Risk-sensitive safety specifications for stochastic systems using Conditional Value-at-Risk

TL;DR

This paper introduces a risk-sensitive safety analysis method for stochastic systems using Conditional Value-at-Risk, allowing a tunable safety level that balances worst-case and average-case perspectives.

Contribution

It defines a new risk-sensitive safe set based on CVaR, and provides an approximation method using CVaR-Markov Decision Processes for linear and non-linear systems.

Findings

The CVaR-based approach effectively balances safety and risk.

An approximate solution can be obtained via value iteration.

The method is applicable to complex, real-world systems like stormwater management.

Abstract

This paper proposes a safety analysis method that facilitates a tunable balance between the worst-case and risk-neutral perspectives. First, we define a risk-sensitive safe set to specify the degree of safety attained by a stochastic system. This set is defined as a sublevel set of the solution to an optimal control problem that is expressed using the Conditional Value-at-Risk (CVaR) measure. This problem does not satisfy Bellman's Principle, thus our next contribution is to show how risk-sensitive safe sets can be under-approximated by the solution to a CVaR-Markov Decision Process. We adopt an existing value iteration algorithm to find an approximate solution to the reduced problem for a class of linear systems. Then, we develop a realistic numerical example of a stormwater system to show that this approach can be applied to non-linear systems. Finally, we compare the CVaR criterion to the exponential disutility criterion. The latter allocates control effort evenly across the cost distribution to reduce variance, while the CVaR criterion focuses control effort on a given worst-case quantile--where it matters most for safety.