Safe Probabilistic Invariance Verification of Stochastic Dynamical Systems

Core Concepts

This paper introduces a comprehensive framework for verifying the safe probabilistic invariance of both discrete-time and continuous-time stochastic dynamical systems over an infinite time horizon. The objective is to compute lower and upper bounds on the liveness probability, which represents the likelihood of the system remaining within a safe set indefinitely.

Abstract

The paper proposes a framework for safe probabilistic invariance verification of stochastic dynamical systems. It covers both discrete-time and continuous-time systems.
For discrete-time systems:

Two sets of optimizations are presented to compute lower and upper bounds on the liveness probability.
The first set adapts stochastic barrier certificates from prior work on safety and reachability verification.
The second set is inspired by an equation that can precisely characterize the probability of reaching a target set while avoiding unsafe states.
It is shown that the two sets of optimizations are equivalent when computing lower bounds.
For continuous-time systems:

The framework is extended to handle continuous-time stochastic dynamical systems.
Two sets of optimizations similar to the discrete-time case are proposed to compute lower and upper bounds on the liveness probability.
The first set adapts barrier certificates, while the second set is derived from an equation that can precisely characterize the reachability probability.
It is demonstrated that the second set of optimizations is more effective than the first set when computing upper bounds.
Several numerical examples are provided to illustrate the performance and effectiveness of the proposed optimizations using semi-definite programming tools.

Stats

The paper does not contain any explicit numerical data or statistics. It focuses on presenting a theoretical framework and optimization-based approaches for safe probabilistic invariance verification.

Quotes

"This paper introduces a comprehensive framework for the safe probabilistic invariance verification of both discrete- and continuous-time stochastic dynamical systems over an infinite time horizon."
"The objective is to ascertain the lower and upper bounds of the liveness probability for a given safe set and set of initial states."
"To address this problem, we propose optimizations for verifying safe probabilistic invariance in discrete-time and continuous-time stochastic dynamical systems."

Key Insights Distilled From

A Framework for Safe Probabilistic Invariance Verification of Stochastic Dynamical Systems

by Taoran Wu,Yi... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.09007.pdf

Deeper Inquiries

How can the proposed framework be extended to handle more complex system dynamics, such as hybrid systems or systems with time-varying parameters

To extend the proposed framework to handle more complex system dynamics, such as hybrid systems or systems with time-varying parameters, several modifications and enhancements can be considered:

Hybrid Systems: For hybrid systems that exhibit both continuous and discrete behaviors, the framework can be extended by incorporating techniques from hybrid system verification. This may involve developing algorithms to handle the discrete transitions and interactions between different modes of operation.

Time-Varying Parameters: Systems with time-varying parameters can be addressed by introducing stochastic processes that model the evolution of these parameters over time. This would require adapting the framework to account for the variability in system dynamics and updating the verification algorithms accordingly.

Advanced Modeling Techniques: Utilizing advanced modeling techniques, such as stochastic differential equations or Markov decision processes, can provide a more accurate representation of complex system behaviors. By integrating these models into the framework, it can better capture the dynamics of the system under consideration.

Simulation and Testing: Extending the framework to include simulation and testing capabilities can help validate the verification results for complex system dynamics. By running simulations with varying parameters and scenarios, the framework can be tested for robustness and accuracy in handling diverse system behaviors.

Overall, by incorporating these enhancements and adaptations, the framework can be effectively extended to handle more intricate system dynamics, ensuring robust and reliable verification results.

What are the potential limitations of the barrier certificate and equation-based approaches, and how can they be further improved

The barrier certificate and equation-based approaches, while effective in certain scenarios, may have potential limitations that can be addressed for further improvement:

Conservatism: One limitation of barrier certificates is their inherent conservatism, which can lead to overly pessimistic results. To mitigate this, refining the barrier functions and optimizing the constraints can help reduce conservatism while maintaining safety guarantees.

Complexity: The complexity of the verification process using these approaches can be a limitation, especially for large-scale systems. Developing scalable algorithms and optimization techniques can streamline the verification process and improve efficiency.

Handling Nonlinearities: Both approaches may struggle with highly nonlinear systems, where traditional methods may not be directly applicable. Introducing nonlinear analysis techniques and adaptive algorithms can enhance the capability to handle nonlinear dynamics more effectively.

Verification Rigor: Ensuring the rigor and correctness of the verification results is crucial. Incorporating formal methods and rigorous mathematical proofs can enhance the reliability of the verification process and provide stronger guarantees of system safety.

By addressing these limitations through advanced algorithmic developments, optimization strategies, and rigorous verification techniques, the barrier certificate and equation-based approaches can be further improved for safe probabilistic invariance verification of complex systems.

What are the connections between the safe probabilistic invariance verification problem and other related problems in the field of formal verification, such as temporal logic verification or robust control

The safe probabilistic invariance verification problem is closely related to other problems in the field of formal verification, such as temporal logic verification and robust control:

Temporal Logic Verification: Temporal logic verification deals with specifying and verifying properties of systems over time. The safe probabilistic invariance verification problem can be seen as a probabilistic extension of temporal logic verification, where the focus is on ensuring system safety with a certain probability. Techniques from temporal logic verification, such as model checking and property specification, can be adapted for probabilistic invariance verification.

Robust Control: Robust control aims to design controllers that can handle uncertainties and disturbances in system dynamics. The safe probabilistic invariance verification problem addresses the likelihood of a system remaining within a safe set under stochastic disturbances. By integrating concepts from robust control theory, such as robust stability analysis and controller synthesis, the verification of safe probabilistic invariance can benefit from robustness guarantees and control strategies.

Model Checking: Model checking is a formal verification technique that systematically checks whether a model satisfies a given property. The safe probabilistic invariance verification problem can leverage model checking algorithms and tools to verify the probabilistic invariance properties of stochastic systems. By combining techniques from model checking with probabilistic analysis, a more comprehensive verification framework can be developed.

By exploring the connections between safe probabilistic invariance verification and these related problems, insights and methodologies from different areas of formal verification can be integrated to enhance the verification process and ensure the safety and reliability of complex systems.