Reference Dependent Invariant Sets: Sum of Squares Based Computation and Applications in Constrained Control

TL;DR

This paper introduces a systematic SOS-based method to compute reference dependent invariant sets for constrained systems, with applications in MPC and ERG schemes, demonstrated through simulations.

Contribution

It presents a novel polynomial and SDP-based approach for invariant set computation tailored to reference-dependent constraints.

Findings

Successfully computed invariant sets for example systems.

Enhanced MPC and ERG schemes with the proposed invariant set computation.

Demonstrated effectiveness through simulation results.

Abstract

The goal of this paper is to present a systematic method to compute reference dependent positively invariant sets for systems subject to constraints. To this end, we first characterize these sets as level sets of reference dependent Lyapunov functions. Based on this characterization and using Sum of Squares (SOS) theory, we provide a polynomial certificate for the existence of such sets. Subsequently, through some algebraic manipulations, we express this certificate in terms of a Semi-Definite Programming (SDP) problem which maximizes the size of the resulting reference dependent invariant sets. We then present the results of implementing the proposed method to an example system and propose some variations of the proposed method that may help in reducing the numerical issues of the method. Finally, the proposed method is employed in the Model Predictive Control (MPC) for Tracking scheme to compute the terminal set, and in the Explicit Reference Governor (ERG) scheme to compute the so-called Dynamic Safety Margin (DSM). The effectiveness of the proposed method in each of the schemes is demonstrated through a simulation study.