Continuous Abstraction of Nonlinear Systems using Sum-of-Squares Programming

TL;DR

This paper introduces a method using sum-of-squares programming to create low-dimensional abstractions of high-dimensional nonlinear systems, enabling effective control design with bounded error guarantees.

Contribution

It proposes a novel control design framework that combines sum-of-squares programming with abstraction techniques for nonlinear systems.

Findings

Successfully computes low-level controllers with bounded relative error.

Provides a systematic way to generate low-dimensional system abstractions.

Ensures boundedness of the system's deviation from its abstraction.

Abstract

We present a control design procedure for nonlinear control systems in which we represent a potentially high dimensional system with a low dimensional continuous-state abstraction. The abstraction generates a reference which the original system follows with a low level controller. We propose sum-of-squares programming as a tool to design this controller and to provide an upper bound on the relative error between the system and its abstraction. We compute the low level controller simultaneously with a simulation function that gives the boundedness guarantee for the relative error.