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

Kandala, Shanti Swaroop and Uchida, Thomas K. and Vyasarayani, Chandrika Prakash (2021) Pole Placement for Delay Differential Equations With Time-Periodic Delays Using Galerkin Approximations. Journal of Computational and Nonlinear Dynamics, 16 (9). pp. 1-10. ISSN 1555-1415

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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. Copyright © 2021 by ASME.

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IITH Creators:

IITH Creators	ORCiD
Vyasarayani, Chandrika Prakash	http://orcid.org/0000-0002-3396-0484

Item Type:

Article

Additional Information:

• C.P.V. gratefully acknowledges the Department of Science and Technology for funding this research through the Inspire fellowship (Grant No. DST/INSPIRE/04/2014/000972). The funders had no role in study design, data collection and anal-ysis, decision to publish, or preparation of the paper.

Uncontrolled Keywords:

Control system; Delay differential equation; Floquet theory; Galerkin approximation; Stability; Time-delay system

Subjects:

Physics > Mechanical and aerospace

Divisions:

Department of Mechanical & Aerospace Engineering

Depositing User:

. LibTrainee 2021

Date Deposited:

14 Sep 2022 14:28

Last Modified:

14 Sep 2022 14:28