An effective computational algorithm for suboptimal singular and/or bang‐bang control II. Applications to nonlinear lumped and distributed systems

Abstract

kl kZ a pure catalyst which catalyzes the reaction A Ft B only.The value of performance index, that is, the mole fraction of the substance C present in the mixture at the reactor exit at t = 1, with the pure bang-bang policy used is 0.047918. Although this policy is only a suboptimal one, it compares quite favorably with the optimal singular solution obtained by Jackson (1968), who gives a value of performance index of only 0.3% smaller, that is, 0.048065. 1966), a two-stage continuous stirred-tank reactor (CSTR) system (Siebenthal and Aris, 1964;Edgar and Lapidus, 1972b; Luus, 1974~1, 1974b, and a six-plate gas absorber with nonlinear gas-liquid equilibrium relationship (Lapidus and Luus, 1967;Weber and Lapidus, 1971). The determination of the optimal catalyst activity distribution policy in a distributed parameter, nonisothermal tubular reactor with radial heat and mass diffusion is also studied. These four test problems represent the typical chemical engineering optimal control problems with high state dimensionality, extreme nonlinearity, and multiple controls. The practical implications of the computational results are discussed, and the comparisons of the proposed algorithm with several existing techniques are given. CONCLUSIONS AND SIGNIFICANCEFrom examination of the numerical and graphical results for the chosen test problems presented in this paper, it can be concluded that the application of proposed algorithm based on the piecewise maximization of the Hamiltonian and a limiting process utilizing a penalty function of the control variables yields very acceptable, suboptimal singular and/or bang-bang control solutions with little expenditure of computer storage and computa-

Time-Optimal Control by Iterative Dynamic Programming

“…After some preliminary runs using P = 5 time stages, we used for our first run a multipass method of 30 iterations each, the number of state grid points N = 5, and the number of allowable values for control and stage length R = 15. As initial time stage length we chose v(k)i0> = 0.4 with an initial region size s(k)<0) = 0.1, for k = 1, 2,..., P. Rapid convergence of the weighted performance index to a value of 0.3296 X 10-4 was obtained as can be seen in The violation of the final state constraints are negligible, and the improvement over the minimum time of 2.8 reported by Nishida et al (1976) is quite significant. As is shown in Figure 7 the convergence to the optimum was obtained in only 5 IDP passes.…”

“…Example 3: Startup of an Autothermic Reaction System Let us consider the optimization of the start-up procedure for an autothermic reaction where a first order reversible reaction A *=2 B is occurring. This nonlinear system was first presented by Jackson (1966) and used for optimal control studies by Nishida et al (1976). It is interesting to investigate this system because of the multitude of solutions reported by Jackson (1966) and by Nishida et al (1976).…”

Evaluation of Penalty Functions for Optimal Control

“…This control policy compares favorably to the policy reported by Luus (1974a), but the control U1 increases more gradually than the policies reported by Rosen and Luus (1991), and by Bojkov and Luus (1994). However, the optimal control policy is substantially different from the control policies obtained by Edgar and Lapidus (1972), Luus (1974b) , and Nishida et al (1976). The trajectories of the first four state variables are given in Figure 4.8.…”

The application of an on-line model error tolerant sub-optimal controller to a chemical reactor

“…However, the common, and fundamentally limiting, factor of these techniques is the accuracy, both parametrically and structurally, of the associated mathematical model. Indeed, many authors (Foss, 1973;Sargent, 1975;Nishida et al, 1976;Bristol, 1975) have commented on the deficiencies o f such control schemes, particularly in the case of chemical systems which are notorious for the complexity of their underlying physico-chemical mechanisms, for their nonlinearities and for the number of, and interaction between, process variables. Evidently, then, in such systems the process model is likely to be poorly understood, and inaccurate.…”