Safety Verification of Stochastic Systems: A Repetitive Scenario Approach

Abstract

In this paper, we develop a data-driven approach for the safety verification of stochastic systems with unknown dynamics. First, we use a notion of barrier certificates in order to cast the safety verification as a robust convex program (RCP). Solving this optimization program is difficult because the model of the stochastic system, which is unknown, appears in one of the constraints. Therefore, we construct a scenario convex program (SCP) by collecting a number of samples from trajectories of the system. Then, we develop a repetition-based scenario framework to provide an out-of-sample performance guarantee for the constructed SCP. In particular, we iteratively solve an SCP for a given number of samples, and then check its feasibility using a certain number of new samples after substituting the optimal decision variables from solving the SCP. We continue the iterations until a desired violation error is achieved. Eventually, a safety condition is checked on top of the feasibility problem. If the safety condition is fulfilled, then we can provide a lower bound on the probability of safety satisfaction for the original stochastic system by leveraging the optimal solution of the successful iteration. We illustrate the effectiveness of the proposed results through a two-tank system case study, where the safety objective is to ensure that the water levels in both tanks are within some safe zones.

Probabilities Are Not Enough: Formal Controller Synthesis for Stochastic Dynamical Models with Epistemic Uncertainty

“…By contrast to our setting, these approaches consider models with non-stochastic dynamics. A few recent exceptions exist (Salamati and Zamani 2022;Lavaei et al 2023), albeit requiring more strict assumptions (e.g., discrete input sets) than our model-based approach.…”

Probabilities Are Not Enough: Formal Controller Synthesis for Stochastic Dynamical Models with Epistemic Uncertainty

“…By contrast to our setting, these approaches consider models with nonstochastic dynamics. A few recent exceptions also consider aleatoric uncertainty (Salamati and Zamani 2022;Lavaei et al 2022), but these approaches require more strict assumptions (e.g., discrete input sets) than our model-based approach.…”

Recovery Type a Posteriori Error Estimation of an Adaptive Finite Element Method for Cahn--Hilliard Equation

“…Additionally, in [24], safety verification for stochastic systems has been investigated, and the results have been extended to the synthesis problem in [25]. Furthermore, the wait-and-judge approach [26] and the repetitive approach [27] have also been used to improve the sample efficiency of the safety verification problem.…”

“…• We first collect N data to formulate the scenario convex program in order to obtain a solution; • Then, we collect N 0 independent validation data as posterior information to test the obtained solution such that the confidence bound can be further improved. In contrast to [26] and [27] for the verification problem, where the information of support constraints number and the information of violation frequency in validation data are used independently, here we not only consider the synthesis problem, but also use these two posterior information jointly. Therefore, our main result is an overall performance bound that connects all three information: the original sample data, support constraints, and validation data, in a uniform manner.…”