Biblio

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).

Stormwater infrastructure is required to safely manage uncertain precipitation events of varying intensity, while protecting natural ecosystems, under restricted financial budgets. In practice, candidate designs for stormwater detention or retention systems are commonly evaluated assuming that a given system operates independently from nearby systems and is initially empty prior to an extreme storm event. In recent work, we demonstrate the use of a control-theoretic method, called reachability analysis, to provide a more realistic design-phase indicator of system performance [1]. In particular, reachability analysis predicts the response of a dynamically-coupled stormwater storage network to a deterministic storm event under a wide range of initial conditions simultaneously [1]. The outcomes of this analysis can be viewed as measures of system robustness that inform the evaluation of safety-critical design choices [1]. Here we discuss how to extend the recent work to incorporate the stochastic nature of surface runoff. We represent surface runoff as a random disturbance to a dynamic system model of a stormwater storage network. Using a probability distribution of surface runoff derived from a Monte Carlo method, we apply an existing algorithm [2] for stochastic reachability analysis to the problem of designing robust stormwater storage systems. We discuss particular advantages and disadvantages of using stochastic reachability analysis, deterministic reachability analysis, or random sampling to assess system robustness.