Linear model predictive control based on polyhedral control Lyapunov functions: theory and applications

TL;DR

This paper introduces a novel linear model predictive control approach using polyhedral control Lyapunov functions to maximize the domain of attraction, ensure stability, and allow formulation as quadratic programs, with demonstrated robustness and practical benefits.

Contribution

It develops new MPC formulations based on PCLFs that guarantee the entire controllable set as the domain of attraction, improving stability and robustness over traditional quadratic Lyapunov approaches.

Findings

Guarantees maximal domain of attraction as the entire controllable set.

Ensures closed-loop stability with PCLF-based terminal conditions.

Formulates control schemes as quadratic programming problems.

Abstract

Polyhedral control Lyapunov functions (PCLFs) are exploited in finite-horizon linear model predictive control formulations in order to guarantee the maximal domain of attraction (DoA), in contrast to traditional formulations based on quadratic control Lyapunov functions. In particular, the terminal region is chosen as the largest DoA, namely the entire controllable set, which is parametrized by a level set of a suitable PCLF. Closed-loop stability of the origin is guaranteed either by using an "inflated" PCLF as terminal cost or by adding a contraction constraint for the PCLF evaluated at the current state. Two variants of the formulation based on the inflated PCLF terminal cost are also presented. In all proposed formulations, the guaranteed DoA is always the entire controllable set, independently of the chosen finite horizon. Closed-loop inherent robustness with respect to arbitrary, sufficiently small perturbations is also established. Moreover, all proposed schemes can be formulated as Quadratic Programming problems. Numerical examples show the main benefits and achievements of the proposed formulations.