Fed-batch trajectory optimal control problems with cubic splines

Abstract

Optimal control problems in fed-batch fermentation processes are often described by sets of nonlinear differential and algebraic equations. The presence of constraints in the state and control variables presents an additional challenge for the optimization of these processes. In the feed trajectory planning optimization problem the tradicional approach consists in computing a linear spline that best fits the optimal feed trajectory. The resulting trajectory is thereof non-differentiable and problems can arise in its practical implementation, resulting in a possible discrepancy between the simulated and real performance. In this paper we develop a technique to obtain a cubic spline for the approximate trajectory, leading to a smooth approximation function. We provide numerical results for a set of case studies where the best trajectory approximation and initial dynamic system conditions are computed. The AMPL modeling language, CVODE ordinary differential equations solver and a particle swarm algorithm were used to obtain the numerical results.