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Computationally Efficient Sampling-Based Algorithm for Stability Analysis of Nonlinear Systems

Péter Antal, Tamás Péni, Roland Tóth

TL;DR

This paper tackles the challenge of estimating the domain of attraction for general nonlinear systems by embedding dynamics into the Lyapunov function parametrization and solving a sequence of tractable problems. It recasts Lyapunov synthesis as an $\ell_1$-regularized linear program to maximize an invariant set on a state-space grid, and uses a learner–verifier loop to ensure Lyapunov conditions over the continuous domain. Scalability is further enhanced with ADMM, enabling parallelized constraint handling and enabling a 5D example to be analyzed. Across 2D, 3D, and 5D nonlinear systems, the approach yields accurate DOA estimates with significantly reduced computation times compared to existing methods, demonstrating practical viability for higher-dimensional stability analysis.

Abstract

For complex nonlinear systems, it is challenging to design algorithms that are fast, scalable, and give an accurate approximation of the stability region. This paper proposes a sampling-based approach to address these challenges. By extending the parametrization of quadratic Lyapunov functions with the system dynamics and formulating an $\ell_1$ optimization to maximize the invariant set over a grid of the state space, we arrive at a computationally efficient algorithm that estimates the domain of attraction (DOA) of nonlinear systems accurately by using only linear programming. The scalability of the Lyapunov function synthesis is further improved by combining the algorithm with ADMM-based parallelization. To resolve the inherent approximative nature of grid-based techniques, a small-scale nonlinear optimization is proposed. The performance of the algorithm is evaluated and compared to state-of-the-art solutions on several numerical examples.

Computationally Efficient Sampling-Based Algorithm for Stability Analysis of Nonlinear Systems

TL;DR

This paper tackles the challenge of estimating the domain of attraction for general nonlinear systems by embedding dynamics into the Lyapunov function parametrization and solving a sequence of tractable problems. It recasts Lyapunov synthesis as an -regularized linear program to maximize an invariant set on a state-space grid, and uses a learner–verifier loop to ensure Lyapunov conditions over the continuous domain. Scalability is further enhanced with ADMM, enabling parallelized constraint handling and enabling a 5D example to be analyzed. Across 2D, 3D, and 5D nonlinear systems, the approach yields accurate DOA estimates with significantly reduced computation times compared to existing methods, demonstrating practical viability for higher-dimensional stability analysis.

Abstract

For complex nonlinear systems, it is challenging to design algorithms that are fast, scalable, and give an accurate approximation of the stability region. This paper proposes a sampling-based approach to address these challenges. By extending the parametrization of quadratic Lyapunov functions with the system dynamics and formulating an optimization to maximize the invariant set over a grid of the state space, we arrive at a computationally efficient algorithm that estimates the domain of attraction (DOA) of nonlinear systems accurately by using only linear programming. The scalability of the Lyapunov function synthesis is further improved by combining the algorithm with ADMM-based parallelization. To resolve the inherent approximative nature of grid-based techniques, a small-scale nonlinear optimization is proposed. The performance of the algorithm is evaluated and compared to state-of-the-art solutions on several numerical examples.
Paper Structure (15 sections, 24 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 15 sections, 24 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Sampling-based DOA estimation with $\ell_1$ optimization
  4. Lyapunov function parametrization
  5. Learner based Lyapunov synthesis
  6. Verifier optimization
  7. Improving scalability
  8. Examples
  9. Two-dimensional examples
  10. Van der Pol oscillator
  11. Example 2
  12. Example 3
  13. Three-dimensional example
  14. Five-dimensional example with ADMM
  15. Conclusion

Figures (4)

  • Figure 1: Estimated domain of attraction for multiple 2D systems. Green and red points denote the elements of $\mathcal{X}_0$ and $\mathcal{X}_\infty$, respectively, while the solid black and dashed blue lines correspond to the 1 level set of the Lyapunov function and the true DOA of the system. Left: Van der Pol system given by \ref{['eq:vanderpol']}, center: Example 2 given by \ref{['eq:sys2']}, right: Example 3 given by \ref{['eq:sys3']}.
  • Figure 2: DOA estimation of the 3D example given by \ref{['eq:sys4']}. The green surface depicts the level set of the LF, and the blue mesh shows the DOA.
  • Figure 3: DOA estimation of the 5D example given by \ref{['eq:5d']}. Each plot shows a slice of the level set $\mathcal{V}(x)=1$ and the DOA, depicted by solid black and dashed blue lines, respectively.
  • Figure : Sampling-based iterative DOA estimation