Znd. Eng. C h e m . Res. 1988,27, 1443-1449
1443
Shinskey, F. G. Process Control Systems, 2nd ed.; McGraw-Hill: New York, 1979. Smith, C. L.; Corripio, A. B.; Martin, J., Jr. “Controller Tuning from Simple Process Models”. Instrum. Technol. 1975,22(12), 39-44.
Latour, P. R.; Koppel, L. B.; Coughanowr, D. R. ”Time-Optimum Control of Chemical Processes for Set-Point Changes”. Ind. Eng. Chem. Process Des. Dev. 1967, 6(4), 452-460. Latour, P. R.; Koppel, L. B.; Coughanowr, D. R. “Feedback TimeOptimum Process Controllers”. Ind. Eng. Chem. Process Des. Dev. 1968, 7(3), 345-353. Lenells, M. “Adaptive Start Up Control”. Ph.D. Thesis, Lund Institute of Technology, Lund, Sweden, 1982.
Received for review May 11, 1987 Revised manuscript received February 23, 1988 Accepted March 5, 1988
Adaptive Control Strategies for Achieving Desired Temperature Profiles during Process Start-up. 2. Experimental Application Susan F. Phillips*and Dale E. Seborg* Department of Chemical & Nuclear Engineering, University of California, Santa Barbara, California 93106
Kenneth J. Legalt E r x o n Research and Engineering Company, Clinton Township, Annandale, N e w Jersey 08801
This paper describes the automated tuning of a temperature control strategy for the start-up of a fluidized sani bath by using adaptive control techniques. In the proposed algorithm, five parameters required t o tune the control strategy were determined during the heat-up to the operating point, eliminating the need for time-consuming trial-and-error tuning techniques. T h e method has been shown t o work well in simulation runs and in experiments on a pilot-scale sand bath. 1. Experimental Apparatus Exxon Research and Engineering (ER&E) has many pilot-scale reactors requiring a constant-temperature environment which is provided by custom-built sand baths. Tuning the existing dual mode temperature controller is a time-consuming procedure because of the long time constants for heating and cooling. It was proposed that adaptive control techniques could reduce the time required to obtain the five necessary control parameters: the switching time tBw;the PID controller settings K,, Ti,and Td;and the proper initial value for the PID controller U. The sand bath used for the experiments is located at the ER&E Clinton Township facility in New Jersey. This sand bath was smaller than the one simulated in part 1 but was otherwise quite similar. It has a steady-state gain of 4.9 “C/ % and a dominant time constant of about 50 min. The proposed control strategy was based on adaptive control techniques and should be applicable to other sand baths and similar start-up problems. The computer system used to monitor and control this sand bath as well as many others in the pilot reactor laboratory was the Exxon EPIC system which has been described by Wang et al. (1983). The system is a three-tier pyramid structure using Hewlett-Packard computers of appropriate sizes a t each level. The algorithm described in this paper runs on the Operator Station (the lowest level computer, a Hewlett-Packard 2250 Intelligent Measurement and Control Front End) which is responsible for all data acquisition and control functions. A sampling period of 10 s or longer was used in these experiments, although it was possible to use a sampling period of 1 s. Typically, several hundred variables can be recorded by the Operator Station. Variables from the algorithm, such as parameter * T o whom all correspondence should be addressed. ‘Now Senior Systems Engineer at Texas Instruments, Inc., Iselin, N J 08830. Now Applications Scientist, Ametek Computer Research Division, Monrovia, CA 91016.
*
0888-5885/88/2627-1443$01.50/0
Table I. Design Parameters set point, yr, OC sampling period, At, min assumed time delay, k A t , min forgetting factor, @ PIDSTC tunging factor, Y polynomial P,, polynomial Pd input excitation error limit, “C
for the Base Case Run 350 1.0 6.0
1.0 1.0
1.0-0.992-’ 1.0-0.902-’ PRBS with a base frequency of 5 min, changes between 80% and 100% 3.0
estimates, were recorded, as well as process measurements and the controller signal to the heater. For each run in this paper, about 35 variables were recorded a t every sampling instant including the fluidized sand bath temperatures, the controller output, parameter estimates, estimation errors, traces of the covariance matrices, calculated PID settings, the predicted maximum temperature, the recommended initial output of the PID controller, and the estimated parameters for the continuous-time process models. The flexibility of the EPIC system allowed both the standard Exxon control package and the proposed algorithm to run simultaneously, meeting the needs of the authors and the operators. Both the proposed algorithm and the standard control package for the sand bath were used in all of the experiments summarized in this paper. In this manner the standard control package, which includes safety features and is designed for proper display on the operator’s screen, was always in place. Two versions of the proposed strategy exist: an active version and a passive version. In the active version, the recommendations for the five parameters were used to replace the corresponding parameters by overriding constants in the standard control package, such as the PID settings, the switching temperature, and the initial output of the PID controller. In the passive version, the five parameters were displayed to the operator as recommendations only. The passive version had no effect on the standard control package, unless the operator chose to 0 1988 American Chemical Society
1444 Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 __ ,-*
_ _
-
~
0 0 , _ _-___--____I ________ ~_ __~-0
1GO
200
SOP
Time (min)
Figure 3. Parameter estimates for prediction error model (run 6-4B).
Figure 1. Sand-bath start-up for base case run (run 6-4B).
1
*I 4
i : 0 i
3k
2
2
-1
,
h--L 3
,
, - ~ _ _ _ __
. L__
100
200
_ & I
-
1
300
Time ( m i n )
Figure 2. Estimation errors of prediction error model and discrete-time model (run 6-4B).
implement the recommendations manually. In this paper, the results from 28 runs using the active version and three user runs using the passive version will be discussed. 2. Base Case Run The design parameters for the base case run are summarized in Table I, while the experimental results are shown in Figures 1-4. Most of these parameters were set at a value in the middle of the available range. In this paper the effect of changing each of these parameters will be investigated. Figure 1 shows the actual temperature, the predicted maximum temperature, and the controller output signal for the base case run. As observed in the simulation runs, the predicted maximum temperature was erratic at the beginning of the run since it was dependent on parameter estimates which had not fully converged. The predicted maximum reached the set point, and the switch from 100% heating to no heating occurred at t = 128 min. This is clearly noted on the figure when the controller output drops to zero and the sand-bath temperature is allowed to coast up to the set point. Although the switching time turned out to be slightly early, the PID controller was able to bring the temperature to set point
no2
'
-
0
,
L_-.-_ 200
100
30C
Time ( m i n )
Figure 4. Parameter estimates for discrete-time model (run 6-4B).
easily. After the switch to PID control at t = 135 min, 100% heating was called for, but the PRBS input mistakenly overrode the signal sent to the heater from t = 135 to 150 min. The estimation errors for the one step ahead prediction error model and for the second-order discretetime model are shown in Figure 2. The estimation errors were quite small in general, and always well within the user-specified error limit of 3 "C, the upper limit for acceptable estimation errors. The parameter estimates for the one step ahead prediction error model and the discrete-time model are shown in Figures 3 and 4, respectively. The parameter estimates converged rapidly but were perturbed when the switch was made to no heating at t = 128 min and then to PID control at t = 135 min. All runs began with initial parameter estimates of zero to provide a severe test of the proposed strategy. 3. Accuracy of the Estimated Switching Time
The accuracy of the estimated switching times is indicated in the summary shown in Table 11. Of the 28 experimental runs, only five showed an undershoot of more than 15 OC. Fifteen of the runs had an overshoot or un-
Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 1445 Table 11. Accuracy of the Predicted Switching Time no. of runs with overshoot, all less than 6 "C no. of runs with no undershoot or overshoot no. of runs with undershoot less than 10 " C no. of runs with undershoot greater than 10 "C but less than 15 " C no. of runs with undershoot greater than 15 "C total no. of experimental runs
4 5 6 8 5 28
Table 111. Influence of Sampling Period
sampling period, A t , min switching time, t,, min switching temp, Taw," C no. of samples at switch overshoot (undershoot), " C run duration, min no. of samples a t end of run
9-5 5.0 120 332 24 (12) 685 137
9-3 2.0 116 323 58 (25) 454 227
Table IV. Influence of Sampling Period on PID Settings from Second-Order Continuous-Time Models run 9-5 9-3 6-4B 9-6 sampling period, A t , min 5.0 2.0 1.0 0.167
PiD Settings from Second-Order Models (at Time of Switch)
K,, % / " C Ti,min Td,min
run 6-4B 1.0 116 320 116 (27) 220 220
9-6 0.167 129 350 774 6 274 1644
dershoot of 10 OC or less. Although, the switching algorithm was quite successful, it was not as reliable as the currently used control package which depends on a trial-and-error determination of the switching temperature. However, the new strategy should perform better for runs which retain information about previous runs. The longrange prediction of the switching time was affected by the quality of the estimated parameters in the prediction error models. These parameter estimates were affected in turn by a number of key design parameters as indicated in the next section.
4. Design Parameters The influence of the following seven design parameters will be summarized: sampling period At; assumed time delay kat; forgetting factor 0;tuning constant for the PIDSTC algorithm v; polynomials P, and Pd for the STC; and the level of input excitation. The key considerations for these design parameters were how each affects the calculated switching time t,,, the initial PID controller output U , and the quality of the regulatory PID control at the end of each run. The design parameters for the base case run in Table I were used for each run unless otherwise noted. All calculated PID controller settings are calculated by using the PIDSTC algorithm, unless otherwise stated. 4.1. Sampling Period. Table I11 shows that, by the time the heater was switched off, there were between 24 and 774 samples available if the sampling period varied between 5 min and 10 s, respectively. For a sampling period (At) of 10 s, the predicted maximum temperature over the next 100 samples (about 17 min) was consistently underestimated. The low estimates of 2"- were an indication that the prediction error models were inaccurate. In part, this can be understood because the estimated minor time constant was small, and thus the second-order system model was effectively a first-order system. For a first-order system, the predicted maximum temperature would occur one time delay after the heater was switched off. Thus, the time of the predicted maximum was underestimated for the 10-s sampling period. At the larger sampling periods, the margin for error was smaller, meaning that if the switch was performed too early or too late, the process temperature would change more before the next sampling instant. Therefore, more undershoot or overshoot occurred for the larger sampling periods if the switching time was incorrect. In order to calculate the switching time (t,) in a timely fashion, it was necessary to obtain some minimum number of samples in a time interval equal to three or four major time constants. In general, the models improved if more
n.a.a n.a. n.a.
neg. neg. 206.0
1.05 109 8.50
2.13 149 0.065
?ID Settings from Second-Order Models (at End of Run) Kc, % / " C 2.32 neg. 2.85 2.19
Ti, min
Td,min
137.0 0.0
neg. 131.0
192 9.90
151 0.078
Not available.
Table V. Influence of Sampling Period on PIDSTC Settings run 9-5 9-3 6-4B 9-6 2.0 1.0 0.167 Gt, min 5.0 68.2 9.34 Kc, % / " C 10.0 (default) 109.0 96.1 1.87 Ti, min 10.0 (default) 296.0 2.44 0.003 Td, min 1.0 (default) 7.16 control conservative oscillatory, good oscillatory perforsensitive to noise mance Table VI. Influence of Assumed Time Delay on Overshoot and Undershoot run 10-4 8-5 10-2 assumed time delay, k A t , min 1.0 6.0 15.0 129 124 123 switching time, t,,, min switching temp, Taw," C 350 339 334 predicted temp change, A T = ?mu - PBw, "C 2.1 11.5 16.3 overshoot (undershoot), " C 4 0 (10)
samples were available, yet the sampling period could not be so small that the signal was obscured by noise. Thus, the choice of a sampling period involved an engineering tradeoff. In general, it would be expected that both extremes, the large sampling period and the small sampling period, would lead to poor parameter estimation models. The effect of the sampling period on the calculated controller settings is shown in Table IV. In the experimental study, the estimated process gain (derived from the one step ahead prediction error model) K , is consistent and accurate for all sampling periods. Thus, the determination of the initial PID controller output ( U )was also consistent. The estimated process gain and time constants were used in the calculation of PID controller settings by the method of Smith et al. (1975), which requires a second-order continuous-time model of the process. When the sampling period was large, the intermediate terms of these calculations, y1and yz,were complex, which meant that the time constants and hence the PID settings could not be calculated. The calculated integral time and derivative time derived from the PIDSTC algorithm varied widely with the sampling period as is apparent in Table V. The control strategy used default values when PID controller settings could not be calculated. Based on the experimental experience, there is a wide range of acceptable sampling periods for this application. 4.2. Assumed Time Delay. Overestimating the time delay by using a large assumed time delay in the process model led to undershoot due to underestimation of the switching time (taw). It also led to inaccurate model parameters and erroneous predictions of Tma.In this in-
1446
Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988
Table VII. Influence of Assumed Time Delay on Calculated PID Settings run 10-4 6-4B 10-2 assumed time delay, k A t , min 1.0 6.0 15.0 PID Settings from Second-Order Models (Not Used) 10.4 1.05 neg. 116.0 109.0 neg. Ti, min 3.58 8.50 neg. Td,min
K,,%/"C
PIDSTC (Used for Control) 94.3 68.2 99.2 96.1 min Td, min 1.22 2.44 control performance good good
K,,%/"C
T,,
53.4 184.0 5.75 sluggish
Table VIII. Influence of Forgetting Factor on Calculated PID Settings run 6-4B 7-4 7-3 forgetting factor, (3 1.0 0.99 0.98 K , at switch, "C/ 70 4.85 neg. 2.03 PIDSTC (Used for Control) 68.2 57.4 96.1 101.0 T,, m'n T,, min 2.44 1.45 good sluggish control performance
Kc,%/"C
47.8 114.0 0.180 sluggish
vestigation, results were obtained for assumed time delays ( k a t ) of between 1 and 15 min where the actual process time delay is almost 6 min. In Table VI an assumed time delay of k a t = 1 min was too small and resulted in an overshoot of 4 "C. On the other hand, an assumed time delay of kAt = 15 min was too large and the early switch caused an undershoot of 10 "C. Thus, overestimating the process time delay can be used as a conservative approach for avoiding overshoot. When the assumed time delay was incorrect, the parameter estimation models were not accurate and the predictions were not reliable. Thus, in addition to the influence of the assumed time delay on the timing of the switch, there can be an effect on the accuracy of the switch. Table VI1 indicates that the assumed time delay has a strong influence on the calculated PID controller settings. The calculated PID controller settings from the PIDSTC algorithm shown in Table VII, especially the controller gain ( K J ,are influenced by the assumed time delay. In general, the conservative approach of overestimating the time delay would be expected to lead to more conservative controller settings. It was observed in the simulation study by Phillips et al. (1988) that more conservative control does result for larger assumed time delays. _Table VI1 indicates that the calculated controller gain (K,) decreases as the assumed time delay increases. As noted above, overestimating the assumed time delay caused conservative switching times and undershoot. 4.3. Forgetting Factor. The use of a constant forgetting factor which discounted past measurements (i.e., p < 1)was investigated but the disadvantages outweighed the benefits. The estimated steady-state gain in Table VI11 was not correct when was equal to 0.98 to 0.99. This inaccuracy caused a large initial PID controller output ( U ) to be used. The recovery from this error was slow in both runs. In general, for /3 < 1, the convergence of the parameter estimates was slow, which was detrimental because the time available to achieve good models was limited by the switching time. In addition, the estimated steady-state gain was inaccurate for /3 < 1 as noted above. Table VI11 indicates that the small values of the forgetting factor
Table IX. Influence of Tuning Constant Y on Calculated PID Settings run 8-5 6-4B 9-1 tuning constant, u 0.5 1.0 2.0 K , at switch, "C/% 5.80 4.85 5.78 PIDSTC (Used for Control) 27.4 68.2 min 108.0 96.1 T d ,min 1.17 2.44 control performance very good good
Kc,% / " C
T,,
116.0 115.0 1.62 noisy
produced more conservative controller settings. A well-known disadvantage of using a constant forgetting factor with /3 < 1 is that the covariance matrix tends to increase during periods of little input excitation, eventually leading to blow-up of the estimation algorithm (Seborg et al., 1986). By contrast, using p = 1 can result in the estimation "going to sleep" if the run is quite long. The potential problem was not a concern in this study due to the relatively short runs. Thus, a value of /3 = 1 was deemed to be the best choice. 4.4. Input Excitation. Adequate input excitation to the process was found to be essential for good parameter estimation. For example, a single step change in the input at the beginning of the run did not provide adequate excitation. Consequently, a pseudo random binary sequence (PRBS) was used consisting of step changes between 80% and 100% in the control signal to the heater during the initial heat-up. Since 100% heating was not used exclusively during the start-up, the use of a PRBS input increased the time for the sand bath to reach the set point from about 90 min to about 120 min. The PRBS input improved the quality of the parameter estimation in two ways: by providing more accurate parameter estimates via better input excitation and by allowing more samples to be taken before the switching time. The frequency of the step changes did not affect the accuracy of the estimated process model as long as there were at least three to five step changes during the heat-up. 4.5. Tuning Factor. The effect of the PIDSTC tuning constant ( u ) on the calculated PID settings is shown in Table IX. The calculated controller gain varied linearly with the tuning constant (v) as would be expected from the theoretical equations for Y in part 1. The other controller settings were not significantly affected. It was difficult to determine an a priori choice for the tuning constant (u). Cameron and Seborg (1983) proposed an approach based on a stability analysis. Following their approach, which required a priori knowledge of the approximate process gain and major time constant, the stability limit was predicted to occur for u = u,, where 1
= exp(-At/TJ
(2)
where K , is the process gain, T~ is the process time constant, and At is the sampling period. Using Kp = 5 "C/ %, At = 1 min, and T~ = 50 min, these equations gave ,,v = 0.2082, which was clearly a conservative estimate of v because values of 0.5, 1.0, and 2.0 were used with no stability problems. (See Table IX.) Fortunately, v can be easily tuned because it affects the calculated controller gain directly without affecting the calculated integral time or derivative time, as noted above. 4.6. Polynomials P , and P d . The user-specified polynomials P, and p d were critical to the quality of control
Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 1447
t
Table X. Influence of P , Polynomial on Calculated PID Settings run 5-2 5-3 6-4B coeff pnl 0.90 0.95 0.99 PIDSTC (Used for Control) 88.6 45.3 Ti, min 1.55 1.60 Td,min 0.048 0.054 control performance oscillatory oscillatory
Kc, %/OC
'I'jl"'
I
"
,
"
iil I
68.2 96.1 2.44 good
20 -
Table XII. Accuracy of Estimated Process Gain no. of runs with estimated process gain greater than 6 "C/ 70 no. of runs with process gain between 4 and 6 "C/ % no. of runs with estimated process gain less than 4 OC/% total no. of experimental runs
I
-L
PIDSTC (Used for Control) 68.2 154 Ti, min 96.1 104.0 Td,min 2.44 1.72 control performance good somewhat sluggish, sensitive to noise
I
Kc, %/"C
0
,
,
~
I
I
__
,
200
I00
300
Time ( m i n )
Figure 5. Estimated steady-state process gain from prediction error model (run 6-4B).
14 9 5 28
achieved. In particular the calculated PID settings were found to be sensitive to small changes in polynomials P, and Pd, as shown in Tables X and XI. For example, the closer coefficient pnlwas set to -1.0, and the calculated PID settings became less conservative since the controller gain (K,) increased and the integral time (Ti)decreased. The integral time was affected more strongly than the controller gain. Because of this sensitivity, it was desirable to have a priori estimates of reasonable values for P, and P d . An alternative approach would be to modify these polynomials on-line based on the estimated process model, thus reducing the number of design parameters to be specified by the user. The effect of P = P , / P d can be deduced from the characteristic equation of the closed-loop system for the PIDSTC. An a priori estimate of pnlcan be obtained by choosing pnl to give an approximate closed-loop time constant equal to one half of the major time constant ( T ~ ) : = exp(-2At/d
"
20 -
Table XI. Influence of PdPolynomial on Calculated PID Settings run 6-4B 6-2 coeff P d l 0.90 0.95
Pnl
I
(3)
However, for the experimental runs of this investigation, polynomial P,, was specified by the user in an ad hoc fashion. The effect of design parameter ( P d ) on the control system parameters and perfor-mance is shown in Table XI. Both the controller gain (K,) and the integral time (Ti) increased as the term P d l approached unity, but the effect of increasing P d l was more significant on K,. At the present time, it is not apparent how P d l should be specified a priori. However, it may be possible to calculate a reasonable value for P d based on an assumed or estimated process gain. 5. PID Controller Settings Calculated from Second-Order Continuous-Time Process Models Table XI1 shows-the accuracy of the estimated steadystate process gain (K,) that was calculated at the switching time. The actual steady-state process gain for the sand bath was about 4.9 "C/ %. Design parameters such as the forgetting factor and the assumed time delay influenced
Table XIII. Change in the Calculated Controller Parameters from Switching Time to the End of the Run no. of runs with less than 5% change no. of runs with 5-10% change no. of runs with greater than 10% change no. of runs with very poor calculated controller settings (full run) total no. of experimental runs
8 5 10 5 28
the quality of the estimated model parameJers and thus the estimated steady-state process gain ( K J . Figure 5 shows a typical trend for the steady-state process gain determined from the one step ahead prediction error model during the base case run. A singularity was observed consistently about 45 min after the start of the heat-up. It was observed also in the simulation runs. The singularities in Figure 5 at t = 140 min and at t = 165 min were a result of the switch to no heating and later to PID control. To determine the initial PID controller output ( V ) accurately, the steady-state process gain must be accurately estimated prior to the switch to PID control. The accuracy of the estimated process gain and time constants also affected the quality of the calculated PID settings based on the second-order continuous-time process models. 6. PID Self-Tuning Controller There were several ways to implement the calculated PID settings that were calculated from the PIDSTC algorithm. The most conservative approach was to view the settings as recommendations, to be entered manually if accepted by the operator. Alternatively, the strategy could be made fully automatic, resulting in true "self-tuning control". Most of the runs reported in this paper were of the latter type. A third alternative would be to use the calculated settings that were available a t the time of the switch to PID control as constant controller settings. This approach avoids updating the PID settings at every sampling instant. These alternatives raise the issue of how often the PID settings were suitable at the time of the switch and how much they were likely to change during the remainder of the run. Table XI11 provides a summary of the changes observed in the calculated controller settings from the time of the switch to the end of the run. In the tabulation, only the controller gain (K,) and the integral time (Ti)were considered, and the greater absolute amount of change of either term was used for the classification.
1448 Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 Table XIV. Influence of Set Point on Prediction of and Controller Settings run 7-1 6-4B set point, "C 250 350 116 switching time, t,, min 72 320 switching temp, T,, "C 241 overshoot (undershoot) (5) (27)
Switch
7-2 450 216 449 0
PID Settings for Second-Order Models (Not Used for Control) Kc, %/OC neg. 1.05 2.13 q, min neg. 109.0 144.0 Td, min 26.2 8.50 3.12 PIDSTC (Used for Control) 71.0 Ti, min 115.0 Td, min 2.82 control performance good, slightly sensitive to noise
Kc,9o,/'C
68.2 96.1 2.44 good
56.0 89.0 1.20 good I
0
100
I
_
i
200 Time
300
400
531
(rrin)
Figure 7. Temperature and controller behavior during sensor failure (run 10-16). i
L
i
; 1(!3
Table XV. Design Parameters for Run 10-12 set point, yr, "C 200, 350 sampling period, At, min 0.167 assumed time delay, k i l t , min 6.0 forgetting factor, /3 1.0 PIDSTC tuning factor, u 1.0 1.G0.99t-' polynomial P, polynomial p d 1.o-o. 902-1 input excitation PRBS with a base frequency of 5 min, changes between 80% and error limit, OC
3
-
I
n1
200 3G( T m e (min,
4ro
500
Figure 6. PIDSTC control compensating changes in flow rate of fluidizing air (run 10-10).
The closed-loop control in many of the runs was quite good. Some of the runs, including the base case run, exhibited a control signal that was sensitive to noise due to the derivative term in the PID controller acting on an unfiltered temperature measurement. The calculated PID settings were sensitive to the design parameters as noted earlier. The PIDSTC tuning algorithm is promising, but additional work is needed to improve it. Although negative values of the calculated derivative time ( T d )were reported by Cameron (1982), none were observed in this investigation. 7. Influence on Set Point At higher set points, more samples were available before the switch to PID control was required. As can be seen in Table XIV, for a 1-min sampling period, 216 samples were obtained before the switch a t 450 "C, whereas only 116 samples were obtained when the set point was 350 "C. Consequently, the switching algorithm tended to be more accurate at high temperatures. 8. Disturbances and Measurement Errors Load disturbances such as a large change in the flow rate
of the fluidizing air affected the sand-bath operation. By contrast other types of disturbances such as changes in the temperature of the fluidizing air were generally damped
100% 3.0
out due to the large heat capacity of the sand bath. Disturbances that occurred after the switch to PIDSTC control were handled via feedback control, especially if selftuning was employed. For example, in Figure 6 the PIDSTC controller compensated for a 25 % decrease in the flow rate of the fluidizing air about 1h after the switch to PID control and also accommodated a 75% increase about 2 h after the switch. The design parameters were the same as those used for the base case run. Measurement problems such as an electrical short in a thermocouple affected the discrete-time model parameters that were developed on-line. In Figure 7 the thermocouple was shorted out for three brief periods of time. Major disturbances or severe measurement errors that occurred during the initial heat-up disrupted the parameter estimation so that the long-range predictions and calculated PID settings were unreliable. A noise spike filter would have improved the performance but was not used in order to provide a severe test of the proposed strategy. The performance of this strategy depends on the effectiveness of the jacketing software which has been described by Phillips et al. (1988). 9. User Acceptance
As stated previously, the proposed algorithm can be used to advise the operator about the switching time and calculated PID settings without actually implementing the recommended changes. This version of the program was used for several runs. The design parameters for one of these runs are given in Table XV. The set point was originally 200 "C, but 30 min after the temperature reached 200 "C, the set point was changed to 350 "C. The PID
Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 1449 settings from the second-order continuous-time models as proposed by Smith et al. (1975) were reasonable, but the PIDSTC did not give good recommendations due to a very small sampling period of 10 s. 10. Conclusions
A series of 28 experimental runs were made in order to evaluate an automated start-up strategy for the temperature control of a pilot-scale fluidized sand bath. The 28 runs covered a wide variety of numerical values of the design parameters in order to investigate the versatility of the proposed adaptive control strategy. Twenty-three of the 28 runs were successful in the sense that the adaptive control strategy calculated satisfactory values for the switching time and PID controller settings. The other five runs were unsuccessful due to inappropriate numerical values for a design parameter such as the sampling period. The sampling period ( A t ) was an especially crucial parameter in this study since only a limited period of time was available for on-line parameter estimation before the heater had to be switched off. Satisfactory results were obtained for a sampling period between 0.5 and 2.0 min. Jacketing software played a crucial role in the success of the adaptive control strategy. A number of recommendations are made concerning the influence and a priori selection of the other design parameters. The adaptive control strategy worked well for a reasonable range of assumed time delays, 1-15 min. For short experimental runs like the ones in this investigation, the forgetting factor should be set equal to one in order to promote rapid convergence of estimated model parameters. Other design parameters such as v and polynomials P, and p d affect the calculated PID controller settings, but it is difficult to specify satisfactory values a priori. User reaction t o the results of this experimental study was mixed. In general, the operating personnel were pleased with the utility of the proposed strategy, the promising results, and the additional process information provided via the estimated model parameters. However, they were not satisfied with the sometimes inappropriate calculated controller settings and the need to specify unfamiliar design parameters. The users were also concerned about the sensitivity of the results to the specified sampling period.
Acknowledgment Financial support from the National Science Foundation is gratefully acknowledged. Experimental data, equipment access, technical guidance, and financial support provided by the FED (formerly RTSD) division of Exxon Research and Engineering Company in Linden, NJ, and Clinton, NJ, are gratefully acknowledged. The assistance of J. Robert Sims is especially appreciated. Parts 1 and 2 of this paper were presented at the National Meeting of the American Institute of Chemical Engineers, Houston, TX, April 1, 1987.
Nomenclature A = system polynomial B = input polynomial kAt = assumed time delay, min K , = controller gain, %/“C K, = calculated controller gain, % / “ C K p = process gain, “C/% Kp = estimated process gain, “(21% p d = denominator polynomial of P P,, = numerator polynomial of P P d l = coefficient in polynomial p d pnl = coefficient in polynomial P , P = polynomial in the PIDSTC algorithm Q = polynomial in the PIDSTC algorithm R = polynomial in the PIDSTC algorithm r d = derivative time, min Td = estimated derivative time, min Ti = integral time, min 3 = estimated integral time, min T, = estimated maximum temperature after switch, “C AT = predicted rise in temperature after heater is switched off, “C A t = sampling period, min T,, = temperature at time of switch, “C T,, = estimated temperature at time of switch, “C t,, = time of switch, min U = initial PID controller output, % yI = set point, “C Greek Symbols
p = forgetting factor yi = intermediate result in Ti and vmlULcalculations v = tuning parameter for the PIDSTC vmm = stability limit for v 7 1 = major time constant, min i, = estimated major time constant, min r2 = minor time constant, min j 2 = estimated minor time constant, min q = arbitrary time constant, min
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Received f o r review May 11, 1987 Revised manuscript received February 23, 1988 Accepted March 5, 1988
