A First-Order Closed-loop Methodology for Nonlinear Optimal Control

This thesis is focused on state-of-art numerical optimization methods for nonlinear (discrete-time) optimal control. These challenging problems arise when dealing with complex tasks for autonomous systems (e.g. vehicles or robots) which require the generation of a trajectory that satisfies the system dynamics and, possibly, input and state constraints due to, e.g, actuator limits or safety region of operation. A general formulation is proposed that allows the implementation of different descent optimization algorithms on optimal control problems exploiting the beneficial effects of state feedback in terms of efficiency and stability. The main idea is the following: at each iteration a new (infeasible) state-input curve is conveniently updated by any descent method, e.g, gradient descent or Newton methods, then a nonlinear feedback controller maps the curve to a trajectory satisfying the dynamics. Thanks to its inherent flexibility, this strategy provides the opportunity to speed-up the resolution of optimization problems by conveniently choosing the descent method. This thesis proposes, for example, to exploit the Heavy-ball method to speed up the convergence. It is important to underline that this methodology enjoys recursive feasibility during the algorithm evolution, i.e. at each iteration a system trajectory is available. This feature is extremely important in real-time control schemes since it allows one to stop the algorithm at any iteration and yet have a (suboptimal) system trajectory. Furthermore, tasks which require the introduction of state and input constraints can be managed introducing an approximate barrier function which embeds the constraints within the cost function. The second main contribution of this thesis is an original Python toolbox developed in order to implement and compare different optimization methods. Moreover, thanks to a modular approach, with just few adjustments it is possible to change system, cost function and constraints.