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Global Extremum Seeking With Double Integrators in the Presence of Local Extrema

Raik Suttner, Christian Ebenbauer, Sergey Dashkovskiy

Abstract

We study the problem of global extremum seeking in the presence of local extrema. We investigate two different perturbation-based methods: 1) a well-known classical extremum seeking scheme for steady-state output optimization, and 2) a source seeking scheme for a two-dimensional point mass. In each of these two scenarios, the closed-loop system involves a damped double integrator subject to an oscillatory force. An averaging analysis reveals that the respective averaged system is again a damped double integrator, but now subject to a potential force. The potential force is given by the gradient of a locally averaged objective function. Such a function is less prone to have undesired local extrema and is therefore better suited for global optimization. We provide sufficient conditions for semi-global practical uniform asymptotic stability of the closed-loop systems. The sufficient conditions only involve assumptions on the averaged objective function but not the original one.

Global Extremum Seeking With Double Integrators in the Presence of Local Extrema

Abstract

We study the problem of global extremum seeking in the presence of local extrema. We investigate two different perturbation-based methods: 1) a well-known classical extremum seeking scheme for steady-state output optimization, and 2) a source seeking scheme for a two-dimensional point mass. In each of these two scenarios, the closed-loop system involves a damped double integrator subject to an oscillatory force. An averaging analysis reveals that the respective averaged system is again a damped double integrator, but now subject to a potential force. The potential force is given by the gradient of a locally averaged objective function. Such a function is less prone to have undesired local extrema and is therefore better suited for global optimization. We provide sufficient conditions for semi-global practical uniform asymptotic stability of the closed-loop systems. The sufficient conditions only involve assumptions on the averaged objective function but not the original one.
Paper Structure (11 sections, 4 theorems, 44 equations, 5 figures)

This paper contains 11 sections, 4 theorems, 44 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A new interpretation of a classical extremum seeking scheme
  4. Problem statement and control law
  5. Analysis and theoretical results
  6. Simulation example
  7. Global source seeking with a two-dimensional double integrator
  8. Problem statement and control law
  9. Analysis and theoretical results
  10. Simulation example
  11. Conclusions and outlook

Key Result

Theorem 1

Suppose that assumption:1assumption:2 are satisfied. Suppose there exists $\bar{\theta}_a^\ast\in\mathbb{R}$ such that the double integrator eq:16:a is GAS w.r.t. $[\bar{\theta}_a^\ast,0]^\top$. Then the extremum seeking system eq:09 is SGPUAS w.r.t. $[l(\bar{\theta}_a^\ast),\bar{\theta}_a^\ast,0,z(

Figures (5)

  • Figure 1: A classical extremum seeking scheme from Krstic2000.
  • Figure 2: Plot of $\bar{\psi}_a$ (blue) in \ref{['eq:19']} and a trajectory of the state component $\hat{\theta}$ (green) in \ref{['eq:06:b']} for $a=0.4$ (left) and $a=0.7$ (right).
  • Figure 3: Plot of the averaged steady-state output function $\bar{\psi}_a$ in \ref{['eq:19']}. The trajectory in green represents the state component $\hat{\theta}$ in \ref{['eq:06:b']} with constant $a>0$ replaced by the amplitude dynamics $\dot{a}(t)=-\epsilon^2\,a(t)$, $a(0)=1$.
  • Figure 4: Left: Plot of the signal $\psi=\lim_{a\downarrow0}\bar{\psi}_a$ given by \ref{['eq:36']}. Center and Right: Plot of the averaged signal function $\bar{\psi}_a$ (red) given by \ref{['eq:35']} and a trajectory of the double integrator point (green) given by \ref{['eq:25:b']}, \ref{['eq:25:c']} for $a=1/2$ (center) and $a=1$ (right).
  • Figure :

Theorems & Definitions (14)

  • Definition 1
  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Proposition 1
  • Example 1
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 4 more