Dynamics and Optimal Control of State-Triggered Affine Hybrid Systems

TL;DR

This paper investigates the dynamics and control of linear and affine hybrid systems with state-triggered resets, highlighting pathologies like Zeno behavior and proposing a hybrid Pontryagin maximum principle for optimal control.

Contribution

It introduces a comprehensive analysis of hybrid system trajectories, explores the Pontryagin maximum principle for affine systems, and discusses optimal control amidst complex behaviors like Zeno and beating.

Findings

Linear spatially-triggered systems exhibit beating and blocking trajectories.

Affine systems can include Zeno trajectories.

The hybrid Pontryagin maximum principle is developed for affine hybrid systems.

Abstract

A study of the dynamics and control for linear and affine hybrid systems subjected to either temporally- or spatially-triggered resets is presented. Hybrid trajectories are capable of degeneracies not found in continuous-time systems namely beating, blocking, and Zeno. These pathologies are commonly avoided by enforcing a lower bound on the time between events. While this constraint is straightforward to implement for temporally-triggered resets, it is impossible to do so for spatially-triggered systems. In particular, linear spatially-triggered hybrid systems always posses trajectories that are beating and blocking while affine systems may also include Zeno trajectories. The hybrid Pontryagin maximum principle is studied in the context of affine hybrid systems. The existence/uniqueness of the induced co-state jump conditions is studied which introduces the notion of strongly and weakly actuated resets. Finally, optimal control in the context of beating and Zeno is discussed. This work concludes with numerical examples.