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Non-Local Extremum Seeking Based on the Divergence Theorem

Raik Suttner, Christian Ebenbauer, Sergey Dashkovskiy

Abstract

We propose a new design strategy for extremum seeking control for a multi-dimensional single-integrator system in the presence of local extrema. The proposed method employs suitably designed sinusoidal dither signals, which force the single-integrator to a spherical motion. Over time, this spherical motion gives approximate access to an integral of the objective function over a sphere. Using the divergence theorem, we identify the integral over the sphere as the gradient of an integral over the enclosed ball. This integral over the ball defines a locally averaged objective function. The proposed extremum seeking method drives the system state into the gradient direction of the averaged objective function. Such a local average of the objective function can eliminate undesired local extrema and is therefore beneficial for global optimization. Under the assumption that the averaged objective function has no undesired critical points, we prove practical asymptotic stability of the closed-loop system. Our theoretical analysis takes sufficiently small $L_\infty$-measurement errors of the objective function into account.

Non-Local Extremum Seeking Based on the Divergence Theorem

Abstract

We propose a new design strategy for extremum seeking control for a multi-dimensional single-integrator system in the presence of local extrema. The proposed method employs suitably designed sinusoidal dither signals, which force the single-integrator to a spherical motion. Over time, this spherical motion gives approximate access to an integral of the objective function over a sphere. Using the divergence theorem, we identify the integral over the sphere as the gradient of an integral over the enclosed ball. This integral over the ball defines a locally averaged objective function. The proposed extremum seeking method drives the system state into the gradient direction of the averaged objective function. Such a local average of the objective function can eliminate undesired local extrema and is therefore beneficial for global optimization. Under the assumption that the averaged objective function has no undesired critical points, we prove practical asymptotic stability of the closed-loop system. Our theoretical analysis takes sufficiently small -measurement errors of the objective function into account.
Paper Structure (13 sections, 5 theorems, 69 equations, 8 figures)

This paper contains 13 sections, 5 theorems, 69 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem statement and underlying idea
  3. Control law and closed-loop system
  4. Main result
  5. Numerical examples
  6. Example 1
  7. Example 2
  8. Example 3
  9. Conclusions and outlook
  10. Proof of Proposition \ref{['proposition:1']}
  11. Approximation for large omega
  12. Approximation for large k
  13. Proof of Proposition \ref{['proposition:1']}

Key Result

Proposition 1

Let $\bar{K}$ and $\tilde{K}$ be compact subsets of $\mathbb{R}^n$ such that $\bar{K}$ is contained in the interior of $\tilde{K}$. Let $\delta,\rho>0$ such that $|J(\tilde{x})+ a\,s| + \delta\leq\rho$ for every $\tilde{x}\in\tilde{K}$ and every $s\in\partial\mathbb{B}$. Then, for every arbitrary la and $|\eta(t)|\leq\rho$ for every $t\in[t_0,t_0+T]$.

Figures (8)

  • Figure 1: Plot of the radially symmetric objective function $J$ in \ref{['eq:03']}. This function attains its global maximum value at the origin. In addition, local extrema occur in spherical sets around the origin.
  • Figure 2: Illustration of spherical coordinates in dimension $n=3$. The rectangle on the left-hand side is the set $W$ in \ref{['eq:07']}. The restriction of the map $\phi$ in \ref{['eq:06']} to the set $W$ parametrizes the entire sphere up to a set of measure zero.
  • Figure 3: Plots of the curve $U_k$ in \ref{['eq:12']} for $k=3,4$ and $n=3$.
  • Figure 4: Sketch of the proposed control scheme. The method also works without the high-pass filter.
  • Figure 5: The three plots are generated for the objective function $J$ in \ref{['eq:03']} and dimension $n=3$. The transformed closed-loop system \ref{['eq:18']} is considered for $a=b=1$, $\omega=\frac{1}{20}$, $k=4$, and initial condition $\tilde{x}(0)=[0,0,2.62]^\top$, $\eta(0)=0$. The left-most plot shows \ref{['eq:19']} as a function of $t\in[0,\frac{2\pi}{\omega}]$. The center plot shows \ref{['eq:20']} for $t=0$ as a function of $\tau\in[0,2\pi]$. The right-most plot shows \ref{['eq:21']} for $t=0$ as a function of $s\in\partial\mathbb{B}$. The different colors only have the intention to improve the visualization of the three-dimensional objects.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Remark 1
  • Proposition 1
  • Remark 2
  • Proposition 2
  • Theorem 1
  • Remark 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 3 more