Reachability Estimation of Stochastic Dynamical Systems by Semi-definite Programming

Liu, Kairong; Li, Meilun; She, Zhikun

doi:10.1109/cdc40024.2019.9029192

“…Recently, a moment-based method, which is also a convex programming based method, was proposed for studying reach-avoid problems over finite time horizons for SDEs in [33], and a semi-definite programming method derived from Feynman-Kac formula was proposed for analysing avoid problems (without the requirement of reaching target sets) over finite time horizons in [20] that is algebraically over-and underapproximating the staying probability in a given safety area. Different from the above two methods, our method in this paper addresses the reach-avoid problem over open time horizons rather than finite time horizons.…”

Section: Related Workmentioning

confidence: 99%

“…We also make comparisons between our method and the method in [16] based on these two examples. Note that the semi-definite program (20) in [16] involves the choice of discount factors β i , i = 1, ..., n β . In these computations, we take n β = 2 and use the semi-definite program (20) with (β 1 = 1, β 2 = 2) and (β 1 = 1, β 2 = 0.01) in [16] as instances to illustrate its performance.…”

Section: Example 3 (Harmonic Oscillator) Consider a Two-mentioning

confidence: 99%

Reach-Avoid Analysis for Delay Differential Equations

Bai

¹

,

Bai

²

,

Zhan

³

et al. 2021

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

View full text Add to dashboard Cite

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.

show abstract

“…Grid-based numerical approaches, e.g., the finite difference method in [16] and the level set method in [22], are traditionally used to solve these equations, leading to the fact that the Hamilton-Jacobi reachability method only scales well to systems of special structures. More recently, a novel constraint solving-based method has been proposed in [20] for algebraically over-and under-approximating the reachability probability, which is nevertheless limited to bounded-time safety verification. In addition to the abovementioned methods, we refer the readers to [7] for a Dirichlet form-based method for stochastic hybrid systems featuring "nice" Markov properties, while to [6,18,39] and [1,17] respectively for related contributions in statistical and discrete/numerical methods for stochastic verification and control.…”

Section: Contributionsmentioning

confidence: 99%

“…Here, the constraints 20 Remark 5. If Λ is chosen as a non-negative matrix, the combination of condition (20) and (22) will force V a (x) = 0 for x ∈ ∂X , whereof the strict equality may be violated due to numerical computations in SDP. In practice, however, this issue can be well addressed by looking for a barrier certificate of the form g(x)V (x), where g(x) satisfies ∂X ⊆ {x | g(x) = 0}, namely, an overapproximation of the boundary of X .…”

Section: Synthesizing Stochastic Barrier Certificates Using Sdpmentioning

confidence: 99%

Unbounded-Time Safety Verification of Stochastic Differential Dynamics

Feng

¹

,

Chen

²

,

Bai

³

et al. 2020

Computer Aided Verification

View full text Add to dashboard Cite

In this paper, we propose a method for bounding the probability that a stochastic differential equation (SDE) system violates a safety specification over the infinite time horizon. SDEs are mathematical models of stochastic processes that capture how states evolve continuously in time. They are widely used in numerous applications such as engineered systems (e.g., modeling how pedestrians move in an intersection), computational finance (e.g., modeling stock option prices), and ecological processes (e.g., population change over time). Previously the safety verification problem has been tackled over finite and infinite time horizons using a diverse set of approaches. The approach in this paper attempts to connect the two views by first identifying a finite time bound, beyond which the probability of a safety violation can be bounded by a negligibly small number. This is achieved by discovering an exponential barrier certificate that proves exponentially converging bounds on the probability of safety violations over time. Once the finite time interval is found, a finite-time verification approach is used to bound the probability of violation over this interval. We demonstrate our approach over a collection of interesting examples from the literature, wherein our approach can be used to find tight bounds on the violation probability of safety properties over the infinite time horizon.

show abstract

“…Grid-based numerical approaches, e.g., the finite difference method in [16] and the level set method in [22], are traditionally used to solve these equations, leading to the fact that the Hamilton-Jacobi reachability method only scales well to systems of special structures. More recently, a novel constraint solving-based method has been proposed in [20] for algebraically over-and under-approximating the reachability probability, which is nevertheless limited to bounded-time safety verification. In addition to the abovementioned methods, we refer the readers to [7] for a Dirichlet form-based method for stochastic hybrid systems featuring "nice" Markov properties, while to [39,6,18] and [1,17] respectively for related contributions in statistical and discrete/numerical methods for stochastic verification and control.…”

Section: Introductionmentioning

confidence: 99%

Unbounded-Time Safety Verification of Stochastic Differential Dynamics

Feng

¹

,

Chen

²

,

Bai

³

et al. 2020

Preprint

0

View full text Add to dashboard Cite

In this paper, we propose a method for bounding the probability that a stochastic differential equation (SDE) system violates a safety specification over the infinite time horizon. SDEs are mathematical models of stochastic processes that capture how states evolve continuously in time. They are widely used in numerous applications such as engineered systems (e.g., modeling how pedestrians move in an intersection), computational finance (e.g., modeling stock option prices), and ecological processes (e.g., population change over time). Previously the safety verification problem has been tackled over finite and infinite time horizons using a diverse set of approaches. The approach in this paper attempts to connect the two views by first identifying a finite time bound, beyond which the probability of a safety violation can be bounded by a negligibly small number. This is achieved by discovering an exponential barrier certificate that proves exponentially converging bounds on the probability of safety violations over time. Once the finite time interval is found, a finite-time verification approach is used to bound the probability of violation over this interval. We demonstrate our approach over a collection of interesting examples from the literature, wherein our approach can be used to find tight bounds on the violation probability of safety properties over the infinite time horizon.

show abstract

Reachability Estimation of Stochastic Dynamical Systems by Semi-definite Programming

Cited by 5 publications

References 32 publications

Reach-Avoid Analysis for Delay Differential Equations

Reach-Avoid Analysis for Delay Differential Equations

Unbounded-Time Safety Verification of Stochastic Differential Dynamics

Unbounded-Time Safety Verification of Stochastic Differential Dynamics

Contact Info

Product

Resources

About