Lopez, A. M.. Smith, C. L., Murril, P. W., Brit. Chem. Eng., 14 ( I l ) , 1553 (1969). McAvoy, T. J.. Johnson, E. F., lnd. Eng. Chem., Process Des. Dev., 6, 440 (1967). McAvoy, T. J., lnd. Eng. Chem., Process Des. Dev., 11,72 (1972). Mellichamp, D. A., lnd. Eng. Chem., Process Des. Dev., 9, 494 (1970). Meyer, J. R., Whitehouse, G. D., Smith, C. L., Murril, P. W.. lnstr. Contr. Syst., 40 (12), 76 (1967). Miller, J. A.. Lopez, A. M., Smith, C. L., Murril, P. W., Contra/ Eng., 14 (1I), 72 (1967). Paraskos. J. A.. McAvoy. T. J., AlChE J., 16 (9,754 (1970). Pontryagin, L. S.,Boltyanskiy, V. G., Gamkrelidze, R . V., Mischenko, E. F., "The Mathematical Theory of Optimal Process", Wiley, New York, N.Y., 1962. Rovira, A. A,, Murrii, P. W., Smith, C. L., lnstr. Control Syst., 42 (12), 67 (1969). Shinsky. F. G., "Process Control Systems," McGraw-Hill, New York, N.Y., 1967.
Smith, 0. J. M.. Chem. Eng. Prog., 53 (5),217 (1957). Smith, 0. J. M., iSA J., 6 (2), 28 (1959). Smith, 0. J. M., "Feedback Control Systems," Chapter 10, p 325, McGrawHill. New York, N.Y., 1958. Smith, C. L., Murril, P. W., /SA J., 13 (9, 50 (1966). Weigand, W. A.. Kegeris, J. E.. hd. Eng. Chem., Process Des. Dev., 11, 86 (1972). Williams, T. J., et. al., Fourth Joint Automatic Control Conference, Minneapolis, 1963. Wills, D. M., Control Eng., 9 (4), 104 (1962a). Wills, D. M., Control Eng., 9 (8). 93 (1962b).
Received for review July 2 5 , 1974 Accepted September 5 , 1975
Linear Feedback vs. Time Optimal Control. II. The Regulator Problem Alan H. Bohl' and Thomas J. McAvoy' Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 0 1002
A realistic formulation of feedback time optimal regulation of second-order dead time systems is proposed and solved. The solution is used as a benchmark against which linear feedback controllers are compared. An ITAE tuned controller is concluded to be a good approximate solution to the time optimal regulator problem. Correlations for determining ITAE settings for both interacting and noninteracting PID controllers are given. Finally, the combined regulator-servo problem is briefly treated.
Introduction In Part I (Bohl and McAvoy, 1976) approximate time optimal servo mechanism control of second-order dead time systems was examined. Proportional derivative control with ideal preload (PD/IP) was shown to be very close to time optimal. In this paper approximate time optimal regulation of second-order dead time processes is treated. The mathematical model of these processes is
A discussion of the utility of this model as well as a literature survey of control studies done using it has been presented in Part I.
Formulation of the Time Optimal Regulator Problem A realistic formulation of the feedback time optimal regulator problem for second-order dead-time systems requires that a dead zone be used around the process output, The reason for this requirement is that time optimal control invariably calls for full on, full off forcing. With no dead zone the slightest amount of noise entering the system would cause the controller to chatter and actually amplify the noise. Effectively the dead zone discriminates between significant loads which cause large deviations and call for time optimal action and insignificant ones which do not. Inside the dead zone standard PID control can be used. When the process output crosses the dead zone, time optimal control action is taken but because of the dead time
'
Present address: McNeil Labs, Camp Hill Rd., Fort Washington, Pa. 19034. 30
Ind. Eng. Chem., Process Des. Dev., Vol. 15.No. 1, 1976
its effect is not felt until one dead time later on. Thus the proper initial conditions to use in solving the time optimal regulator problem are the values of Y and Y' which occur one dead time after the response crosses the dead zone. See Koppel and Latour (1965) and Latour et al. (1968) for a discussion of shifting initial conditions ahead in time. The most important part of the formulation of the time optimal regulator problem is the specification of the functional form of the load, since this must be known in order to solve the problem (Bohl, 1974). In general it will be extremely difficult if not impossible to determine this functional form in a feedback mode during actual operation. In this paper the time optimal regulator problem is solved for step loads only. I t is assumed that the magnitude of the step can be predicted from the process response if it is known, a priori, that the load is a step. Because the nature of loads is not known in practice, a linear feedback controller will not approximate time optimal control as closely for the regulator problem as it did for the servo problem. In the servo problem the nature of the forcing function, namely the set point change, is precisely known and this is a significant difference between the two problems. To sum up, the time optimal regulator problem for step loads can be defined by adding three modifications to the standard mathematical formulation. (1) T o differentiate between noise deviations and those caused by a load, a dead zone is placed around the setpoint. When Y leaves the zone, time optimal control action is taken. (2) The initial conditions are the values of Y and Y' that occur one dead time after Y has left the dead zone. (3) It will be assumed that the magnitude, L , of the step load can be accurately estimated from the process response. With these modifications, the problem can be solved
I
I
O i 0 ,
b.05
1
IO 0 i e c L:030 K:l00 i s 00 ~
:
i
IIE.
0
1
I
I
5
I0
I5
1
20
I 25
1 30
1 35
I 40
- 005 45
T I M E , SECONDS
o
10
40 I
I
20
30
, 30 1 40
206
IAE = 7 SI
2 96
I
I
50 50 TIME, SECONDS
1
I
1
70
a0
90
1 loo
Figure 1. Ziegler-Nichols and ITAE PID control compared to time optimal control for dead zone of e b = 0.
Figure 2. ITAE PID control compared to time optimal control with various nonzero dead zones.
using Koppel and Latour’s (1965) feedback time optimal control solution, in the form of switching curves. The limits on available forcing action must be modified to K - L and h - L. Finally, the final controller output is -L, to cancel the load. See Bohl (1974) for details of these calculations.
problem. An algorithm, called pulse preloading, was found to work well for step loads, but failed for limited ramps (Bohl, 1974). This algorithm was actually more complicated than simply using the switching curves of Koppel and Latour (1965) in the case of step loads. It was concluded after trying many modifications of linear feedback algorithms that an ITAE tuned controller represents a practical and workable approximate solution to the time optimal regulation of second-order dead time processes. The ITAE algorithm probably cannot be improved upon because of one’s inability to know the precise nature of load upsets while using only feedback control. Finally, although the integral mode caused trouble in the servo problem, it is a very effective search technique for the regulator problem.
Comparison of Time Optimal a n d Linear Feedback Regulation Lopez (1968) and Lopez et al. (1967, 1969) have compared ITAE (integral of time multiplied by absolute value of error), ISE (integral square error) and IAE (integral of absolute value of error) tuning to one another and to various empirical schemes including Ziegler Nichols’ control. The conclusion of these studies was that an ITAE tuned controller is the best. Typical curves presented for secondorder dead time systems (Lopez, 1968) show that the settling time for an ITAE controller is smaller than any of the other controllers considered. Thus an ITAE tuned controller will compare the best with time optimal control. This should not be surprising since time is included in the kernel of the performance index. It is interesting to see just how good the comparison is. Figure 1 compares the theoretical time optimal response (with zero dead zone) to the responses obtained using a PID controller. In one case the PID controller is tuned with the Ziegler-Nichols ultimate period-ultimate gain rules while in the other ITAE tuning is used. The ITAE-tuned PID response is clearly much better than the Ziegler-Nich01s response. It can be noted here that for less severe forcing than step forcing the difference between the ITAE response and the Ziegler-Nichols response would not be as great as that shown in Figure 1. The ITAE tuning method provides a response whose oscillations die out quickly, making it qualitatively similar to the time optimal response. The time optimal response is far better than the ITAE tuned PID response; however this time optimal response will not be achievable in practice, since a zero dead zone is used. Figure 2 shows the ITAE PID response compared to the time optimal response for dead zones of various sizes. Inside the dead zone, ITAE PID control is used, while outside it, time optimal control action is taken. Dead zone sizes of lebl = 0.0, 0.0010, 0.0025, and 0.0050 are used. As a larger and larger dead zone is used the time optimal response begins to approach the ITAE PID response; but even with the largest dead zone used, the theoretical time optimal response is much better than the ITAE PID response. Because of this difference an attempt was made to find a feedback algorithm to achieve the improvement predicted by the time optimal results. Initial efforts centered around using a P D controller with preload outside the dead zone, since this type of control had worked well for the servo
Determining ITAE P I D Settings Because of the excellent performance of an ITAE PID controller it is desirable to present a method for obtaining the controller settings easily. Lopez (1968) has presented graphs for P, PI, and PID controller settings using an ITAE criterion. These graphs have also been published elsewhere (Lopez et al., 1969), but a t a significantly reduced scale which makes them difficult to read. In his work Lopez used the following noninteracting form of a PID controller
However, almost all pneumatic controllers now in use are of the interacting type (Shinskey, 1967) (3) I t will be shown below that significantly different optimum PID settings are obtained for an interacting controller than for a noninteracting controller. Thus the noninteracting results cannot be used for the interacting controller. Noninteracting Controller To provide a method of finding optimal values of K,, T d , and T I ,in eq 2 a series of correlations are presented. These are based on 25 optimizations (minimizing the ITAE of the response) of the PID controller, performed on the analoghybrid system discussed in Part 1 (Bohl and McAvoy, 1975). The approximation of PD control, plus integral control, used in Part 1, was again used in this set of simulations. Values of a = 0.1, 0.2, 0.3, 0.4, 0.5 were used in combination with values of b = 0.12, 0.3, 0.5, 0.7, 0.9, thus yielding the 25 cases. Lopez’ graphs gave a good starting point for each of the optimizations, but the optimal values of K,, TI, and T d often differed somewhat from the values Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
31
Table I. Correlation Parameters for Predicting ITAE PID (Noninteracting) Controller Parameters Kc * Kp TITI Td IT
10.a
Kc*Kp
-
2.3934 1.211 -2.230
-1.2096 -0.7750 0.6025
0.1044 -0.17 0 1 0.4531
0.1760 0.1026 -0 .O 624
0.1806 -0.0092 -0.0479
-0.2071 -0.0081 0.0128
Ku * Kp
0.8193
2.0302
1.3135
0.9553
0.1809
0.5219
Ku*Kp
1.312
0.0517
-0.5 9 14
0.0643
0.0832
-0.0687
-4.9945
-4.261 3
-0.2 658
-1.1100
10-b 10. b 10-b
10.a 10.a
UT
P U
UT P,.
PUITI
a?
Ku.Kp
p,
TdlPu
-1.1436
-1.5855
Table 11. Correlation Parameters for Predicting ITAE PID (Interacting) Controller Parameters
z
V
P. Kp
D
R
-orT T D
10-b
1.7053
10.a
10-b
0.6249
UT
P-Kp
-
R
-or pu pu
In (C)
W
10.a
Ku*Kp
-0.6055
KU*K,
-3.2366
P
0.1648
r
4
0.1709
0.1100
0.6212
0.3144
-0.0760
1.2264
1.2190
0.4568
-0.07 0 3
-0.622 1
-0.2 142
-0.0062
0.0085
-0.0958
-0.1007
-0.1 125
-0.2944
P U
UT -
-1.9314
pu
read from the graphs. The correlations are of the form = C Vrn+pln ( V I . W n + q l n ( W ) + r l n ( V )
.
0201
(4)
Interacting Controller Equation 3 can be cast into the form of eq 2 by defining the following effective settings
K,* = P ( l + D / R I ) TI* = D
+RI
(5) (6)
For the interacting controller the ratio of Td* to TI* is given by
-Td* TI*
(DIRI) ( D / R I 1)*
+
(8)
By using simple calculus it can be shown that Td*/TI* has a maximum value of 0.25 and this occurs when D = RI. Thus Td* can never exceed of TI*. Unfortunately, for all of the 25 cases previously optimized by the ITAE criterion Td* is greater than T1*/4. T o provide optimal parameters that can be applied to an interacting controller, 25 more optimizations were done; the format was the one used previously. A noninteracting controller was used, but constrained so that Td* IT1*/4. I t was quickly found that the minimum ITAE response invariably occurred on the constraint Td* = T1*/4. McAvoy and Johnson (1967) noted that the same constraint applies when an ISE criterion is used. This constraint was used in the optimization, reducing it to a two-dimensional search. Again a correlation of the form of eq 4 was made using a least-square fit. The parameters in that correlation are shown in Table I1 and the prediction of the data is excellent. Figure 3 compares the minimum ITAE response for noninteracting control to the minimum ITAE response for inInd. Eng. Chem., Process Des. Dev., Vol. 15,No. l , 1976
1 I
The values of the parameters obtained by a least-square fit are given in Table I. Note that the PID parameters can be found in terms of ultimate period (P,) and ultimate gain (K,) as well as in terms of the normalized dead time ( a ) and minor time constant ( b ) . These correlations agree within f10% with the data used in developing them.
32
n
rn
-1.1445
0 I5
‘A
0 1 O r
0
-0051 0
I
I
5
10
I I5
20 25 TIME,SECONDS
,
30
1 35
40
Figure 3. Interacting vs. noninteracting ITAE PID control. teracting control. The system parameters are L = 0.30, T = 10 sec, a = 0.1, b = 0.5. While the initial peaks are about the same height, the response with the interacting controller does not die out as quickly, making the ITAE significantly larger. This is caused by a lack of sufficient derivative action; the constraint that Td* = T1*/4.0 means that a compromise between derivative and integral action must be effected. For the noninteracting parameters, K , = 20.95, T d = 1.99 sec, and TI = 4.06 sec. Thus Td/TI = 0.49; however, for the interacting controller, Kc* = 22.7, Td* = 1.56 sec, and TI* = 6.24 sec. Servo vs. Regulator Problem Approximate time optimal solutions have been found for both the servo problem (Bohl and McAvoy, 1975) and the regulator problem. In the former case the solution is PD/IP (proportional derivative with ideal preload) control while for the latter it is ITAE tuned PID control. The controller settings required for these two schemes are substantially different. The PD/IP controller uses a significantly smaller gain and a somewhat smaller derivative time than the ITAE tuned controller. For many processes the regulator mode of operation is the normal mode. A question which arises is how well would the regulator tuned controller do for the servo problem if it were given the benefit of preload. As discussed in part I a PID controller without modification is poor for the servo problem. Figure 4 shows typical comparisons between regulator and servo tuned controllers. As can be seen, the regulator tuned controller shows considerably more oscillation in response. As the dead time of the process increases the regulator tuned controlled be-
7 7 ,+TAE
+ Preload
i
K,:272
R = 10
uO
10
20
30
40
50
60
70
SECONDS ____
TIME,
Figure 4. Servo-tuned vs. regulator-tuned change; both controllers use preload.
presented. The time optimal solution for step loads has been used as a basis to compare linear feedback controllers. An ITAE tuned controller has been shown to represent a good approximate time optimal controller. Potential for improvement over ITAE control exists, as shown by the time optimal results. However, the inability of one to know the nature of load upsets through feedback alone probably precludes obtaining this improvement. It has been shown that interacting PID controllers have different ITAE settings than noninteracting controllers. Correlations have been presented to facilitate obtaining these settings. Finally, it has been shown that regulator tuned settings cannot be used for the servo problem if one desires a near time optimal response.
controller
for s e t p o i n t
comes much poorer in comparison to the PD/IP controller. The main reason for this is the substantial difference in the settings of the two schemes. Two additional modifications suggested by Shinsky (1967) were tried to improve the servo response of the ITAE tuned controller: (1) using derivative action on the measurement rather than on the error; and (2) introducing a first-order lag into the setpoint circuit. While these modifications improved the servo response, the improvement was small (see Bohl, 1974). Thus it must be concluded that although the above modifications plus preload will improve the servo response of a regulator tuned PID controller, servo tuned PD/IP control should be used for the servo problem if approximate time optimal responses are required. (To achieve both automatic ITAE regulation and a PD/IP servo response a digital computer would be required to change controller settings as the situation warranted.) The reason that McAvoy (1972) obtained a good servo response when he simply used preload was that he used the same proportional and derivative settings given by Latour and Koppel (1965). These settings were K , = 10.8 and T d = 2.87 as compared with the ITAE settings K , = 20.95 and T d = 1.99. The PD/IP settings for the same process are K , = 6.42 and T d = 2.99. Thus the settings used by McAvoy were actually close to the PD/IP settings. Finally, it can be mentioned that the results presented in this paper were verified on the laboratory heat exchanger discussed in Part I. For details of the experimental work see Bohl (1974).
Conclusions A realistic formulation of the time optimal regulator problem for second-order dead time processes has been
Nomenclature a = normalized dead time b = normalized minor time constant D = derivative setting for interacting controller e = error e b = sizeofdeadzone I P = ideal preload K , = gain for noninteracting controller K , = ultimategain L = magnitude of load upset M = manipulated variable P = proportional setting for interacting controller P, = ultimate period PD = proportional derivative control PID = three mode control R I = reset setting for interacting controller s = Laplacevariable T d = derivative time for noninteracting controller TI = integral time for non-interacting controller Y = process output ' = differentiation * = effective settings for interacting controller Literature Cited Bohl. A. H . . M.S. Thesis, University of Massachusetts, Amherst. Mass., 1974. Bohl, A. H., McAvoy, T. J., Ind. Eng. Chern., Process Des. Dev., 15, 24 (1976). Koppel. L. B., Latour, P. R., Ind. Eng. Chern., Fundarn., 4, 463 (1965). Latour, P. R., Koppel, L. B., Coughanowr, D. R., h d . Eng. Chern., Process Des. Dev., 7, 345 (1966). Lopez, A. M., Ph.D. Thesis, Louisiana State University, Baton Rouge, La., 1968. Lopez, A. M., Miller, J. A., Smith, C. L., Murrill. P. W., Instr. Tech., 14 (1l), 57 (1967). Lopez, A. M., Smith, C. L.. Murrill. P. W., Brit. Chern. Eng., 1553-1555 (Dec 1969). McAvoy, T. J., Ind. Eng. Chern., Process Des. Dev.. 11, 71-78 (1972). McAvoy, T. J.. Johnson, E. F., h d . Eng. Chern., Process Des. Dev., 6, 440446 (1967).
Receiued for reuieu; July 25, 1974 Accepted September 5 , 1975
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
33
