Chemical Engineering Science, Vol.51, No.3, 449-459, 1996
Online Optimization of Batch Processes with Nonlinear Manipulated Input
The classical approach to end-point optimization of batch processes leads to a two-point boundary value problem. The numerical solution of this two-point boundary value problem provides the time profile of the manipulated input which has to be implemented in an open-loop fashion. However, due to batch-to-batch variations in the system parameters and initial conditions (which is a typical characteristic of many commercial batch processes), the implementation of this open-loop profile can lead to suboptimal performance. Thus it is desirable to have a feedback which provides the optimal input profile. In a previous paper, we had synthesized optimal state feedback laws for the case where the state model is a linear function of the manipulated input. In this paper, optimal state feedback laws for end-point optimization of dynamic systems are derived where the state model is a nonlinear function of the manipulated input and the system states. Such problems arise when one has to calculate the optimal temperature or pH profiles in batch reactors. The necessary conditions for optimality are cast in terms of the system Lie Brackets and the adjoint states. An optimal state feedback law is derived which is independent of adjoint states. The nature of the optimal state feedback laws (static or dynamic) is characterized in terms of the system dynamics. As illustrative examples of application of the proposed methodology, two end-point optimization problems in batch chemical reactors are considered.
Keywords:FERMENTATION;REACTOR