of the synthesis process. An insufficiency of ammonia will produce ammonium bisulfate or triammonium hydrogen sulfate-for example, the theoretical yield of ammonium sulfate can be calculated when 0 . 2 7 of the sulfur dioxide in flue gas is completely oxidized to sulfur trioxide, and 0.4% of the ammonia is added to the system at 200°C. The equilibrium pressure of ammonia a t 200°C. calculated from Equation 12 is atm., which is equal to 0.014.Therefore, 0.39% of the ammonia will react, leaving behind 0.01”C a t 200°C. Thus 0.19% of the sulfur dioxide will change to ammonium sulfate, and the residual 0.OlLCwill change to ammonium bisulfate. Miller’s (1955) process for pickling iron with ammonium bisulfate, Holowaty’s (1959) process for recovering ammonia from coke oven gas, and the many industrial processes mentioned above can be investigated in detail by utilizing Equations 10, 11, 12, and Figure 8.
Acknowledgment
The authors express sincere thanks to H. Kuronuma and G. Uwanishi of the laboratory for their useful suggestions.
Literature Cited
Bonfield, J. H., Bohn, F. L., U. S.Patent 3,282,646 (1966). Datin, R. C., U.S. Patent 3,172,751 (1965). Dixon, P., Nature 154, 706 (1944). Duval, C., Wadier, C., Servigne, Y., Anal. Chim. Acta 20, 263 (1959). Erdey, L., Gal, S., Liptay, G., Talanta (London) 11, 913 (1964). Haeter, L., Reichau, U., Ger. Patent 1,151,492 (1963); C A 59, 8386h (1963). Holowaty, M. O., U. S. Patent 2,899,277 (1959). Ito, S., Uchida, S., J . Chem. SOC.Japan. Pure Chem. Sec. 73,348 (1952). John, C. M., U. S. Patent 3,047,369 (1962). Kiyoura, R., Chem. Erg. News 44 (26), 23; (27), 36 (1966a). Kiyoura, R., J . Air Pollution Control Assoc. 16, 488 (1966b). Lange, N. A., Ed., “Handbook of Chemistry,” 10th ed., p. 219, McGraw-Hill, New York, 1961. McCullough, R . F., U. S. Patent 2,927,001 (1960). Miller, C. O., U. S. Patent 2,700,004 (1955). Montgomery, J. C., U. S. Patent 3,047,369 (1962). Rafal’skii, N. G., Ostrovskaya, L. E., Geterogennye Reaktsii i Reakts. Sposobnost 1964, 95 (1964); C A 64, 15057h (1966).
RECEIVED for review December 18, 1968 ACCEPTED April 25, 1970
Model Predictive Time-Opti mal Control of Second-Order Processes D. A. Mellichamp Department o f Chemical and Nuclear Engineering, University of California, Santa Barbara, Calif. 931 06
Set point changes for second-order processes are implemented using a hybrid computer and a fast-time model of the controlled process t o search out the time-optimal trajectory.
Process outputs are used as initial conditions t o monitor the process transition in continuous, closed loop fashion. Model scan rates up to 100 per second (with convergence t o the optimal solution in milliseconds) furnish a continuous updating of predicted switch times for the controlled process a n d make possible the multiplexing of a single hybrid controller t o a large number of controlled processes. The controller has excellent characteristics for set point changes a n d also as a regulator with both analog simulated processes a n d an experimental two-tank liquid level system.
C o n t r o l methods that use predictive techniques are applied in many critical aerospace and military situations. The ability to predict and program system changes in advance is a well-recognized practice in the supervision of space probes and satellite orbit changes, and control of high performance aircraft and submarines. Most predictive techniques rely on a rather sophisticated mathematical model of the controlled system; a computer is used to scan a large number of possible control schemes, from which the most useful is selected. Alternatively, a large 494
Ind. Eng. Chern. Process Des. Develop., Vol. 9, No. 4,1970
body of mathematical theory has been developed which can be used with a linear system model to generate a control procedure that will be optimal in some sense. Procedures differ, depending on the immediacy of the situation: The rocket burn time for a deep space probe obviously can be computed months in advance; a similar calculation for satellite rendezvous may have to be made as approach occurs. I n either case, although minor corrections will have to be made during the transition to take into account discrepancies between the actual system and
the mathematical model, the initial estimates are of primary importance in effecting the change. Until the advent of digital control computers, there was little foreseeable application for such techniques in the process industries. At this point, however, it seems a great waste of computational ability to use a control computer to mimic a flock of two- and three-mode controllers with a nod in the direction of steady-state optimization. Several procedures for dynamic optimization which can be implemented on control computers have been published. Latour et al. (Koppel and Latour, 1965; Latour et al., 1967, 1968), in particular, have suggested the use of minimum time transitions from one set of steady-state operating conditions to a new desired operating state. The rationale for such a time-optimal change is simply to obtain the new steady-state conditions (presumably selected as more desirable than the present situation) as quickly as possible. Minimum time transition changes are obtained by operating the manipulated variable or control input a t the extreme limits (saturation) during transition. Some control engineers feel that this overdisturbs the process; however, for any given process, it is possible to choose some artificial maximum and minimum values for the input variable which would not be exceeded so long as the desired process output remained reachable. Such “interior constraints” could be selected t o give satisfactory transition times and be modified periodically as desired. The literature contains many references to the timeoptimal control problem. Koppel and Latour (1965) present a good bibliography of the theoretical development. A number of papers deal largely with computational aspects. Neustadt (1960) and Eaton (1962) developed theoretical algorithms for obtaining time-optimal solutions. Several papers outside the chemical engineering literature have dealt with computer implementation of the general time-optimal problems; Gilbert (1963) treats the hybrid computer solution of Neustadt’s algorithm and Nishimura et al. (1966) discuss an analog computer implementation of a general geometrical search procedure (with applications to a second-order system). Within the chemical engineering literature Latour et al. (1967, 1968) have covered the application of timeoptimal control techniques to second-order systems with dead time. They derived the equations for appropriate switch times (process initially a t steady state) which can be implemented to furnish a programmed time-optimal manipulated variable, obtained expressions for the switching curves, and showed how to implement the timeoptimal feedback control law (no assumption of process initial conditions required) (Latour et al., 1968). This paper describes the design and evaluation of a controller which continuously predicts the response of a second-order process during set point changes and specifies the input sequence required to make the remaining transition time-optimal. The method is an extension of a technique developed by Coales and Noton (1956) and applied to the time-optimal control of a positioning servomechanism. Because the method is predictive, it extends in a relatively straightforward way to second-order processes that include dead time. The method combines the advantage of programmed input time-optimal procedures (input sequence known in advance) with those of feedback methods (relative insensitivity t o model inaccuracies and independence of
starting conditions). Additionally, the future state of the process throughout the transition is known. As might be expected, these advantages are obtained a t the expense of increased computational requirements. Some convenient geometrical properties of time-optimal trajectories in the state variable space (for second-order systems, the state plane) are used in the searching process. Hence, we look a t some characteristics of the timeoptimal solution curves before considering details of the computational procedures. Background
Two equivalent formulations of the second-order process are considered. Ordinarily, the process response is obtained experimentally and found to be approximately secondorder. Hence, the process equations (equivalently, the gain and time constants) are obtained empirically. The process may be represented as in Figure 1 (upper), where only one output variable is measured (measurable). Occasionally it is possible to measure two process outputs directly-for example, by measurement a t an intermediate stage in a series of stirred tank reactors or in a distillation column-and to relate these output variables to a single manipulated variable (Figure 1, lower). For the system given in Figure 1 (upper) we have
bT2Z + ( b + 1)TZ + 2 = K(X + L )
(1)
or defining a pair of state variables 2 1=
z
(2)
2, = 2 we have
2
0
1
(2:) = I-l,bT’ - ( b
z
+ l),bT’l (Z:l
+
0
[ K / b T ’ l ( X + L ) (3) 2, and 22 ordinarily are called phase variables and, since the time-optimal control of a second-order system requires the measurement (or estimation) of two output quantities, the formulation of the problem in terms of the process output and its first derivative is referred to subsequently as the phase variable approach. Momentarily disregarding the effect of the load variable, L , we note that a plot of Z 2 us. 2, (the phase plane) reveals two LOAD
IL MANI PULATED VARIABLE
OUTPUT VARIABLE
Figure 1. Second-order process block diagrams Upper. Lower.
Single output variable measured Output a n d intermediate variables measured
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 4, 1970 495
stable nodes corresponding to the two allowable inputs on X, X L and XH, where the subscripts refer to the low and high values of the manipulated variable. The phase plane for a typical system ( b = 0.667, K = 0.6, XL = 0.25,XH = 0.9) is plotted in Figure 2 . The lines represent trajectories of the state variables toward the two possible steady states. The horizontal axis represents all possible set points: The system is a t rest (zero derivative). For a given set of initial conditions (point 1) and a desired set of different final conditions (point 3) the time-optimal trajectory consists of two curve segments. Segment 1 to 2 is obtained by applying the input X H to the process until the process reaches point 2, then switching to X L until the process reaches point 3. For any given process initial condition, the time-optimal trajectory requires one switch a t most to reach 3. The locus of the switch points constitutes two switching lines which consist of the initial portions of the two trajectories passing through 3 approaching the two stable nodes. The switching curves for final Z1 = 0.36 are shown as the heavy lines in Figure 2 . An alternative formulation of the problem uses two measurable state variables (or independent process outputs) (Figure 1, lower). For this case we have
+ Y1 = K1(X + L1) Try?+ Y ?= Kl(Y1 + L?) TIYI
(4)
or in vector matrix form
. -
+02
L
z, 0
I
I
1 J
~
I
0
I 0.2
I 0.4
\
0.6
1 0.8
I
i 1
I .0
Figure 2. Phase plane trajectories for a typical secondorder process ( b = 0.667, K = 0.6) with two allowable input levels ( X L = 0.25, X H = 0.9) 496
Figure 3. State plane time-optimal trajectories for a secondorder process ( T ? / T i = 0.667, Ki = 1, K? = 1, 11 = 1~ = 0 , X i = 0.25, X H = 0.9)
look similar to phase plane optimal paths; for the state plane case all possible final set points lie on the straightline segment connecting the two steady states corresponding i o XL and XH. Method
The state variable trajectories are plotted in Figure 3 (Ti/Ti = 0.667, Ki = 0.6, K ? = 1.0, L1 = LI = 0 , XI, = 0.25, XH = 0.9, and final Yz = 0.36). For this case, only the actual time-optimal trajectories are shown for several starting (initial) conditions. The switching curves are drawn in and the two steady states corresponding to the two possible inputs, XL and XH, are indicated. Time-optimal trajectories in the state plane
+04
y2
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970
The control procedure utilizes the established mathematical fact that a linear second-order system with a constrained manipulated variable requires a t most one switch from maximum to minimum input (or vice versa) to move from one set point (set of initial conditions) to another in minimum time. A second switch to some intermediate value of the manipulated variable, X 7 , is required to retain the process a t the new (terminal) conditions, or the process may be controlled a t the new conditions by rapid switching of the input between constraints (on-off or relay control). Therefore, for a second-order system a controller needs only to compute which input comes first (high or low) and the time a t which the switch should be made to reach the desired final conditions. The resulting trajectory must be, mathematically, the minimum time trajectory. A relatively straightforward procedure can be developed to synthesize a second-order time-optimal controller using a hybrid analog computer. A “fast time” analog computer model of the controlled process is used to predict process trajectories for various programmed time sequences of the manipulated variable. A small amount of binary logic is required to direct the analog computer search for the input sequence which yields the optimal future trajectory. The computational algorithm is discussed for the situation where two successive state variables can be measured. Where only one output variable is measurable, the output time derivative must be calculated. Mathematically, the treatments are identical. I n subsequent discussions we have to differentiate process quantities from analogous model quantities, To facilitate writing the equations, upper case letters are used to designate actual process variables or parameters. Equivalent terms in the mathematical model are written in lower case. For time-optimal control the appropriate set of model
equations (neglecting load terms for the moment) is given by
I xzx,
INITIAL CONDITIONS
TERMINAL CONDITIONS
INPUT
x=
x,o 5 t
< ts {x,t s 5 t < t 7
t
where subscripts 0 and T denote initial and terminal conditions and correspond to the immediate present and desired (set point) values of the process state variable, respectively. The variables t and T denote model time and process time. respectively. Subscript S denotes the time a t which the input switches from one extreme to the other. Initially the model input, x , is equivalent to the process input, X , and a t the switching time jumps t o “not X ” or the alternate possible input value. The problem a t this point is in two-point boundary value format with an open condition on the terminal time, t T . In addition, the switching time, t s , is not known. Several methods may be employed to determine both these quantities-for example, the closed form solution of Equation 6 subject to input 9 can be obtained and searched numerically for t s and tr. A very efficient procedure, which takes advantage of the geometry of the state plane solution trajectories and can be implemented easily on a small analog computer, was used by Coales and Noton (1956) to actuate a positioning servomechanism time optimally. The state plane geometry of the solution curves for this electromechanical system [operation described approximately by T(d’Z/dt’) + d Z / d t = x ] resembles that for the overdamped second-order system sufficiently that the method may be extended to the more general second-order case. If all time-optimal solutions are considered (Figure 3 where the coordinates are viewed as model outputs), all trajectories cross the horizontal line yl = y l T a t least a t the final or terminal point (sometime after switching the input). If the terminal time, t T , is defined as t ( y l = y i r , t > t , ) , any error in the assumed switch time, ti., will result in an error in y? a t t7 (Figure 4). The error, y r ( t ~ -) y27, can be used to adjust T u in such a way as to decrease the value of the error in the next successive calculation of the model trajectory. Figure 4 shows one typical iteration, indicating that the terminal error will be decreased during the following cycle by a slight increase in the value of T , (or t s ) during the adjustment period ( t i , t ~ ) Initial . conditions are reset during the final part of the iteration ( t K ,t(.). The approach, then, is to implement a model of the process on a high speed analog computer along with the small amount of logic required to direct an iterative search for the optimal switching time. An equation of the form
x,
Ts
MODEL T I M E , t
-
Figure 4. Search for time-optimal solution, one iteration
with A a controller parameter of appropriate sign and magnitude, is satisfactory for this purpose. Additional logic to initiate the search and to change the input to the process whenever the computed switch time, T,, becomes less than zero is easily implemented. Figure 5 , a schematic diagram for the model predictive controller, shows explicitly that the process states are used as model initial conditions. Ts is predicted continuously throughout the process transition with the input to the process switched as required. The net effect of operating the model 10’ to lo4 times faster than the process is to achieve a form of true feedback control. I n this approach the controller always functions to compute the next switch in the process input. Because of small delays in the computation, the controller always overshoots the switching curve slightly, yielding t s < t 7 , t., N tT along the final portion of the trajectory. Similarly, on reaching the terminal point, the process is maintained in a tight limit cycle about ylT, y L T . Proper choice of controller parameters ( A in Equation 10 and the initial choice for TL inserted whenever the process input is switched) results in time-optimal transitions for set point changes and excellent regulation a t the final control point. Details of the predictive controller including the extension to systems which include time delay are covered in a recent paper (Mellichamp, 1970b). Evaluation
Set Point Changes. The controller was tested for set point transitions using a phase variable formulation of the process and model. The process was modeled on the computer with first time constant (TI)of 30 seconds and gain ( K ) equal to 1. Transitions of Z 1 , Z , between (0,O) and (0.5,O) were made. The input variable was normalized to Xi = 0, XH = 1. The effect of the second process time constant on the time to the first switch ( T , ) was Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970 497
Y PROCESS
y,
- 1
II
I
I I
:kA',!:a;----i fi INPUT SWITCH
Figure 5 . Block d i a g r a m of model predictive controller
studied in the range 0.1 5 6 5 1.0. A time scale factor of 500 for the model time constants and an iteration time t, = 125 ms (8 solutions per second) were selected (Figure 6). The initial estimate of Ts/T1 (the value obtained just as the transition t o the new set point is started) is plotted us. 6. ATs/T1,a measure of the difference between the initial estimate of Ts and the measured time to the first switch during the closed loop control transition, is also plotted. A certain amount of error involved in measuring TS is reflected in the scatter of the ATs points. However, the average discrepancy is less than +0.01, or better than 1% accuracy in closed loop estimation of the switching time. This small dynamic lag could be easily compensated, but the resulting overshoot of the switching curve eliminates "chattering" or switching as the process deviates slightly from side to side of the switching curve. The resulting slight increase in the total transition time is negligible.
1
1
1
'
1
x
0.010
/
\
I t"
0.2
0.4
0.6
0.8
1.0
PROCESS SECOND TIME CONSTANT, b
Figure 6. Effect of process second time constant on the time to first switch
498
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970
The estimated T,'s are essentially identical with values calculated from theoretical equations. We conclude that, for set point changes, the controller functions timeoptimally, well within the limits set by our ability to model the process. Regulator Response Characteristics. Time-optimal controllers described in the literature to date are designed to function only during transitions (set point changes). Some authors recommend replacing the time-optimal controller with standard regulating equipment such as a threemode controller upon conclusion of the transition. The dual mode switching device described by Latour regulated satisfactorily but the resulting limit cycle in the process output variable was judged too large for precise regulation. The regulating characteristics of the predictive controller were very good over a wide range of process parameters and load disturbances. Although such a controller ordinarily would not be used except to predict and maintain process transitions, it is capable of functioning independently of auxiliary control schemes. At the end of a process transition, the controller puts the process into a tight limit cycle about the desired operating point. For a process operating approximately a t mid-range (the range of 2 reachable by means of the available inputs X H and X L equally spaced about ZT), the amplitude of the limit cycle is about 0.001 times the range of Z with no load applied. For example, in a thermal process with a range of process output of 100" to 200°C the controller could regulate the process in the vicinity of 150"C within 10.1'C temperature limits. For a set point located a t 90% of the reachable output variable range, the limit cycle amplitude deteriorates by a factor of about 3. For the above example process operating a t 110" or 190°C we could expect to maintain the temperature within 1 0 . 3 " C . These data are obtained with a controller adjusted for best time-optimal transitions. Much higher regulation can be obtained by decreasing T,,, the initial choice for Tr, and increasing A ; however, the resulting controller tends to chatter in the vicinity of the switching curves during set point changes and consequently is not time-optimal. The application to the process of unmeasured and rather severe loads results in a slight increase in limit cycle amplitude and period, as shown in Figure 7 for the
operating point located a t mid-range (process described above with b = 0.1). As Latour points out, the applied load must be measured as a fraction of the total control effort available. For this case a load L / ( X H - X,)= 0.5 corresponds to an asymptotically reachable situation for which the limit cycle period is infinity. For set point located a t 90% of the reachable output variable range, we find the situation illustrated in Figure 8. Application of a positive load to the process actually
0.30357
00030
1 F
li J
0
LL
t
0
0
t 2
-
LL
0
LL
W
improves the output limit cycle (by effectively increasing the availability of positive control input). The similarity of Figures 7 and 8 indicates that the limit cycle amplitude and period are predominantly functions only of the distance between the desired value of 2 , ZT, and the Z which would result with balanced control effort. Variation in the second process time constant was shown to have virtually no effect on regulating operation of the predictive controller as shown in Table I . Also shown is the offset from Z r which results from application of the load. The observed offset is small, a function of the magnitude and direction of the load, and consequently could be used to adjust automatically parameter L in the model Equation 2 to compensate for long-term ambient changes. In all of these tests the load was applied as a step input to the process. Even for a large unmeasured step change in load, the process output change is practically nil (Figure 9). Hence, we conclude that the predictive controller yields time-optimal set point changes and a t the same time is capable of regulating step load changes far better than the most critically adjusted three-mode controller.
w
n
d >-
2
Table I. Effect of Second Time Constant on System load Response
0
L / (Xx- XI.) = 0.25
1
0.0005
ZT = 0.5
n
Limit Cycle Amplitude
w 0.I
0
0.2 L/(X,
0.4
0.3
-xL)
Figure 7. Regulator response of controller Limit cycle amplitude and period vs. applied b o d with set paint a t midrange
b
No load
Load
Limit Cycle Offset Under Load
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.o
0.0016 0.0013 0.0014 0.0013 0.0014 0.0017 0.0018 0.0019 0.0018 0.0020
0.0019 0.0017 0.0015 0.0016 0.0017 0.0022 0.0021 0.0022 0.0024 0.0024
0.0024 0.0024 0.0021 0.0023 0.0020 0.0027 0.0022 0.0024 0.0029 0.0029
2E
c
u
0
-
02
L
c
d 04
06
. /\X,-X,)
Figure 8. Regulator response of controller Set point at 90% of reachable output range
08
0.6
0.0
I
.c
I .2
1.4
T/T Figure 9. Effect of modeling error Transition time vs. relative error in first or second time constant
Ind. Eng. Chem. Process Des. Develop., Vol. 9 , No. 4, 1970
499
Effect of Modeling Errors. Obviously, there would be little purpose in using a predictive controller if a useful model of the process were not available. As with optimal control solutions in general, the effort expended in determining the “best” control is only as good as the mathematical model of the process. On the other hand, we expect a control system to functici I despite minor discrepancies and to be somewhat inseri itive to them. Whether the model agrees perfectly with the process, we can a t least expect it to be self-consistent. I n particular, the values of the gains, the set value of the manipulated variable, and the desired or terminal values of the outputs should be consistent. The predictive controller was tested to determine the effect of modeling inaccuracies, using a state variable formulation of the process and model equations (Equation 4 ) . I n this form the model requires specification of several additional parameters: I n particular, estimates of intermediate process gain and process load variables are required. A wide range of parameters was tested; results are presented for the specific process conditions given in Table 11. I n all cases, data are reported for deviations of the process parameters from the parameters given in Table I1 which were appropriately scaled and used for the model. The model solution rate was set a t 100 repetitions per second (t, = 10 ms) and a time scale factor of 5 x lo3 was used. This was the fastest model scan rate used in these studies and was limited by the reset capability of the sampleihold units. There appears t o be no practical reason why repetition rates of lo3 to lo4 solutions per second cannot be achieved. However, such speeds are hardly necessary unless one contemplates multiplexing 100 or more control loops to the same predictive controller. Figure 9 illustrates the results of varying process time constants on the time required to transition from (0,O) to the final limit cycle about (0.36,0.36). A certain amount of scatter in the points results from the presence of extra switches in the input. These spurious switches are caused by discrepancy between the model-predicted trajectory and the actual process path. For process time constants below the model value, the transition time is lower than the time predicted by the controller; however, in these cases it is not the minimum for that particular process configuration. As an example, if T I is only 60% of the modeled first time constant, the transition takes 25.1 seconds rather than 31.5 seconds as predicted. The theoretical minimum time for the process with T I = 9 is 24.2 seconds. The important conclusion here is that modeling inaccuracies do not disable the controller; in fact, for deviations within engineering accuracies of i10% the performance is roughly equivalent. Figure 10 presents a similar plot of transition time as a function of deviations between process and model gain parameters. I t should be possible to determine the over-all process gain fairly accurately; hence, one curve is for the product K 1K 2 constant a t the theoretical value
Table II. Values of Process Parameters
T I = 15 sec T p= 10 sec X L = 0.25 XM = 0.90
Ki = 0.6 K2 = 1.0 X r = 0.60
L1 = 0 Lp = 0 Yir = 0.36 Ypr = 0.36
I\\
C.6
0.0
2
IO
K/
+ 24.1 0.622
Y I ,Y , in chart units: 20 units X in inch’/min:
= 1-inch level
X L = 189
XH = 285 T in minutes 1.33 minutes; the actual switch occurred a t 1.39 minutes. Both the above changes were initiated with the process a t steady state, although the controller functions equally well for all transitions. Regulation of the process output Y y a t the final steady state is maintained 1 0 . 5 chart unit corresponding to a level deviation of about *0.025 inch. Load changes imposed on the process (by changing the steady process inputs) are scarcely detected; however, these changes would have to be incorporated in the process model for subsequent set point changes. Ordinarily, the predictive controller would be replaced by a conventional controller after the transition, in order to minimize wear on the process valve while regulating the process a t the new conditions. Extension to Higher Order Systems (Including Time Delay)
The optimal control of an nth order process requires the measurement or estimation of n independent state variables. This requirement can be met in theory by measuring a single process output and extracting n 1 successive time derivatives; however, in the usual experimental environment, random and cyclic process noise limits this approach to the estimation of a single derivative.
X
X,
rn . u
TANK I
t
TRANSMITTER
TO CONTROLLER
TANK 2
P/ E TRANSDUCER I
i TO CONTROLLER
Figure 12. Schematic diagram of liquid level system
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970 501
TIME, MINUTES
*Ot
0
I
2
3
4
5
I
I
I
I
I
1
Figure 13. Time-optimal level changes Upper.
2-inch increase in tank 2 level
Lower.
1-inch decrease in tank 2 level
Furthermore, although the procedures outlined above for designing a predictive controller can be extended to systems of third-order or higher, the problems of implementing the high-speed switching logic are considerable. For these reasons, a second-order-plus-time-delay system controller appears to be a very useful compromise. Koppel and Latour have cited the widespread empirical application of this model to processes of order much higher than 2. The utility of this process model is due largely to the time-delay term into which the high-order process time constants may be empirically lumped. Because knowledge of the process future trajectory is generated in advance with the predictive method, the extension of the second-order controller to the case including time-delay is straightforward. In fact, there is no necessity to model the time-delay term; a simple binary logic circuit is used to implement switching of the process input in advance of the predicted times (by the amount of the time delay). Details of the extended controller and its application to simulated processes are given by (Mellichamp, 1970b). Results are also reported (Mellichamp, 1970a) for application of the predictive method to the control of a bench-scale heat exchange process. In this case, the process contained a significant transportation lag, and the output required heavy filtering to remove medium and high frequency noise prior to derivative extraction. However, by lumping the filter
502
Ind.
Eng. Chem. Process Des. Develop., Vol. 9, No. 4,
1970
dynamics empirically in the second-order-plus-time-delay model, the resulting fifth- or sixth-order system was driven by the predictive controller in a manner that resulted in close to the minimum time transition for the original (unfiltered) system. Literature Cited
Coales, J. F., Noton, A. R. M., Inst. Elect. Eng., Proc. Paper M1895 (1956). Eaton, J. H., J . Math. Anal. Appl. 5, 329 (1962). Gilbert, E . G., “Proceedings of Spring Joint Computer Conference,” p. 197, Spartan Books, Baltimore, 1963. Koppel, L. B., Latour, P. R., Ind. Eng. Chem. Fundam. 4, 463 (1965). Latour, P. R., Koppel, L. B., Coughanowr, D. R., IND. DES. DEVELOP.6, 452 (1967). ENG. CHEM. PROCESS Latour, P. R., Koppel, L. B., Coughanowr, D. R., IND. ENG. CHEM. PROCESS DES. DEVELOP.7, 345 (1968). Mellichamp, D. A., AIChE 67th National Meeting, Paper 42c (February 1970a). Mellichamp, D. A., Simulation 14, 27 (January 1970b). Neustadt, L. W., J . Math. Anal. A p p l . 1, 484 (1960). Nishimura, M., Fujii, K., Kuroda, E., Tech. Repts. Osaka Uniu. 16, 247 (1966).
RECEIVED for review December 30, 1968 ACCEPTED May 12, 1970
