Continuous Optimization for Control of Hybrid Systems with Hysteresis via Time-Freezing

TL;DR

This paper introduces a novel time-freezing reformulation that transforms hybrid systems with hysteresis into piecewise smooth systems, enabling high-accuracy numerical optimal control using nonlinear optimization techniques without integer variables.

Contribution

The paper presents a new reformulation method for hybrid systems with hysteresis, facilitating the use of nonlinear optimization and existing tools for control problems.

Findings

Reformulation enables the application of Filippov system analysis tools.

Allows high-accuracy control without mixed-integer programming.

Demonstrates effectiveness through a time optimal control example.

Abstract

This article regards numerical optimal control of a class of hybrid systems with hysteresis using solely techniques from nonlinear optimization, without any integer variables. Hysteresis is a rate independent memory effect which often results in severe nonsmoothness in the dynamics. These systems are not simply Piecewise Smooth Systems (PSS); they are a more complicated form of hybrid systems. We introduce a time-freezing reformulation which transforms these systems into a PSS. From the theoretical side, this reformulation opens the door to study systems with hysteresis via the rich tools developed for Filippov systems. From the practical side, it enables the use of the recently developed Finite Elements with Switch Detection [Nurkanovic et al., 2022], which makes high accuracy numerical optimal control of hybrid systems with hysteresis possible. We provide a time optimal control problem example and compare our approach to mixed-integer formulations from the literature.