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Extremum Seeking Boundary Control for Euler-Bernoulli Beam PDEs

Paulo H. F. Biazetto, Gustavo A. de Andrade, Tiago Roux Oliveira, Miroslav Krstic

TL;DR

The paper addresses online optimization of an unknown static map applied to a 1D Euler–Bernoulli beam PDE with end-boundary actuation, where the input to the map is the uncontrolled end displacement $\Theta(t)$. It introduces an extremum-seeking controller that uses demodulation signals and additive probing within a backstepping framework, leveraging a Schrödinger-equation representation to compensate actuation dynamics and achieve a stable, averaged convergence to a neighborhood of the unknown optimum. The main contributions are the design of a two-step backstepping compensation, the formulation of implementable averaged ES laws with gradient and Hessian estimates, and a rigorous stability analysis showing local exponential convergence under averaging theory, complemented by simulations that validate performance. The work advances ES for infinite-dimensional PDE systems and provides a practical control strategy for optimizing unknown static maps in flexible beam applications, with potential extensions to other boundary conditions and PDE–ODE hybrids.

Abstract

This paper presents the design and analysis of an extremum seeking (ES) controller for scalar static maps in the context of infinite-dimensional dynamics governed by the 1D Euler-Bernoulli (EB) beam Partial Differential Equation (PDE). The beam is actuated at one end (using position and moment actuators). The map's input is the displacement at the beam's uncontrolled end, which is subject to a sliding boundary condition. Notably, ES for this class of PDEs remains unexplored in the existing literature. To compensate for PDE actuation dynamics, we employ a boundary control law via a backstepping transformation and averaging-based estimates for the gradient and Hessian of the static map to be optimized. This compensation controller leverages a Schrödinger equation representation of the EB beam and adapts existing backstepping designs to stabilize the beam. Using the semigroup and averaging theory in infinite dimensions, we prove local exponential convergence to a small neighborhood of the unknown optimal point. Finally, simulations illustrate the effectiveness of the design in optimizing the unknown static map.

Extremum Seeking Boundary Control for Euler-Bernoulli Beam PDEs

TL;DR

The paper addresses online optimization of an unknown static map applied to a 1D Euler–Bernoulli beam PDE with end-boundary actuation, where the input to the map is the uncontrolled end displacement . It introduces an extremum-seeking controller that uses demodulation signals and additive probing within a backstepping framework, leveraging a Schrödinger-equation representation to compensate actuation dynamics and achieve a stable, averaged convergence to a neighborhood of the unknown optimum. The main contributions are the design of a two-step backstepping compensation, the formulation of implementable averaged ES laws with gradient and Hessian estimates, and a rigorous stability analysis showing local exponential convergence under averaging theory, complemented by simulations that validate performance. The work advances ES for infinite-dimensional PDE systems and provides a practical control strategy for optimizing unknown static maps in flexible beam applications, with potential extensions to other boundary conditions and PDE–ODE hybrids.

Abstract

This paper presents the design and analysis of an extremum seeking (ES) controller for scalar static maps in the context of infinite-dimensional dynamics governed by the 1D Euler-Bernoulli (EB) beam Partial Differential Equation (PDE). The beam is actuated at one end (using position and moment actuators). The map's input is the displacement at the beam's uncontrolled end, which is subject to a sliding boundary condition. Notably, ES for this class of PDEs remains unexplored in the existing literature. To compensate for PDE actuation dynamics, we employ a boundary control law via a backstepping transformation and averaging-based estimates for the gradient and Hessian of the static map to be optimized. This compensation controller leverages a Schrödinger equation representation of the EB beam and adapts existing backstepping designs to stabilize the beam. Using the semigroup and averaging theory in infinite dimensions, we prove local exponential convergence to a small neighborhood of the unknown optimal point. Finally, simulations illustrate the effectiveness of the design in optimizing the unknown static map.

Paper Structure

This paper contains 18 sections, 1 theorem, 81 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Euler-Bernoulli Beam Mathematical Model
  4. Control Problem
  5. Extremum Seeking Boundary Control Design
  6. Demodulation Signals
  7. Additive Probing Signal
  8. Estimation Errors and PDE-Error Dynamics
  9. Euler-Bernoulli Beam Compensation via Backstepping Boundary Control
  10. Target System
  11. Invertibility of the Transformations
  12. Target System of the Euler-Bernoulli Beam Equation
  13. Control Laws
  14. Achieving Regulation to zero
  15. Implementable Extremum Seeking Control Law
  16. ...and 3 more sections

Key Result

Theorem 1

Consider the control system in Figure fig:euler, with control laws $U_{1}$ and $U_{2}$ given in eq:control_law_filtered-eq:control_law_filtered_2, respectively. There exists $\overline{c}^{*}>0$ such that, $\forall \overline{c}\geq \overline{c}^{*}$, $\exists \omega^{*}(\overline{c})>0$ such that, $ Furthermore where $r(\omega) = \frac{|a(\cosh(\sqrt{w})+\cos(\sqrt{w}))|}{2}$.

Figures (3)

  • Figure 1: Block diagram of the ES control loop applied to the EB beam problem.
  • Figure 2: The closed-loop response of the EB beam PDE with ES compensating controller.
  • Figure 3: Time evolution of the beam displacement.

Theorems & Definitions (3)

  • proof
  • Theorem 1
  • proof