Distributionally Robust Lyapunov Function Synthesis for Uncertain Dynamical Systems

Core Concepts

This paper develops methods for proving Lyapunov stability of dynamical systems subject to disturbances with an unknown distribution, using only a finite set of disturbance samples.

Abstract

The paper addresses the problem of synthesizing a valid Lyapunov function (LF) for a dynamical system with model uncertainty. The authors assume that the system dynamics are affected by an uncertain disturbance term, but only a finite set of disturbance samples is available, and the true online disturbance distribution is unknown.
The key contributions are:

The authors formulate a distributionally robust version of the Lyapunov function derivative constraint, which ensures that the LF conditions are satisfied for all distributions in an ambiguity set around the empirical distribution of the available disturbance samples.

For polynomial systems, the authors show that the distributionally robust constraint can be reformulated as multiple sum-of-squares (SOS) constraints, allowing the LF synthesis with uncertainty to remain an SOS polynomial optimization problem.

For general nonlinear systems, the authors propose a distributionally robust neural network approach for learning Lyapunov functions, which minimizes a loss function that accounts for the uncertainty in the system dynamics.

The paper evaluates the proposed SOS-based and neural network-based approaches on a polynomial system and a pendulum system, demonstrating that the distributionally robust formulations outperform the standard approaches that do not consider the uncertainty.

Stats

The polynomial system has the form: ẋ1 = -1/2x1^3 + 1 - 3/2x2^2, ẋ2 = 1 - x2 - 6x1x2, with two cases of model uncertainty.
The pendulum system has the form: θ̇ = ω, ω̈ = -mgl sin(θ) - bω/ml^2 + ξ, with perturbations in the damping and length.

Quotes

"We assume only a finite set of disturbance samples is available and that the true online disturbance realization may be drawn from a different distribution than the given samples."
"We formulate an optimization problem to search for a sum-of-squares (SOS) Lyapunov function and introduce a distributionally robust version of the Lyapunov function derivative constraint."
"For general systems, we provide a distributionally robust chance-constrained formulation for neural network Lyapunov function search."

Key Insights Distilled From

Distributionally Robust Lyapunov Function Search Under Uncertainty

by Kehan Long,Y... at arxiv.org 05-01-2024

https://arxiv.org/pdf/2212.01554.pdf

Deeper Inquiries

How can the proposed distributionally robust Lyapunov function synthesis approaches be extended to handle time-varying or state-dependent uncertainty distributions

The proposed distributionally robust Lyapunov function synthesis approaches can be extended to handle time-varying or state-dependent uncertainty distributions by incorporating adaptive mechanisms in the learning process. One way to achieve this is by updating the uncertainty set or ambiguity ball based on real-time data or feedback. By continuously adjusting the uncertainty set using new information, the Lyapunov function synthesis can adapt to changing uncertainty distributions over time or across different states. This adaptive approach ensures that the learned Lyapunov function remains robust and stable even in the presence of varying or evolving uncertainties.

What are the theoretical guarantees on the stability and robustness of the learned Lyapunov functions under different types of model uncertainty and disturbance distributions

Theoretical guarantees on the stability and robustness of the learned Lyapunov functions under different types of model uncertainty and disturbance distributions can be analyzed using tools from robust control theory and optimization. For instance, the distributionally robust Lyapunov function synthesis approaches provide guarantees on the probabilistic constraints being satisfied with high probability, even in the presence of uncertain or shifting distributions. By formulating the Lyapunov function search as a distributionally robust optimization problem, the resulting Lyapunov functions are designed to be valid under a wide range of uncertainty scenarios. Theoretical analyses can establish bounds on the performance and robustness of the learned Lyapunov functions, ensuring stability and safety in uncertain dynamical systems.

Can the distributionally robust Lyapunov function search be integrated with control synthesis to obtain robust and stable control policies for uncertain dynamical systems

The distributionally robust Lyapunov function search can be integrated with control synthesis to obtain robust and stable control policies for uncertain dynamical systems by leveraging the learned Lyapunov functions as safety certificates or stability guarantees. By combining the distributionally robust Lyapunov functions with control synthesis techniques such as model predictive control or reinforcement learning, robust control policies can be designed that ensure stability and performance in the presence of uncertain disturbances. The integration of distributionally robust Lyapunov functions with control synthesis enables the development of adaptive and resilient control strategies that can handle varying uncertainty distributions while maintaining stability and safety in complex dynamical systems.