Efficient Synthesis of Piecewise Constant Stochastic Barrier Functions for Safety Verification of Complex Stochastic Systems

Core Concepts

This work introduces a novel framework for synthesizing piecewise constant stochastic barrier functions (PWC-SBFs) that can efficiently provide probabilistic safety guarantees for complex stochastic systems with nonlinear dynamics and non-convex safe sets.

Abstract

This paper presents a novel approach for safety verification of stochastic systems based on piecewise (PW) stochastic barrier functions (SBFs). The key contributions are:

A general formulation for PW-SBFs, with a particular focus on PW-Constant (PWC) SBFs, which provide a simple yet effective framework for safety verification of general stochastic systems.

A derivation of the PWC-SBF synthesis problem as a minimax convex optimization problem.

Three efficient and scalable computational methods to solve the PWC-SBF synthesis problem:

An exact LP duality-based approach that provides the optimal solution to the minimax problem.
A Counter-Example Guided Synthesis (CEGS) algorithm that iteratively refines the solution and is proven to converge in finite time.
A Gradient Descent (GD) algorithm that offers significant reduction in computation time.

Extensive case studies demonstrating that the proposed PWC-SBF methods outperform state-of-the-art techniques based on sum-of-squares (SOS) and neural barrier functions (NBFs), both in terms of safety probability bounds and computational efficiency, especially for high-dimensional systems.

The key insight is that by using constant functions for the PW-SBF, the challenging operations of expectation and function composition required by the martingale condition can be significantly mitigated, enabling both simplicity and scalability. The authors prove that the PWC-SBF synthesis problem reduces to a minimax optimization problem, for which they introduce the three efficient computational methods.

Stats

The paper provides the following key metrics:

For the 2D example system, the safety probability bound obtained using the SOS SBF is 0.075, the NBF is 0.93, and the PWC-SBF is 0.93.
The computation times for the 2D example are: SOS SBF - 197s, NBF - 3600s, PWC-SBF - 69s.

Quotes

"Our benchmarks demonstrate that PWC-SBFs outperform state-of-the-art methods, namely sum-of-squares and neural barrier functions, and can scale to eight dimensional systems."
"The power of PWC-SBFs is illustrated in Figure 1d, where the same safety probability bound of 0.93 as NBF is achieved with 69 seconds computation time, i.e., an order of magnitude (50×) faster in computation."

Key Insights Distilled From

Piecewise Stochastic Barrier Functions

by Rayan Mazouz... at arxiv.org 04-29-2024

https://arxiv.org/pdf/2404.16986.pdf

Deeper Inquiries

How can the proposed PWC-SBF framework be extended to handle time-varying or partially observable stochastic systems

The proposed PWC-SBF framework can be extended to handle time-varying or partially observable stochastic systems by incorporating dynamic updates to the barrier functions and adapting to the changing system states. For time-varying systems, the PWC-SBFs can be updated at each time step based on the current system state and the evolution of the dynamics. This can involve re-optimizing the barrier functions to account for the changing conditions and ensure continued safety guarantees.
In the case of partially observable systems, where not all states are directly observable, the PWC-SBF approach can be modified to incorporate estimation techniques such as Kalman filters or particle filters to infer the unobserved states. By integrating these estimation methods with the barrier function synthesis, the PWC-SBFs can provide safety guarantees even in scenarios with partial observability.

What are the potential limitations or drawbacks of the PWC-SBF approach compared to other SBF synthesis methods, and how can they be addressed

One potential limitation of the PWC-SBF approach compared to other SBF synthesis methods is the reliance on a partitioning of the safe set, which may introduce discretization errors and lead to suboptimal safety guarantees. To address this limitation, adaptive partitioning techniques can be employed to dynamically adjust the partition boundaries based on the system dynamics and the evolution of the safe set. This adaptive approach can help improve the accuracy of the safety guarantees provided by the PWC-SBFs.
Another drawback of the PWC-SBF approach is the potential for over-conservatism in the safety guarantees, especially in high-dimensional systems where the partitioning may lead to a large number of regions. To mitigate this issue, advanced optimization techniques such as mixed-integer programming or reinforcement learning can be used to optimize the partitioning and the barrier functions simultaneously, aiming to reduce conservatism while maintaining safety.

Can the PWC-SBF synthesis techniques be integrated with reinforcement learning or other control synthesis methods to provide safety guarantees for autonomous systems operating in complex environments

The PWC-SBF synthesis techniques can be integrated with reinforcement learning or other control synthesis methods to provide safety guarantees for autonomous systems operating in complex environments. By combining the safety certification capabilities of PWC-SBFs with the adaptive learning and decision-making abilities of reinforcement learning, autonomous systems can ensure both safety and performance in uncertain and dynamic environments.
One approach is to use reinforcement learning to train a policy that respects the safety guarantees provided by the PWC-SBFs. The reinforcement learning agent can learn to navigate the system while avoiding unsafe states based on the barrier functions, ensuring that the system operates within the specified safety bounds.
Additionally, the PWC-SBF synthesis techniques can be integrated into model predictive control (MPC) frameworks to enable real-time safety verification and control synthesis. By incorporating the PWC-SBFs as constraints in the MPC optimization problem, the autonomous system can proactively avoid unsafe states while optimizing its performance objectives. This integration can provide a comprehensive safety assurance mechanism for autonomous systems operating in complex environments.