Approximately symbolic models for a class of continuous-time nonlinear systems

TL;DR

This paper introduces a new approach to approximate symbolic modeling for continuous-time nonlinear systems using only state-space abstraction, ensuring safety and stability without time sampling.

Contribution

It proposes a novel stability concept and a control interface linking continuous and discrete systems, enabling approximate simulation without time discretization.

Findings

The method guarantees safety through approximate simulation.

A new stability notion is introduced for continuous-time systems.

Simulation example demonstrates effectiveness.

Abstract

Discrete abstractions have become a standard approach to assist control synthesis under complex specifications. Most techniques for the construction of discrete abstractions are based on sampling of both the state and time spaces, which may not be able to guarantee safety for continuous-time systems. In this work, we aim at addressing this problem by considering only state-space abstraction. Firstly, we connect the continuous-time concrete system with its discrete (state-space) abstraction with a control interface. Then, a novel stability notion called controlled globally asymptotic/practical stability with respect to a set is proposed. It is shown that every system, under the condition that there exists an admissible control interface such that the augmented system (composed of the concrete system and its abstraction) can be made controlled globally practically stable with respect to the given set, is approximately simulated by its discrete abstraction. The effectiveness of the proposed results is illustrated by a simulation example.