Numerical Methodes for Optimal Control and Model Predictive Control

Abstract

This thesis addresses, via numerical and optimisation methods, the control of nonlinear systems whose inputs or trajectories are subject to constraints. Nevertheless, we review and apply theoretical results, such as conditions of optimality, to characterise the optimal trajectory and to validate numerical results obtained using our proposed methods. We overview most used software packages for solving optimal control problems, including numerical solvers which invoke local search methods and interfaces with distinct features. A benchmark involving a differential drive robot with state constraints is presented in order to compare the performances of the solvers. We propose and develop an optimal control algorithm based on a direct method with adaptive refinement of the time–mesh. When using direct methods to solve nonlinear optimal control, regular time meshes having equidistant spacing are most frequently used. However, in some cases, these meshes cannot cope accurately with nonlinear behaviour and increasing uniformly the number of mesh nodes may lead to a more complex problem, resulting in an incoherent solution. We propose a new adaptive time–mesh refinement algorithm, considering different levels of refinement and several mesh refinement criteria. This technique is applied to solve an open– loop optimal control problem involving nonholonomic vehicles with state constraints, which is characterized by presenting strong nonlinearities and by having discontinuous controls, and a compartmental model for the implementation of a vaccination strategy. This algorithm leads to results with higher accuracy and yet with lower overall