Regulation of lumped periodic processes

Abstract

When a periodic process is disturbed, control action may be necessary to maintain satisfactory performance. Regulating controls to achieve this objective can be synthesized by solving a linear-quadratic optimization problem with periodic coefficients. Generalized proportional and proportional-integral feedback regulators developed in this manner are tested by simulating a cyclic reactor. Very satisfactory control is achieved for 10% step disturbances in reactor feed conditions although the results are sensitive to the choice of manipulated variables. J . 4 . SCOPESignificant recent research interest has been focused on possible advantages of intentional unsteady state processing Among the potential benefits of periodic operation are increased reactor selectivity (see Bailey, 1974) and enhanced separation (see, for example, Robinson and Engel, 1967;Wilhelm et al., 1968). All previous work, however, has considered only the question of periodic process design: how should some manipulated variable be oscillated to achieve optimum performance? Once a suitable design has been developed, another problem which we call periodic process regulation must be resolved: What control manipulations are necessary when the operating cyclic system is disturbed by an unexpected change in its environment? This paper presents one approach to this problem for systems described by ordinary differential equations, By adopting the objective of maintaining design conditions and assuming that deviations from these conditions are sufficiently small, regulating controls can be synthesized in feedback form by solving a time-varying linearquadratic optimization problem (see Edgar et al., 1973). Employing this approach, generalized proportional (P) and proportional-integral (PI) regulators are developed. A continuous-flow stirred tank reactor (CSTR) in which parallel reactions occur is employed to test these regulating controls. In the absence of disturbances, periodic variations in heat flux to the reactor provide greater selectivity for the desired product than can be obtained by steady state operation.Step disturbances in feed conditions can unfavorably alter the reactor's performance. The effectiveness of flow rate or heat flux manipulation in counteracting these upsets is investigated using simulation studies. CONCLUSIONS AND SIGNIFICANCEA generalized proportional-integral control employing heat flux as the regulating variable effectively returns the cyclic reaction system to its design specifications following 10% step decreases and increases in feed concentration and temperature. This can be seen by comparing the response with this controller, denoted PI-Heat, to the periodic fluctuations of the reference design (undisturbed) process in Figures 2, 4, and 5. Proportional control using heat flux regulates the reactor almost as well but requires more extreme fluctuations in the manipulated variable as shows that flow rate is far less effective than heat flux in maintaining reference design conditions. Without a regulating control these distu...