Risk-Aware Stability of Discrete-Time Systems

TL;DR

This paper introduces a comprehensive stability framework for stochastic discrete-time systems using risk functionals, enabling nuanced analysis of system variability beyond traditional average-based methods.

Contribution

It develops a generalized risk-aware stability theory for nonlinear and linear systems using diverse risk measures, including coherent and mean-conditional-variance functionals.

Findings

New risk-aware stability conditions for linear systems.

Revealed noise-to-state stability properties reflecting risk measures.

Illustrated robustness and control design implications.

Abstract

We develop a generalized stability framework for stochastic discrete-time systems, where the generality pertains to the ways in which the distribution of the state energy can be characterized. We use tools from finance and operations research called risk functionals (i.e., risk measures) to facilitate diverse distributional characterizations. In contrast, classical stochastic stability notions characterize the state energy on average or in probability, which can obscure the variability of stochastic system behavior. After drawing connections between various risk-aware stability concepts for nonlinear systems, we specialize to linear systems and derive sufficient conditions for the satisfaction of some risk-aware stability properties. These results pertain to real-valued coherent risk functionals and a mean-conditional-variance functional. The results reveal novel noise-to-state stability properties, which assess disturbances in ways that reflect the chosen measure of risk. We illustrate the theory through examples about robustness, parameter choices, and state-feedback controllers.