Enclosing the Sliding Surfaces of a Controlled Swing

TL;DR

This paper introduces an interval-based method to accurately approximate the sliding surface in hybrid systems with non-continuous controllers, helping to identify potential chattering effects that could damage the system.

Contribution

It presents a novel application of thick sets for outer approximation of sliding surfaces, accounting for uncertainties in hybrid system trajectories.

Findings

Efficient computation of outer approximations of sliding surfaces.

Application to verify a child's swing controller.

Identification of trajectories prone to chattering effects.

Abstract

When implementing a non-continuous controller for a cyber-physical system, it may happen that the evolution of the closed-loop system is not anymore piecewise differentiable along the trajectory, mainly due to conditional statements inside the controller. This may lead to some unwanted chattering effects than may damage the system. This behavior is difficult to observe even in simulation. In this paper, we propose an interval approach to characterize the sliding surface which corresponds to the set of all states such that the state trajectory may jump indefinitely between two distinct behaviors. We show that the recent notion of thick sets will allows us to compute efficiently an outer approximation of the sliding surface of a given class of hybrid system taking into account all set-membership uncertainties. An application to the verification of the controller of a child swing is considered to illustrate the principle of the approach.