Model Predictive Control: Shortcomings and Resolutions

Abstract

This thesis is concerned with model predictive control (MPC); a method that approximates solutions of an infinite-horizon optimal control problem by considering a sequence of finite-horizon optimal control problems implemented in a receding horizon fashion. In the standard scheme, one control input is implemented in each iteration, and a new initial state is set for the next finite-horizon. One critical issue within the MPC setup is guaranteeing stability, and much of the literature on the topic is devoted to this, mainly by using classical tools from Lyapunov theory. Such results often rely on suitable terminal ingredients, whose design significantly impacts the performance of MPC. As we explore in this thesis, the issue mentioned is more critical than only performance, and leads to major shortcomings. The main objective of this work is to mathematically describe the source of some of these issues, and provide some resolutions. In particular, we present a novel MPC scheme with relaxed stability criteria, based on generalized control Lyapunov functions. Most notably, this scheme allows for implementing a flexible number of control inputs in each iteration, in a computationally attractive manner, while guaranteeing recursive feasibility and stability. The advantages of our flexible-step implementation are demonstrated on nonholonomic systems, switched systems and lastly, in the setting of adaptive control. Specifically, we provide a systematic method for constructing generalized control Lyapunov functions for the novel MPC scheme in the case of linear systems. When the true linear system is unknown, generating terminal conditions is not possible. In fact, even when the estimation of the unknown system matrices is done through solving a least-squares problem, existing results that guarantee stabilizing controls rely on unnatural choices of terminal costs. We present an extension of our MPC scheme to the unknown setting and show convergence to the origin of the closed-loop system.