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A classic reachability problem for safety of dynamic systems is to compute the set of initial states from which the state trajectory is guaranteed to stay inside a given constraint set over a given time horizon. In this paper, we leverage existing theory of reachability analysis and risk measures to devise a risk-sensitive reachability approach for safety of stochastic dynamic systems under non-adversarial disturbances over a finite time horizon. Specifically, we first introduce the notion of a risk-sensitive safe set asa set of initial states from which the risk of large constraint violations can be reduced to a required level via a control policy, where risk is quantified using the Conditional Value-at-Risk(CVaR) measure. Second, we show how the computation of a risk-sensitive safe set can be reduced to the solution to a Markov Decision Process (MDP), where cost is assessed according to CVaR. Third, leveraging this reduction, we devise a tractable algorithm to approximate a risk-sensitive safe set and provide arguments about its correctness. Finally, we present a realistic example inspired from stormwater catchment design to demonstrate the utility of risk-sensitive reachability analysis. In particular, our approach allows a practitioner to tune the level of risk sensitivity from worst-case (which is typical for Hamilton-Jacobi reachability analysis) to risk-neutral (which is the case for stochastic reachability analysis). 

Effective stormwater management requires systems that operate safely and deliver improved environmental outcomes in a cost-effective manner. However, current design practices typically evaluate performance assuming that a given system starts empty and operates independently from nearby stormwater infrastructure. There is a conspicuous need for more realistic design-phase indicators of performance that consider a larger set of initial conditions and the effects of coupled dynamics. To this end, we apply a control-theoretic method, called reachability analysis, to produce a more objective measure of system robustness. We seek two primary contributions in this work. First, we demonstrate how the application of reachability analysis to a dynamic model of a stormwater network can characterize the set of initial conditions from which every element in the network can avoid overflowing under a given surface runoff signal of finite duration. This is, to the authors’ knowledge, the first published application of reachability analysis to stormwater systems. Our second contribution is to offer an interpretation of the outcomes of the proposed reachability analysis as a measure of system robustness that can provide useful information when making critical design decisions. We illustrate the effectiveness of this method in revealing the trade-offs of particular design choices relative to a desired level of robustness. 

A classic reachability problem for safety of dynamic systems is to compute the set of initial states from which the state trajectory is guaranteed to stay inside a given constraint set over a given time horizon. In this paper, we leverage existing theory of reachability analysis and risk measures to devise a risk-sensitive reachability approach for safety of stochastic dynamic systems under non-adversarial disturbances over a finite time horizon. Specifically, we first introduce the notion of a risk-sensitive safe set as a set of initial states from which the risk of large constraint violations can be reduced to a required level via a control policy, where risk is quantified using the Conditional Value-at-Risk (CVaR) measure. Second, we show how the computation of a risk-sensitive safe set can be reduced to the solution to a Markov Decision Process (MDP), where cost is assessed according to CVaR. Third, leveraging this reduction, we devise a tractable algorithm to approximate a risk-sensitive safe set, and provide theoretical arguments about its correctness. Finally, we present a realistic example inspired from stormwater catchment design to demonstrate the utility of risk-sensitive reachability analysis. In particular, our approach allows a practitioner to tune the level of risk sensitivity from worst-case (which is typical for Hamilton-Jacobi reachability analysis) to risk-neutral (which is the case for stochastic reachability analysis).