Optimal Feedback Control of Batch Reactors with a State Inequality Constraint and Free Terminal Time

h this paper we derive optimal state feedback laws for end-point optimization of a dynamic system where the final time is free and the system has a scalar inequality constraint. The existence of a singular region as well as the nuture of the state feedback law (static or dynamic) is completely characterized in terms of the system dynamics. Explicit synthesis formulae for the state feedback laws are presented. Once the state feedback laws for end-point optimization have been derived, issues on how these laws can be implemented as part of a closed-loop scheme are discussed. As illustrative examples of application of the proposed methodology, several end-point optimization problems in batch chemical reactors are considered. INTRODUCTION Batch and semi-batch processes are of great importance to the chemical industry. A wide variety of speciality chemicals such as antibiotics and polymers are produced in batch reactors. Since batch reactors produce low-volume, high value products, optimal operation is very important. In a previous paper (Palanki et al., 1!293), we had synthesized optimal feedback laws for end-point optimization of batch processes with fixed final time. In this paper we will study the general problem of synthesizing optimal state feedback laws which guarantee optimality when the system has a state inequality constraint and the final time of the batch is left free. State inequality constraints are usually physical constraints to the system. For example, in a semibatch reactor one can feed only a finite amount of substrate due to a volume constraint on the reactor. Similarly, one may not want to operate a reactor beyond a certain maximum temperature for safety reasons. In a fed-batch bioreactor there could be constraints on the cell mass concentration (beyond which oxygen transfer is limited) or the substrate concentration (beyond which undesirable side reactions occur). Due to batch-to-batch variation in yield, the final time may not be fixed a priori. The reactor operation is stopped when the optimum yield is achieved. Thus, it is important to consider yield optimization problems with state inequality constraints and free terminal time. The purpose of this paper is to develop state feedback laws for such optimization problems. +Present address: Department of Chemical Engineering, Florida A & M University/Florida State University, College of Engineering, Tallahassee, FL 32316, U.S.A. Author to whom correspondence should be addressed. FORMULATION OF THE END-POINT OPTlMIZATION PROBLEM: THE CLASSICAL OPTIMAL CONTROL PERSPECTIVE The end-point optimization problem can be math ematically formulated as follows: Minimize the performance index, J = W(tl), C/) (1 subject to the dynamics i =.f(x) + 57(X)% 0 d t B tl Gin < u G u,.. and the scalar state inequality constraint: C(x) < 0. (3) In this formulation II is the scalar manipulated input bounded by umin and u,,., x is the n-vector of states, t/ is the free final time, f(x) and g(x) are smooth vector functions and I#+) is a smooth scalar function such that g(x) # coo.. .O] a4 dx#[OO...O] for all x. The part of the optimal solution where the state constraint is hit (C(x) = 0) is called the constrained arc or the boundary arc. The part of the solution where C(x) < 0 is called the unconstrained arc or the interior arc. Necessary conditions for optimality on an unconstrained arc By Pontryagin’s Principle, the minimization problem (1) is equivalent to minimizing the Hamiltonian: H(x, 1, u) = nTf(x) + ITg(x)u (5)