Lydersen, A. L., Greenkorn, R. A,. Hougen, 0. A,, "Generalized Thermodynamic Properties of Pure Fluids." Eng. Expt. Station Report No. 4, U. Wis.. Madison, Wis., 1955. Pitzer, K. S., Hultgren, G. O., J. Am. Chem. SOC.,80, 4793 (1958). Redlich, O., Ackerman, F. J., Gunn, R. D., Jacobson, M., Lau. S., lnd. Eng. Chem., Fundam., 4,369 (1965). Rowlinson. J. S.. in "Applied Thermodynamics," pp 1-8, American Chemical Society, Washington, D.C., 1968. Scott, R. L., J. Chem. Phys., 25, 193 (1956). Van Ness, H. C., "Classical Thermodynamics of Non-electrolyte Solutions," p 90, Pergamon Press, 1964. Winnick, J.. Prausnitz, J. M., Chem. Eng. J., 2, 241 (1971).
White, M. G., "Principle of Corresponding States for Liquid Mixtures." M.S. Thesis, Purdue University, 1973. Yesavage. V. D.. Katz. D. L.. Powers, J. E., AlChEJ., 16, 867 (1970).
Received for review January 23, 1974 Accepted F e b r u a r y 24,1975 T h i s work was supported by N a t i o n a l Science F o u n d a t i o n t h r o u g h G r a n t GK-42051.
Secondary-V aria ble Control Chorng-hour Yang and Thomas J. Ward* Chemical Engineering Department, Clarkson College of Technology, Potsdam, New York 73676
The qualitative structural features of various secondary-variable control configurations are illustrated with signal flow diagrams to provide some comparison of the control logic involved.
The measurement of a secondary process variable to provide a n auxiliary control actuation path is an appealing possibility for process systems. However, there are many possible configurations for such secondary-variable control. In addition to the switching (override) and sequential (interlock) logic used for startup, constraint, and safety, there have been several continuous logic methods proposed for the regulator control of staged contactors and distributed processes. The qualitative structural features of these continuous secondary-variable configurations are examined here. The control logic is illustrated with signal flow diagrams in which nodes represent the variables and directed branches represent the functions relating the nodes.
Series Cascade Control The series cascade arrangement, known simply as cascade control, has received considerable attention (Ziegler, 1954; Franks and Worley, 1956; Gollins, 1956; Wills, 1960; Harriott, 1964; Perlmutter, 1965; Shinskey, 1967; Koppel, 1968; Clay and Fournier, 1973). A distillation example (Luyben, 1973) of series cascade control is illustrated in Figure l a . In this example, a tray temperature YC is the controlled variable and the steam flow controller output XM is the manipulative input. The secondary variable is the measured steam flow Yh. From the control configuration shown in Figure l b , the secondary variable obviously is on the primary X,M YCcontrol path and serves as the measured variable for an inner feedback compensation. This secondary or inner feedback loop can be viewed in terms of four possible logic roles. ( a ) Inner Loop Compensation. The secondary controller FS modifies the troublesome dynamics associated with the secondary process function GS and consequently simplifies the design of the primary feedback controller F (Koppel, 1968). (b) Dynamic System Modifier. The two-loop combination can be designed to give overall dynamics that would be difficult to achieve with a single feedback controller (Perlmutter, 1965). ( c ) Partial Disturbance Compensation. The secondary
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controller can initiate some manipulative effort to comYS disturbance transmission path pensate for the XC. before that part of the disturbance reaches y c (Pollard, 1971). (a) Primary Feedback. The secondary loop can be viewed as the primary feedback control, while the outer loop serves as a trimming or setpoint correction for the inner loop. (This approach, with an intermittent manual compensator in the outer loop, is characteristic of some process control cases.) While these various roles might seem to be inconsistent, each is a valid and related viewpoint. The suitability of a particular view depends on the process situation and requires some skill on the part of the designer. In any case, it is important to realize that the inner loop is an inherent part of the overall feedback control, as emphasized in role (b) above, and will modify the characteristic equation and dynamic response of the system.
Parallel Cascade Control Luyben (1973) has proposed a parallel cascade control scheme for the distillation example shown in Figure 2a. The overhead vapor composition YC is controlled by manipulating the reflux rate X M . The secondary variable is the tray temperature Yh. From the corresponding control configuration, shown in Figure 2b, it appears that the tray temperature Yc:does not affect the overhead composition YC through the process model structure used in the design. However, the Y5 measurement is an internal measure of the effect of the disturbance Xr.. In this sense, the secondary Fh controller is a "partial disturbance compensator" for the primary control loop. In other -words, the secondary controller is providing a partial feedforward role. Since the secondary loop significantly modifies the system characteristic equation, the design is based on feedback methods similar to those used for series cascade control. Series-Parallel Cascade Control Slight process model changes can significantly influence the control logic. For example, if the disturbance in the Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
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Figure 1. Series cascade control: A, distillation bottoms-end example; B, control logic.
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Figure 2. Parallel cascade control: A, distillation light-end example; B, control logic.
Luyben distillation example were specified as a feed disturbance, the light-end process model of Figure 2 might be developed to show the disturbance Xr- affecting the tray temperature Ys and this, in turn, affecting the overhead composition Ye. The cascade control of this different light-end model, shown in Figure 3a, would exhibit a dual series-parallel nature. This is illustrated in Figure 3b. This dual series-cascade control is based on a more general process structure of the light-end distillation case cited above. If the designer neglects the parallel XM YC path, the contrd reduces to series cascade. Alternately, if the YS YC path is assumed to be zero, then the control reduces to the parallel cascade of Luyben. This underscores the need for suitable model selection before a control system is synthesized. Since the external form of the control for these three c a x a d e configurations is the same, it is quite possible that the parallel X M -* YC path used by Luyben has been neglected in some earlier series cascade control applications.
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Figure 3. Series-parallel cascade control: A, distillation light-end example; B, control logic.
Static Estimation Control A form of intermittent manual cascade control based on role (d) above has been widely used for process situations in which the controlled variable is not readily measured on a continuous basis. For example, automatic tray temperature control loops have been used to manipulate a light-end variable such as distillate rate or reflux rate. Periodic distillate composition measurements were then used to manually adjust the temperature-loop setpoint. An interesting modification of this sampled-continuous cascade control has been presented by Weber and Brosilow (1972). Consider the light-end distillation case shown in Figure 4a. Instead of using a single tray temperature, several , Y53 are noise-filtered tray temperatures Y S I , Y s ~ and measured, along with the manipulated reflux input X M . These provide the input to a n on-line algorithm that generates an estimate YCEof the overhead composition. The algorithm is based on a prior determination of the static sensitivities relating the manipulative input XM and selected disturbances such as the feed disturbance X r to both the tray temperatures and the overhead composition. In other words, the measurements provide a partial feedforward compensation of the disturbance transmission path. A design calculation generates the size and identity of an acceptable measurement set so as to minimize the amplification of modeling and measurement noise. In this example, the calculation gives the number and location of tray temperature measurements to be used in the estimator algorithm. The overhead composition estimate YCEis the secondary variable used in the continuous F S control path, as shown in Figure 4b. The primary F loop, shown as a dotted line, is an automatic sampled path analogous to the manual trimming correction mentioned above. In the example, the sampled measurements of the overhead composition Yc are compared with the estimate YCE at the sampling instant k to generate an “estimate corrector” (Ye - Y c E ) ~This . corrector signal, a constant between sampling instants, can be operated on by the logic function F to provide a setpoint adjustment for the continuous secondary loop. The “effective setpoint” for the continuous secondary loop is YeSET+ F ( Y m - Y c ) ~The . function F was selected as the unit constant so as to provide the true error (YcSET - Y c ) as ~ the input to the secondary
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B Figure 5. Practical two-point control: A, heat exchanger example; B, control logic.
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Figure 4. Static estimation control: A, distillation light-end example; B, control logic; C, F = 1case.
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controller a t the sampling instant. This can be seen in Figure 4c. The cited reference presents a multiloop generalization of the above static estimation logic.
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Two-Point Practical Control
A distributed system control study (Koppel et al., 1970) has been based on the heat exchanger example shown in Figure 5a. The exit tube-side temperature YCis controlled by manipulating the tube-side flowrate XM,with the jacket temperature assumed constant. The tube-side temperature YS at an internal point is the secondary variable. This is used in a feedfonvard role, as illustrated in Figure 5b, for compensation of the tube-side inlet temperature disturbance X I , . The process model structure, shown as the solid lines of Figure 5b, is nearly the same as that used in the parallel cascade case of Figure 2b. However, the proposed “twopoint practical control” is quite different. Instead of cascading or multiplying the control functions, these investigators use an additive combination of the control functions. This permits a n analytical design of the secondary controller FS based on the invariance principle. The primary F controller is viewed as a trimming correction for the secondary controller and is designed by feedback methods. It is interesting to note that the heat exchanger example of Koppel et al. could also be modeled by a modified process structure identical to the solid-lined model of Figure 3b, for which a simplified two-point practical control is available (Yang and Ward, 1974). Internal Variable Control Haskins and Sliepcevich (1965) and Greenfield and Ward (1967) used a secondary variable approach called
B Figure 6. Internal variable control: A, stirred-tank heater example; B, control logic.
“internal variable control.” The jacketed stirred-tank heater example of Haskins and Sliepcevich is illustrated in Figure 6a. The exit temperature YC is controlled by manipulating the feed rate XM, with the tank wall temperature YS being utilized as an internal measure of the coolant rate disturbance Xu, The control logic, shown in Figure 6b, is feasible for process structures in which the secondary variable Y.~i can be assumed to be only on the disturbance transmission path X U YCand not on the primary control path X M Yc. The secondary controller FS is serving as a partial feedforward disturbance compensator for the primary loop and the design of FS is based on the invariance principle.
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The independent primary loop design is based on conventional single-loop methods. Note that the control functions are combined in an additive manner, rather than being cascaded. This design requires a reasonably valid model and knowledge of the secondary variable setpoint. Discussion Several characteristics of these secondary-variable contvd methods can-be summarized. 1. The secondary loop tends to provide some kind of feedforward control role relative to the primary feedback loop. For example, consider a feed disturbance entering the light-end distillation example. If the temperature sensor were located near the feed tray, the secondary controller would serve nearly as a feedforward controller. If the sensor is on the top tray, the role would be close to that of the primary feedback controller. Intermediate tray locations, which practically are more useful, provide a partial feedforward role. 2. The process model can have a significant effect on the control logic selected for the design. For example, the light-end distillation process model shown as the solid lines of Figure 3b can be reduced to the process model used for series cascade, parallel cascade, and internal-variahle control by neglecting, in turn, the XM Yc, YS Y( , or X ~ I Y. paths. 3 . The series, parallel, and series-parallel cascade control configurations all have the same external control structure. However, the differences in the process model can significantly affect the analytical control design. 4 . The cascade configurations, when compared with the additive combinations, probably are less sensitive to process model errors. While the analytical feedback design of cascade systems can be cumbersome, it is often possible t o utilize simple controller forms and semiempirical tuning uethods to obtain a practical result. 5. The additive controller configurations offer the possibility of independent, serial design procedures using the invariance principle for the secondary controller and single loop feedback methods for the primary loop. The disadvantages of these additive methods are sensitivity to process model error, the need for a secondary setpoint input. and controller realizability limitations.
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This study has highlighted the role of structure in recent secondary control configurations. While all of the methods use additional process measurements to augment the primary feedback control, the logic and design can be very different. As a result, effective secondary control design requires careful analysis and thorough quantitative methodology .
Nomenclature F = primarycontroller FS = secondary controller G = transfer function relating indicated variaJles X U = disturbance input X M = manipulative input YC = controlled output YS = secondary variable E = estimator algorithm YcSET= controlled output setpoint YsSET= secondary variable setpoint k = subscript denoting the sampling instant YCE= estimate of YC Literature Cited Clay, R. M.. Fournier, C. D., "Cascade vs. Single-Loop Control of Processes With Dead Time," Proceedings, 1973 Joint Automatic Control Conference, 363, IEEE, New York, N. Y., 1973. Franks, R. G., Worley, C. W., Ind. Eng. Chem., 48, 1074 (1956). Gollins, N. W., ControIEng., 3, 7, (1956). Greenfield, G. G., Ward, T. J., Ind. Eng. Chem., Fundam., 6, 564 (1967). Harriott, P., "Process Control," pp 154-166, McGraw-Hill, New York, N. Y., 1964. Haskins. D. E., Sliepcevich. C. M., Ind. Eng. Chem., Fundam., 4, 241 (1965). Koppel, L. B., "Introduction to Control Theory With Applications To Process Control," pp 25-27, Prentice-Hall, Englewood Cliffs, N. J., 1968. Koppel, L. B., Kamrnan, D. T., Woodward, J . L., Ind. Eng. Chem., Fundam., 9, 198 (1970). Luyben, W. L., Ind. Eng. Chem., Fundam., 12,463 (1973). Perlmutter, D. D., "Introduction to Chemical Process Control," pp 155-156, Wiley, NewYork, N. Y., 1965. Pollard, A,, "Process Control," pp 351-352, American Elsevier, New York, N. Y., 1971. Shinskey, F. G., "Process Control Systems." pp 154-160, McGraw-Hill, New York, N. Y., 1967. Weber, R.,Brosilow, C., AlChEJ., 18, 614 (1972). Wills, D. M., "Cascade Control Applications & Hardware," Bull. TX-119-1 Minneapolis-Honeywell Co., Philadelphia, Pa., 1960. Yang, C. H., Ward, T. J., Ind. Eng. Chem., Fundam., 13, 160 (1974). Ziegler, J. G., "Cascade Control Systems," Symposium on Instrumentation for the Process Industries, Texas A&M, College Station, Texas, 1954. Received for review October 15, 1974 Accepted M a r c h 3, 1975
