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Prescribed-Time and Hyperexponential Concurrent Learning with Partially Corrupted Datasets: A Hybrid Dynamical Systems Approach

Daniel E. Ochoa, Jorge I. Poveda

TL;DR

The main result establishes convergence rates faster than any exponential while guaranteeing uniform global ultimate boundedness in the presence of disturbances, with an ultimate bound that shrinks to zero as the magnitude of measurement disturbances and corrupted data decreases.

Abstract

We introduce a class of concurrent learning (CL) algorithms designed to solve parameter estimation problems with convergence rates ranging from hyperexponential to prescribed-time while utilizing alternating datasets during the learning process. The proposed algorithm employs a broad class of dynamic gains, from exponentially growing to finite-time blow-up gains, enabling either enhanced convergence rates or user-prescribed convergence time independent of the dataset's richness. The CL algorithm can handle applications involving switching between multiple datasets that may have varying degrees of richness and potential corruption. The main result establishes convergence rates faster than any exponential while guaranteeing uniform global ultimate boundedness in the presence of disturbances, with an ultimate bound that shrinks to zero as the magnitude of measurement disturbances and corrupted data decreases. The stability analysis leverages tools from hybrid dynamical systems theory, along with a dilation/contraction argument on the hybrid time domains of the solutions. The algorithm and main results are illustrated via a numerical example.

Prescribed-Time and Hyperexponential Concurrent Learning with Partially Corrupted Datasets: A Hybrid Dynamical Systems Approach

TL;DR

The main result establishes convergence rates faster than any exponential while guaranteeing uniform global ultimate boundedness in the presence of disturbances, with an ultimate bound that shrinks to zero as the magnitude of measurement disturbances and corrupted data decreases.

Abstract

We introduce a class of concurrent learning (CL) algorithms designed to solve parameter estimation problems with convergence rates ranging from hyperexponential to prescribed-time while utilizing alternating datasets during the learning process. The proposed algorithm employs a broad class of dynamic gains, from exponentially growing to finite-time blow-up gains, enabling either enhanced convergence rates or user-prescribed convergence time independent of the dataset's richness. The CL algorithm can handle applications involving switching between multiple datasets that may have varying degrees of richness and potential corruption. The main result establishes convergence rates faster than any exponential while guaranteeing uniform global ultimate boundedness in the presence of disturbances, with an ultimate bound that shrinks to zero as the magnitude of measurement disturbances and corrupted data decreases. The stability analysis leverages tools from hybrid dynamical systems theory, along with a dilation/contraction argument on the hybrid time domains of the solutions. The algorithm and main results are illustrated via a numerical example.

Paper Structure

This paper contains 9 sections, 6 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Prescribed-Time and Hyperexponential Concurrent Learning with Switching datasets
  4. The Estimation Problem
  5. The "Data-Querying" Automaton
  6. Closed-Loop System and Main Result
  7. Analysis
  8. Numerical Example
  9. Conclusions

Key Result

lemma 1

(Dynamic Gains) Consider the family of gain ordinary differential equations (ODEs): where $F_{\mu,\ell}$ is given in eq:muFlowmaps. For each $\mu_0\in\mathbb{R}_{\geq1}$, the unique solutions to muODE from $\mu_0$ are given by: for all $t\in\left[0,T_{\mu_0,\ell}\right)$, where $T_{\mu_0,0}\coloneqq\infty$, $T_{\mu_0,\ell}\coloneqq\Upsilon\mu_0^{(1-\ell)/\ell}$ for $\ell>1$, $T_{\mu_0,\infty}\co

Figures (2)

  • Figure 1: Block diagram of the proposed Switching Prescribed-Time and Hiperexponential Concurrent Learning algorithm. Recorded measurements (data) are transmitted remotely yet they can be susceptible to tampering or corruption.
  • Figure 2: Comparison between standard concurrent learning with fixed dataset $\Delta_2$, and switching prescribed-time ($\ell=\infty$) and hyperexponential ($\ell=1$) concurrent Learning with informative data $\mathcal{Q}_s = \{1,2\}$, uninformative data $\mathcal{Q}_i=\{3\}$, and corrupt data $\mathcal{Q}_c=\{4\}$. The true parameter is $\theta^{\star} = (1,-2,1)$, and the prescribed-time $T_{\mu_0,\infty}=8$ for the $\ell=\infty$ case.

Theorems & Definitions (26)

  • lemma 1
  • remark 1
  • definition 1
  • remark 2
  • remark 3
  • definition 2
  • remark 4
  • definition 3
  • lemma 2
  • remark 5
  • ...and 16 more