Stochastic reachability of a target tube: Theory and computation

Vinod, Abraham P.; Oishi, Meeko

doi:10.1016/j.automatica.2020.109458

Cited by 16 publications

(7 citation statements)

References 30 publications

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“…Verification of Continuous-time Systems. Reachability analysis is a verification approach that provides safety guarantees for a given continuous dynamical system (Gurung et al 2019;Vinod and Oishi 2021). Most dynamical systems in safety-critical applications are highly nonlinear and uncertain in nature (Lechner et al 2020).…”

Section: Related Workmentioning

confidence: 99%

GoTube: Scalable Statistical Verification of Continuous-Depth Models

Gruenbacher

Lechner

Hasani

et al. 2022

AAAI

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We introduce a new statistical verification algorithm that formally quantifies the behavioral robustness of any time-continuous process formulated as a continuous-depth model. Our algorithm solves a set of global optimization (Go) problems over a given time horizon to construct a tight enclosure (Tube) of the set of all process executions starting from a ball of initial states. We call our algorithm GoTube. Through its construction, GoTube ensures that the bounding tube is conservative up to a desired probability and up to a desired tightness. GoTube is implemented in JAX and optimized to scale to complex continuous-depth neural network models. Compared to advanced reachability analysis tools for time-continuous neural networks, GoTube does not accumulate overapproximation errors between time steps and avoids the infamous wrapping effect inherent in symbolic techniques. We show that GoTube substantially outperforms state-of-the-art verification tools in terms of the size of the initial ball, speed, time-horizon, task completion, and scalability on a large set of experiments. GoTube is stable and sets the state-of-the-art in terms of its ability to scale to time horizons well beyond what has been previously possible.

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Section: Related Workmentioning

confidence: 99%

GoTube: Scalable Statistical Verification of Continuous-Depth Models

Gruenbacher

Lechner

Hasani

et al. 2022

AAAI

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“…In the literature, discrete-time Markov chains and stochastic differential equations (SDEs) are among the most commonly used models for stochastic processes. The p-reach avoid problem of the former over both finite time horizons and open time horizons has been studied in, e.g., [2], [11], [38], [41]. The quest for generalizations to continuous-time dynamical system models, especially SDEs, remains largely unanswered.…”

Section: Introductionmentioning

confidence: 99%

Reach-Avoid Analysis for Delay Differential Equations

Bai

Zhan

et al. 2021

2021 60th IEEE Conference on Decision and Control (CDC)

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In this paper we propose a novel semi-definite programming approach that solves reach-avoid problems over open (i.e., not bounded a priori) time horizons for dynamical systems modeled by polynomial stochastic differential equations. The reach-avoid problem in this paper is a probabilistic guarantee: we approximate from the inner a p-reach-avoid set, i.e., the set of initial states guaranteeing with probability larger than p that the system eventually enters a given target set while remaining inside a specified safe set till the target hit. Our approach begins with the construction of a bounded value function, whose strict p super-level set is equal to the p-reach-avoid set. This value function is then reduced to a twice continuously differentiable solution to a system of equations. The system of equations facilitates the construction of a semi-definite program using sum-ofsquares decomposition for multivariate polynomials and thus the transformation of nonconvex reach-avoid problems into a convex optimization problem. The semi-definite program can be solved efficiently in polynomial time via interior point methods. There are many existing powerful algorithms and off-the-shelf software packages to solve it. It is worth noting here that our approach can straightforwardly be specialized to address classical safety verification by, a.o., stochastic barrier certificate methods and reach-avoid analysis for ordinary differential equations. Several examples demonstrate theoretical and algorithmic developments of the proposed method.

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“…In real world applications, uncertainties may arise due to model discrepancies, sensing limitations, and the influence of external agents (for example, humans) acting on the system. Stochastic verification can be used to quantify the likelihood of achieving a desired specification with a minimum desired probability, while respecting system dynamics and control bounds [1][2][3]. A theoretical framework for addressing stochastic reachability and viability problems provides a dynamic programming solution.…”

Section: Introductionmentioning

confidence: 99%

Sampling-free enforcement of non-gaussian chance constraints via fourier transforms

Vinod

Sivaramakrishnan²,

Oishi³

2019

Proceedings of the Fifth International Workshop on Symbolic-Numeric Methods for Reasoning About CPS and IoT

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Verification of stochastic dynamical systems can often be formulated as chance-constrained optimization problems -maximize probability of satisfaction of safety/reachability objectives subject to dynamics and control bounds. For linear systems perturbed by Gaussian noise, chance-constraint techniques have proven to be highly efficient. With the goal of extending this approach to non-Gaussian disturbances, this short paper focuses on tractable approaches to enforce chance constraints involving non-Gaussian random vectors. After reviewing existing techniques, we propose a novel approach to enforce chance constraints for arbitrary disturbances using Fourier transforms that is sampling-free and provides tight approximations. We demonstrate the efficiency of our approach in a simple example. CCS CONCEPTS• Theory of computation → Stochastic control and optimization; Convex optimization; • Computing methodologies → Control methods; Computational control theory;

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Stochastic reachability of a target tube: Theory and computation

Cited by 16 publications

References 30 publications

GoTube: Scalable Statistical Verification of Continuous-Depth Models

GoTube: Scalable Statistical Verification of Continuous-Depth Models

Reach-Avoid Analysis for Delay Differential Equations

Sampling-free enforcement of non-gaussian chance constraints via fourier transforms

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