind. Engr, Chern. Proc. Des. and Dev., 6, 452 (1967).

(7) Latour, P. R., "Time-Optimum Control of Cbemical Processes",Ph.D. Thesis, Purdue Univ., 1966.

(8) Knudsen, H. K., "Iterative Procedure for Computing Time-OptimalComrols", Westcon Tech Papers (Auto. Contr., Computers, Info.Theory) , 7, pt. 4, 12.4 (1963).

(9) Gilbert, E. G., "Hybrid Computer Solution of Time Optimal Con-trol Problems", Spg. Joint Computer Conf., Detroit, p. 197, May1963) .

(10) Trombetta, M. L., "Optimal Design of Chemical Processes - Varia-tional Methods", Chern. Eng, Progr. Symp, Ser. 55, 61, 42 (1965).

Manuscript received December 12, 1967; accepted May 13, 1968.

* * *384 The Canadian Journal of Chemical Engineering, Vol. 46, October, 1968

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING

Volume 46, Number 5Devoted to the publication ofChemical Engineering Science

Industrial PracticeApplizd Chemistry

October1968

L. W. SHEMIL T, Editor

D. D. KRIST1VIA1\SO:'-J, Assistant Editor

The University of TCW Brunswick, Fredericton, N.B., Canada

R. H. CLARK, Queen's University,Kingsron, ant. '

]. L. CORj EILLE, Ecole Poly technique,~\!ontreal, Que.

W . .T. M. DOUGLAS, McGillUniversity, .\-!ontreal, Que.

D. ,\1. ED\VARDS, Nopco ChemicalCanada Limited, Hamilton, ant.

K EPSTEI)';, University of BritishColumbia, Vancouver, B.e.

EDITORIAL BOARD

H. K. RAE, Chairman

Atomic Energy of Canada Limited, Chalk River, ant.

H. B. MARSHALL, Domtar Limited,Senneville, Que.

.T. A. J'vlORRISON, National ResearchCouncil, Ottawa, ant.

I. S. PASTER1\A((, Imperial OilEnterprises Ltd., Sarnia, ant.

D. B. ROBINSON, University ofAlberta, Edmonton, Alta.

D. S. SCOTT, University af Waterloo,Waterloo, ant. .

Ex-Officio

K. J. '\IcCALLU,\J, President, TheChemical Institute of Canada

R. ORD, Chairman of the Board ofDirectors

.f. W. TO'\lECKO, Director ofPublications

,V. '\\. CA :\IPTlf:I.I., Chairman,Pu hlicarions Committee, e.S.Ch.E.

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