Suboptimal feedback control of distributed systems: Part II. Discussion of test systems and numerical results

Abstract

Q Q,N while the LGL method is restricted to systems linear in the control.One definite advantage held by the instantaneous minimization method is the fact that the actual performance functional kernel is used, whereas in a Lyapunov scheme the Lyapunov functional may bear little relation to the actual performance functional, thus degrading the quality of the resulting suboptimal control. Part II. Discussion of Test Systems and Numerical ResultsThe suboptimal control algorithms developed in Part I are tested on four example problems to obtain some indication of their performance. Results are presented and compared with results obtained from the application J. GREGORY VERMEYCHUKof classical open-loop optimal control synthesis algorithms to the same test and LEON LAPIDUS problems. Department of Chemical EngineeringThe suboptimal feedback control algorithms are found to give acceptprinceton university, princeton, J, 08540 able performance on both linear and nonlinear systems at the expense of very little computational effort. SCOPEThe performance of the suboptimal control algorithms developed in Part I of this work is to be evaluated by direct comparison of the effect of suboptimal control to the effect . _ of classically optimal control upon four specific These four classes encompass a great number of the distributed systems encountered in engineering practice and are thus sufficient to provide the indications of performance we desire. test problems.The basic test systems are a linear parabolic system, a linear first-order hyperbolic system, a linear parabolic It should be noted that a rigorously optimal control for a linear system with a quadratic performance functional system with variable coefficients, and a nonlinear para-may be obtained in feedback form but O~Y at the cost of d l i c system. extensive computation. No method for the computation ofConsider the system depicted in Figure 1. The sheet, entering the furnace at velocity a with temperature ug, is to be heated to an exit temperature T. The temperature inside the furnace is u (t), and the heat trans-

New methods for suboptimal feedback control of parabolic systems

“…The classical optimal control serves as a standard of comparison. The scheme based on the minimization of d, presented by Vermeychuk and Lapidus (1973) is also applied to the example for purposes of comparison.…”

“…In this as in previous work (Vermeychuk and Lapidus (1973), we revise the criteria for optimality in such a way as to allow the generation of the control in feedback form without excessive sacrifice of performance.…”

“…Performance of two of these techniques is found to compare favorably with that obtained by use of a rigorous open-loop optimal control computed by the gradient method. It is also shown for linear systems having quadratic performance criteria that the new algorithms are superior to the technique presented by Vermeychuk and Lapidus (1973) in cases where state deviation has a weighting coefficient much greater than that for control energy. …”