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Distributionally Robust Lyapunov Function Search Under Uncertainty

Kehan Long, Yinzhuang Yi, Jorge Cortes, Nikolay Atanasov

TL;DR

The paper addresses stability analysis of uncertain nonlinear systems when disturbance distributions are unknown and only finite samples are available. It introduces distributionally robust counterparts of the Lyapunov derivative constraint and develops two solution pipelines: SOS-based polynomial Lyapunov function search and neural-network Lyapunov function search, leveraging Wasserstein ambiguity sets and CVaR relaxations. For polynomial systems, the DRCC-SOS formulation provides a tractable SOS program, while for general nonlinear systems, DRCC-NN-LF enables scalable, data-driven certificates. Evaluations on nonlinear uncertain systems, including a polynomial example and a pendulum, demonstrate that the proposed DRCC formulations maintain Lyapunov validity under out-of-distribution disturbances, offering robust stability guarantees and open-source tooling.

Abstract

This paper develops methods for proving Lyapunov stability of dynamical systems subject to disturbances with an unknown distribution. 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. We show that this constraint may be reformulated as several SOS constraints, ensuring that the search for a Lyapunov function remains in the class of SOS polynomial optimization problems. For general systems, we provide a distributionally robust chance-constrained formulation for neural network Lyapunov function search. Simulations demonstrate the validity and efficiency of either formulation on non-linear uncertain dynamical systems.

Distributionally Robust Lyapunov Function Search Under Uncertainty

TL;DR

The paper addresses stability analysis of uncertain nonlinear systems when disturbance distributions are unknown and only finite samples are available. It introduces distributionally robust counterparts of the Lyapunov derivative constraint and develops two solution pipelines: SOS-based polynomial Lyapunov function search and neural-network Lyapunov function search, leveraging Wasserstein ambiguity sets and CVaR relaxations. For polynomial systems, the DRCC-SOS formulation provides a tractable SOS program, while for general nonlinear systems, DRCC-NN-LF enables scalable, data-driven certificates. Evaluations on nonlinear uncertain systems, including a polynomial example and a pendulum, demonstrate that the proposed DRCC formulations maintain Lyapunov validity under out-of-distribution disturbances, offering robust stability guarantees and open-source tooling.

Abstract

This paper develops methods for proving Lyapunov stability of dynamical systems subject to disturbances with an unknown distribution. 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. We show that this constraint may be reformulated as several SOS constraints, ensuring that the search for a Lyapunov function remains in the class of SOS polynomial optimization problems. For general systems, we provide a distributionally robust chance-constrained formulation for neural network Lyapunov function search. Simulations demonstrate the validity and efficiency of either formulation on non-linear uncertain dynamical systems.
Paper Structure (10 sections, 3 theorems, 27 equations, 2 figures, 1 table)

This paper contains 10 sections, 3 theorems, 27 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Lyapunov Theory and Sum-of-Squares Optimization
  4. Conditional Value-at-Risk and Distributionally Robust Chance Constraint
  5. Problem Formulation
  6. Lyapunov Function Search For Systems with Model Uncertainty
  7. Sum-of-Squares Approach For Lyapunov Function Search
  8. Neural Network Approach For Lyapunov Function Search
  9. Evaluation
  10. Conclusions

Key Result

proposition 1

Assume $\beta \leq \frac{1}{N}$, the constraint in eq: cvar_explicit_form is equivalent to: Furthermore, if $f$ and $d_i$ are polynomials, the following $N$ SOS conditions are sufficient for eq: proposition_1,

Figures (2)

  • Figure 1: Results from SOS and NN formulations to design LF certificates for the polynomial system with Case 2 perturbations and online uncertainty $\boldsymbol{\xi}^* = [1.9, 3.0]^\top$. The plots display the value of $\dot{V}$ over the domain, where the red areas indicate positive values (violation of the LF derivative requirements).
  • Figure 2: Results from SOS and NN formulations to design LF certificates for a pendulum with perturbation in the damping and length and online uncertainty $\boldsymbol{\xi}^* = [-3.6,1.4]^{\top}$. The plots display the value of $\dot{V}$ over the domain, where the red areas indicate positive values (violation of the LF derivative requirement).

Theorems & Definitions (6)

  • proposition 1: CC-SOS Condition
  • proof
  • proposition 2: DRCC-SOS Condition
  • proof
  • proposition 3: Stability guarantee of DRCC-SOS formulation
  • proof