TIME=OPTIMUM CONTROL OF CHEMICAL PROCESSES FOR SETIPOINT CHANGES PIERRE R. L A T O U R , l L O W E L L
B. K O P P E L , A N D D O N A L D R . C O U G H A N O W R
School of Chemical Engineering, Purdue University, West Lajayette, Ind.
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 b e 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.
competition and technology in the process inprovide economic incentive for optimization. Supervisory digital control is a proved and widely used concept for commercial plants (Crowther et a/., 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 conventional 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. NCREASISG
I dustries
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 improvement 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 \vi11 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 5 rn(t) _< K . I n 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 conPresent address, Shell Oil Co., Houston, Tex. 452
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PROCESS DESIGN A N D DEVELOPMENT
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 timeoptimum control action for linear processes, subject to saturation on r n ( 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, \vith possible switches between these values, during the entire transient. Such nonlinear control action is called extrema1 or bang-bang control. I t 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 pseudoconstraint may be assumed, a t 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 conservative values initially until the controller tests were completed. Also we assume rn 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 procedure 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 interpolation and for many control system designs, simpler empirical models with statistically fitted parameters may suffice. Complex theoretical models are avoided here for the following reasons: Process dynamics are neither well known nor susceptible to accurate measurement. Rigorous models tend to contain nonlinearities and distributed parameters.
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-manipulatedinput single-output processes, because we will show that significant control improvenient can often be achieved based on simpler process dynamics models.
1
rn m
I
For these reasons, a n 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. T h e transfer function of this model is C(s)
M(s)
-
-I
rn
I
I
I”
C
+ l)exp(-dTs) ___ + l)(bTs + 1)
/
/
1
/
STEP
/ /
K,(aTs (Ts
r
where
C = process output to be controlled M = manipulated variable K, = process gain, units c/m T = major time constant bT = minor time constant, 0 _< b 5 1 dT = dead time, d 2 0 U T = 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, 1?161), some heat exchangers (Hougen, 1963), distillation columns (Lupfer and Parsons, 1962; Moczeck et a / . , 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 mathematical 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 enipirically fitted parameters T , 6 , d, a, and K,. Frequent occurrence of this shape of step response is the reason for the relatively general utility of Equation 1. Optimum Control Function
Suppose a t time zero ithe process is a t rest with output equal to the set point (c = ro),, and the supervisory digital computer indicates desirability of operation a t a new steady state with output c = r. By application of Pontryagin’s maximum principle with a phase-plane analysis, the optimum control function m * ( t ) which drives our model output from ro to steady state a t 7 in minimum time can be shown (Latour, 1966)
TIME,
Figure 1. response
t
Example of optimum control function and
to be a t one of the constraints K or k during the entire transient, with one switch between them a t time t z , and a final switch to the new steady-state value m = r/K, a t time t 4 . An example forcing function and time-optimum response when r > ro is shown in Figure 1. Of course, if r < ro, m*(O+) = k instead of K . T h e time-optimum response has no overshoot. For a = 0, the first switch should occur (Latour, 1966) a t time t = t z given by the implicit equation
[
[K
-1
- k - ( r o / K p - k ) exp (-tz/bT) = K - r/Kp K - k - (ro/Kp - k) exp ( - t ? / T ) K - r/K,
(2)
for r < To, 0 < b < 1 . For the particular case b = 1 (or approximately 1, say b > 0.9), and r < To, t z is given by ro/Kp
-k
[ I F F - - k - (K exp($)]
In
Y
K
p
TO/KP
- k) exp (tz/T)} + -K
tz
T exp(tz/T)
-
= 0
(3)
When r > ro interchanging K and k in Equations 2 and 3 gives the correct results. These equations are valid if the system is dc (0) ) To. T h e switching time t z initially a t rest, __ = 0, ~ ( 0 =
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 a n algebraic equation on a digital computer. T h e final switch should occur (Latour, 1966) a t time t = t 4 given by VOL. 6
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453
ln[roIKp t 4 / T
=
ln[r,,/K,
-k
-
(K - k) T/K, - K
-K -
(k
r/K,
- K ) exp(tZ/T) -k
(4)
1
I -
r =
I
0.65
,r>ro
where 0 < b 5 1. t 4 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 steadystate m = r / K , should occur a t t = t 2 = t 4 given by
O u r 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 = K , = 1 were assumed to prepare the graph. The minimum response-Le., the minimum time to drive the output from ro to steady state a t r-is t5 =
t4
+dT
b
K =
r
= K, = i , k =
o
(7)
since a n additional delay time after the return at t 4 to m = r/K, is required to complete the transient. After time ti has elapsed control should be returned to conventional closedloop regulator control. The P I D 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 t 4 the computer should set m ( t ) a t r / K p , set the controller set point at the new desired value, and switch the controller to the automatic mode a t time t 5 . 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 parameters, we can determine the optimum control m*(t) by calculating t 2 and t 4 from Equations 2 and 4. If a n on-line digital computer is not available, the bang-bang control can be implemented manually from a table or graph of t 2 and t 4 , 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. Fitting Model Parameters
Experimental modeling techniques to obtain G,(J.) 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 t 2 and t 4 . These values oft2 and t 4 will not give a n optimum response but they can be easily estimated from Equations 2 and 4 using rough estimates of T , b, and K , 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
MINOR TIME CONSTANT,
Figure 2. Effect of minor time constant on switching times
I & E C P R O C E S S D E S I G N A N D DEVELOPMENT
Some bang-bang responses shown in Figure 3 obtained from a normalized system ( K = K , = T = 1, k = a = 0, b = 0.5, d = 0.1) when t~ is too small and t l is in error, can be compared with the optimum response in Figure 1 ( t 2 = 0.74, t 4 = 0.89, ts = 0.99). Switching times considerably in error (15%) still give responses which are not poorer than the simple step response. I t 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 controller. The model parameters can now be fitted in the time domain to the response data from bang-bang forcing with these incorrect (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 K,. 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 encountered in this work. I n 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 parameters are chosen to minimize the sum of squares of deviation 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
b= 0.65-
--
c
0.5
Q66 0.65
10
II
-
I I/
12
0.67
13 14
0.67 STEP
0.72 0.82 0.90 0.95
O S O W
I
I
1
I
I
I
I
0
I
2
3
4
5
6
TIME, t
Figure 3.
Bang-bang responses with
f2
too small,
t4
incorrect
K = K, = T = 1 , k = a = O , 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 automatically 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-wrying-parameter systems. For nonlinear systems further improvements might be possible by correlating the model parameters with steady-state conditions ro and r . Numerator Dynamics
If a # 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 a t the constraints during the initial transient, but is followed by a specific time variation after the ouput c ( t ) has come to rest a t the desired value r (Athans and Falb, 1966). For practical purposes, this control will be difficult to implement. However, use of the switching times given in Equations 2 and 4 will drive c ( t ) to rest at r , although in a slightly longer time. Hence, for practical purposes this control should suffice.
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, ro = 0.5, r = 0.65 were used unless otherwise noted. ,Marks on the transients indicate t 2 and tl. Second-Order Process. The second-order system with b = ‘/2 exp ( - 2s)
Gp(s) = (20s
+ 1)ilOs i 3
was simulated. The correct switching times for r0 = 0.5, r = 0.65 were calculated from Equations 2 and 4 to be t z = 14.5, t 4 = 17.5
(9)
Switching times t 2 = 17.6, t 4 = 21.5 produced the transient in Figure 4, a. Data points from this transient were computermodeled to obtain the model transfer function exp( - 1.9s) Gm(s) = (23.8s 1)(8.3s f 1)
+
which, upon application of Equations 2 and 4, leads to the revised calculated switching times t2
=
14.9,
t4
=
17.8
(1 1)
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. I n general, the evaluation procedure was as follows. The system was forced with a bang-bang input, using intuitively selected switching timer,, to drive it to a desired new steady state. I n general, these switching times were based on visual observation of the response. T h e resulting transient was modeled to Equation 1, using the least-squares technique described above. T h e resulting model was used to calculate new switching times from Equations 2 and 4. T h e new switching times were used to obtain a new transient, which was examined for “optimality” of performance. Transient responses were recorded on a strip chart. Data points ti, c i were obtained manually and transferred to data cards for digital computer processing.
Figure 4.
Modeling overdamped second-order system K = 1,k = 0
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455
If the switching times are too short, the transient Figure 4, b , can result (tP = 13.0, t 4 = 14.9). This transient was fitted by the model transfer function was considered. The step response of this system was modeled graphically to obtain
exp (- 2.2s) Gm(s) = (21.6s 1)(9.7s f 1)
+
which leads to the revised switching times from Equations 2 and 4 t.2
=
14.9, t 4 = 18.0
(13)
In both cases, these revised switching times would produce a n excellent response similar to Figure 4, c (tz = 14.7, t 4 = 17.8). Curves 4a and 46 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 m ( t ) 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 response 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 conventional set-point responses by a n 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 { = 1.0003). The predicted switching times when the process necessarily differed from the model still produced a bane;-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. Hougen also states that mixing in long tubes and a variety of heat exchangers show second-order, critically damped tendencies (f = 1). Details are given by Latour (1966). Third-Order Process. T h e response of the system
G p ( s ) = (20s
+
exp ( - 2s) 1)(1Os f l)(6s f 1)
to arbitrarily selected switching times was modeled by
Gm(s) =
Tenth-Order System. Staged processing systems (tray columns, extractors, etc.) exhibit high-order dynamics. T o test the modeling procedure and selection of switching times on such a system, the tenth-order system 456
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(18)
The transient resulting from these switching times was fitted digitally to obtain the model
tz = 25.4,
t4 =
31.2, d = 1.89, b = 0.9
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 Ark-Amundson (1358), Grcthlein-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 = k,exp(--/Re)
(20)
where
k = Arrhenius reaction rate constant, set.-' k, = frequency factor, 7.86 X 10l2, sec.-l E = activation energy, 28,000 cal./mole
R
e
= gas constant, 1.987 cal./mole = absolute temperature of reactor,
K. K.
The mass balance on the reactor contents assuming uniform mixing is
exp ( - 6.1s) Gm(s) = __ (19.2s f 1)(13.4s f 1) and yielded switching times which produced a very satisfactory response that was significantly improved over the setpoint response of a conventional P I D controller. Model dead time is larger than process dead time to account for the third time constant. Although switching times d o 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
exp (- 55s) t? = 20.1, t i = 22.9 (40s f 1)(7s f 1)’
V
dE - = FE, dt
- FE - VkE
(21)
where
V E E, F
= material volume, 1000 cc.
concentration of A in exit, moles/cc. concentration of A in inlet, 6.5 X = volumetric flow, IO cc./sec. =
=
mole/cc.
If we assume no heat transfer with the surroundings, and the physical properties of inlet and outlet streams identical, the energy balance is
where p
=
density, 1 g./cc.
K. -- heat capacity, 1 cal./g. = temperature of reactor and exit stream, ' K. e, = temperature of inlet stream, 350' K. - A H = exothermic heat of reaction, 27,000 cal./mole UA = over-all heat transfer coefficient times cooling coil area, 7 cal./sec. - O K. e,, = inlet coolant temperature, 300' K. e, = exit coolant temperature, O K. 5
B
An approximate lumped energy balance for the cooling coil is
t2
-
12.6
5, b
+ + +
10.9
12.1
5,c
12.1
12.7
5,d
1) exp( - 5.7s) (25.7s 1) X 1) (6.9s
-
6,
+ +
(33.0s 1) x (4.6s 1 ) Slight overshoot Step increase in F, -4.4( -0.55 1) exp( -4.0s) (126.4s 1) x (6.2s 1 ) -5.O( -0.7s 1) exp( -5.1s) (141.2s 1) x (3.7s 1) Nearly optimum
a
14.7
15.8
6, b
12.6
20.2
6,c
12.5
17.9
6, d
+
+ +
V , = coil volume, 100 cc. 1 g./cc. = coolant heat capacity, 1 cal./g. -
F,
=
coolant flow, 0
6
F,
6
O
K
e,
= = E, = F,, =
eCs
460' K. 419'K. 0.162 X mole/cc. 5.13 cc./sec.
is considered here. Orent reports responses to i 1 cc. per second steps in F,. H e modeled these by &(s) - - -().lexp(-lls) AF,(s) 70s 1
+
O K .
cc./sec.
(24)
T h e response to a step in F , 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 I. T h e predicted switching times were duplicated 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
Figure
12.3
1 1_ .3 12.3 -
-
12.6
20.1
12.0
16.8
-
-
+
+ +
20, cc./sec.
This cooling coil equation is added to impart higher order dynamic effects. T h e manipulated variable is the bounded coolant flow rate. T h e output temperature was delayed by a second-order Pade circuit to introduce a n additional 6-second dead time, representative of higher-order lags. Only control about the stable high wmperature steady-state
11.1
+
-5.7(0.3s 1) ~exp( -7.0s)_
p , = coolant density 5,
t4
-5.1(0.1s 10.6
-
with
Table I . Modeling Nonlinear Reactor tc Figure Model t2 5. a Step decrease in F ,
demonstrates that modeling nearly optimum transients gives repetition of switching times; in other words, modeling convergence is maintained. Extensive study of convergence requires hybrid computation. The undershoot of Figure 5, c, prompted the slight increase in t?, t 4 for Figure 5, d. I n view of this transient and the accuracy of the time measurements, the model times 11.3 and 12.3 give a n excellent response. The model major time constant is less than half that in Equation 24 for the simple step response. T h e response to a returning step, F , from 4.13 to 5.13, is shown in Figure 6, a. ,4 bang-bang transient from guessed sivitching times is shoizm in Figure 6, b, and the modeling results are in Table I. 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 F,. Bang-bang forcing between full cooling, F, = 20, and adiabatic operation, F , = 0, is very
5. Modeling nonlinear reactor Temperature increases
VOL.
6
NO. 4
OCTOBER
1967
457
Figure 6.
Modeling nonlinear reactor Temperature decreases
severe. Increasing temperature reponses requiring F, = 20 initially are faster than the returning responses requiring F, = 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 460' and 466' 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 F, and the other for decreases. The rigorous time-optimum control of similar nonlinear reactors was studied by Siebenthal and Aris (1964). I n 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 Roquemore and Eddey (1961), the large compartmentalized polymerization 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
I
EXFERMENT IN AUTOMATIC PROCESS CONTROL
FLOW DIAGRAM
l E z Y d dl ER
r----
COMPUTER
,-J
~~~~
Figure 7. 458
~
Experiment in automatic process control flow diagram
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
Figure 8.
Figure 9.
Experimental responses from water temperature process
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
_0 ($1 -_ Q(s)
12.09 _ exp ~ (- 24.7s) (151.8r
+ 1)(81.0s -I- 1)
O
F./kw.
(2 5)
In terms of the parameters of the general model of Equation 1, we have d = 0.162, b == 0.533, a = 0. 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 tz = 125 z8econds,t d = 155 seconds was fitted by the model G,(s)
=
12.6(-0.2s (159.11 ~
+ l)exp(-25.51) + 1)(95.5s + 1)
O
F./kw.
(26)
in excellent agreement with Equation 25. This model yields revised switching times tz = 99, t 4 = 139 for the same temperature change (- 3.1 6' F.). These switching times would undoubtedly reduce the overshoot. An improved response with 1370 overshoot using t 2 = 111, t 4 = 134 is shown in Figure 8, a. This was fitted by the model C,(S)
=
+ 1) exp(-25.9s) (164.6s + 1)(95.4s -l-1)
13.5(1.4s ~
F./ kw.
(27)
which yielded revised switching times ti = 109, t 4 = 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 a n 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 a t the left of each transient. T h e 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-integralderivative control after the final value was reached. Conclusions
Improved set-point responses can be achieved, for processes whose dynamics can be represented as overdamped secondorder 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. T h e model parameters can be obtained from bang-bang type response data (using nonoptimum switching times) by nonlinear regression. The design procedure gave satisfactory results when evaluated with a variety of processes, some differing widely from the assumed model, plus one physical system. I t is anticipated that the method will be particularly useful when incorporated into a supervisory digital control system. VOL. 6
NO. 4
OCTOBER 1967
459
Acknowledgment
P. M . Aiken assembled the experimental equipment. literature Cited
Ark, R., Amundson, N. R., Chem. Eng. Sci. 7 , 131 (1958). Athans, M., Falb, P. L., “Optimal Control,” McGraw-Hill, 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., Chem. Eng. Progr. 57, No. 6, 39 (1961). Gibson, J . E., Johnson, C. D., I E E E Trans. Auto. Contr. AC-8, No. 1, 4 (1963). Gray, R . I., Prados, J. W., A.Z.Ch.E.J. 9,No. 2, 211 (1963). Grethlein, H. E., Lapidus, L., A.Z.Ch.E.J. 9, No. 2, 230 (1963). Harriott. Control.” McGraw-Hill. New York. 1964. ~ ~ . P.. . “Process ~ Hougen,’J. O., Chem. Eng. Progr. Monograph Sei. 4, 60 (1964). Hougen, J. O., Chem. Eng. Progr. 59,49 (1963). Kouuel. L. B., Latour, P. R., IND. ENG.CHEM.FUNDAMENTALS, k,‘463 (1965). Lapse, C. G., I S A J . 3, 134 (1956). Latour, P. R., M.S. thesis, Purdue University, M‘est Lafayette, Ind., June 1964. Latour, P. R., Ph.D. thesis, Purdue University, West Lafayette, Ind., June 1966. ~
~
~
Lefkowitz, I., Chem. Eng. Progr. Symp. Ser. 46, 59, 178 (1963). Lupfer, D. E., Oglesby, M. W., I S A Trans. 1, No. 1 , 72 (1962). Lupfer, D. E., Parsons, J. R., Chem. Eng. Progr. 58, No. 9, 37 (1962). Marquardt, D. LY., Chem. Eng. Progr. 5 5 , 65 (1959). Marr, G. R., Johnson, E. F., Chem. Eng. Progr. Symp. Ser., No. 36, 57, 109 (1961). Mayer, F. X., Rippel, G. R., Chem. Eng. Progr. Symp. Ser., No. 46, 59. 84 11963). Mocieck,‘J. S.’, Otto, R. E., LYilliams, T. J., Chem. Eng. Progr. Symp. Ser.: No. 55, 61, 136 (1965). Oldenbourg R. C., Sartorius, H., Trans. A . S . M . E . , 7 0 , 78 (1948). Orent, H. H., Ph.D. thesis, Purdue University, LYest Lafayette, Ind.. June 1965. Pontryagin, L. S., Boltyanskii, V. G., Gamkralidze, R. V., Mischchenko, E. F., “The Mathematical Theory of Optimal Processes,” Wiley, New York, 1962. Roquemore, K. G.. Eddey, E. E., Chem. Eng. Progr. 57, No. 9, 35 (1961) Siebenthal, C D., Ark, R.. Chem. Eng. Sci. 19, No. 10, 729 (1964). Sproul, J. S., Gerster, J . A., Chem. Eng. Progr. Symp, Ser., No. 46, 59, 21 (1963). Williams, T. J., ZSA J . 12, 9, 76 (1965). RECEIVED for review October 31, 1966 ACCEPTED March 27, 1967 A.1.Ch.E. Meeting, Atlantic City, N.J. Financial assistance received from Purdue University and the National Science Foundation.
NONINTERACTING PROCESS CONTROL SHEAN-LIN
L I U
Central Research Division Laboratory, Research Department, Mobil Oil Gorp., Princeton, N . J . 08540
A new technique for the design of noninteracting control systems can handle constraint conditions on the control variables and 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.
N MULTIVARIABLE
feedback control systems, a change in one
I 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 any one reference input quantity will cause only the one corresponding controlled output variable (such as one state variable) to change. T h e 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. I n all the above references only linear problems were discussed, and constraints on control variables were neglected. Petrov (1960) discussed a very special nonlinear 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 Mesarovic (1964). The present procedure requires a complete dynamic model of the system and a n on-line digital computer. This paper announces two new results: Certain nonlinear, unconstrained processes can (1) be made noninteracting over the entire state variable space in a manner analogous to that for linear systems, and (2) be controlled in a piecewise non460
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
interacting way even when there are constraints on the process input variables. 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 Tiariable. In the second example, noninteracting control of a plate-type absorption column is considered. I t is assumed that there are seven plates in the absorption column and that one component in the gas phase is absorbed by liquid absorbent. I t is demonstrated that, by manipulating the liquid flow rate, one can maintain the gas outlet concentration a t a fixed value even if perturbations in the gas flow rate or the gas inlet concentration occur. Basic Theory
Before discussing the design of noninteracting control systems, the classical approach to control of linear multivariable processes (Freeman, 1958; Kavanagh, 1957) is reviewed, Since the process is linear, the control problem can be discussed in terms of Laplace transformed variables. In a closed loop system, as shown in Figure 1, if P represents the
