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Extremum Seeking for Controlled Vibrational Stabilization of Second Order Mechanical Systems

Ahmed A. Elgohary, Sameh A. Eisa

TL;DR

The paper addresses model-free vibrational stabilization of second-order mechanical systems that involve generalized forces quadratic in velocity, using a one-perturbation extremum-seeking control (ESC) framework. It derives a variation-of-constant (VOC) averaged model to capture the perturbation's effect and proves practical stability by linking the averaged system's asymptotic behavior to the original system via the averaging theorem. The main contributions include extending ESC to quadratic-velocity forces, providing a VOC-based stability analysis with a Lyapunov candidate for convergence, and validating the approach on a mass-spring-damper, an inverted pendulum, and a novel 1D height-seeking task with a flapper. The results demonstrate effective, model-free vibrational stabilization with potentially reduced perturbation energy, and the work opens doors to broader force-laws and cubic-damping extensions in bio-inspired and aeroelastic contexts.

Abstract

This paper presents a novel extremum seeking control (ESC) approach for the vibrational stabilization of a class of mechanical systems (e.g., systems characterized by equations of motion resulting from Newton second law or Euler-Lagrange mechanics). Inspired by flapping insects mechanics, the proposed ESC approach is operable by only one perturbation signal and can admit generalized forces that are quadratic in velocities. We test our ESC, and compare it against approaches from literature, on some classical mechanical systems (e.g., mass-spring and an inverted pendulum systems). We also provide a novel, first-of-its-kind, application of the introduced ESC by achieving a 1D model-free source-seeking of a flapping system.

Extremum Seeking for Controlled Vibrational Stabilization of Second Order Mechanical Systems

TL;DR

The paper addresses model-free vibrational stabilization of second-order mechanical systems that involve generalized forces quadratic in velocity, using a one-perturbation extremum-seeking control (ESC) framework. It derives a variation-of-constant (VOC) averaged model to capture the perturbation's effect and proves practical stability by linking the averaged system's asymptotic behavior to the original system via the averaging theorem. The main contributions include extending ESC to quadratic-velocity forces, providing a VOC-based stability analysis with a Lyapunov candidate for convergence, and validating the approach on a mass-spring-damper, an inverted pendulum, and a novel 1D height-seeking task with a flapper. The results demonstrate effective, model-free vibrational stabilization with potentially reduced perturbation energy, and the work opens doors to broader force-laws and cubic-damping extensions in bio-inspired and aeroelastic contexts.

Abstract

This paper presents a novel extremum seeking control (ESC) approach for the vibrational stabilization of a class of mechanical systems (e.g., systems characterized by equations of motion resulting from Newton second law or Euler-Lagrange mechanics). Inspired by flapping insects mechanics, the proposed ESC approach is operable by only one perturbation signal and can admit generalized forces that are quadratic in velocities. We test our ESC, and compare it against approaches from literature, on some classical mechanical systems (e.g., mass-spring and an inverted pendulum systems). We also provide a novel, first-of-its-kind, application of the introduced ESC by achieving a 1D model-free source-seeking of a flapping system.

Paper Structure

This paper contains 14 sections, 3 theorems, 21 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Extremum Seeking Feedback Control Systems
  3. Vibrational Stabilization
  4. Extremum Seeking and Vibrational Stabilization of Mechanical Systems
  5. Motivation and Contribution
  6. Main Results
  7. Variation-of-Constant Averaging Approach
  8. Stability Results and Design Guidelines
  9. Applications
  10. Classic Mass-Spring-Damper System
  11. Classic Inverted Pendulum
  12. Comparison with ESC Literature
  13. 1 Dimensional Source (Height) Seeking by a Flapper
  14. Conclusion

Key Result

Corollary 1

If $\bar{\mathbf{\textcolor{black}{x}}}(t) \in \mathscr{C}_0$, $\forall t \in [0,t_f/\epsilon]$ with $t_f>0$ and $\bar{\mathbf{\textcolor{black}{x}}}(0)=\mathbf{\textcolor{black}{x}}(0)$, we have $|\mathbf{\textcolor{black}{x}}(t)-\bar{\mathbf{\textcolor{black}{x}}}(t)| = O(\epsilon)$ for $t \in [0,

Figures (6)

  • Figure 1: The proposed ESC for a class of mechanical systems.
  • Figure 2: The flow along $\mathbf{Z} + \mathbf{Y}$ is equivalent to the flow along the vector fields $\mathbf{F}$ and then $\mathbf{Y}$. Starting at $\mathbf{x}(0) = \mathbf{z}(0) = \mathbf{x}_0$, the flow first evolves according to $\mathbf{F}$, and then continues along $\mathbf{Y}$, resulting in $\mathbf{y}(0) = \mathbf{z}(t)$.
  • Figure 3: Top 3 subplots: Trajectories of mass-spring system generalized coordinate, its velocity and ESC input, along with their VOC average. Bottom subplot: Lyapunov function rate.
  • Figure 4: Top 3 subplots: Trajectories of inverted pendulum generalized coordinate, its velocity and ESC input, along with their VOC average. Bottom subplot: Lyapunov function rate.
  • Figure 5: Comparison between the proposed ESC and existing methods from grushkovskaya2021extremum and suttner2022extremum. Top: Mass-spring-damper system with linear damping. Middle: Mass-spring-damper system with cubic damping. Bottom: Inverted pendulum system.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Remark 1
  • Corollary 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Proposition 1