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Initialization-Free Lie-Bracket Extremum Seeking in $\mathbb{R}^n$

Mahmoud Abdelgalil, Jorge Poveda

Abstract

Stability results for extremum seeking control in $\mathbb{R}^n$ have predominantly been restricted to local or, at best, semi-global practical stability. Extending semi-global stability results of extremum-seeking systems to unbounded sets of initial conditions often demands a stringent global Lipschitz condition on the cost function, which is rarely satisfied by practical applications. In this paper, we address this challenge by leveraging tools from higher-order averaging theory. In particular, we establish a novel second-order averaging result with \emph{global} (practical) stability implications. By leveraging this result, we characterize sufficient conditions on cost functions under which uniform global practical asymptotic stability can be established for a class of extremum-seeking systems acting on static maps. Our sufficient conditions include the case when the gradient of the cost function, rather than the cost function itself, satisfies a global Lipschitz condition, which covers quadratic cost functions. Our results are also applicable to vector fields that are not necessarily Lipschitz continuous at the origin, opening the door to non-smooth Lie-bracket ES dynamics. We illustrate all our results via different analytical and/or numerical examples.

Initialization-Free Lie-Bracket Extremum Seeking in $\mathbb{R}^n$

Abstract

Stability results for extremum seeking control in have predominantly been restricted to local or, at best, semi-global practical stability. Extending semi-global stability results of extremum-seeking systems to unbounded sets of initial conditions often demands a stringent global Lipschitz condition on the cost function, which is rarely satisfied by practical applications. In this paper, we address this challenge by leveraging tools from higher-order averaging theory. In particular, we establish a novel second-order averaging result with \emph{global} (practical) stability implications. By leveraging this result, we characterize sufficient conditions on cost functions under which uniform global practical asymptotic stability can be established for a class of extremum-seeking systems acting on static maps. Our sufficient conditions include the case when the gradient of the cost function, rather than the cost function itself, satisfies a global Lipschitz condition, which covers quadratic cost functions. Our results are also applicable to vector fields that are not necessarily Lipschitz continuous at the origin, opening the door to non-smooth Lie-bracket ES dynamics. We illustrate all our results via different analytical and/or numerical examples.
Paper Structure (24 sections, 15 theorems, 147 equations, 5 figures)

This paper contains 24 sections, 15 theorems, 147 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Dynamical Systems and Stability Notions
  5. On Global Stability via Second-Order Averaging
  6. A Global Practical Near-Identity Transformation
  7. Global Stability via Second-Order Averaging
  8. Applications to Extremum Seeking Systems
  9. Main Assumptions
  10. ES Dynamics with Linear Growth
  11. ES Dynamics with Bounded Control
  12. Proofs
  13. Proof of \ref{['prop:near_identity_transformation']}
  14. Proof of \ref{['thm:GUES_implies_GUPES_timevarying']}
  15. Proof of \ref{['cor:GUES_implies_GUPES_timevarying_1']}
  16. ...and 9 more sections

Key Result

Lemma 1

Let $\delta_2> \delta_1$. Then, the function $\varphi$ is $\mathcal{C}^\infty$ on $\mathbb{R}^n$, all of its derivatives have the compact support $[\delta_1,\delta_2]$, and it satisfies:

Figures (5)

  • Figure 1: Block diagram description of system \ref{['eq:es_system_open_loop']}. In the diagram, the matrix $B_i=[b_{i,1},\,b_{i,2}]$ multiplies the vector $u_i(J(x),\tau)=(u_{i,1}(J(x),\tau),\,u_{i,2}(J(x),\tau))$.
  • Figure 2: Visual depiction of $\varphi$ and the sets $\mathcal{M}_j$ for $j\in\{1,2,3\}$.
  • Figure 3: Visual depiction of the local effect of $\Psi$ on the integral curves of $f_{\varepsilon}$.
  • Figure 4: Numerical results for Example 1. The initial conditions are: $x(0)=(10^{6},-10^{6})$, $v(0)=(10^{3},-10^{3})$. The parameters are: $\varepsilon=1/\sqrt{8\pi}$, $\gamma_1=\frac{3}{4}$, $\gamma_2=1$.
  • Figure 5: Numerical results for Example 4 (left) and Example 5 (right). The insets in the top right of the figures depict the quasi-steady state in the vicinity of $x^\star$.

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • Remark 1
  • Proposition 1
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 25 more