(340b) A Mixed-Integer Dynamic Optimization Approach for Singular Optimal Control

Singular optimal control problems are frequent in process engineering applications such as: batch reactor control, dynamic optimization of batch processes  and the start-up of chemical reactors and distillation columns. From a mathematical point of view, singular optimal control problems arise in optimization formulations where the manipulated variable appears linearly both in the dynamic model and in the performance index leading to a situation where no enough information is available for the computation of the optimal solution.  Several numerical strategies have been suggested for handling singular optimal control problems based on: (a) solving the singular control problem as a series of non-singular control problems and (b) the use of Lie algebra for dealing with high-index DAE problems. However, both solution strategies become cumbersome for large-scale problems. Recently [1] an approach, based in recognizing that singular optimal control problems are ill-conditioned systems, has been proposed that attempts to regularize the ill-conditioned system.  The problem is regularized trough the construction of a simple objective function that attempts to enforce continuity conditions on the derivative of the Hamiltonian while monitoring certain error residuals.  In this work we extend this regularization approach to deal with switches and singular arcs where the system can be ill-conditioned in certain parts of the solution domain both well behaved in other parts of the control trajectory.  To switch between ill and well conditioned models we use binary variables leading to a mixed-integer dynamic optimization (MIDO) problem where disjunctions allow us to take into account only the part of the model that applies instead of the full model. In this a way a robust optimization formulation is obtained since only the parts of the formulation that apply need to be converged. For solving the dynamic optimization problem we use the transcription approach. The MIDO singular optimal control formulation was tested using several problems such as catalyst mixing  and tubular reactors.