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Time-Transformation-Based Analysis of Systems with Periodic Delay via Perturbative Expansion

Jungbae Chun, Sengiyumva Kisole, Matthew M. Peet, Peter Seiler

Abstract

It is difficult to analyze the stability of systems with time-varying delays. One approach is to construct a time-transformation that converts the system into a form with a constant delay but with a time-varying scalar appearing in the system matrices. The stability of this transformed system can then be analyzed using methods to bound the effect of the time-varying scalar. One issue is that this transformation is non-unique and requires the solution of an Abel equation. A specific time-transformation typically must be computed numerically. We address this issue by computing an explicit, although approximate, time-transformation for systems where the delay has a constant plus small periodic term. We use a perturbative expansion to construct our explicit solutions. We provide a simple numerical example to illustrate the approach. We also demonstrate the use of this time-transformation to analyze stability of the system with this class of periodic delays.

Time-Transformation-Based Analysis of Systems with Periodic Delay via Perturbative Expansion

Abstract

It is difficult to analyze the stability of systems with time-varying delays. One approach is to construct a time-transformation that converts the system into a form with a constant delay but with a time-varying scalar appearing in the system matrices. The stability of this transformed system can then be analyzed using methods to bound the effect of the time-varying scalar. One issue is that this transformation is non-unique and requires the solution of an Abel equation. A specific time-transformation typically must be computed numerically. We address this issue by computing an explicit, although approximate, time-transformation for systems where the delay has a constant plus small periodic term. We use a perturbative expansion to construct our explicit solutions. We provide a simple numerical example to illustrate the approach. We also demonstrate the use of this time-transformation to analyze stability of the system with this class of periodic delays.
Paper Structure (10 sections, 3 theorems, 51 equations, 3 figures)

This paper contains 10 sections, 3 theorems, 51 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Main result: Perturbative expansions for $h$
  4. First-Order Expansion
  5. Second-Order Expansion
  6. Pipeline for the Stability Analysis
  7. Illustrative Examples
  8. Example With Sinusoidal Delays
  9. Numerical Example
  10. Conclusion and Future work

Key Result

Theorem 1

Let $\tau \in C^1(\mathbb{R}, \mathbb{R}_{+})$ be a bounded function with $\dot{\tau}(t) < 1$ for all $t \in \mathbb{R}$. For a given $\tau^* \in \mathbb{R}_{+}$, there exists a strictly increasing, invertible function $h \in C^1([-\tau^*, \infty), \mathbb{R})$ which satisfies Moreover, $x$ satisfies eq:DDEwithTVdelay for a given initial function $x_0 \in C([-\tau(0),0], \mathbb{R}^{n_x})$ if and

Figures (3)

  • Figure 1: First-order (solid blue) and second-order (dashed red) approximations of $\dot{h}(\lambda)$ with $\tau^* = \tau_0 = 3$, $\omega = 5$, and $\epsilon=0.01$. The numerical fit (dashed black) is also shown.
  • Figure 2: First-order (solid blue) and second-order (dashed red) approximations of $\dot{h}(\lambda)$ with $\tau^* = \tau_0 = 3$, $\omega = 5$, and $\epsilon=0.1$. The numerical fit (dashed black) is also shown.
  • Figure 3: Error $e$, the difference between the approximation and an actual time transformation, versus $\epsilon$.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof