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Adaptive Extremum Seeking Control via the RMSprop Optimizer

Patrick McNamee, Zahra Nili Ahmadabadi

TL;DR

This work proposes the use of the RMSprop optimizer for ESCs as RMSprop is an adaptive gradient-based optimizer which attempts to have a normalized convergence rate in all parameters.

Abstract

Extremum Seeking Control (ESC) is a well-known set of continuous time algorithms for model-free optimization of a cost function. One issue for ESCs is the convergence rates of parameters to extrema of unknown cost functions. The local convergence rate depends on the second, or sometimes higher, order derivatives of the unknown cost function. To mitigate this dependency, we propose the use of the RMSprop optimizer for ESCs as RMSprop is an adaptive gradient-based optimizer which attempts to have a normalized convergence rate in all parameters. Practical stability results are given for this RMSprop ESC (RMSpESC). In particular notability, the proof of practical stability uses Lyapunov function based on observed contracting, attractive sets. Versions of this Lyapunov function could be applied to other areas of applications, in particular for interconnected systems.

Adaptive Extremum Seeking Control via the RMSprop Optimizer

TL;DR

This work proposes the use of the RMSprop optimizer for ESCs as RMSprop is an adaptive gradient-based optimizer which attempts to have a normalized convergence rate in all parameters.

Abstract

Extremum Seeking Control (ESC) is a well-known set of continuous time algorithms for model-free optimization of a cost function. One issue for ESCs is the convergence rates of parameters to extrema of unknown cost functions. The local convergence rate depends on the second, or sometimes higher, order derivatives of the unknown cost function. To mitigate this dependency, we propose the use of the RMSprop optimizer for ESCs as RMSprop is an adaptive gradient-based optimizer which attempts to have a normalized convergence rate in all parameters. Practical stability results are given for this RMSprop ESC (RMSpESC). In particular notability, the proof of practical stability uses Lyapunov function based on observed contracting, attractive sets. Versions of this Lyapunov function could be applied to other areas of applications, in particular for interconnected systems.
Paper Structure (8 sections, 4 theorems, 34 equations, 1 figure)

This paper contains 8 sections, 4 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. RMSprop Extremum Seeking Control
  4. Quadratic Functions
  5. Semiglobal Practical Stability
  6. Numerical Example
  7. Conclusions
  8. Properties of Estimates in the Average System

Key Result

Theorem 1

For a cost function ${J}:\mathbb{R}^n \to \mathbb{R}$ satisfying the Assumptions asmp:continuously-differentiable-asmp:cost-function-radially-unbounded, the RMSpESC defined by Eqs eq:rmsprop:parameter-ode-eq:rmsprop:washoutfilter is sGPUAS to the point $\left(\theta_{ *},0,{J}\left(\theta_{ *}\right

Figures (1)

  • Figure 1: Parameter estimates for the example quartic scalar cost function with different initial washout filter states $\xi_0$. Simulations parameters where $a=0.02$, $\omega=10$, $\omega_{\xi}=1$, $\omega_{l} = \frac{1}{4}$, $\varepsilon=0.05$, and ${k}=1$. The initial conditions for the other states were $\hat{\theta}_0 = 2$ and $\hat{v}_0=0.81$.

Theorems & Definitions (11)

  • Theorem 1: RMSpropESC is sGPUAS
  • Lemma 1
  • proof
  • Remark 1
  • proof
  • proof : Proof of Theorem \ref{['thm:sgpuas']}
  • Lemma 2
  • proof
  • Corollary 1
  • proof
  • ...and 1 more