This plot also supports the argument advanced previously that it is the relative velocity V, which determines the axial mixing characteristic of a bubble column, provided that the gas-phase flow velocity is sufficiently high. The experimental data indicate that a t low gas flow velocities D L is dependent on V, as well as on Vs,d, the absolute gas velocity. However, the importance of the latter as a factor diminishes as the flow rate is increased, it finally disappears when Vs,d becomes greater than 0.43 cm/sec, leaving the relative velocity as the only factor controlling DL.This phenomenon is also true for 10 mesh/in. screen cylinders used in this study.
L = aerated column height, cm Lo = static water level, cm M = uL/PDL, dimensionless Q = volumetric flow rate, ml/sec t = time, sec u = V,,J(l - t ~ =) interstitial liquid velocity, cm/sec v b = Vs,d/tG = average bubble rise velocity, cm/sec V, = slip or relative velocity, cm/sec V,,c = superficial liquid velocity, cm/sec Vs,d = superficial gas velocity, cm/sec z = axial distance, cm t~ = gas holdup, fractional 0 = average residence time, sec
Summary a n d Conclusions Axial mixing and gas holdup were determined in a bubble column containing screen cylinders of different size and mesh number. The data show that the screen cylinder can reduce axial mixing, increase gas holdup, and effect an even distribution of bubbles within the column. It is possible to investigate the effect of liquid phase flow on axial mixing and gas holdup by using screen cylinders in a bubble column. This study confirms that the relative velocity of the two phases, rather than the liquid phase velocity, should be one of the controlling factors for axial mixing in unpacked bubble columns.
Literature Cited
Acknowledgment The author acknowledges the assistance of J. W. Hines in the experimental work. Nomenclature A = column cross section, cm2 C = tracer concentration, g/ml Co = initial concentration, g/ml D L = axial dispersion coefficient, cm2/sec
Brenner, H.. Chem. Eng. Sci., 17, 229 (1962). Bridge, A. G., Lapidus, L., Elgin, J. C.. A./.Ch.E. J., I O , 619 (1964). Carleton, R . J., Flain, R. J., Rennie, J., Valentine, H. H., Chem. Eng. Sci., 22, 1839 (1967). Chen, B. H., Vallabh, R., ind. Eng. Chem.. Process Des. Dev., 9, 121 (1970). Chen, B. H., Manna, B. B., Hines, J. W.. ind. Eng. Chem., Process Des. Dev., 10, 341 (1971). Chen, B. H., Douglas, W. J. M., Can. J. Chem. Eng., 47, 113 (1969). Chen, B. H., Osberg, G. L., Can. J. Chem. Eng., 45, 90 (1967). Chen, B. H., Can. J. Chem. Eng., 50, 436 (1972). Chen, B. H., Brit. Chem. Eng., 16, 197 (1971). Deckwer, W., Graeser, V., Langernann, H., Serpernen, Y. Chem. Eng. Sci,, 28, 1223 (1973). Hofman, H., Chem. Eng. Sci., 14, 193 (1961). Hoogendoorn, C. J., Lips, J., Can. J. Chem. Eng., 43, 125 (1965). ishii, T., Osberg, G. L., A.i.Ch.E.J., 11, 279 (1965). Kato, Y., Nishiwaki, A,, Int. Chem. Eng., 12, 182 (1972). Mashelkar, R. A., Sharma, M. M., Trans. inst. Chem. Eng., 15, 162'(1970). Sahay, 8. N., Sharma, M. M., Chem. Eng. Sci., 28, 2245 (1973). Singh, S., B. Eng. Thesis, Nova Scotia Technical College, Halifax, N. S., Canada, 1969. Sutherland, J. P., Vassilatos. G., Kubota, H., Osberg, G. L., A.i.Ch.E. J., 9, 437 (1963). Tadaki, T., Maeda. S., Chem. Eng. (Japan),2 , 195 (1965). Voyer, R. D., Miller, A. I., Can. J. Chem. Eng., 46, 335 (1968). Webber, H. H., Adv. Chem. Eng., 7, 105 (1968).
Received for reuiew June 12, 1974 Accepted May 27,1975
Linear Feedback vs. Time Optimal Contol. I. The Servo Problem Alan H. Bohl' and Thomas J. McAvoy' Deparfment of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 0 1002
The effectiveness of linear feedback controllers containing proportional, integral, and derivative modes is compared to time optimal control for the servo response of second-order dead-time processes. A proportional derivative controller with ideal preload is shown to be nearly time optimal. Such a controller represents a practical and economic solution to the time optimal servo problem. Because of the undesirable features of the integral mode a standard PID controller is shown to be poor for the servo problem.
Introduction Linear feedback control (see Figure 1) using various combinations of the proportional, integral, and derivative modes is commonly used in the chemical process industries. I t is straightforward in its implementation and requires a minimum of information about the process. A more recent development is time optimal control, which gets the process from some given initial conditions, to the 'Present address: McNeil Labs, Camp Hill Rd., Fort Washington, Pa. 19034. 24
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
desired final conditions, in minimum time. According to Pontryagin's Maximum Principle (Pontryagin, 1962), this can be accomplished (for a linear, single-input, single-output system) by using a combination of full-on, full-off forcing. This study uses linear feedback control to obtain approximate solutions to the time optimal problem. In Part I the servo mechanism problem is considered and in Part I1 the regulator problem is treated. To solve the time optimal problem analytically, the limits on available control action and the mathematical model of the process must be known. For linear systems, the pro-
cess model can be expressed as a transfer function; one transfer function that has been extensively studied is the second-order with dead time model, which is very general. This is the model used in this study. Because the second-order (overdamped) model contains two time constants, it can account for process inertia common to higher order systems (a first-order model does not exhibit inertia); the dead time can represent either transportation lag or the excessive inertia found in very high order systems, such as distillation columns (Williams et al., 1963). McAvoy and Johnson (1967), Weigand and Kegeris (1972), and Lopez et al. (1969) all confirmed that a secondorder with dead time model is superior to the first-order with dead time model often proposed (Miller et al., 1967; Rovira et al., 1969). Thus the second-order with dead time model represents a good compromise between flexibility and simplicity. Furthermore, various methods of modeling a process as second order with dead time are available; for example, Meyer et al. (1967) give a graphical technique, while Lopez (1968) gives a numerical method. Because it is simple yet flexible, the second-order with dead time model has been applied to a wide variety of processes such as a cracking furnace (Lapse, 1956), two tank water heating process (Latour, 1964), heat exchangers, pneumatic lines, agitated mixing vessels (Biery and Boylan, 1963), as well as distillation columns (Williams et al., 1963). Furthermore, many people have used this model in optimizing P, PI, and PID controllers for both setpoint and disturbance upsets (van der Grinten, 1963a, 1963b; Lopez et al., 1967; Lopez et al., 1969; Smith and Murril, 1966; Wills, 1962a, 1962b). Finally, much work has been done using nonconventional algorithms on this model; these efforts include time optimal control (Hsu et al., 1972; Koppel and Latour, 1965; Latour, 1967; Latour et al., 1968), feedforward control (Koppel and Aiken, 1969), and model predictive techniques (Mellichamp, 1970), including the Smith Linear Predictor (Smith, 1957, 1958, 1959). While feedforward, time optimal, and model predictive control methods not only require accurate and extensive knowledge of the process and its dynamics, they are also more costly to implement, than linear feedback control. This paper compares time optimal control to both P D and PID control for the servo problem in an attempt to determine if time optimal control and its implementation provide an improvement over conventional control that is sufficient to warrant the use of the more complicated, expensive time optimal control scheme. To understand the importance of this study, it is necessary to examine some of the work done previously.
Background Rovira et al. (1969) compared disturbance and setpoint tuning and found that disturbance tuning required a larger gain and shorter integral time for both P I and PID control of a first-order dead time process based on an IAE criterion (minimum integral of absolute value of error). As a result, disturbance tuning for setpoint changes (and vice versa) produced non-optimum responses. Koppel and Latour (1965) compared PID control, theoretical time optimal control, and a feedback implementation of time optimal control (abbreviated FBTO, for feedback time optimal) for step setpoint changes; their criterion for comparison was 5% settling time, abbreviated t j s . However, they tuned the PID controller for step load upsets, using a modified IAE criterion; this did n o t give the best possible PID results for step setpoint changes. Moreover, their implementation of time optimal control, FBTO, was significantly worse when 5% settling time is compared
FEEDBACK CONTROL SCHEME
- PID
CONTROL
SECOND ORDER DEAD TIME PROCESS -0VERDAMPED
L
l
BASIC VARIABLES :
-
R Set Point or Reference Value L - Load e error - Basis for feedback action Y - Controlled variable M - Manipulated variable
Figure 1. Simple feedback control system.
to the minimum time for theoretical time optimal control. Thus, for the servo problem it may be possible to find a better implementation of time optimal control, and better values of the PID parameters. McAvoy (1972) presented an alternative to FBTO control for setpoint changes, P D control with preload. When the setpoint is changed, a new steady-state controller output is required; integral control will search out the value of this new output and hence eliminate offset (Shinsky, 1967). However, the new steady-state output can also be calculated
M:! = M i 4- R / K ,
(1)
where Mz is the new steady-state controller output, M i is the old steady-state controller output, R is the setpoint change, and K , is the process gain. Instead of using integral control for a setpoint change, a constant value of R / K , can be added to the controller output; this will keep the process a t the new steady state. This technique is called preloading the controller; a discussion of nonideal preloads can be found in Shinsky (1967). The value of R / K , can be defined as the ideal preload. McAvoy used Koppel and Latour’s PID settings for a step setpoint change; then he used the same settings but replaced the integral mode by the ideal preload, giving PD control with ideal preload, PD/IP. Comparing the two responses, he found not only that the PD/IP was much better than the PID, but that the PD/IP response was almost as good as the theoretical time optimal one. He concluded that it was the inefficiency of the integral search technique which caused the PID response to be so poor. There are two subtle but important points that must be made about McAvoy’s work. First, while the integral search technique was inefficient in McAvoy’s simulation, it was not tuned for setpoint changes and hence it cannot be concluded that this technique is inefficient i n general. Second, the PD/IP controller was not optimized at all, yet it gave excellent results. If optimized for step setpoint changes, the PD/IP controller might give a response that approaches the time optimal one even more closely. As stated before, linear feedback control is simpler and less costly to implement than time optimal control. Furthermore, McAvoy’s work showed that a PD/IP controller can give results close to theoretical time optimal for step setpoint changes. For these reasons, it was desirable to determine just how well PD/IP and PID control could do when compared to theoretical time optimal control and its implementation, FBTO. Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976 25
Mathematical Definitions. The overdamped secondorder with dead time model can be represented as exp(-aTs) (Ts l)(bTs 1) X(s) where Y ( s )and X ( s ) are the normalized, transformed output and input variables, respectively, T is the major time constant, a is the normalized dead time (0 < a ) , and b is the normalized minor time constant (0 < b < 1.0). Throughout this study, the noninteracting form (see Shinsky, 1967, for interacting form) of PID controller was used. In Laplace transform notation, this is given by
Y ( s )--
+
+
where M ( s ) and e ( s ) are the transformed, normalized manipulated variable and error respectively, K, is the controller gain, TI the integral time, and T d is the derivative time. Figure 1 shows a PID controller and a second-order proCESS with dead time in a feedback control loop. Limits of K and k represent the upper and lower limits on the manipulated variable M ; these make the problem realistic, since there are always limits on the manipulated variable in any real process.
Servo Problem Koppel and Latour’s study (1965) of time optimal vs. PID control was used as a basis for this study of the servo problem. Latour (1964) states that he tuned the PID controller for step load upsets because tuning it (using a minimum 5% settling time as the criterion) for step setpoint changes would cause saturation of the manipulated variable. However, when he used his load-tuned PID parameters for step setpoint changes, saturation occurred anyway. Furthermore, it can be shown that saturation cannot be disregarded. Saturation. Step setpoint forcing is a common occurrence; if derivative action, based on the error, is used, and the setpoint is step changed, then the manipulated variable will saturate for even small changes in setpoint. Ideal derivative action would respond to a step setpoint change by suddenly increasing to infinity, remaining there for an infinitesimal time, and then returning to zero. While ideal derivative action is impossible to get, reasonably good nonideal derivative action is available. Coughanowr and Koppel (1965) give the transfer function of a nonideal proportional-derivative controller as (4) Here y determines the approach to ideal controller action; as it approaches infinity, this controller approaches an ideal PD controller. Coughanowr and Koppel (1965) state that a value of y = 10 is found for many industrial controllers. The response of this controller to a step setpoint change of R units is given by the following expression
M ( t )= K , .R[1
+ (7- 1)exp(-t/Td)]
(5)
A t t = 0+, the output is
M(O+) = K ,
R
(6)
If the limits on the manipulated variable are given as f 1 . 0 , then R / K , represents the fraction of the upper (or lower) limit on the available control action that is necessary to keep the process a t the new setpoint. The fraction of control action necessary for a given setpoint change depends on the process gain as well as on the amount of con26
Ind. Eng. Chem., Process Des. Dev., Vol. 1 5 , No. 1 , 1976
trol action available. Because these things vary from system to system, the manipulated variable limits were standardized to f l . O , while the process gain was normalized to unity. If the calculated value of M exceeds f 1 . 0 then the controller output is set equal to f l . O to simulate saturation. Thus the results of this study can be readily generalized to any process. Since K,.r.R = f l . O will produce saturation of the manipulated variable, and since y is 10 for many industrial controllers
-
K , R = -10
(7)
will produce saturation. If K , has a modest value of 5.0, then R = 0.02 will produce saturation. That is, a setpoint change that requires only 2% of the upper (or lower) limit on the manipulated variable will cause saturation. Furthermore, calculations show that this is true of nonstep setpoint changes, provided that the setpoint is changed in a time much less than the major time constant of the process (Bohl, 1974). Finally, the nonideality of the P D controller caused its output to slowly decay from its initial maximum value (see eq 5 ) . Thus saturation from a step setpoint change is prolonged rather than momentary. Because derivative action on the error causes saturation to occur for even small step setpoint changes, the control problem is nonlinear. Therefore, there is no guarantee that any one set of linear (PID, PD) controller parameters will give optimal or near optimal responses for the full range of possible values of setpoint change. I t is important to note that previous work by Wills (1962a, 1962b) and Rovira e t al. (1969) on tuning PID controllers for step setpoint changes completely ignored the question of saturation. Since saturation for step setpoint changes is the rule rather than the exception, it was necessary to consider it in finding the optimal parameters for a PID or PD/IP controller. I t was hoped that one particular value of R could be found, such that tuning the controllers for this value would produce near-optimum responses for all values of R . With this end in mind, a preliminary study was conducted. Simulations were carried out on an Applied Dynamics AD-80 analog computer which was interfaced with a Digital Equipment Corporation PDP-8 digital computer. The analog simulated the process and controller dynamics, while the digital simulated the dead time. Coughanowr and Koppel’s approximation of P D control (see eq 4), plus the integral mode, was used in the simulations. Preliminary Study. In the preliminary study, the system parameters a = 0.1, b = 0.5, T = 10 sec were used, while the process gain was unity. The limits on the manipulated variable were held constant a t f 1 . 0 and the PD/IP and PID controllers were tuned for step setpoint changes. The PID and PD/IP controllers were tuned to give a response which reached the new steady state in minimum time with no observable overshoot. (See Bohl, 1974, for details of the tuning.) Setpoint changes of 0.03,0.10, and 0.30 were used. With the PD/IP controller, the tuned values of K, and Td varied somewhat, depending on the value of R R
KC
0.30 0.10 0.03
8.00 6.60 3.21
Td,
sec
2.79 3.18 4.48
Saturation Yes Yes N O
Because of the saturation of the manipulated variable, there were two problems that could not be eliminated, overshoot of the new steady state, and sluggishness of the response. When the controller was tuned for R = 0.30, and then used for R = 0.10 or 0.03, overshoot occurred. Similar-
Table I. Simulation Results Variable
_ _ _ _ _ ~ _ _
Test
Set point
Minor time constant
R
1
0.90
2 3 4 5 6 7 3
0.60
13
Asymmetry
15 16 17 3
3
18
19 20
0.05 0.10
0.20 0.50 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.30
0.30
C U R V E S , LEFT TO R I G H T : R
i
. 0 5 , .IO, . 5 0 , .eo; Open Loop Response t o Load of " R " (Preload)
a :0.1 b = 0.5 T = IOssc
K:100 k
0
10
20
30
40
50
: -1.00
60
70
Bo
90
T I M E , SECONDS
Figure 2. PD/IP control for various values of setpoint change.
ly, when the controller was tuned for R = 0.10, and used for 0.03, overshoot again resulted. Tuning the PD/IP controller for R = 0.03 did not produce saturation of the manipulated variable; as a consequence, tuning for R = 0.03 will not cause overshoot even for cases where R < 0.03. However, when the values of K , and Td obtained from tuning for R = 0.03 were used for R = 0.10 and 0.30, the responses were sluggish. Thus overshoot and sluggishness cannot both be eliminated for all values of R. A compromise value of R = 0.10 was used in tuning the PD/IP controller. This meant that overshoot would occur only in cases where R < 0.10; for processes with a moderate gain and a moderate amount of control action available, the setpoint change for R < 0.10 will be small, and the overshoot will not be significant. Furthermore, tuning the PD/IP controller for R = 0.10 gave results for R > 0.10 that were very satisfactory Figure 2 shows the responses for various values of R , using a PD/IP controller tuned for R = 0.10; all the responses but R = 0.05 are critically damped and reach the new steady state in near-minumum time. Thus the effect of saturation is to cause the tuned values of K , and Td to be a function of R ; fortunately, they are a weak function. As a result, near-optimal responses for all values of R (-1.0 6 R < 1.0) can be obtained by tuning the PD/IP controller for R = 0.10; the worst responses occur for the smaller setpoint changes, where overshoot will be '
k
-0.40
0.50 0.70 0.90 0.50 0.50 0.50 0.50 0.50
1.00 1.oo 1.00 1.00 1.00 1.00 1.oo 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.50
0.70
-0.70
0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
1.00
-0.40 -0.70
0.30
0.1 0.1 0.0
0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30
12 14
0.1 0.1 0.1 0.1 0.1
0.30
11 3
0.5 0.5 0.5 0.5 4.5 0.12
0.1 0.1 0.1
0.30 0.10 0.03 0.30 0.30 0.30 0.30
9 10
Saturation leyel
K
b
-
8
Dead time
a
1.30 1.60 1.30 1.00
0.70 0.40 1.00
-0.40 -0.40 -0.40
-0.40 -0.40 -0.40 -0.40 -0.40 -0.40 -0.40 -0.40 -0.40
-0.40 -0.40
-1.00 -0.10
-0.40 -0.70 -1.00 -1 .oo
PDIIP,
FBTO,
T5
t5 7 G
t 5 92
3.09 1.67 1.03 0.63
2.70 1.55 1.05 0.65
4.56 2.35 1.72
0.40
1.00
0.63 0.84 1.03 1.19 1.33 0.93 0.98 1.03 1.13 1.43
1.32 1.03
0.88 0.79 1.00 1.03
1.23 2.12 0.98
0.65 0.85 1.05 1.20 1.25 0.83 0.90 1.05 1.20 2.10 1.25 1.05 0.95 0.90 0.95 1.05 1.25 1.95 1.05
1.30 1.14 1.18
1.57 1.72 1.81
1.90 0.91 0.97 1.72 2.41 4.25 2.02 1.72 1.57 1.45 1.64 1.72 1.90 3.17 1.65
tolerable. It should be noted, however, that for a process with a large gain and/or a large amount of control action available, that values of R < 0.10 may be common. For such a process, it might be better to tune the PD/IP controller for a smaller value of R , say 0.05; however, such cases are not considered in this paper. The PID controller was also tuned for R = 0.03, 0.10, and 0.30. While it could be tuned to give excellent results for a particular value of R , no one set of PID parameters would give good responses for all values of R . This result is attributable to the nonlinearity of the problem, which caused the integral control mode to be an inefficient search technique. Thus, while the tuned proportional and derivative settings are a weak function of R , the integral setting is a very strong function of R. For this reason only PD/IP control was considered in the more extensive study. (See Bohl, 1974, for further details of the problems using integral control for this nonlinear problem.) PD/IP vs. Time Optimal Control. Koppel and Latour considered five variables in their 1965 study: magnitude of setpoint change, R ; normalized dead time, a ; minor time constant, b; difference in forcing limits, K - k; and asymmetry of the two forcing modes, K - R and k - R. These variables were changed one at a time, keeping the others constant a t a base value. The major time constant was 1.00 sec, and the process gain was unity. The format of the twenty simulations is shown in Table I. PD/IP control, tuned for R = 0.10, was used in all cases considered by Koppel and Latour. The 5% settling times for Koppel and Latour's FBTO control and the PD/IP 5% settling times from this study are compared in this table. In addition, the time for theoretical time optimal control to reach the new steady state, Ts, is shown. The PD/IP controller gives 5% settling times that are quite close to the theoretical minimum time to reach steady state in all but a few cases, and better than the FRTO 5% settling time in all cases. In the first set of simulations, where magnitude of setpoint change was the variable, t 5 % for the PD/IP controller is actually less than T5 for the case of R = 0.90 and R = 0.60. This is possible since the PD/IP controller was tuned to give a critically damped resonse and t 5 % occurred at 0.95R. By contrast, the theoretical time optimal response Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 1, 1976
27
K : 1.00 k : -1.00 R = 0.30
T 5 , Time Optimol Control
bs0.5 T = 10 sec
CURVES, LEFT TO R I G H T
[r
r
0.67
0
I1
0.0
0
e1
0.1
40.00 6.60
3. 18
31 41
0.3 0.5
1.45
4.50
0.81
5.00
E
0.33
>0 OT
1.43 scc
I
1
I
lO+oT
2OlaT
30toT
i
,“: pb”: k
W+OT
TIME, SECONDS
Figure 3. PD/IP control compared to theoretical time optimal
control. was required to reach LOOR (see Figure 3). For the smaller values of R, t5% for PD/IP control is longer than T,s;the relatively poor showing of PD/IP control in the case of R = 0.03 is due to overshoot of the new steady state, as discussed previously. In the second set of simulation results, we see the effect of the minor time constant on the 5% settling time. As b increases, the 5% settling time for PD/IP control approaches Tg, the theoretical minimum time, until a t b = 0.9, t5% is less than Ts. This trend is caused by the fact that as b increases, the inertia of the system increases, allowing the use of a larger controller gain. The third set of simulation results shows that dead time has a very important effect on the 5% settling time for the PD/IP controller. While the 5% settling time for PD/IP control is quite close to T5 for a = 0.1, T5 is much shorter than the PD/IP t 5 % for a = 0.5. This occurs because a much smaller gain must be used to maintain critical damping for large values of dead time. This relationship is clearly seen in Figure 3. With all responses shifted one dead time, the theoretical minimum time appears as a single point (Koppel and Latour, 1965). Clearly, PD/IP control gets worse in comparison to theoretical time optimal control as “a” increases. When “a” is large, a feedforward implementation of time optimal control using Koppel and Latour’s switching curves should be considered (1965). The last two sets of results can be described together. Comparing T.=,for cases 3 and 20, where K = +1.00 and k = -0.40 and -1.00, respectively, we see that decreasing the lower limit on the manipulated variable has only a small effect on T5. For case 3, T5 is 1.03 sec, while for case 20 it is 0.98 sec. Thus a change in the lower limit from -0.40 to -1.00, 150%, makes only a 5% change in Tg. It is not surprising then, to see that t 5 % for PD/IP control is the same for cases 3 and 20. One value for t 5 % for PD/IP control is also found for cases 15 and 17 and a single value is again found for cases 14 and 18; for each of these pairs of cases, K , the upper saturation limit, is constant. Thus the value of k does not affect the 5% settling time for PD/IP control. A look a t Figure 4 illustrates why. The manipulated variable (for R = 0.30, k = -1.00) does not come near the lower saturation limit when PD/IP. control is used. This is true in general of PD/IP control. The manipulated variable saturates at the limit in the direction of setpoint change, but never nears zero, much less the opposite saturation limit. From the above discussion, it is easy to understand why PD/IP control can give results close to theoretical time optimal control. While the time optimal method has the advantage of using full off (opposite the setpoint change) forcing, which allows longer full on forcing, this advantage is small. All that must be done to make PD/IP control nearly as good, is to tune it to saturate for the appropriate length of time, a time only slightly shorter than the full on saturation time for theoretical time optimal control. 28
OPTIMAL
Ind. Eng. Chern., Process Des. Dev., Vol. 15,No. 1, 1976
:-1.00
- 1.00L 0
IO
PO 33 TIME, SECONDS
40
M
Figure 4. Manipulated variable transients. PD/IP control compared to theoretical time optimal control. ( T = 10.0 sec.)
Table I also shows a very remarkable fact: t j % for PD/IP control is less than t 5 % for Latour and Koppel’s implementation of time optimal control, FBTO, for all 20 cases considered. Moreover, the improvement of PD/IP control over FBTO is quite large in most cases. Thus, PD/IP control is a much better way to achieve results close to theoretical time optimal than is FBTO control. In any feedback implementation of time optimal control two difficulties can arise. First, the switching curves are nonlinear. If they are approximated to avoid an iterative solution, then switching from full on to full off may not take place at the correct time. Small errors can cause deterioration of the response when saturation forcing is used. Secondly, time optimal control requires an accurate process model plus, in this case, the derivative of the controlled variable; while both of these are readily available in an analog simulation, they will not be as available in the control of a real process. As pointed out above, little improvement in response is gained by switching the controller from full on to full off. For these reasons it is much better to use a PD/IP controller than to try to implement time optimal control in the feedback sense. Furthermore, the PD/IP controller will be less expensive. As noted previously, for large values of “ a ” , feedforward time optimal control should be considered. Optimization of PD/IP Parameters. To allow prediction of optimal values of K , and T d , an extensive optimization study was done. For all cases, R = 0.10 and K = +1.00, k = -1.00 were used. 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,O.g. This gave rise to 25 cases. Optimal values of K , and Td were found using the hybrid system and a manual search (Bohl, 1974). Correlations were developed to allow calculation of K , and Td if the values of T , K,, a , and b are known. In addition, a second set of correlations was developed; this second second set is based on knowing the ultimate period, P,, the ultimate gain, K,, the dead time, UT, and K,. The general form of the correlation used was
z = C . vrn + p In ( V ) . Wn + q In ( W )+ r In ( V )
(8)
where C, m, n, p , q, and r are constants whose values are given in Table 11. These were determined from a leastsquares fit of the data. The variables used for 2, V, and W are also shown in Table 11. These correlations predict values of K , and Td that agree within f10% of the results of the simulations. Experimental Verification. The PD/IP algorithm was used to control the outlet temperature of a double pipe steam-water heat exchanger. Details of the experimental apparatus have been given by Paraskos and McAvoy (1970). Steam pressure was held constant a t 12.5 psig and the flow rate of water was manipulated t o control the outlet temperature. Both the graphical technique of Meyer et al.
Table 11. Correlation Parameters for Predicting Optimal PD/IP Controller Parameters
z K,.Kp
V
W
TdlT
10.a 10.a aT
10. b 10. b
KC.KP
p,, -
Ku'. K ,
In (C) 1.501 -2.262 -1.4045
m
n
P
q
-1.240 0.3664 -0.4986
-0.0252 0.7797
-0.0773
0.0270
0.3875
0.1409 -0.0476 -0.7510
-0.2667
-3.1111
-6.1523
--1.4494
-,1.4285
(1967) and a nonlinear least-square technique were used to fit a second-order dead time model to the process reaction curves. Different reaction curves were obtained, depending on the direction of the flow rate change. As a result, different model parameters were gotton from each reaction curve. Change in flow rate
Increase Decrease
K;, 0.07 0.165
T , sec
a
1.56 2.38
0.32 0.32
172
I
I
I
r
0.1542
-0.1616
0.0402
I
i
-0.4602
I
I
I i
Z
I
lli
1 1
b 1.0 1.o
Substituting the two sets of values for K,, T , a, and b into the correlations for K,.K, and Td/T (see Table 11) gave rise to the following values. Change in flowrate
K,.K,,
Increase Decrease
1.83 1.83
TdIT 0.625 0.625
K, 26.1 11.0
sec 0.98 1.49
Td,
To give the PD/IP control a fair chance against time optimal control, the controller gain was set equal to 26.1; that is, the controller was tuned for an increase in flow rate through the heat exchanger (equivalent to tuning for a step decrease in the setpoint of the controlled variable, outlet temperature). However, in the interest of avoiding any possible stability problems-a powerful consideration in any industrial application-the more stable value of T d = 1.49 sec was chosen. Perhaps a slightly better response would have been obtained if T d = 0.98 had been used; however, the response actually obtained was quite satisfactory. Figure 5 compares the theoretical time optimal response of the exchanger with the experimental PD/IP response. The ideal preload was determined experimentally by making the set point change using PID control. The ideal preload was set equal to the final output minus the initial output of the integral mode of the PID controller. As can be seen, the PD/IP controller compares very favorably with the theoretical time optimal response. Conclusions The time optimal servo problem for second-order dead time processes has been solved approximately by using proportional-derivative control with ideal preload. Saturation has been shown to be important in the servo problem and it cannot be ignored. Simulation results indicate that the PD/IP response is very close to the theoretical time optimal response. The major reason for this closeness is the relative lack of significance of full off forcing after full on forcing. A standard PID controller is poor for the servo problem because of the sensitivity of the optimum integral setting to the magnitude of the change in the set point. Correlations have been presented to facilitate tuning PD/IP controllers. Lastly, an experimental study on a laboratory heat exchanger has been given to verify the simulation results. Nomenclature a = normalized dead time b = normalized minor time constant
e = error FBTO = Feed Back Time Optimal, Koppel's implementation of time optimal control IAE = Integral of Absolute Value of Error ITAE = Integral of Time times Absolute Value of Error lz = lower limit on manipulated variable K = upper limit on manipulated variable K , = proportional gain, noninteracting control K , = processgain L = magnitude of load upset A4 = manipulated variable P D = proportional-derivative control PD/IP = proportional-derivative control with ideal preload PID = three mode control R = magnitude of setpoint change T = major time constant T d = derivative time, noninteracting control T I = integral time, noninteracting control T;i = final time, to reach new setpoint, time optimal control t 5 % = five percent settling time-time to get within and remain within 5% of desired setpoint X = forcing Variable Y = controlled Variable y = nonideality factor for derivative control Literature Cited Biery. S. C.. Boylan, D. R., Ind. fng. Cbem., Fundam., 2, 44 (1963). Bohl, A., M.S. thesis, University of Massachusetts,Amherst, Mass., 1974. Coughanowr, D. R., Koppel, L. B.. "Process Systems Analysis and Control." p 282, McGraw-Hill, New York. N.Y., 1965. van der Grinten, P. M. E. M., ControIEng., 10 ( I O ) , 87 (1963a). vander Grinten, P. M. E. M., ControIEng., 10, (12), 51 (1963b). Hsu, E. H., Bacher, S., Kauffman, A,. AICb€ J.. 18, (6),1133 (1972). Koppel, L. B., Latour, P. R., Ind. Eng. Cbem., Fundam., 4 (4), 463 (1965). Koppel, L. B., Aiken, P. M., Ind. Eng. Chem., Process Des. Dev., 8 , 174 (1969).
Lapse, C. G., /SA J., 3, 134 (1956). Latour. P. R., M.S. Thesis, Purdue University, W. Lafayette, Ind., 1964. Latour, P. R., Ind. Eng. Cbem., Process Des. Dev., 6, 452 (1967). Latour. P. R.. Koppel. L. B., Coughanowr, D. R., Ind. Eng. Chem., Process Des. Dev., 7, 345 (1968). Lopez A. M.. Miller, J. A,, Smith, C. L., Murril, P. W., hsfr. Tech., 14 (11). 57 (1967).
Lopez, A. M., Jr., Ph.D. Thesis, Louisiana State University, Baton Rouge,
La.,
1968.
Ind. Eng. Chem.,Process Des.
Dev., Vol. 15, No.
1, 1976
29
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 (1 I), 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
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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
