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Extremum Seeking Control for Wave-PDE Actuation with Distributed Effects

Elisio Juvenal Muchave, Pedro Henrique Silva Coutinho, Tiago Roux Oliveira, Miroslav Krstić

TL;DR

This work addresses gradient-based extremum seeking control for systems whose actuation dynamics are governed by distributed wave PDEs. The authors solve a trajectory-generation problem to inject a probing signal and design a backstepping-inspired boundary controller to compensate wave-propagation effects, establishing exponential stability of the averaged closed-loop system. By applying averaging theory for infinite-dimensional systems, they prove convergence to a neighborhood of an unknown extremum, and they provide a real-time implementable control law with a low-pass filter. Numerical simulations demonstrate effective convergence of the output to the unknown optimum and validate the boundary ESC approach for distributed wave PDE actuation.

Abstract

This paper deals with the gradient-based extremum seeking control (ESC) with actuation dynamics governed by distributed wave partial differential equations (PDEs). To achieve the control objective of real-time optimization for this class of infinite-dimensional systems, we first solve the trajectory generation problem to re-design the additive perturbation signal of the ESC system. Then, we develop a boundary control law through the backstepping method to compensate for the wave PDE with distributed effects, which ensures the exponential stability of the average closed-loop system by means of a Lyapunov-based analysis. At last, by employing the averaging theory for infinite-dimensional systems, we prove that the closed-loop trajectories converge to a small neighborhood surrounding the optimal point. Numerical simulations are presented to illustrate the effectiveness of the proposed method.

Extremum Seeking Control for Wave-PDE Actuation with Distributed Effects

TL;DR

This work addresses gradient-based extremum seeking control for systems whose actuation dynamics are governed by distributed wave PDEs. The authors solve a trajectory-generation problem to inject a probing signal and design a backstepping-inspired boundary controller to compensate wave-propagation effects, establishing exponential stability of the averaged closed-loop system. By applying averaging theory for infinite-dimensional systems, they prove convergence to a neighborhood of an unknown extremum, and they provide a real-time implementable control law with a low-pass filter. Numerical simulations demonstrate effective convergence of the output to the unknown optimum and validate the boundary ESC approach for distributed wave PDE actuation.

Abstract

This paper deals with the gradient-based extremum seeking control (ESC) with actuation dynamics governed by distributed wave partial differential equations (PDEs). To achieve the control objective of real-time optimization for this class of infinite-dimensional systems, we first solve the trajectory generation problem to re-design the additive perturbation signal of the ESC system. Then, we develop a boundary control law through the backstepping method to compensate for the wave PDE with distributed effects, which ensures the exponential stability of the average closed-loop system by means of a Lyapunov-based analysis. At last, by employing the averaging theory for infinite-dimensional systems, we prove that the closed-loop trajectories converge to a small neighborhood surrounding the optimal point. Numerical simulations are presented to illustrate the effectiveness of the proposed method.
Paper Structure (14 sections, 3 theorems, 66 equations, 4 figures)

This paper contains 14 sections, 3 theorems, 66 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Extremum Seeking Control Formulation
  3. Nonlinear Maps with Wave-PDE Actuation and Distributed Effects
  4. Demodulation Signals
  5. Trajectory Generation for the Motion Planning of the Additive Perturbation Signal
  6. Estimation Errors and Error Dynamics
  7. Control Objective
  8. Boundary Control and Closed-Loop System
  9. Backstepping Design in Infinite Dimensions
  10. Average Closed-Loop System
  11. Implementable Control Law
  12. Stability and Convergence Analysis
  13. Numerical Results
  14. Conclusion

Key Result

Lemma 1

If the frequency $\omega > 0$ satisfies then, the solution of the trajectory generation problem defined by equations eq:28--eq:25 is In particular, the additive perturbation signal $S(t)$ is

Figures (4)

  • Figure 1: Extremum seeking control system with distributed wave PDEs.
  • Figure 2: Closed-loop simulation of the ESC system with actuation dynamics governed by the wave PDE with distributed effect using the controller \ref{['filteredcontrolreal2026']}.
  • Figure 3: Signals $\theta(t)$ and $\Theta(t)$ of the closed-loop simulation.
  • Figure 4: Convergence of the state $\alpha(x,t)$ in a three-dimensional space. The signal in red is $\theta(t)=\alpha(D,t)$, while the signal in blue is $\alpha(0,t)$, both reaching a neighborhood of $\Theta^\ast = 2$.

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Theorem 1