Forward Invariance of Sets for Hybrid Dynamical Systems (Part II)

TL;DR

This paper develops tools for designing control laws that ensure hybrid dynamical systems remain within a set despite disturbances, extending previous analysis with synthesis methods and practical examples.

Contribution

It introduces sufficient conditions and control Lyapunov functions for robust forward invariance, along with methods for designing continuous state-feedback laws.

Findings

Conditions for robust controlled forward invariance are established.

Synthesis methods for state-feedback laws are proposed.

Applications to bouncing ball and robotic manipulator demonstrate effectiveness.

Abstract

This article presents tools for the design of control laws inducing robust controlled forward invariance of a set for hybrid dynamical systems modeled as hybrid inclusions. A set has the robust controlled forward invariance property via a control law if every solution to the closed-loop system that starts from the set stays within the set for all future time, regardless of the value of the disturbances. Building on the first part of this article, which focuses on analysis (Chai and Sanfelice, 2019), in this article, sufficient conditions for generic sets to enjoy such a property are proposed. To construct invariance inducing state-feedback laws, the notion of robust control Lyapunov function for forward invariance is defined. The proposed synthesis results rely on set-valued maps that include all admissible control inputs that keep closed-loop solutions within the set of interest. Results guaranteeing the existence of such state-feedback laws are also presented. Moreover, conditions for the design of continuous state-feedback laws with minimum point-wise norm are provided. Major results are illustrated throughout this article in a constrained bouncing ball system and a robotic manipulator application.