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Global Output-Feedback Extremum Seeking Control with Source Seeking Experiments

Nerito Oliveira Aminde, Tiago Roux Oliveira, Liu Hsu

TL;DR

This work tackles real-time optimization of an unknown nonlinear map $y=\Phi(z)$ for systems with uncertain relative degree, using an output-feedback extremum seeking controller based on a monitoring function, a norm observer, and a time-scaling technique to guarantee global convergence to the unknown maximum. The method integrates a carefully designed monitoring function to handle unknown control direction, and a modulation function to ensure finite-time convergence to a small neighborhood of the extremum, while allowing passage through local extrema. It further extends the approach to uncertain and arbitrary relative degrees via singular-perturbation/time-scaling, providing scaled controller parameters and proving global convergence bounds that scale with a small parameter $\mu$. The approach is validated experimentally in a one-dimensional light-source seeking task using a cart-track servomechanism with only output measurements from a photosensor, demonstrating robust localization and tracking despite disturbances and unmodeled dynamics. The results indicate practical relevance for navigation and source-localization tasks under significant model uncertainties, with potential extensions to higher-dimensional optimization leveraging gradient/Hessian information.

Abstract

This paper discusses the design of an extremum seeking controller that relies on a monitoring function for a class of SISO uncertain nonlinear systems characterized by arbitrary and uncertain relative degree. Our demonstration illustrates the feasibility of achieving an arbitrarily small proximity to the desired optimal point through output feedback. The core concept involves integrating a monitoring function with a norm state observer for the unitary relative degree case and its expansion to arbitrary relative degrees by means of the employment of a time-scaling technique. Significantly, our proposed scheme attains the extremum of an unknown nonlinear mapping across the entire domain of initial conditions, ensuring global convergence and stability for the real-time optimization algorithm. Furthermore, we provide tuning rules to ensure convergence to the global maximum in the presence of local extrema. To validate the effectiveness of the proposed approach, we present a numerical example and apply it to a source-seeking problem involving a cart-track linear positioning servomechanism. Notably, the cart lacks the ability to sense its velocity or the source's position, but can detect the source of a light signal of unknown concentration field.

Global Output-Feedback Extremum Seeking Control with Source Seeking Experiments

TL;DR

This work tackles real-time optimization of an unknown nonlinear map for systems with uncertain relative degree, using an output-feedback extremum seeking controller based on a monitoring function, a norm observer, and a time-scaling technique to guarantee global convergence to the unknown maximum. The method integrates a carefully designed monitoring function to handle unknown control direction, and a modulation function to ensure finite-time convergence to a small neighborhood of the extremum, while allowing passage through local extrema. It further extends the approach to uncertain and arbitrary relative degrees via singular-perturbation/time-scaling, providing scaled controller parameters and proving global convergence bounds that scale with a small parameter . The approach is validated experimentally in a one-dimensional light-source seeking task using a cart-track servomechanism with only output measurements from a photosensor, demonstrating robust localization and tracking despite disturbances and unmodeled dynamics. The results indicate practical relevance for navigation and source-localization tasks under significant model uncertainties, with potential extensions to higher-dimensional optimization leveraging gradient/Hessian information.

Abstract

This paper discusses the design of an extremum seeking controller that relies on a monitoring function for a class of SISO uncertain nonlinear systems characterized by arbitrary and uncertain relative degree. Our demonstration illustrates the feasibility of achieving an arbitrarily small proximity to the desired optimal point through output feedback. The core concept involves integrating a monitoring function with a norm state observer for the unitary relative degree case and its expansion to arbitrary relative degrees by means of the employment of a time-scaling technique. Significantly, our proposed scheme attains the extremum of an unknown nonlinear mapping across the entire domain of initial conditions, ensuring global convergence and stability for the real-time optimization algorithm. Furthermore, we provide tuning rules to ensure convergence to the global maximum in the presence of local extrema. To validate the effectiveness of the proposed approach, we present a numerical example and apply it to a source-seeking problem involving a cart-track linear positioning servomechanism. Notably, the cart lacks the ability to sense its velocity or the source's position, but can detect the source of a light signal of unknown concentration field.
Paper Structure (16 sections, 4 theorems, 56 equations, 14 figures)

This paper contains 16 sections, 4 theorems, 56 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and problem formulation
  3. Output-feedback extremum seeking controller via monitoring function
  4. Error Dynamics
  5. Monitoring Function Design
  6. Modulation Function Design
  7. Global Convergence Result
  8. Multiple Extrema Envisage
  9. Illustrative example
  10. Uncertain and Arbitrary Relative Degrees
  11. Singular Perturbation Analysis
  12. Time-Scaling for Control Redesign
  13. Scaled Controller Parameters
  14. Global Convergence Result
  15. Light Source Seeking Experiments
  16. ...and 1 more sections

Key Result

proposition 1

Consider the system (plant_inverse2)--(saidameasured), reference model (modref) and control law (control_law). Outside the $\Delta$-vicinity $\mathcal{D}_\Delta$, if $\rho$ in (control_law) is designed as then, while $z \notin \mathcal{D}_\Delta$, one has: (a) the monitoring function switching stops, (b) no finite-time escape occurs in the system signals ($t_M \to +\infty$), and (c) the error $e$

Figures (14)

  • Figure 1: Extremum seeking controller using a monitoring function.
  • Figure 2: The trajectories of $\varphi_m(t)$ and $|e(t)|$.
  • Figure 3: Performance map $\Phi(z)$.
  • Figure 4: Parameter $z$ converges to $z^*=5$ that maximizes $y$ using different initial conditions $z(0)$.
  • Figure 5: Time history of the output plant $y$ (solid line) and the output model $y_m$ (dashed line). The output plant tends to the maximum value $y^*=1.5$.
  • ...and 9 more figures

Theorems & Definitions (9)

  • remark 1: Main Changes in the Monitoring Function:
  • proposition 1
  • remark 2: Modulation Function Reset
  • theorem 1
  • remark 3: Chattering Alleviation:
  • proposition 2
  • remark 4: Tuning Rules:
  • theorem 2
  • remark 5: Uncertain Relative Degree: