Optimal Control of a Continuous Distillation Column David H. Pike*’ and Michael E. Thomas Industrial Engineering Department, University of Tennessee, Knoxville, Tennessee 379 76 and Department of lndustrial and Systems Engineering, University of Florida, Gainesville, Florida
In recent years, many researchers have demonstrated the desirability of applying optimal control techniques to chemical processes. However, most of the experimental results were obtained by using computer simulations of the controlled system. The research reported herein is an attempt to “bridge the gap” between theory and practice by applying optimal control techniques to the actual system rather than a computer simulation of the system. A linear discrete time model of the process was used to obtain the optimal control law. The optimal control law was obtained by a dynamic programming solution procedure. The control is an explicit function of the state of the system and of future upsets to the system. Statistical models were developed which allow nonstationary stochastic upsets to be considered.
Introduction Many investigators have shown the potential for cost savings when using optimal control techniques. The investigations of Lapidus (1965) as well as Rafal and Stevens (1968) are examples. Most of these investigators have relied on computer simulations of the controlled system for experimental results. However, the use of a computer simulation avoids many of the problems associated with implementing the optimal control scheme. The work of Brosilow and Handley (1968) illustrates some of the problems which may be encountered. Their control law was a feedback control law which required, among other variables, the liquid compositions on each plate. Such information is not readily available and they were forced to estimate these compositions from temperature measurements. Had they used a computer simulation of the distillation column they would not have had to face this problem. In summary, many optimal control techniques may look promising when applied to computer simulations, yet they may not be practical because implementation on the actual system may be difficult if not impossible. In previous investigations, very little attention has been devoted to random upsets which are often encountered in chemical processes. There are many models which account for noise in the measurements. However, these models are inadequate to describe a sustained upset, such as ramp, which is frequently encountered with chemical prccesses. A model which is adequate to describe such upsets is presented herein. The objectives of this research were (1) to demonstrate that an optimal control scheme can be obtained for a distillation process using a linear model which is only in terms of the input variables, control variables, and upset variables; (2) to present statistical models to account for upsets of long duration; and (3) to demonstrate the applicability of the control scheme and upset models by applying the control law to control a distillation process. The process chosen for study was a 12-plate bubble cap distillation column. The tower was well insulated. The reboiler was a steam-jacketed kettle with an overflow outlet, a bottom pump, and a recirculation pump. The overhead vapors were condensed by passing them through a heat exchanger with water on the tube side. The condensed liquid was then fed into an accumulator. The liquid in the accumulator was split into a product stream and a reflux stream.
The feed mixture consisted of methanol and tert-butyl alcohol. This mixture was chosen because data were available from prior research which used this mixture.
Model of the System An optimal control law for a distillation column is both difficult to derive and difficult to implement. One reason for this is that a complete model requires a state vector which is not only of large dimension but has many components which are difficult to measure. For example, a complete model would require that the liquid and vapor compositions as well as the liquid holdup on each plate be measured. Such data are not as readily measurable as the related external variables such as top product flow rate. The authors of this paper felt that if an optimal control scheme were to be practical, the state vector should consist only of external variables. This approach to modeling the system is similar to that suggested by Zahradnik, et al. (1963). One of the objectives of this research was to derive an optimal control law for a distillation column using a linear model of the system. The linear model would have a state vector consisting only of external variables. Such a model would necessarily sacrifice accuracy to achieve a computationally feasible control law. The remainder of this section is devoted to describing such a model. A discrete time linear model of the following form was obtained X(t 1) = MX(t) NU(t) OF(t) QU(t 1) RF(t 1) (1) The state vector, X ( t ) , is a four-component vector. The control vector is U(t), a two-component vector. F(t) is a two-component vector of upsets. All variables in this model represent deviations from a steady-state operating condition. Table I gives a physical description of the components of each vector, the steadystate operating condition, and the units in which measurements were made. A complete derivation of the model is too lengthy to present here. A discussion of the pertinent assumptions will be given. The reader may obtain a detailed discussion in Eschenbacher’s Ph.D. Dissertation (1970). Eschenbacher assumed that the transfer function of the system was of the form
+
+
+
+ + +
+
’ Permanent address, University of Tennessee Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2,1974
97
TABLE I. Variables in the Model Variable State
Component 1
2 3
4
ConB~ol Upset
1 2 1 2
Description Top product
flow rate Bottom product flow rate Top product composition Bottom product composition Steam flow rate Reflux rate Feed composition Feed flow rate
Steady state
Units
135
Ib/hr
140
Ib/hr
10
% methanol
90
% methanol
185 220 50
lb/hr lb/hr % methanol
275
lb/hr
for i~{1,2,3,4\ and jc(1,2,3,4] where X i (s) = Laplace transform of state variable i, I?, (s) = Laplace transform of either a control or upset variable, K,, = constant, ,511 = constant, and Ti, = time constant. Using the principle of superposition, one can obtain
(3)
The model as described above is essentially a “fit” of an assumed model to an observed response under a very special set of conditions. A question naturally arises as to the accuracy of such a model. To analyze the model, the column was monitored while operating under conventional control. The model’s effectiveness in predicting the state vector (using eq 1) was analyzed by comparing the predicted state vector to the observed state vector. Fifty data points were used in this analysis. The results are shown in Table 11. Using the average deviation between the predicted and actual value of the state variable as the criterion, the authors felt that the linear model was sufficiently accurate to justify its use in obtaining an optimal control law.
Optimal Control Law A quadratic cost functional was assumed N
Ccu‘(X(i)’AX(i) 1-0
+ U(i)’BU(i))
(7 )
,
The optimal control problem was to minimize eq 7 subject to the constraints X(t 1) = MXit) NU@) OF(t) QU(t 1) RF(t 1) (8)
+
+
+
+ + +
+
and the boundary conditions as the Laplace transform of the response for variable i, ic{1,2,3,4}.It is then possible to find a common denominator for the right-hand side of eq 3. This will yield a fourthorder transfer function. An optimal control law derived from a fourth-order model is a function of the four most recent observations of the state vector. This requires, of course, more data storage capacity in the control computer and requires more time to calculate the optimal control. A first-order model would, therefore, be more desirable than a fourth-order model. To obtain a first-order linear model, the time constants T i j appearing in eq 3 were replaced with a common time constant Ti. The common time constant was an average time constant (4)
X(0)= U(0)
= F(0) = 0
The time interval over which control is applied is NAt. It is assumed that the steady-state operating condition is optimal as determined by an appropriate objective function. By making A and B positive definite matrices, the control will tend to be maintained about the desired operating condition ( i e . , about X ( t ) = U ( t ) = 0). This follows since if A and B are positive definite matrices then any X ( t ) # 0 causes X ( t ) ’ A X ( t )to be positive (similarly U(t) # 0 causes U(t)’BU(t) to be positive). Dynamic programming was used to solve the optimal control problem. The optimal control law at time ( t + 11, u(t l),was obtained as
+
U(t
+ 1) = G,(t + l)X(t) + G,(t + l)U(t) +
2 G 3 ( j , t+ 1)F(j) (10)
1’1
Equation 3 then becomes
1‘1
By taking the inverse transform of eq 5 and discretizing the resultant differential equation, one obtains eq 1. There is only one variation in the model described above. Due to turbulence in the reboiler, accurate measurements in the bottom product flow rate, X z ( t ) , were difficult to realize. More accurate results were obtained when instead of eq 5, the following mass balance equation was used
X,(t) Fdt) - Xi(t)
(6)
To determine the unknown constants, eq 3 was used. Step upsets were made one at a time, holding all other control and upset variables constant. The column response was recorded. The responses to the one-at-the-time step upsets were used to determine the unknown constants ( K i j , Li,, and Ti]). It was from a consideration of this data that A t = 72 sec was chosen. It was felt that 72 sec allowed an adequate time for controls to be calculated, implemented, and for their resultant effect to be experienced. Certain time lags in the observed system response were also accounted for using this value of A t . 98
(9 )
Ind. Eng. Chern., Process Des. Develop., Vol. 13, No. 2, 1974
Details of the derivation are given by Pike (1970). The matrices appearing in eq 10 are determined by solving a set of simultaneous difference equations. The constant (Y which appears in eq 7 is a constant greater than zero and less than one. Use of this constant, called a “discount factor,” enables one to obtain an asymptotic solution. It has been shown by Bellman (1957) thatwithO j . Making use of this fact enables one to develop a realistic statistical model of the upset process, It will be assumed in the discussion which follows that the components of the upset process Fl(t) and F*(t)are independent random variables. A general component, F q ( t ) , of F ( t ) will be used to describe the process. The discussion is equally applicable to any component of F(t ) . It is assumed that a class of functions, c = If&): k = 1, 2, . . . , m),exists. The elements of C may be any finite valued functions. The upset process randomly assumes .one of the elements of C at random instants of time. That is, if F,(t) assumed the kth element of C y stages ago, then
1 0
o iJ
and
The second case, the “low control cost case,” was obtained by multiplying the E matrix by 0.1. Figure 1 shows the components of the G3G) matrix as a function of j . Two conclusions can be drawn from this data. First, the future upsets have more of an effect on the control law for the low control cost case than for the equal cost case. Second, the future upsets, in either case, have a significant effect on the control for a t only about 30 stages into the future. That is lim G3G) = 0, (a null matrix) and the series may be regarded as having converged for j > 30.
Models for the Upset Process One of the distinguishing characteristics of upsets for chemical processes might be termed a “persistence phenomenon.” That is, once a change has occurred in the upset process, it or the trend it establishes is likely to continue for some time. For example, if the composition of the feed of a distillation column is observed to change suddenly, it is very likely to remain at the new level for some time to come. In order to adequately develop the persistence concept, the existence of functional trends
In the research reported herein, the functions f k ( y ) were assumed to be constant functions. Two probability distributions must be defined. One distribution is for the length of time between changes. That is to say, if Wl, Wz, . . . is the sequence of times between element changes, then
P,(W, 5. a ) = $(a)
for i = 4 2 , ...
(13)
must be specified. $ ( a ) is simply the probability that the time between changes in “a” stages. The other probability distribution, which must be specified, gives the distribution on element changes given that a change occurs. A distribution, G ( s ) ,is defined, where G(s) = P,{F,(t) IS / a change
OCCU~S~
(14)
In summary, one must specify the distribution defined by eq 13 and the distribution defined by eq 14. In application it is necessary to define a random variable Y ( t ) . Y ( t ) assumes nonnegative integer values and denotes the “age” of the process (the number of stages which have elapsed since the last change). For values of kt {1,2. . .) the expected value of Fq(t k ) depends upon F,(t) and Y(t). The following equation may be used to determine E[F, ( t + k ) ]
+
E[F,(t
+ k)/F,(t) = b, Y(t)= T I =
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2,1974
99
where
H ( a )= 1 - $(a) and A
1.1 = s + m s dG(s)
(16)
-m
$
The derivation of eq 15 is given in the Appendix. Experimental Procedure a n d Results Each experiment was initiated by bringing the column to the steady state given in Table I. Each optimal control result was compared to a “conventional control” scheme which controlled the top and bottom product composition. In controlling top production composition, a refractometer was used to monitor the composition. An air signal from the refractometer was used to drive the reflux controller which in turn controlled the reflux rate. The bottom composition was controlled by monitoring the bottom product stream with a refractometer. The air signal from the refractometer was used to drive the steam controller. The steam controller was used to control the steam rate to the reboiler. The controllers were pneumatic set point controllers with integral and proportional modes. When performing optimal control experiments, the configuration detailed in Figure 2 was used. The compositions were monitored by means of electrical signals from refractometers. Flow rates were monitored by air signals from transmitting rotameters. The air signals were transformed to electrical signals by using P - I transducers. All electrical signals were then connected to an IBM-1070 system. The IBM-1070 system served as an interface between the distillation column and the process control computer. The process control computer was an IBM-360/65 computer used in the time sharing mode. The gain matrices in eq 11 were stored in the computer. At each instant of time, the state was measured and stored. The previous state, previous control, and estimates of future upsets were used to calculate the current control. The controllers on the process were pneumatic set point controllers with integral and proportional modes. The controllers were equipped with stepping motors which could control the set point. The control law was implemented by sending a signal to the IBM-1070 system. The interface would convert this signal to a pulse train of appropriate duration which caused the set point of the controllers to be changed. The series of experiments was performed using the stochastic model described previously for both components of the upset vector. The distributions on length of time in a form were assumed to be
