The concept of the optimal synthesis of dynamic process systems with uncertain parameters is introduced. A structure parameter approach is used to theoretically derive the necessary condition for the optimal performance system structure, and an effective algorithm for implementing the synthesis method is presented. The results are applied to the optimal synthesis of a reactor-separator system for the dynamic start-up of two reaction systems. SCOPESignificant progress has been made in the past few years on the synthesis of chemical process systems. The determination of the optimal type and design of processing units, as well as their optimal interconnections within a process flowsheet, can now be done with some success by many synthesis techniques. A good description of the useful techniques has been given by Hendry, Rudd, and Seader (1973), and a latest review of the state of the art of process synthesis can be found in a report by Mah (1975).One serious limitation with most recent studies of process synthesis is that they are mainly concerned with the synthesis of steady state process systems. A wide variety of important chemical process operations, such as the start-up or shutdown of equipment, the batch or semibatch operation of processing units, etc., however, fall into the category of dynamic process systems. Furthermore, a large number of recent studies in chemical engineering have shown that certain types of processing units, such as the parametric pumping and the cycling zone absorption, etc., can give better performance by periodic (dynamic) operations. Therefore, it is of practical interest in many situations to consider the problems of dynamics and control at the stage of process synthesis (Mah, 1975;Henley and Motard, 1972) rather than merely synthesizing the process by assuming a steady state operation and then compensating the effect of dynamics by means of control after the synthesis is completed. Unfortunately, except for the recent studies by the authors (Nishida and Ichikawa, 1975; Nishida, Liu, and Ichikawa, 1975a), this important aspect of including the dynamics and control considerations in the process synthesis problem has been neglected in the literature. Another important aspect of process synthesis which has not been given sufficient attention is the effect of uncertainty in process parameters. In the literature, only the optimal synthesis of steady state process systems with uncertainty has been studied recently (Nishida, Tazaki, and Ichikawa, 1974).The objective of this work is to extend the recent results on the optimal synthesis of dynamic process systems (Nishida and Ichikawa, 1975; Nishida, Liu, and Ichikawa, 1975n) to include the effect of process parameter uncertainty. The problem considered may be stated briefly as: "Given the process dynamics, how to synthesize the process flow sheet and to specify the process design and control variables subject to the uncertainty in process parameters so as to achieve an optimal dynamic operation of the process?" The approach taken is to co...

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-

Theoretical Developments and Applications to Linear Lumped SystemsA new and simple computational algorithm for obtaining suboptimal singular and/or bang-bang solutions of typical lumped and distributed parameter control problems is proposed. The algorithm is based on the piecewise maximization of the Hamiltonian and a limiting process utilizing a penalty function of the control variables. Theoretical developments and computational applications of the algorithm to several linear lumped parameter control problems are presented. Extensions and applications of the algorithm to nonlinear and distributed parameter systems are given in Part II. SCOPEThe application of optimal control theory to dynamic systems of interest to the chemical process industry is often hampered by the existence of constraints, in addition to the common characteristics of these systems of having high state dimensionality, extreme nonlinearity, and multiple controls. The maximum principle formulation (Pontryagin et al., 1962) of such optimization problems leads to a nonlinear, two-point, boundary value prob-Correspondence concerning this paper should be addressed to Y. A.

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