This document presents a method for driving a chemical process output to a new operating level in minimum time using bang-bang control. The method involves: 1) Modeling the process using a second-order model with time delay and fitting the model parameters to process response data. 2) Calculating the switching times between maximum and minimum input levels using the model to achieve an optimal response time. 3) Implementing the bang-bang control by switching the input at the calculated times to drive the process output to the new level, then returning to conventional control. The method provides improved set-point responses for processes compared to conventional control, without requiring detailed process dynamics information.

1. TIME-OPTIMUM CONTROL OF CHEMICAL
PROCESSES FOR SET-POINT'CHANGES
School of Chemical Engineering, Purdue University, J17estLafayette, Ind.
PIERRE R. LATOUR,! LOWELL B. KOPPEL, AND DONALD R. COUGHANOWR
A procedure to drive the process output to a new operating level in minimum time is proposed for a wide class
of single-manipulated-input, single-output processes subject to input saturation. The bang-bang control
based upon switching times can be implemented in a programmed sense either manually or with direct
digital control for set-point changes without detailed process dynamics information. A technique for fitting
the model to bang-bang response data allows possible adaptation.
I
NCREASING competition and technology in the process in-
dustries provide economic incentive for optimization.
Supervisory digital control is a proved and widely used con-
cept for commercial plants (Crowther et al., 1961; Williams,
1965). Typical operation is as follows: A digital computer
performs a steady-state optimization perhaps hourly based on
current operating conditions. Each optimization results in a
new set of set-point values for the plant loops. The computer
makes the required step changes in set points of the conven-
tional controllers, which must therefore be adjusted for a
compromise between good set-point and load responses.
The results given in the present paper provide an improved,
practical technique for achieving good set-point responses,
while allowing the conventional controllers to be tuned for
good regulator action so that load disturbances will be well
compensated. This will significantly improve the performance
of such supervisory control systems.
Criterion of Optimality
For commercial processes, maximum return on investment
or profit is perhaps the most common criterion function for
optimum engineering design. Since the economic criterion
functions are unique to each problem, and may be difficult to
state or use mathematically, we propose a minimum time
criterion as a useful general criterion of optimality for the servo
problem in process control. The reasons for this choice are:
A particularly simple implementation results, the responses
which are achieved will generally provide significant improve-
ment over conventional controller set-point responses by any
criterion of optimality, and the process dynamics are only
infrequently known to sufficient accuracy to warrant use of a
more complex criterion function.
Bang-Bang Control
Intuitively one feels that the harder a system is forced the
faster it will respond. However, there is always a physical
limit to the available energy for control. Saturation occurs
in an automatic valve when it is wide open or closed tight.
The resulting limits on the available energy must be considered
and utilized in forcing the system as rapidly as possible,
This constraint on the manipulated variable will be written
as k :::; m(t) :::; K.. In conventional regulator design methods,
error perturbations are assumed to be small, and linear theory
is applied to design linear controllers. Although errors are
kept small during the design calculations to avoid saturation
deliberately and ensure that linearity is preserved, such con-
i Present address, Shell Oil Co., Houston, Tex.
452 1& E CPR 0 C E S S DES I G NAN D D EVE lOP MEN T
trollers do not take proper account of the saturation which
inevitably occurs in operation.
From Pontryagin's maximum principle (Pontryagin et al.,
1962), Gibson and Johnson (1963) have shown that the time-
optimum control action for linear processes, subject to satura-
tion on m(t), requires the manipulated variable to be at the
bounds of its limits throughout the transient-that is, m(t) will
be at either k or K, with possible switches between these
values, during the entire transient. Such nonlinear control
action is called extremal or bang-bang control.
It is understood that if driving the process with m(t) at its
actual physical constraint can cause mechanical damage or
undesirable conditions in the system, a less severe pseudo-
constraint may be assumed, at the expense of response time,
of course-that is, the values of k and K. need not represent
true physical limits, and in practice would be assigned con-
servative values initially until the controller tests were com-
pleted.
Also we assume m can be switched instantaneously between
K. and k. This is not a severe restriction if valve dynamics
are lumped with the process.
Mathematical Model
Rational design of a process controller requires some sort of
mathematical description of the process dynamics. Absolute
optimum control of a physical process can seldom be achieved;
only optimum control of its mathematical model is possible.
Obviously, if the model is a good mirror of reality, the control
designed to be optimum on the model will approach the true
optimum for the physical process. A comparison between
the open-loop responses of the model and process is only an
indirect measure of the model's validity. A direct measure
is the performance of the resulting controller when it is applied
to the physical system.
Synthesis of a mathematical model for process dynamics, as
in statistical regression, requires determination of both the
model form and the model parameters. Lefkowitz (1963)
has stated that the model complexity and the analysis pro-
cedure used are highly dependent upon the purpose intended
for the model. For process scale-up and extrapolation,
complex models based upon fundamental laws of physics
and chemistry are usually required. However, for inter-
polation and for many control system designs, simpler empirical
models with statistically fitted parameters may suffice. Com-
plex theoretical models are avoided here for the following
reasons:
Process dynamics arc neither well known nor susceptible to
accurate measurement.
Rigorous models tend to contain nonlinearities and distrib-
uted parameters.

2. Control theory is not well developed for such models and
each design tends to be unique to its process.
Rigorous models have many parameters that are unknown or
difficult to measure. Few state variables can be measured
directly for feedback, so higher-order models are of limited
utility even if the process dynamics are theoretically of high
order. Also, optimal control algorithms for higher-order
models have not been obtained from present-day theory.
Detailed process analyses and complex controllers are
difficult to justify economically for most single-manipulated-
input single-output processes, because we will show that signif-
icant control improvement can often be achieved based on
simpler process dynamics models.
For these reasons, an overdamped second-order model with
dead time and fitted parameters is assumed to describe the
process dynamics to sufficient accuracy that the optimum
controller for this model performs significantly better than a
conventional one, on the actual process. Note the significant
difference between this statement and the statement that the
dynamic response of the process may be described to some
specified level of accuracy by that of the model. The transfer
function of this model is
C(s)
M(s)
Kp(aTs + 1)exp( -dTs)
(Ts+ 1)(bTs+ 1)
where
C process output to be controlled
}J manipulated variable
Kp = process gain, units e/m
T major time constant
b T minor time constant, °S b S 1
dT dead time, d ;:::0
aT numerator time constant
This model can be used to represent the dynamic response
of liquid-liquid and gas-liquid extractors (Biery and Boylan,
1963; Gray and Prados, 1963), mixing in agitated vessels
(Marr and Johnson, 1961), some heat exchangers (Hougen,
1963), distillation columns (Lupfer and Parsons, 1962; Moczeck
et al., 1965; Sproul and Gerster, 1963), and some chemical
reactors (Lapse, 1956; Lupfer and Oglesby, 1962; Mayer
and Rippel, 1963; Roquemore and Eddey, 1961). Clearly,
however, there exist processes which cannot usefully be
described by such dynamics, and the specific results of this
paper are not directly applicable to such processes, although
the general methods may be applicable with more mathe-
matical effort. We will nevertheless demonstrate that even a
nonlinear exothermic chemical reactor may sometimes be
usefully treated by using Equation 1 for a model.
An example step response of this model is shown in Figure 1.
Any process whose step response resembles the long sigmoidal
shape of the dashed curve would be a candidate for modeling
by Equation 1 with empirically fitted parameters T, b, d, a,
and Kp. Frequent occurrence of this shape of step response
is the reason for the relatively general utility of Equation 1.
Optimum Control Function
Suppose at time zero the process is at rest with output equal
to the set point (c = To), and the supervisory digital computer
indicates desirability of operation at a new steady state with
output c = T. By application of Pontryagin's maximum
principle with a phase-plane analysis, the optimum control
function m*(t) which drives our model output from To to
steady state at r in minimum time can be shown (Latour, 1966)
m f - - -- -
0
-
c
~~~
-- -".
----
"c·
,-
;' /STEP
/
/
/
I
/
/
/
0
K
m
m
k
r
c
(1)
r
Figure 1.
response
TIME,
Example of optimum control function and
to be at one of the constraints K or k during the entire transient,
with one switch between them at time 12, and a final switch to
the new steady-state value m = T / Kp at time t«. An example
forcing function and time-optimum response when T > To
is shown in Figure 1. Of course, if T < To,m*(O+) = k instead
of K. The time-optimum response has no overshoot.
For a = 0, the first switch should occur (Latour, 1966) at
time t = t2 given by the implicit equation
[
&K - k - (ro/Kp - k) exp (-tdbT)Jb =
K - r/Kp
K - k - (ro/ K; - k) exp (-t2/T)
(2)
K - T/Kp
for r < 1'0' ° < b < 1. For the particular case b = (or
approximately 1, say b > 0.9), and T < To,t2 is given by
[
To/Kp - k _
K - k
(
t2)J {To/Kp - k - (K - k) exp (t2/T)}
exp T In TjKp - K +
When T > Tointerchanging K and k in Equations 2 and 3 gives
the correct results. These equations are valid if the system is
dc(O)
initially at rest, -- = 0, c(O) = To' The switching time 12
dt
does not depend upon d. These implicit equations can be
solved easily using a "halving-the-increment" method (Latour,
1966) for finding the root of an algebraic equation on a digital
computer. The final switch should occur (Latour, 1966) at
time t = t4 given by
VOl. 6 NO. 4 0 C TO BER 1 9 67 453

3. t41T =
[
ro/ Kp - k - (K - k) eXP(t2IT)]
In rlKp-K ,r<ro
[
'olKp - K - (k - K) exp(t'/T)]
In rlKp-k ,r>ro
where ° < b ::;; 1. t, is also independent of d. Equations
2, 3, and 4 are obtained from solutions of the differential
equations defined by Equation 1 for m = K and m = k,
Derivations are given by Koppel and Latour (1965).
For a first-order system (b = 0) with dead time, no switching
reversal is required, and the single switch to the final steady-
state m = rIK p should occur at t = t2 = t ; given by
[rolKp - kJ r < ro (5)In I '
t21T =
r Kp - k
[rolKp - KJ (6)
In rlKp - K ' r > '0
Our experience suggests that Equations 5 and 6 can be used for
second-order systems when b < 0.1.
The effect of the minor time constant on switching times is
shown in Figure 2 for two set-point change magnitudes.
Values of K = 1, k = 0, T = Kp = 1 were assumed to prepare
the graph.
The minimum response-i.e., the minimum time to drive
the output from r, to steady state at r-is
t5 = t; + dT (7)
since an additional delay time after the return at I; to m =
rlKp is required to complete the transient. After time t5
has elapsed control should be returned to conventional closed-
loop regulator control. The PID controller should not
integrate the error during bang-bang control. This may be
achieved easily on the newer conventional controllers equipped
with "bumpless transfer" from manual to automatic modes.
During the bang-bang forcing, the computer should switch the
controller to the manual mode. At i, the computer should
set met) at rl Kp, set the controller set point at the new desired
value, and switch the controller to the automatic mode at
time ts. If in addition to the supervisory action, direct digital
control is used in place of the conventional instrument, a
similar implementation is easily achieved.
Knowing the constants ro, r, K, and k, and the model pa-
rameters, we can determine the optimum control m*(t) by
calculating t2 and t4 from Equations 2 and 4. If an on-line
digital computer is not available ..the bang-bang control can be
implemented manually from a table or graph of t2 and t;, as
functions of ro and r prepared off-line. Analog timing circuits
(Latour, 1964) which solve these equations can be used to
generate m*(t) directly.
FiHing Model Parameters
Experimental modeling techniques to obtain Gp(s) have been
generally based on step, pulse, or frequency response data
(Hougen, 1964). However, for expedience and to simplify
computations, we propose that the process be forced in a
bang-bang fashion as in Figure 1 with guessed values of t2
and I;. These values of 12 and t4 will not give an optimum
response but they can be easily estimated from Equations
2 and 4 using rough estimates of T, b, and Kp perhaps obtained
from a step response (Harriott, 1964; Latour, 1966), or chosen
during the transient by a skilled process operator using his
"feel" for the process response to guide his switching times.
454 I & E CPR 0 C E S S DES I G NAN DOE VEL 0 P MEN T
(4)
r • 0.65
roO 0.5
1.2
0.8
t~ __ -
,/
/" -,/ --,..."...---
/--- »>
#' .--
/f.-/ /'
r
/
O~----L- ~ ~ -L ~
o 0.2 0.4 0.6 0.8 1.0
MINOR TIME CONSTANT, b
Figure 2. Effect of minor time constant on
switching times
K = T = Kp = 1, k = 0
Some bang-bang responses shown in Figure 3 obtained from a
normalized system (K = Kp = T = 1, k = a = 0, b = 0.5,
d = 0.1) when t2 is too small and t; is in error, can be com-
pared with the optimum response in Figure 1 (t2 = 0.74, t4 =
0.89, 15 = 0.99). Switching times considerably in error (15%)
still give responses which are not poorer than the simple step
response. It is not likely that the switching times intuitively
chosen by a skilled operator would cause a serious deterioration
in the test response compared to that of a conventional con-
troller.
The model parameters can now be fitted in the time domain
to the response data from bang-bang forcing with these in-
correct (but known) switching times by nonlinear statistical
regression. The model response to the known forcing can
be written analytically in terms of the unknown parameters
T, b, a, d, and Kp. To start the regression analysis, one can
obtain reasonable estimates of the parameters from a step
response by means of graphical techniques such as the familiar
one of Oldenbourg and Sartorius (1948). These estimates
were so close that convergence problems were never en-
countered in this work. In our work, a digital computer
program (Latour, 1966) based upon Marquardt's (1959)
nonlinear regression was used to obtain the fitted model
parameters from data curves similar to Figure 3. The pa-
rameters are chosen to minimize the sum of squares of devia-
tion of the physical process response from Equation 1. Revised
switching times can then be calculated from the model using
Equations 2 and 4.
This procedure is directly related to the control objective of
improved set-point responses, since the model and process
are made to agree on the type of input forcing which will be
used for set-point control. Also, modeling can be a repetitive
procedure if an on-line digital computer is available. New
set-point commands r are determined by the supervisory
computer (a steady-state optimizer). With current estimates

4. b » 0.5
d = 0.1
t2 t4
10 0.66 0.72
C II 0.65 0.82
12 0.67 0.90
13 0.67 0.95
14 STEP
o 234 5 6
TIME, t
Figure 3. Bang-bang responses with t2 too small, t4 incorrect
K = Kp = T = 1, k = a = 0, b = 0.5, d = 0.1
of the model parameters, switching times can be rapidly
calculated to obtain the correct m*(t), which is then applied to
the process. The resulting process response can be auto-
matically fitted by the computer to revise the model for the
next set-point command. Since the model is repeatedly
updated to agree with the most recent process response, the
controller design automatically adapts itself as the process
evolves in time. This adaptive control increases the power
of the design approach and extends the variety of processes
for which Equation 1 might lead to improved responses to
include slowly time-varying-parameter systems. For non-
linear systems further improvements might be possible by
correlating the model parameters with steady-state conditions
To and T.
Numerator Dynamics
If a r" 0, the switching times given by Equations 2 and 4
are not theoretically time-optimum. The true time-optimum
control for systems with numerator dynamics is at the con-
straints during the initial transient, but is followed by a
specific time variation after the ouput e(t) has come to rest at
the desired value T (Athans and Falb, 1966). For practical
purposes, this control will be difficult to implement. How-
ever, use of the switching times given in Equations 2 and 4 will
drive e(t) to rest at T, although in a slightly longer time. Hence,
for practical purposes this control should suffice.
Evaluation
A variety of analog-computer-simulated systems, plus one
physical system (water temperature process), were employed to
evaluate the proposed modeling and control technique. In
general, the evaluation procedure was as follows. The
system was forced with a bang-bang input, using intuitively
selected switching times, to drive it to a desired new steady
state. In general, these switching times were based on visual
observation of the response. The resulting transient was
modeled to Equation 1, using the least-squares technique
described above. The resulting model was used to calculate
new switching times from Equations 2 and 4. The new
switching times were used to obtain a new transient, which was
examined for "optimaIity" of performance. Transient re-
sponses were recorded on a strip chart. Data points ti, Ci
were obtained manually and transferred to data cards for
digital computer processing.
With the exception of the reactor simulation and the water
temperature process, models were restricted to a = 0 and four
fitted parameters. Also, values of K = 1, k = 0, To = 0.5,
T = 0.65 were used unless otherwise noted. Marks on the
transients indicate t2 and ts,
Second-Order Process. The second-order system with
b = 1/2
exp( -25)
Gp(S) = (205 + 1)(105 + 1) (8)
was simulated. The correct switching times for To 0.5,
T = 0.65 were calculated from Equations 2 and 4 to be
t2 = 14.5, t4 = 17.5 (9)
Switching times t2 = -17.6, t4 = 21.5 produced the transient
in Figure 4, a. Data points from this transient were computer-
modeled to obtain the model transfer function
exp( -1.95)
Gm(s) =
(23.85 + 1)(8.35 + 1)
(10)
which, upon application of Equations 2 and 4, leads to the
revised calculated switching times
t2 = 14.9, t4 = 17.8 (11)
, 1 -,
- ,
•-.
0- • _.
- .. - .
• to.:- -I-
:., ,.'"
, !--:.-s
E 1"
- •. J
~-
1f".L =f=i= :!-
=t
b~' -I??',=,=:_
"33·-
Figure 4. Modeling overdornpe d second-order system
K=I,k=O
VOl. 6 NO. 4 0 C TO B E R 1 967 455

5. If the switching times are too short, the transient Figure 4, b,
can result (t2 = 13.0, t4 = 14.9). This transient was fitted by
the model transfer function
exp( -2.2s)
(12)
Gm(S) = (21.6s + 1)(9.7s + 1)
which leads to the revised switching times from Equations 2
and 4
t2 = 14.9, t. = 18.0 (13)
In both cases, these revised switching times would produce an
excellent response similar to Figure 4, c (tz = 14.7, t. = 17.8).
Curves 4a and 4b resulting from positive and negative switching
time errors were modeled by transfer functions somewhat
different from the analytical simulation (b = 0.35 and 0.45,
respectively). However, the revised switching times agree
with each other and the analytical values, and result in the
improved transient in Figure 4, c. Switching times could
not be reproduced manually with a stop watch as accurately
as they could be measured from the chart recording of met)
afterward; hence, the slight differences between predicted
values and those actually used in Figure 4, c. The procedure
is therefore seen to converge to essentially the optimum re-
sponse after one modeling step in both cases. An earlier
paper (Latour, 1964) showed that the response of Figure 4, c
and similar time optimum responses are faster than con-
ventional set-point responses by an amount on the order of
T, the major process time constant.
An underdamped second-order system with ( = 0.707 was
simulated. Its response was modeled with b = 0.95 (or
I = 1.0003). The predicted switching times when the
process necessarily differed from the model still produced a
bang-bang response much improved over the step response.
Modeling behavior for this process is of interest because
Hougen (1964) states that mixing a nonvolatile dye on plates
of a commercial distillation column shows second-order
dynamics with 0.8 damping coefficient plus dead time. Hou-
gen also states that mixing in long tubes and a variety of heat
exchangers show second-order, critically damped tendencies
«( = 1). Details are given by Latour (1966).
Third-Order Process. The response of the system
exp( -2s)
Gp(s) = (20s + 1)(10s +-1-) (-6-s-+-1-) (14)
to arbitrarily selected switching times was modeled by
exp(-6.1s)
(15)
Gm(s) = (19.2s + 1)(13.4s + 1)
and yielded switching times which produced a very satis-
factory response that was significantly improved over the set-
point response of a conventional PID controller. Model dead
time is larger than process dead time to account for the third
time constant. Although switching times do not depend upon
d, including this parameter really allows the model to achieve
more realistic time constants and a better fit. ,
Similarly satisfactory results were obtained (Latour, 1966)
for the third-order system
(16)
(r + 1)3
Tenth-Order System. Staged processing systems (tray
columns, extractors, etc.) exhibit high-order dynamics. To
test the modeling procedure and selection of switching times
on such a system, the tenth-order system
456 I&EC PROCESS DESIGN AND DEVELOPMENT
1
Gp(s) = (10s + 1)10 (17)
was considered. The step response of this system was modeled
graphically to obtain
exp( -55s)
Gm(s) = (40s + 1)(7s + 1)' t2 = 20.1, t4 = 22.9 (18)
The transient resulting from these switching times was fitted
digitally to obtain the model
Gm(s) = exp(-51s)
(27s + 1)(24s + 1)
t2 = 25.4, t4 = 31.2, d = 1.89, b = 0.9
(19)
The transient using these switching times was considerably
improved over the step response and the response based on the
first set of switching times in Equation 18. This improvement
may be described in terms of the 10% settling times as defined
by Coughanowr and Koppel (1965). The settling times for
the step response and the response based on the switching times
of Equation 18 are 142 and 153, respectively; the settling time
for the response based on the switching times of Equation 19 is
90. Further details on these responses are given by Latour
(1966).
Nonlinear Exothermic Reactor. Further study of the
utility and limitations of the design procedure was attempted by
modeling a highly nonlinear, exothermic reactor simulation.
Orent (1965) simulated and controlled a modified version of the
Aris-Amundson (1958), Crethlein-Lapidus (1963) continuous
stirred-tank reactor, for the irreversible exotheric reaction
k
A -+ B with first-order kinetics. The modification involved
addition of cooling coil dynamics. The reaction rate constant
is
k koexp (- E/ Re) (20)
where
k Arrhenius reaction rate constant, sec."?
ko frequency factor, 7.86 X 1012, sec.-1
E activation energy, 28,000 cal./mole
R gas constant, 1.987 cal./mole - a K.
e absolute temperature of reactor, a K.
The mass balance on the reactor contents assuming uniform
mixing is
dE
V - = FE - FE - VkE
dt 0
(21)
where
V material volume, 1000 cc.
E concentration of A in exit, moles/cc.
Eo concentration of A in inlet, 6.5 X 10-3 mole/cc.
F volumetric flow, 10 cc./sec.
If we assume no heat transfer with the surroundings, and the
physical properties of inlet and outlet streams identical, the
energy balance is
de
VPCT - = FpCT(eo - e)
dt
_ !:::.HVkE _ UA(ec ~_!!co)
In(~)e - ec
(22)
where
p density, 1 g./cc.

6. e
= heat capacity, 1 cal./g. - 0 K.
temperature of reactor and exit stream, 0 K.
temperature of inlet stream, 3500
K.
exothermic heat of reaction, 27,000 cal./mole
over-all heat transfer coefficient times cooling coil
area, 7 cal./sec. - 0 K.
inlet coolant temperature, 3000
K.
exit coolant temperature, 0 K.
e,
-!1H =
UA
An approximate lumped energy balance for the cooling coil is
(23)
with
Ve coil volume, 100 cc.
Pe coolant density, 1 g./cc.
rre coolant heat capacity, 1 cal./g. 0 K.
Fe coolant flow, 0 ~ Fe ~ 20, cc./sec.
This cooling coil equation is added to impart higher order
dynamic effects. The manipulated variable is the bounded
coolant flow rate. The output temperature was delayed by a
second-order Pade circuit to introduce an additional 6-second
dead time, representative of higher-order lags. Only control
about the stable high temperature steady-state
4600
K.
4190
K.
0.162 X 10-3 mole/cc.
5.13 cc./sec.
is considered here.
second steps in Fe.
Orent reports responses to ± 1 cc. per
He modeled these by
6.1exp(-11s) OK.
70s + 1 cc.y'sec,
(24)
The response to a step in Fe from 5.13 to 4.13 is shown in
Figure 5, a. A bang-bang transient from guessed switching
times is shown in Figure 5, b, and the modeling results are
listed in Table 1. The predicted switching times were du-
plicated closely for the improved response in Figure 5, c, which
was fitted by a somewhat different transfer function but
yielded essentially the same revised switching times. This
Table I. Modeling Nonlinear Reactor
t, t , Figure Model I, 14
5, a Step decrease in F,
-5.1(0.ls + 1)
10.6 12.6 5, b
exp( -5.7s)
11.1 12.3
(25.7s + 1) X
(6.9s+ 1)
-5.7(0.3s + 1)
10.9 12.1 5, c
exp( -7.0s)
11.3 12.3
(33.0s + 1) X
(4.6s + 1)
12.1 12.7 5, d Slight overshoot
6, a Step increase in Fc
-4.4(-0.5s + 1)
14.7 15.8 6, b
exp( -4.0s)
12.6 20.1
(126.4s + 1) X
(6.2s + 1)
-5.0( -0.7s + 1)
12.6 20.2 6, c
exp( -5.1s)
12.0 16.8
(141.2s + 1) X
(3.7s + 1)
12.5 17.9 6, d Nearly optimum
demonstrates that modeling nearly optimum transients gives
repetition of switching times; in other words, modeling con-
vergence is maintained. Extensive study of convergence re-
quires hybrid computation. The undershoot of Figure 5, c,
prompted the slight increase in t-i, t.l for Figure 5, d. In view
of this transient and the accuracy of the time measurements,
the model times 11.3 and 12.3 give an excellent response.
The model major time constant is less than half that in Equation
24 for the simple step response.
The response to a returning step, Fe from 4.13 to 5.13, is
shown in Figure 6, a. A bang-bang transient from guessed
switching times is shown in Figure 6, b, and the modeling
results are in Table 1. The predicted switching times gave the
improved response in Figure 6, c, which in turn was modeled
and predicted switching times 12.0 and 16.8. In view of
Figure 6, d, obtained by trial, these times should be further
improvements. The model major time constant now is twice
that for the simple step response.
This nonlinear reactor can have three possible steady states
(one is unstable) for a single Fe. Bang-bang forcing between
full cooling, Fe = 20, and adiabatic operation, Fe = 0, is very
~~.~~i-
j--"ji-
~:t:::::- ~~ .. ;~3,.lc
:C-L_
; ~.--I
Figure 5. Modeling nonlinear reactor
Temperature increases
VOL. 6 N O. 4 0 C T 0 B ER 1 9 67 457

7. Figure 6. Modeling nonlinear reactor
Temperature decreases
severe. Increasing temperature reponses requiring Fe = 20
initially are faster than the returning responses requiring Fe =
20 initially.
This work illustrates a limitation of the method if design is
restricted to fixed model parameters, and the system is highly
nonlinear, since the parameters depend upon direction.
Nevertheless, the bang-bang response of this reactor between
4600
and 4660
K. can be fitted to a second-order dead time
transfer function which predicts improved switching times.
Although modeling convergence is not quite complete in one
step as for the linear systems, the response is acceptable and
much superior to the simple step response. The adaptive
procedure described above would require a simple modification
to allow the computer to use two different models, one for
increases in Fe and the other for decreases. The rigorous
time-optimum control of similar nonlinear reactors was studied
by Siebenthal and Aris (1964).
In general, the nonlinear effects of temperature and chemical
kinetics make reactor models complex and optimum control
difficult. However, in some cases the nonlinearities are not
pronounced, and a simple linear method such as that suggested
here might be adequate for process control purposes. The
cracking furnace studied by Lapse (1956), the commercial
polymerization reactor for synthetic rubber studied by Roque-
more and Eddey (1961), the large compartmentalized poly-
merization reactors tested by Mayer and Rippel (1963), and
the catalytic polymerization in a solvent reported by Lupfer
and Oglesby (1962) are good examples for application. Even
?r EXPERIMENT IN AUTOMATIC PROCESS CONTROL
.~~. fLOW DIAGRAM
TO
DRAiN'-
~
N'AOT_ FL~ ,,,[
~rc- ~
0 SUPPlY r r"
T~
= ...:~.:.
- c::c:J
~
POWER
I--
- COHTROLl..ER
V -IIO~TEll -
TC2
10==
POWER
METER
SUPPLY
Q~~1
~ r----
== II DEADTIME
.~
l1IHK I
WATER
LJ~
©r=
ANALOG
MANIFOLD ~ COMPUTER
220 "-
POWER
'~
r- •._J
l.OlD I£XfER METER
VARIAC •.• ~.I'. ....-- I
8AYCHET RESlSWOCE
I
LEGEND
HEATER I~~B
~
@r =kj"-rou-i f-
=><== GATE 'oIALVE
=-= G~OBE VALVE
110 ""
--- INS~'" LIlIES ~
== POWER LINES POTENTIOMETER RECORDER
_TC3 THERMOCOUP!.E LOAD HEATER
TANKZ ~oo~ TRANSMITTER CONTROLl..ER
. VARIAC
POWER toIETER
Figure 7. Experiment in automatic processcontrol flow diagram
458 I & E CPR 0 C E S S 0 E S I G NAN DOE VEL 0 PM E N T

8. I ; : ; : :...:..;_~~~ : : ; : .
.:':: ~;:~-i
Figure 8. Experimental responses from water temperature process
Figure 9. Optimum bang-bang responses from water temperature process
for the highly nonlinear reactor discussed above, acceptable
results are obtained from the linear techniques.
Water Temperature Process. A temperature control
system for a water flow process with two agitated vessels and a
connecting pipeline (Figure 7) was used to test this control
technique. A.-c. power to a 2-kw. heater in the first tank was
manipulated to control the temperature of water leaving the
second tank. Water volumes and flows were maintained
approximately constant. A theoretical model obtained from
differential lumped energy balances and physical properties is
(J(s) 12.09 exp( -24.7s) 0 /
- F. kw.
Q(s) (151.8s + 1) (81.0s + 1)
(25)
In terms of the parameters of the general model of Equation 1,
we have d = 0.162, b = 0.533, a = O. Time constants are in
seconds. More details of the process are given by Latour
(1966).
A response with 29% overshoot to bang-bang forcing (not
shown) with 12= 125 seconds, 14= 155 seconds was fitted by
the model
G ()
_ 12.6( -0.2s + l)exp( -25.5s) 0 /
m s - F. kw.
(159.1s + 1)(95.5s + 1)
(26)
in excellent agreement with Equation 25. This model yields
revised switching times 12 = 99, 14 = 139 for the same tem-
perature change (-3.160
F.). These switching times would
undoubtedly reduce the overshoot. An improved response
with 13% overshoot using 12= 111,14 = 134 is shown in Figure
8, a. This was fitted by the model
13.5(1.4s + 1) exp( -25.9s) 0
Gm(s) = ( F.j kw. (27)
164.6s + 1) (9S.4s + 1)
which yielded revised switching times 12 = 109, I, = 149.
Because steady-state conditions are difficult to repeat exactly,
the procedure followed in the simulation tests, wherein the
revised switching times were used in an otherwise identical
test, could not be used in these experimental tests. However,
the revised switching times are clearly in the right direction,
and would tend to reduce the slight overshoot in Figure 8, a,
which is already a distinct improvement over the simple step
response of Figure 8, b. The final steady-state values are
indicated at the left of each transient.
The excellent responses in Figure 9 were obtained from the
process using switching based upon the theoretical model
Equation 25 and returning to closed-loop proportional-integral-
derivative control after the final value was reached.
Conclusions
Improved set-point responses can be achieved, for processes
whose dynamics can be represented as overdamped second-
order with dead time, by driving the manipulated variable
in a bang-bang fashion using two switching times. Algebraic
equations for these switching times (one is implicit) are given
in terms of the model parameters (two time constants and
gain), the manipulated variable saturation constraints, and
the starting and desired values of the output. The model
parameters can be obtained from bang-bang type response
data (using non optimum switching times) by nonlinear re-
gression. The design procedure gave satisfactory results when
evaluated with a variety of processes, some differing widely
from the assumed model, plus one physical system. It is
anticipated that the method will be particularly useful when
incorporated into a supervisory digital control system.
VOl. 6 NO. 4 0 CT0 8 ER '9 6 7 459

9. Acknowledgment
P. M. Aiken assembled the experimental equipment.
Literature Cited
Aris, R., Amundson, N. R., Chern. Eng. Sci. 7, 131 (1958).
Athans, M., Falb, P. L., "Optimal Control," McGraw-Hili,
New York, 1966.
Biery, S. C., Boylan, D. R., IND.ENG.CHEM.FUNDAMENTALS,2,
44 (1963).
Coughanowr, D. R., Koppel, L. B., "Process Systems Analysis and
Control," McGraw-Hill, New York, 1965.
Crowther, R. H., Pitrak, J. E., Ply, E. N., Chern. Eng. Progr. 57,
No.6, 39 (1961).
Gibson, J. E., Johnson, C. D., IEEE Trans. Auto. Contr. AC-8,
No.1, 4 (1963).
Gray, R.I., Prados, J. W., A.l.Ch.E.J. 9, No.2, 211 (1963).
Grethlein, H. E., Lapidus, L., A.l.Ch.E.J. 9, No.2, 230 (1963).
Harriott, P., "Process Control," McGraw-Hill, New York, 1964.
Hougen, J. 0., Chern. Eng. Progr. Monograph SeT. 4,60 (1964).
Hougen, J. 0., Chern. Eng. Progr. 59, 49 (1963).
Koppel, L. B., Latour, P. R., IND. ENG. CHEM.FUNDAMENTALS,
4,463 (1965).
Lapse, C. G., ISA J. 3, 134 (1956).
Latour, P. R., M.S. thesis, Purdue University, West Lafayette,
Ind., June 1964.
Latour, P. R., Ph.D. thesis, Purdue University, West Lafayette,
Ind., June 1966.
Lefkowitz, I., Chern. Eng. Progr. Symp . Ser, 46, 59, 178 (1963).
Lupfer, D. E., Oglesby, M. W., ISA Trans. 1, No.1, 72 (1962).
Lupfer, D. E., Parsons, J. R., Chern. Eng. Progr. 58 No.9 37
(1962). ', ' ,
Marquardt, D. W., Chern. Eng. Progr. 55, 65 (1959).
Man, G. R., Johnson, E. F., Chern. Eng. Progr, Symp, Ser. No. 36
57, 109 (1961). ' ,
Mayer, F. X., Rippel, G. R., Chern. Eng. Progr. Syrnp. Ser . No. 46
59,84 (1963). ' ,
Moczeck, J. S., Otto, R. E., Williams, T. J., Chern. Eng. Progr.
Symp . Ser., No. 55, 61, 136 (1965).
Oldenbourg R. C., Sartorius, H., Trans. A.S.M.E., 70, 78 (1948).
Orent, H. H., Ph.D. thesis, Purdue University, West Lafayette,
Ind., June 1965.
Pontryagin, L. S., Boltyanskii, V. G., Gamkralidze, R. V., Misch-
chenko, E. F., "The Mathematical Theory of Optimal Proc-
esses," Wiley, New York, 1962.
Roquemore, K. G., Eddey, E. E., Chern. Eng. Progr. 57, No.9,
35(1961).
Siebcnthal, CD., Aris, R., Chern. Eng. Sci.19, No. 10,729 (1964).
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RECEIVEDfor review October 31, 1966
ACCEPTEDMarch 27, 1967
i.I.Ch.E. Meeting, Atlantic City, N.J. Financial assistance
received from Purdue University and the National Science Founda-
tion.
NONINTERACTING PROCESS CONTROL
SHEAN-LIN LIU
Central Research Division Laboratory, Research Department, Mobil Oil Corp., Princeton, N. J. 08540
A new technique for the design of noninteracting control systems c~n handle constraint conditions on the
control variabl~~~d can be applied to nonlinear problems. Two examples illustrate the design method.
The first concerns a nonisothermal chemical reactor. The second deals with the control of a plate-type
absorption column. It is demonstrated that one state variable can be moved from one point to another
without affecting the other state variables.
IN MULTlVARIABLEfeedback control systems, a change in one
reference variable will usually affect all output variables.
In some applications (temperature control in a chemical
reactor, for example), one desires to design a noninteracting
control-that is, a system in which a variation of anyone
reference input quantity will cause only the one corresponding
controlled output variable (such as one state variable) to
change. The design of such systems was considered by
Boksenbom and Hood (Tsien, 1954). Using matrix algebra,
Kavanagh (1957), Freeman (1958), and Morozovskii (1962)
discussed the transfer matrix of noninteracting control systems.
Chatterjee (1960) considered noninteracting process control
using an analog computer and standard three-mode process
controllers. In all the above references only linear problems
were discussed, and constraints on control variables were
neglected. Petrov (1960) discussed a very special non-
linear problem without constraints. .
Although noninteracting control is potentially a powerful
tool for reducing the complexity of control systems, it has
several limitations, as discussed by Morgan (1958) and Mesa- Basic Theory
rovic (1964). The present procedure requires a complete Before discussing the design of noninteracting control
dynamic model of the system and an on-line digital computer. systems, the classical approach to control of linear multi-
This paper a~nounces two new results: Certain nonlinear, variable processes (Freeman, 1958; Kavanagh, 1957) is re-
unconstrained processes can (1) be made noninteracting over viewed. Since the process is linear, the control problem can
the entlre·statevanablespac·e in-a--~anner analogous to th:.::a:.:.t__ be discussed in terms of Laplace transformed variables. In a
-foriinear-sysTems, and(2)bec?n~!:.;.!!.edin-~p.iiCe~':-~~~-~9n- closed loop system, as shown in Figure 1, if P represents the
460 I & E CPR 0 C E S S DES I G NAN D D EVE ·L0 P MEN T
interacting way even when there are constraints on the process
~.!lR-~_tyarI~l:iics::-----·-·---··--·-·-- ... - ..--------
Two examples illustrate the present method. The first deals
with a nonisothermal chemical reactor in which a second-order
irreversible chemical reaction, 2A -.. B, takes place. The
concentration of component A and the temperature are to be
controlled by manipulating the flow rates of reactant and
coolant. Either state variable, temperature or concentration,
can be moved from one point to another without affecting the
other state variable.
In the second example, noninteracting control of a plate-type
absorption column is considered. It is assumed that there
are seven plates in the absorption column and that one com-
ponent in the gas phase is absorbed by liquid absorbent.
It is demonstrated that, by manipulating the liquid flow rate,
one can maintain the gas outlet concentration at a fixed value
even if perturbations in the gas flow rate or the gas inlet con-
centration occur.