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Parameterization of Seed Functions for Equivalent Representations of Time-Varying Delay Systems

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

Abstract

Abel's classic transformation shows that any well-posed system with time-varying delay is equivalent to a parameter-varying system with fixed delay. The existence of such a parameter-varying constant delay representation then simplifies the problems of stability analysis and optimal control. Unfortunately, the method for construction of such transformations has been ad-hoc -- requiring an iterative time-stepping approach to constructing the transformation beginning with a seed function subject to boundary-value constraints. Moreover, a poor choice of seed function often results in a constant delay representation with large time-variations in system parameters -- obviating the benefits of such a representation. In this paper, we show how the set of all feasible seed functions can be parameterized using a basis for $L_2$. This parameterization is then used to search for seed functions for which the corresponding time-transformation results in smaller parameter variation. The parameterization of admissible seed functions is illustrated with numerical examples that contrast how well-chosen and poorly chosen seed functions affect the boundedness of a time transformation.

Parameterization of Seed Functions for Equivalent Representations of Time-Varying Delay Systems

Abstract

Abel's classic transformation shows that any well-posed system with time-varying delay is equivalent to a parameter-varying system with fixed delay. The existence of such a parameter-varying constant delay representation then simplifies the problems of stability analysis and optimal control. Unfortunately, the method for construction of such transformations has been ad-hoc -- requiring an iterative time-stepping approach to constructing the transformation beginning with a seed function subject to boundary-value constraints. Moreover, a poor choice of seed function often results in a constant delay representation with large time-variations in system parameters -- obviating the benefits of such a representation. In this paper, we show how the set of all feasible seed functions can be parameterized using a basis for . This parameterization is then used to search for seed functions for which the corresponding time-transformation results in smaller parameter variation. The parameterization of admissible seed functions is illustrated with numerical examples that contrast how well-chosen and poorly chosen seed functions affect the boundedness of a time transformation.
Paper Structure (11 sections, 3 theorems, 49 equations, 2 figures)

This paper contains 11 sections, 3 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Equivalence Between Variable- and Fixed-Delay DDEs
  4. Time Transformations
  5. Recursive Construction of the Time Transformation
  6. Parameterization of Seed Functions
  7. Numerical Comparison of Seed Functions
  8. Example 1
  9. Example 2
  10. Analysis
  11. Conclusion

Key Result

Lemma 1

Suppose $\tau^*\in \mathbb{R}_+$ and $\tau \in \mathcal{C}^1(\mathbb{R}, \mathbb{R}_+)$ with $\dot{\tau}(t) < 1$. Let $h \in \mathcal{C}^1([-\tau^*,\infty), [-\tau(0), \infty))$ be a strictly increasing unbounded function, with $h(0)=0$ and where For any $\zeta \in \mathcal{C}[-\tau(0),0],$ if $x(t)$ satisfies then $\bar{x}(\lambda)= x(h(\lambda))$ and $\bar{\zeta} = \zeta (h(\lambda))$ satisfy

Figures (2)

  • Figure 1: The derivative $h'(\lambda)$ of the time transformation for (a) quadratic seed, (b) affine plus sinusoidal seed, and (c) exponential seed, with different maximum values of $h'$ for horizon of 100-- illustrating the impact of seed function choice on stability margins for a delay $\tau(t) = (\frac{1}{2\pi} - 0.001) \sin(2\pi t) + (\frac{1}{2\pi} + 0.001)$.
  • Figure 2: The derivative $h'(\lambda)$ of the time transformation for (a) quadratic seed, (b) affine plus sinusoidal seed, and (c) exponential seed, with different maximum values of $h'$ for horizon of 100 for the delay $\tau(t)=1+0.3sin(t)$.

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Definition 1
  • Theorem 2
  • proof
  • Remark 1
  • Corollary 2.1
  • proof