Znd. E n g . Chem. Res. 1987,26, 1117-1127
1117
An Experimental Evaluation of Saturation Algorithms for Advanced Digital Controllers Pui Man Wong, Paul A. Taylor,* and Joseph D. Wright? Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7
This paper investigates some of the problems caused by implementing advanced digital controllers in applications where the manipulated variable saturates; it then proposes and evaluates four solutions to these problems. The reset windup problem in advanced digital controllers can be solved by employing either antireset windup or simple clamping methods. However, our simulation and experimental results show that these methods produce a much slower process response than the algorithms proposed in this paper. Four saturation algorithms were studied, ranging from an improved antireset windup method to an on-line constrained optimization, with the aim of improving the process response during periods of manipulated variable saturation. The simulations and experiments were performed on a pilot-scale stirred-tank process involving dead time. They show that it is worthwhile to include these saturation algorithms in advanced digital controller designs. Saturation occurs when the final control element, the physical device acting on the manipulated variable, is temporarily unable to respond to the signal from the control&. For the three-term controller (PID), many antireset windup algorithms have been used to handle the saturation problem. The typical strategy is to use conditional integration; i.e., integrate the error only when it is appropriate. Since the PID controller is expressed explicitly in Proportional, Integral, and Derivative terms, it is easy to implement the conditional integration algorithm. However, these terms are usually collapsed together in advanced digital controllers and saturation protection is more often implemented as a simple clamping of the controller output. By way of an example, the experimental stirred tank process shown in Figure 1was simulated with a DAHLIN controller (Dahlin, 1968) as the feedback controller. Figure 2 shows this simulation with a simultaneous failure of the final control element for 10 control periods and a set-point change. Curve ii is the response when the conventional clamping algorithm is used for saturation protection, while curve i is the response without saturation protection. Although the control signals in curve i were greater than 10 V for some periods, the final control element saturated at 10 V and control signals greater than this value were of no use in achieving the desired closed-loop performance. This figure shows the need for a saturation protection algorithm to prevent windup. However, the consequence of using conventional clamping is that the optimality of the advanced digital controller may be lost. This is illustrated in the experiment shown in Figure 3a. By comparing the desired closed-loop response (curve i in Figure 3a) with the closed-loop responses produced with our best saturation algorithm (curve ii in Figure 3a) and with the conventional clamping method (curve iii in Figure 3a), one can see that the conventional clamping method results in a sluggish output response. One can conclude that two aspects should be considered in dealing with the saturation problem; controller windup and closed-loop optimality. The conventional clamping algorithm solves only the windup problem. The objective of this work is to find saturation algorithms such that the combined control and saturation algorithm c b handle both the windup and the optimality problem. Simulations and experiments were designed to *Author to whom correspondence is to be addressed. Research Centre of Canada, Mississauga, Ontario,
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show not only the improvement in the closed-loop performance by incorporating these saturation algorithms into the advanced digital control algorithm, but also the robustness of these saturation algorithms to the severity of the control problem and to model mismatch. In addition, we report an experimental application of an on-line constrained optimization algorithm, QIMC (Quadratic Internal Model Control), and demonstrate its superiority when constraints on the value and rate of change of the manipulated variables are important operational considerations.
Saturation Algorithms Four saturation algorithms are presented in the following sections. They differ in the extent to which they make use of process model information. Since advanced digital controllers already make use of a process model in the controller design equations, the accompanying saturation algorithm can also make use of this information. They are designed to solve the saturation problem for single-input, single-output systems (SISO) and are arranged in the sequence of increasing computational effort. A Heuristic algorithm represents the idea of conserving the control action prescribed by the advanced digital controller. A New Antireset algorithm freezes the integration of the output error only when the saturation is a result of the integral action. An Optimal Switch algorithm makes use of the process model to explicitly calculate the optimal period for the control action to stay at the saturation limit. The last algorithm, QIMC, treats the saturation problem as a constrained optimization problem. It will provide the ultimate performance for a system with saturation, but will require the greatest computation. The DAHLIN digital controller (Dahlin, 1968) was chosen as an example of an advanced digital controller. A desired closed-loop trajectory is specified in the design of this controller, but the actual closed-loop trajectory will not follow the desired trajectory when saturation occurs. Therefore, the sum of squared deviations of the actual closed-loop trajectory from this desired trajectory was used as a measure of the effectiveness of the saturation algorithm. Conventional Saturation Protection Many papers have proposed solutions of the windup problem (Astrom and Wittenmark, 1984; Smith, 1972; Khandria and Luyben, 1976). The conventional solution is to freeze the integration of the output error whenever 1987 American Chemical Society
1118 Ind.
Eng. Chem. Res., Vol. 26, No. 6, 1987 3.5
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The process actually had asymmetric dynamics, and a comparison of eq 28 and 29 with the nominal process model, eq 25, shows mismatch in all three parameters. The experimental responses to a 3 "C change in set point are shown in Figure 11. In order to determine whether or not the oscillations were due to model mismatch in the controller design or to mismatch in the saturation algorithms, a further experiment was performed. Instead of the 3 OC set-point change, which caused saturation, a smaller set-point change (1 "C) was designed so that saturation would not occur. If oscillations were generated by this smaller set-point change, then they must come from the mismatched controller and not from the saturation algorithm. The results for the DAHLIN controller for both the 3 and 1 "C set-point changes are shown in Figure 12a and the corresponding results for QIMC in Figure 12b. These experiments suggest that the oscillation is caused by the model mismatch in the controller and not by any mismatch in the saturation algorithm. The simulation and experimental results indicate that QIMC produces more oscillations than the DAHLIN controller under model mismatch conditions. This is due to the fact that QIMC uses more model information than the DAHLIN controller. This result corroborates the observation by Kestenbaum et al. (1976) that model-based controllers are more sensitive to model mismatch than controllers, such as PID, which uses process information to a lesser extent; the question of the optimality of these two types of controllers not withstanding. From the simulation and experimental results, we observed that only the initial stage of the transient response is significantly affected by the saturation algorithm, whereas the overall performance is dictated by the controller and the extent of model mismatch. In any particular experiment, the performance is governed more by the type of controller and its tuning than by the saturation algorithm. There are examples of a poorly tuned controller and a poor saturation algorithm (e.g., the conventional clamp), giving better performance during saturation than the same poorly tuned controller with a better saturation algorithm (e.g., new antireset windup). In this case the better saturation algorithm does a better job of preserving the control actions from the poorly tuned controller and therefore allows more of the wrong control action to affect the process. Of course, the poorly tuned controller degrades the performance when the manipulated variable is not saturating, and we achieved better overall performance
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with a properly tuned controller and the best saturation algorithm.
Conclusion The simulation and experimental runs show that the proposed algorithms not only handle the windup problem but also performed much better than the conventional clamping algorithm. Although more computation is required for these new algorithms, the increase in speed of digital computers means that these algorithms can be applied on-line; this was demonstrated by the experimental work. The Heuristic and New antireset algorithms can be implemented easily without too much extra computer memory; therefore, they may be incorporated into frontend computer systems for low level control. The Optimal Switch algorithm requires a little more memory, and computation and is more suitable for implementation cin high-level control algorithms in the host computer. The QIMC algorithm is a sequence of local, constrained optimizations which is equally applicable to SISO and multivariable processes; we presented an experimental application for a SISO process. It provided the greatest flexibility in specifying the constraints on the process input and output and produced the best performance. It required considerably more computation than the other algorithms and is best suited for higher level, supervisory control. The proposed saturation algorithms show considerable improvement over the conventional clamping
1126 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987
SUM5=K(5)*(E(1)-4.O*E(2)+6.O*E(3)-4.O*E(4)+
algorithm; they give better performance as well as windup protection. It is therefore worthwhile to incorporate these saturation algorithms into advanced digital control algorithms.
Acknowledgment The research reported in this paper was supported by Operating Grants A7017 and A6075 from the Natural Sciences and Engineering Research Council of Canada. P.M.W. acknowledges the support of McMaster University through the award of a Clifton W. Sherman Graduate Scholarship.
Nomenclature Epr,(i+l:i+P) = vector (P X 1)of prediction errors composed of the best available forecast of the combined model uncertainty and process disturbances over the i + 1to i + P prediction period g(m) = controller impulse weight at lag m; g(m) = 0 if m < 0 or m > Ng h(l) = process impulse weight at lag I ; h(1)= 0 if 1 < 1 or 1
>N
M = input horizon (number of intervals into the future over which the process input is to be computed) N = number of impulse weights in process model Ng = number of impulse weights defining the controller P = prediction horizon (number of intervals into the future over which the process output is to be optimized) Ufutur,(i:i+M-l)= vector (M x 1) of future process inputs composed of Uti), U(i+l),..., U(i+M-l) Up&-N+l:i-l) = vector ( ( N - 1)x 1)of past process inputs composed of U(i-N+l), U(i-N+P), ..., U(i-1) Y,,,(i+l:i+P) = vector (P x 1)of predicted outputs composed of Y red(i+l), ypred(i+2), ..., y red(i+P) Ydeslred[i+l:i+P) = vector ( P x f) of the desired output responses over the prediction horizon, composed of Ydeslred(i+l)>Ydeslred(i+2),.*.,Ydesired(i+P) Greek Symbols \k = matrix ( P X M) of impulse weights for future inputs; \k, = h(i-j+l), j # Mor i 5 M; \kL,= h(i-j+l) + Czk=Mh(i-rZ), j = Mand i > M @ = matrix ( P x ( N - 1))of impulse weights for past inputs;
1 E(5))
SUM4=K(4)*(E(1)-3.O*E(2)+3.O*E(3)-E(4)) SUM3=K(3)*(E(1)-2.O*E(2)+E(3)) SUMl=K(l)*(E(I)-E(2)) C C ADD THE PROPORTIONAL, DERIVATIVES AND C DEAD-TIME COMPENSATION C CONTRIBUTION TO T H E PAST CONTROL C ACTION
UPD=UPAST+SUM5+SUM4+SUM3+SUMl+ 1 DEAD C C CALCULATE INTEGRAL CONTRIBUTION DELUI=K(2)*E(1) C C SUM TO COMPUTE T H E TOTAL CONTROL C ACTION UPID = UPD + DELUI C C TEST FOR OUTPUT LIMITS C CLAMP UPID ONLY WHEN SATURATION C CAUSED BY INTEGRATION C
IF ((UPD.GE.UMAX).AND.(UPID.GE.UMAX)) 1 THEN
U(1)=UMAX ELSEIF ((UPD.LE.UMIN).AND. (UPID.LE. 1 UM1N))THEN U(l)=UMIN ELSEIF ((UPD.LT.UMAX).AND.(UPID.GE. 1 UMAX))THEN UPID=UMAX U(l)=UMAX ELSEIF ((UPD.GT.UMIN).AND.(UPID.LT. 1 UMIN)) THEN UPID=UMIN U(l)=UMIN ELSE U(1)=UPID ENDIF
ai, = h(i-j+N)
Appendix 1. Fortran Coding for Antireset Windup Algorithm (Velocity Form; Adapted from Segall and Taylor (1986)). C C C C C C C C C C C C
Following eq 5, E ( i ) is the current error. We need U ( i ) ,the new control action. Define error vector E and control action vector U. E vector: E ( i ) ,E(i-l), ...,E(i-4). Uvector: U ( i ) ,U(i-l), ..., U(i-NB-1). U(1) is thus the actual control action applied to the process CALCULATE T H E CONTRIBUTION FROM DEAD-TIME COMPENSATION DEAD=0.0 DO I = l , NB DEAD=B(I)*(U(I+ 1)-U( 1+2))+DEAD ENDDO
C C CALCULATE THE PROPORTIONAL AND C DERIVATIVE CONTRIBUTIONS C ASSUMING NA=4 C
C C UPDATE THE UPAST UPAST=UPID C C Now call a subroutine to shuffle backwards the E and C U vectors ready for the next control interval. 2. Fortran Coding for Heuristic Algorithm. C C C C C C C C
U(i) FROM ANY ADVANCED DIGITAL ALGORITHM THE EXCESS CONTROL ACTION FROM T H E PREVIOUS INTERVALS IS COMP
IF CONTROL ACTION ALREADY SATURATED; DON’T ADD T H E COMPENSATION IF (U.LT.HILIM.AND.U.GT.LOLIM) 1 U=U+COMP
C C SET T H E COMPENSATION IF CONTROL C ACTION AT UPPER LIMIT
Ind. E n g . Chem. R e s . 1987,26, 1127-1132
IF (U.GT.HILIM) THEN COMP=U-HILIM U=U-COMP C C SET T H E COMPENSATION IF CONTROL C ACTION AT LOWER LIMIT ELSEIF (U.LT.LOLIM) THEN COMP=U-LOLIM U=U-COMP C C SET THE COMPENSATION IF CONTROL C ACTION NOT SATURATED ELSE COMP=O.O ENDIF Literature Cited Astrom, K. J.; Wittenmark, B. Computer Control Systems: Theory and Design; Information and System Science Series; PrenticeHall: Englewood Cliffs, NJ, 1984. Clark, D.; Hastings-James, R. Proc. Inst. Electr. Eng. 1971, 118, 1503-1506. Dahlin, E. B. Znstrum. Control Syst. 1968, 41, 77-83. Garcia, C. E.; Morari, M. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 308-323. Garcia, C. E.; Morshedi, A. M. “Solution of the Dynamic Matrix Control Problem Via Quadratic Programming (QDMC)”, Proceedings of the Canadian Industrial Computing Society, Ottawa, Ontario, Canada, 1984.
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Gill, P. E.; Murray, W.; Saunders, M. A.; Wright, M. H., Report SOL 84-6,1984; Systems Optimization Laboratory, Stanford University, Stanford, CA. Harris, T. J.; MacGregor, J. F.; Wright, J. D. Can. J. Chem. Eng. 1982, 60,425-432. Kestenbaum, A.; Shinnar, R.; Thau, F. E. Ind. Eng. Chem. Process Des. Deu. 1976, 15, 1-13. Khandria, J.; Luyben, W. L. Ind. Eng. Chem. Process Des. Deu. 1976, 15, 278-284. Kuest, J. L. Optimization Techniques with Fortran; McGraw-Hill: New York, 1973. Marshall, J. IEE Control Eng. Ser. 1979, 1. Martin, G. D. AZChE J. 1981,27, 748-805. Mosler, H.; Koppel, L.; Coughanowr, D. AIChE J . 1967,13,768-778. Palmor, Z.; Shinner, R. Znd. Eng. Chem. Process Des. Deu. 1979,18, 8-30. Richalet, J.; Rault, A.; Testud, J. L.; Papon, J. Automatica 1978,14, 413-428. Rosenbrock, H. H. Comput. J . 1960,3, 175-184. Segall, N. L., M. Eng. Thesis, McMaster University, 1983. Segall, N. L.; MacGregor, J. F.; Wright, J. D. Report 1008, 1986; Process Control Laboratory, McMaster University, Hamilton, Canada. Segall, N. L.; Taylor, P. A. Znd. Eng. Chem. Process Des. Dev. 1986, 25, 495-498. Smith, C. L. Digital Computer Process Control; Intext Educational: Scranton, PA, 1972. Wong, P. M., M. Eng. Thesis, McMaster University, 1983. Wong, P. M.; Taylor, P. A.; Wright, J. D. Report 1007,1986; Process Control Laboratory, McMaster University, Hamilton, Canada.
Received for reuiew April 24, 1986 Accepted February 17, 1987
A Modified Oldershaw Column for Distillation Efficiency Measurements Mohammad A. Kalbassi and Michael W. Biddulph* Department of Chemical Engineering, University of Nottingham, University Park, Nottingham, U . K .
This paper describes experimental studies using a conventional and a modified 3.8-cm-tray diameter Oldershaw column with external downcomers. The point efficiencies evaluated using the modified column are much lower due to the tendency of the new column not t o support the froth by its wall and to reduce the wetted wall effects. The point efficiencies measured in the modified column are in close agreement with predictions from large-scale tray efficiency measurements and would lead to conservative designs of large distillation columns. T h e surface tension behavior of the systems studied suggests that highly surface tension positive systems exhibit higher point efficiencies than neutral and negative systems. For many years workers in the field of distillation have tried to simulate the behavior of large distillation columns by using laboratory-scale studies. Recent mathematical simulation of the tray models by high-speed digital computer (Biddulph 1975), and also the deficiencies of the prediction methods (Lockett and Ahmed, 1983) together with unavailability of field data, necessitates an easy method to obtain point efficiency data for the design of a distillation column. One of the difficulties associated with the use of small columns is that of “wall supported“ froth which has been referred to by Ashley and Haselden (1972), Thomas and Hag (19761, Lockett and Ahmed (1983), Young and Weber (19721, Standart (19741, Zuiderweg (1969))Sargent et al. (1964), Bubble Tray D e s i g n Manual (1958))Dribika (1986), Pruden et al. (1974), and Finch and Van Winkle (1964).
* Corresponding author. 0888-5885/87/2626-1127$01.50/0
Early studies of surface tension positive and negative systems using laboratory-scale “Oldershaw” columns (Zuiderweg and Harmens 1958) revealed higher efficiencies for positive systems due to the Marangoni effect on froth formation, the lower efficiencies evaluated for the negative system being explained in terms of contraction effects on the froth. However, the reverse was observed in the “spray” regime (Bainbridge and Sawistowski, 1964), which was interpreted in terms of “Marangoni” effects on droplet formation. Other effects such as liquid holdup (Haselden and Thorogood, 1964; Fane and Sawistowki, 1969; Brown and England, 1961; Finch and Van Winkle, 1964; Umholtz and Van Winkle, 1957; Hellums et al., 1958; Jeromin et al., 1969; Sargent et al., 1964) have also been considered. There is evidence that the deep froths formed in conventional Oldershaw columns are due to the wall effects supporting the froth which result in high point efficiencies, often exceeding 100%. Under these conditions, the effi0 1987 American Chemical Society
