A Sum-of-Squares-Based Procedure to Approximate the Pontryagin Difference of Semialgebraic Sets

TL;DR

This paper introduces a sum-of-squares programming method to approximate the Pontryagin difference between semialgebraic sets, facilitating applications in control theory.

Contribution

It presents a novel SOS-based procedure to compute inner approximations of the Pontryagin difference for semialgebraic sets, advancing set-theoretic control methods.

Findings

The method successfully computes inner approximations in computational examples.

The approach is applicable to robust model predictive control.

It demonstrates the effectiveness of SOS programming in set operations.

Abstract

The P-difference between two sets $\mathcal{A}$ and $\mathcal{B}$ is the set of all points, $\mathcal{C}$, such that the addition of $\mathcal{B}$ to any of the points in $\mathcal{C}$ is contained in $\mathcal{A}$. Such a set difference plays an important role in robust model predictive control and in set-theoretic control. In the paper we demonstrate that an inner approximation of the P-difference between two semialgebraic sets can be computed using the Sums of Squares Programming, and we illustrate the procedure using several computational examples.