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Unbiased Extremum Seeking for PDEs

Cemal Tugrul Yilmaz, Mamadou Diagne, Miroslav Krstic

Abstract

There have been recent efforts that combine seemingly disparate methods, extremum seeking (ES) optimization and partial differential equation (PDE) backstepping, to address the problem of model-free optimization with PDE actuator dynamics. In contrast to prior PDE-compensating ES designs, which only guarantee local stability around the extremum, we introduce unbiased ES that compensates for delay and diffusion PDE dynamics while ensuring exponential and unbiased convergence to the optimum. Our method leverages exponentially decaying/growing signals within the modulation/demodulation stages and carefully selected design parameters. The stability analysis of our designs relies on a state transformation, infinite-dimensional averaging, local exponential stability of the averaged system, local stability of the transformed system, and local exponential stability of the original system. Numerical simulations are presented to demonstrate the efficacy of the developed designs.

Unbiased Extremum Seeking for PDEs

Abstract

There have been recent efforts that combine seemingly disparate methods, extremum seeking (ES) optimization and partial differential equation (PDE) backstepping, to address the problem of model-free optimization with PDE actuator dynamics. In contrast to prior PDE-compensating ES designs, which only guarantee local stability around the extremum, we introduce unbiased ES that compensates for delay and diffusion PDE dynamics while ensuring exponential and unbiased convergence to the optimum. Our method leverages exponentially decaying/growing signals within the modulation/demodulation stages and carefully selected design parameters. The stability analysis of our designs relies on a state transformation, infinite-dimensional averaging, local exponential stability of the averaged system, local stability of the transformed system, and local exponential stability of the original system. Numerical simulations are presented to demonstrate the efficacy of the developed designs.
Paper Structure (13 sections, 3 theorems, 55 equations, 4 figures)

This paper contains 13 sections, 3 theorems, 55 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Unbiased ES with Delay
  4. Excitation signals and gradient/Hessian estimates
  5. Parameter estimator design and error dynamics
  6. Stability analysis
  7. Unbiased ES with Diffusion PDE
  8. Perturbation signal
  9. Parameter estimator design and error dynamics
  10. Stability analysis
  11. Numerical Simulation
  12. Conclusion
  13. Proof of Lemma \ref{['lemS']}

Key Result

Theorem 1

Let Assumption assquad holds and the parameters satisfy paramcond. Then, there exists $\bar{\omega}$ and for any $\omega > \bar{\omega}$, the closed-loop system closeddelay1--closeddelay2 is exponentially stable at the origin in the sense of the norm Furthermore, the input $\theta(t)$ and output $y(t)$ exponentially converge to $\theta^*$ and $y^*$, respectively.

Figures (4)

  • Figure 1: Unbiased ES with delay compensator. The design employs exponentially growing multiplicative signals, $M(t)=\frac{2}{a}e^{\lambda t}\sin(\omega t)$, and $N(t)=-\frac{8}{a^2}e^{2\lambda t}\cos(2\omega t)$, as well as exponentially decaying dither signal $\dot{S}(t+D)=\frac{d}{dt}\left(e^{-\lambda (t+D)} a \sin(\omega (t+D))\right)$ to achieve exponential and unbiased convergence to the optimum $\theta^*$ at the rate of the user-defined $\lambda>0$.
  • Figure 2: Unbiased ES with diffusion PDE compensator. The design requires feedback of $\theta$, uses the same excitation signals, $M(t)$ and $N(t)$ as in Fig. \ref{['ESDeBlock']}, and employs a properly designed perturbation signal, $\dot{S}(t)$.
  • Figure 3: The trajectory of input $\theta$ resulting from the application of the delay-compensated uES \ref{['delayuES']} in the presence of a delay of $D=5$ seconds.
  • Figure 4: The trajectory of input $\theta$ resulting from the application of the diffusion PDE-compensated uES \ref{['updHeat']}.

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Lemma 1
  • Theorem 2
  • proof