1. ~--.-.... --. -.' --- ...- - - -.'-.l
, -~ I
D. E. Bourne
O. Figueiredo
M. E. Charles 289
James L. White 294
H. A l-Shahristani
D. G. Andrews 299
A. A. Nicol
J. T. Mc I.ean 304
J. D. Ford
R. W. Missen 309
W. Kozicki
C. Tiu
A. R. K. Rao 313
D. A. Cygan
P. D. Richardson 321
P. Bourgeois
P. Grenier 325
K. V. S. Reddy
R. J. Fleming
J. W. Smith 329
R. D. Voyer
A. T. Miller 335
H. D. Goodfellow
W. F. Graydon 342
W. F. Petryschuk
A. 1. Johnson 34·8
1. H. S. Henderson
S. G. Ladan. 355
1. H. S. Henderson
S. G. Ladan 361
Theodore P. Labuza
Max Rutman 364
B. R. Dickey
L. D. Durbin 369
V. G. Chant
R. Luus 376
Pierre R. Latour 382
.>
M. A. Lusis
G. A. Ratcliff 385
S. L. Hagan
D. A. Ratkowsky 387
The Canadian Journal
of
Chemical Engineering
CONTENTS
Laminar and Turbulent Flow In Annuli of Unit Eccentricity
Motion of Continuous Surfaces Through Stagnant Viscous
Non- ewtonian Fluids
Heat Transfer from Teflon-Treated Surfaces under Flow
Conditions
Boiling Heat Transfer from a Rotating Horizontal Cylinder
On the Conditions for Stability of Falling Films Subject to Surface
Tension Disturbances; the Condensation of Binary Vapors
Filtration of Non-Newtonian Fluids
A Transcendental Approximation for Natural Convection at Small
Prandtl Numbers
The Ratio of Terminal Velocity to Minimum Fluidizing Velocity
for 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 Fluid
Properties
The Mathematical Representation of a Light Hydrocarbon Refining
Network
The Preparation and Structure of Electrodeposited Sponge
Cadmium Electrodes
The Electrolytic "Sintering" of Nickel Powder
The Effect of Surface Active Agents on Sorption Isotherms of a
Model Food System
Continuous and Discrete Time Response Analysis of the Backftow
Cell 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 Initial
Conditions in Optimum Control Theory
Diffusion in Binary Liquid Mixtures at Infinite Dilution
Ole to the Editor
Laminar Flow in Cylindrical Ducts Having Regular Polygonal
Shaped Cores

2. On the Relation Between State and Adjoint Variable
Initial 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 relation
between adjoint initial conditions and switching time sug-
gests that difficulties might arise when boundary-value
search 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 modern
control theory for the optimum dcsign and control of
chemical processe~ arc of a computational narurc"-". Recently
Paynter and Bankoff!" emphasized that much more computing
experience with realistic problems will have to be accumulated
before these methods become routine tools of the chemical
engineer. The computational di fficulties arc associated with
two-point boundary-value problems which result from a calculus
of variations or Pontryagin's maximum principle formulation
of the necessary conditions for the optimum function!".
The most natural method of attack, boundary-condition
iteration, involves guessing some initial conditions, 'integrating
the process and adjoint differential equations using a control
which satisfies the maximum principle to determine terminal
conditions, and adjusting the initial conditions based upon the
deviation between the calculated and specified terminal con-
ditions't-:". For the problem of optimum cooling jacket design
on a nonlinear turbular reactor with recycle, Paynter and Bankoff
found boundary-condition iteration to be notably unsuccessful
because of rhe inherent instability of the adjoint equations and
the multimodal nature of the terminal condition error contour
surface. The optimum forcing function was of bang-bang type.
In supporr of their results, this communication reports the
algebraic relation between state and adjoint variable initial
conditions for time-optimum control of a particular second-
order linear system. The optimum forcing function is also of
bang-bang type. The relation between adjoint initial conditions
and switching time suggests that difficulrics might arise if
numerical search methods employing boundary-condition itera-
tion are used to solve the two-point boundary-value problem in
agreement with the results of Paynter and Bankoff.
Statement of the Problem
Consider ,1 process which has ovcrdamped second-order
dynamics for small excursions from steady-state(5,6,;) represented
bv 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 et
Ies conditions initiales, variables et accessoir es, pour Ie
meilleur reglage du temps dans Ie cas d'un systeme lineaire
particuIier de second ordre, La relation existant entre les
conditions initiales et accessoires et Ie temps de declenche-
ment indique qu'il peut surgir des diff'icrrltes Ioi-squton
utilise des methodes de recherche de la valeur-limite pour
resoudr e des pr-oblernes comportant des valeurs-limites it
denx points; ces resultats analytiques corroborent les con-
clusions exprrmees dans un travail recent fait par Paynter
et Bankoff.
with given initial conditions
c(O) eo
e(O) eo
r - Cn ~ ()
where
e(f) - l' - e(t)
r = set pOlOt, piecewise constant value of the desired
process 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 process
time constant, is used throughout. The optimum control
function, m*(t), which drives the output from Co to r (or e
from eo to zero) in minimum time is bang-bang at the extreme
values 0 or 1 allowed for m with at most one switch at time t,
during the rransicntwv. From a design point of view, a second
switch to the new steady-state 1IZ = r is needed when the final
state (e = e = 0) is reached at the final time, (5, to hold the
process at the new set point thereafter. Using state-variable
notation 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.

3. with initial and final conditions as yet unspecified. Maximizing
a 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) = 1
l0, 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 rhc
adjoint boundary conditions A(O) or A(t.) in the Ponrryagin
formulation of problems with specifi~d initial and terminal
sYStem states (the fixed right-end problem), for this parricu lar
system using the known switching time and switching function
equations, a relation can be written between arbitrary :'0 and
A(0) = Ao= (AIO,A20)', Search procedures have been suggested
by Knud~en(8), Gilbert(9), and Lewallen'v to satisfy all boundary
conditions for more complex optimization problems. T rom-
betta(IO) reports that the solution for met) is often very sensitive
ro the assumed Au, and convergence is not assured (4,l0). This
sensitivity will b~ evident in the results obtained here, and for
some 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 that
both values of AIOand A20are not needed to precisely determine
m*(t) in a mathematical sense, only their ratio and the sign of
one is needed. There is only one free parameter, A"o, cor-
responding ro the single switching time, t.. From the definition
of 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 in
Figure 1. To insure that t, ~ 0, the constraint 0 ~ AnO< b must
be satisfied, a severe restriction on the allowable Ao. It follows
that AlOand A20have the same sign. The sensitivity of t, > 0
to Anois evident in Figure 1, particularly for large switching
time. (The function of Equation (9) is graphed for t, < 0 for
general 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 approaches
Hlllry.
For arbitrary initial conditions, Co, the switching time can
be 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 of
points eo to the left of the phase plane switching curve with the
same time t, before arrival at the switching curve along optimum
trajectories. The equation was obtained directly from the
switching function for a second-order system with dead time(6. 7).
The switching function gives the locus of points which are a
dead time away from the final path to the phase plane origin.
(5)
I
I
I
I
I
I
I
______ ..J _
: b
I
I
Fi~llre I-Relation between adjoint initial condition ratio
and switching tirne,
A companion equation when m*(O+) = 1 to the right of the
switching curve is obtained by replacing r by 1 - r, e by -e,
and e by - e. Substituting t, from Equation (9) gives the direct
relation from eo co Ano:
(9 )
.. (12)
Since this relation is implicit, the difficulty of formulating a
general explicit relation between Ao and eo is evident. If Ano
from this equation satisfies the constraint as-well, then m*(O+) =
o and AIQ,A20> O. If the constraint is not satisfied, the companion
equation can be solved for Ano, implying that 117*(0+) = 1 and
AIO,A20 < o.
The ratio of the adjoint final conditions
.. (13)
provides a second parameter which can be associated with
each point :"0through a one-to-one relarion with the time on the
final 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, and
A"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 that
the relation between A"5 and If of interest for tf > 0 is shown in
Figure I to the left of the origin, (that is, replace t, by - If).
The> Canadian Journal of Chemical Engineering, Vol. 46, October, 1968 383

4. Now a value of An 5 > 1 or < 0 can be assigned to each initial
portion of optimum trajectories since all points on such a curve
have 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 a
design point of view, knowledge of tf- (or t5) is needed as a
stopping criterion for m*(t) to switch to the final steady-state
711 = 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 at
the intersection in the phase plane of the optimum final path
(or switching line) and the slow eigenvector to an extremal
nodc'" . For most initial conditions, including rest :'0 = (eo, 0),
optimum trajectories have tf less than this value. In the limit
as leol becomes arbitrarily large, initial trajectories from rest
approach the slow eigenvector and hence the intersection singu-
larity. These topological characteristics illustrate the nature of
the computational difficulties that might be encountered.
For an ntb order linear process the n-parameter search for
A(O) can be reduced to an n-l paramcter search plus deter-
mination of the polarity of the nth parameter, provided that the
final 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 agreement
with Gilbert's statement'?' for ntb order linear systems with a
single manipulated variable, that m*(t) is independent of the
magnitude 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 - - -
where
m
(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 some
properties of this mapping and shows that it is impossible in
general to evaluate the inverse mapping 0.0, Is = [(:::0) analyti-
cally.
Conclusions
The relation (it is implicit) between the adjoint and state
boundary conditions for time-optimum control of a particular
second-order system was obtained analytically. The two
adjoint initial conditions always have the same sign for positive
switching time, and their ratio is directly related to switching
time. The sign of these initial conditions determines the starting
polarity of the manipulated variable, and only the value of their
ratio is needed to specify the optimum control function and solve
the problem in a mathematical sense. The final value of their
ratio is needed to determine the final time and completely
specify 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 is
needed. The relation between this ratio and switching time has
the form of a decaying exponential over the positive real line,
hence the sensitivity of the forcing function to the adjoint
initial conditions as observed by Trombetta. In view of Figure
1, a general search for adjoint initial conditions could often lead
to difficulty.
The ratio of adjoint final conditions is directly related to the
remaining time; but often, in the vicinity of the singularity,
large changes in the final ratio would have little effect upon the
remaining time. Selection of the adjoint boundary conditions
in certain ranges, b < Ano, An 5 < 1, leads to nonreal functions
for switching and final times, and no switching in met) would
be expected. This behavior was exhibited in the numerical
results of Paynter and Bankoff.
Acknowledgments
The autbor expresses appreciation to L. B. Koppel for guidance and
direction during tbis work. The helpful suggestions of D. R. Cougbanowr
are acknowledged.
Financial support from tbe National Science Foundation in the form
of Graduate and Co- operative Fellowships and from Purdue University
is acknowledged with thanks.
Nomenclature
b
c(t)
eel)
met)
dimensionless minor time constant; 0 < b < 1
process output
error, deviation of output from set point; r - c(t)
manipulated variable, the process input to be deter-
mined; 0 ::S: m(t) ::S: 1
set poin t, piecewise constan t desired val ue of c; 0 < r < 1
dimensionless time, real time/major process time
constant
value of I at which switch in m*(t) occurs
time 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)
r
I
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-Optimal
Comrols", 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, May
1963) .
(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

5. THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING
Volume 46, Number 5
Devoted to the publication of
Chemical Engineering Science
Industrial Practice
Applizd Chemistry
October1968
L. W. SHEMIL T, Editor
D. D. KRIST1VIA1SO:'-J, Assistant Editor
The University of TCW Brunswick, Fredericton, N.B., Canada
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Kingsron, ant. '
]. L. CORj EILLE, Ecole Poly technique,
~!ontreal, Que.
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