Global exponential stabilization of a force- and torque-actuated unicycle by flexible-step MPC

TL;DR

This paper introduces a novel flexible-step MPC approach with state-dependent weights for globally exponentially stabilizing a force- and torque-controlled unicycle model, extending to general nonlinear systems.

Contribution

It presents the first flexible-step MPC protocol with stability guaranteed by a control Lyapunov function, applicable beyond the unicycle example.

Findings

The method guarantees feasibility and stability with explicit parameter rules.

It achieves the best possible exponential stability for the second-order unicycle model.

Numerical simulations confirm effectiveness in continuous-time torque-actuated unicycle applications.

Abstract

We study the problem of global exponential stabilization of a force- and torque-controlled unicycle model in discrete time. To this end, we extend a recently introduced approach to model predictive control (MPC) in which a flexible number of inputs is implemented in every iteration. We present the first flexible-step MPC protocol with state-dependent weights for average descent. Notably, the proposed method relies neither on a suitable design of running or terminal cost functions nor on a suitable choice of terminal constraints. Instead, stability is guaranteed through a generalized discrete-time control Lyapunov function. We establish a new theoretical framework for global exponential stabilization of general nonlinear discrete-time control systems by flexible-step MPC. The obtained results go beyond the unicycle example. However, given the importance of the unicycle dynamics, we make that a focal point of our work. For the particular case of the dynamic (second-order) unicycle model, we show that global exponential stability cannot be attained in the classical sense, but in a slightly weaker sense. The proposed flexible-step MPC method is shown to induce the best possible notion of global exponential stability for this model. We provide explicit rules for the choice of parameters, which guarantee feasibility and global exponential stability. Our numerical simulations show that the discrete MPC method also works very well in applications to a continuous-time torque-actuated unicycle.