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Extremum Seeking Control for Scalar Maps with Distributed Diffusion PDEs

Pedro Henrique Silva Coutinho, Tiago Roux Oliveira, Miroslav Krstic

TL;DR

This work develops gradient extremum seeking for static scalar maps when the actuator dynamics are governed by distributed diffusion PDEs. It combines a backstepping-inspired compensation controller for the PDE with a carefully designed trajectory for the perturbation signal and an averaging-based estimation of the gradient and Hessian, achieving real-time optimization. The authors prove local exponential stability of the averaged error dynamics and, using infinite-dimensional averaging theory, show convergence to a small neighborhood of the extremum with high perturbation frequency ω. Numerical simulations illustrate effective tracking of the optimum and validate the proposed PDE-ESC framework, highlighting its applicability to systems with distributed actuation such as heat diffusion processes.

Abstract

This paper deals with the gradient extremum seeking control for static scalar maps with actuators governed by distributed diffusion partial differential equations (PDEs). To achieve the real-time optimization objective, we design a compensation controller for the distributed diffusion PDE via backstepping transformation in infinite dimensions. A further contribution of this paper is the appropriate motion planning design of the so-called probing (or perturbation) signal, which is more involved than in the non-distributed counterpart. Hence, with these two design ingredients, we provide an averaging-based methodology that can be implemented using the gradient and Hessian estimates. Local exponential stability for the closed-loop equilibrium of the average error dynamics is guaranteed through a Lyapunov-based analysis. By employing the averaging theory for infinite-dimensional systems, we prove that the trajectory converges to a small neighborhood surrounding the optimal point. The effectiveness of the proposed extremum seeking controller for distributed diffusion PDEs in cascade of nonlinear maps to be optimized is illustrated by means of numerical simulations.

Extremum Seeking Control for Scalar Maps with Distributed Diffusion PDEs

TL;DR

This work develops gradient extremum seeking for static scalar maps when the actuator dynamics are governed by distributed diffusion PDEs. It combines a backstepping-inspired compensation controller for the PDE with a carefully designed trajectory for the perturbation signal and an averaging-based estimation of the gradient and Hessian, achieving real-time optimization. The authors prove local exponential stability of the averaged error dynamics and, using infinite-dimensional averaging theory, show convergence to a small neighborhood of the extremum with high perturbation frequency ω. Numerical simulations illustrate effective tracking of the optimum and validate the proposed PDE-ESC framework, highlighting its applicability to systems with distributed actuation such as heat diffusion processes.

Abstract

This paper deals with the gradient extremum seeking control for static scalar maps with actuators governed by distributed diffusion partial differential equations (PDEs). To achieve the real-time optimization objective, we design a compensation controller for the distributed diffusion PDE via backstepping transformation in infinite dimensions. A further contribution of this paper is the appropriate motion planning design of the so-called probing (or perturbation) signal, which is more involved than in the non-distributed counterpart. Hence, with these two design ingredients, we provide an averaging-based methodology that can be implemented using the gradient and Hessian estimates. Local exponential stability for the closed-loop equilibrium of the average error dynamics is guaranteed through a Lyapunov-based analysis. By employing the averaging theory for infinite-dimensional systems, we prove that the trajectory converges to a small neighborhood surrounding the optimal point. The effectiveness of the proposed extremum seeking controller for distributed diffusion PDEs in cascade of nonlinear maps to be optimized is illustrated by means of numerical simulations.
Paper Structure (13 sections, 3 theorems, 59 equations, 7 figures)

This paper contains 13 sections, 3 theorems, 59 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. A motivating example
  4. The standard gradient ESC without PDEs
  5. Gradient ESC with Distributed Diffusion PDEs
  6. Trajectory Generation to the Additive Perturbation Signal
  7. Estimation Error Dynamics
  8. Main Results
  9. Distributed Diffusion PDE Compensation Controller
  10. Controller Design for the Average System
  11. Stability and Convergence Analysis
  12. Numerical Results
  13. Conclusion

Key Result

Lemma 1

The solution of the trajectory generation problem perturbation_beta--initial_cond_beta is where and with

Figures (7)

  • Figure 1: Geometric representation of the temperature profile modeled by the diffusion PDE in \ref{['eq:diffusion-1']}--\ref{['eq:diffusion-2']}, where $\theta(t)$ is the heat/cold flow source.
  • Figure 2: Standard ESC scheme without PDEs.
  • Figure 3: ESC scheme with distributed diffusion PDE governing the actuator dynamics.
  • Figure 4: Illustration of the signals involved in Lemma \ref{['lem:Sdesign']}.
  • Figure 5: Closed-loop simulation of the ESC system with actuation dynamics governed by the distributed diffusion PDE with the controller \ref{['eq:controlador_non_average_3']}.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Lemma 1
  • Theorem 1
  • Theorem 2