Pole Placement for Delay Differential Equations With Time-Periodic Delays Using Galerkin Approximations

Author:

Kandala Shanti Swaroop1,Uchida Thomas K.2,Vyasarayani C. P.1

Affiliation:

1. Department of Mechanical and Aerospace Engineering, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, Telangana 502285, India

2. Department of Mechanical Engineering, University of Ottawa, 161 Louis-Pasteur, Ottawa, ON K1N 6N5, Canada

Abstract

Abstract Many practical systems have inherent time delays that cannot be ignored; thus, their dynamics are described using delay differential equations (DDEs). The Galerkin approximation method is one strategy for studying the stability of time-delay systems (TDS). In this work, we consider delays that are time-varying and, specifically, time-periodic. The Galerkin method can be used to obtain a system of ordinary differential equations (ODEs) from a second-order time-periodic DDE in two ways: either by converting the DDE into a second-order time-periodic partial differential equation (PDE) and then into a system of second-order ODEs, or by first expressing the original DDE as two first-order time-periodic DDEs, then converting into a system of first-order time-periodic PDEs, and finally converting into a first-order time-periodic ODE system. The difference between these two formulations in the context of control is presented in this paper. Specifically, we show that the former produces spurious Floquet multipliers at a spectral radius of 1. We also propose an optimization-based framework to obtain feedback gains that stabilize closed-loop control systems with time-periodic delays. The proposed optimization-based framework employs the Galerkin method and Floquet theory and is shown to be capable of stabilizing systems considered in the literature. Finally, we present experimental validation of our theoretical results using a rotary inverted pendulum apparatus with inherent sensing delays as well as additional time-periodic state-feedback delays that are introduced deliberately.

Funder

Department of Science and Technology, Ministry of Science and Technology

Publisher

ASME International

Subject

Applied Mathematics,Mechanical Engineering,Control and Systems Engineering,Applied Mathematics,Mechanical Engineering,Control and Systems Engineering

Reference39 articles.

1. Galerkin Approximations for Stability of Delay Differential Equations With Distributed Delays;ASME J. Comput. Nonlinear Dyn.,2015

2. Analysis of a System of Linear Delay Differential Equations;ASME J. Dyn. Syst., Meas., Control,2003

3. The Lambert W Function and the Spectrum of Some Multidimensional Time-Delay Systems;Automatica,2007

4. Survey on Analysis of Time Delayed Systems Via the Lambert W Function;Dyn. Contin. Discrete Impulsive Syst.,2007

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Simple Time-Periodic Delay Can Support Complex Dynamics;International Journal of Bifurcation and Chaos;2023-12-11

2. Floquet Theory for Linear Time-Periodic Delay Differential Equations Using Orthonormal History Functions;Journal of Computational and Nonlinear Dynamics;2023-06-14

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