Real-Time Algorithms for Model Predictive Control of Hybrid Dynamical Systems

TL;DR

This paper introduces real-time hybrid MPC algorithms for nonlinear hybrid systems modeled as complementarity systems, addressing discontinuities and infeasibility issues in traditional approaches.

Contribution

It develops three hybrid MPC schemes solving quadratic programs with complementarity constraints, providing bounded approximation errors despite solution discontinuities.

Findings

Algorithms successfully demonstrated on robotic manipulation example.

Proved conditions for continuity and differentiability of MPCC solutions.

Addressed infeasibility of standard nonlinear programming in hybrid systems.

Abstract

Model predictive control (MPC) of hybrid dynamical systems is challenging because the associated optimization problem is nonsmooth and the resulting feedback law is discontinuous. This paper develops real-time MPC algorithms for nonlinear hybrid systems modeled as dynamical complementarity systems. The resulting optimal control problems are formulated as mathematical programs with complementarity constraints (MPCCs). We show that the solution map of parametric MPCCs is discontinuous, and that standard nonlinear-programming-based approaches may become infeasible when the hybrid system switches. To address this, we introduce three real-time hybrid MPC schemes whose feedback phase solves a quadratic program with complementarity constraints per sample, yielding local discontinuous piecewise affine approximations of the MPC feedback law. Moreover, we derive continuity and differentiability results for parametric MPCCs, and establish conditions under which the approximation error of our new hybrid MPC algorithms remains uniformly bounded despite solution discontinuities. The algorithms are demonstrated on a robotic manipulation example, where contact sequences are discovered online.