Risk-Aware Stability, Ultimate Boundedness, and Positive Invariance

TL;DR

This paper develops risk-based definitions of stability, boundedness, and invariance for stochastic systems using CVaR, enabling focus on tail risks and designing controllers with guaranteed bounds.

Contribution

It introduces CVaR-based notions of stability and invariance for stochastic systems and derives event-triggered control strategies ensuring boundedness.

Findings

CVaR-based stability and invariance definitions for stochastic systems.

Event-triggered control strategies guaranteeing boundedness.

Numerical examples demonstrating the effectiveness of the proposed methods.

Abstract

This paper introduces the notions of stability, ultimate boundedness, and positive invariance for stochastic systems in the view of risk. More specifically, those notions are defined in terms of the worst-case Conditional Value-at-Risk (CVaR), which quantifies the worst-case conditional expectation of losses exceeding a certain threshold over a set of possible uncertainties. Those notions allow us to focus our attention on the tail behavior of stochastic systems in the analysis of dynamical systems and the design of controllers. Furthermore, some event-triggered control strategies that guarantee ultimate boundedness and positive invariance with specified bounds are derived using the obtained results and illustrated using numerical examples.