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Technical Brief

Analysis of Multidimensional Time Delay Systems Using Lambert W Function

[+] Author and Article Information
Niraj Choudhary

Department of Electrical Engineering,
Indian Institute of Technology Delhi,
New Delhi 110016, India
e-mail: niraj.choudhary@ee.iitd.ac.in

Janardhanan Sivaramakrishnan

Department of Electrical Engineering,
Indian Institute of Technology Delhi,
New Delhi 110016, India
e-mail: janas@ee.iitd.ac.in

Indra Narayan Kar

Department of Electrical Engineering,
Indian Institute of Technology Delhi,
New Delhi 110016, India
e-mail: ink@ee.iitd.ac.in

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received November 11, 2016; final manuscript received May 10, 2017; published online August 8, 2017. Assoc. Editor: Davide Spinello.

J. Dyn. Sys., Meas., Control 139(11), 114506 (Aug 08, 2017) (6 pages) Paper No: DS-16-1549; doi: 10.1115/1.4036874 History: Received November 11, 2016; Revised May 10, 2017

In this note, the analysis of time delay systems (TDSs) using Lambert W function approach is reassessed. A common canonical (CC) form of time delay systems is defined. We extended the recent results of Cepeda–Gomez and Michiels (2015, “Some Special Cases in the Stability Analysis of Multi-Dimensional Time-Delay Systems Using the Matrix Lambert W Function,” Automatica, 53, pp. 339–345) for second-order into nth order system. The eigenvalues of a time delay system are either real or complex conjugate pairs and therefore, the whole eigenspectrum can be associated with only two real branches of the Lambert W function. A new class of time delay systems is characterized to extend the applicability of the above-said method. Moreover, this approach has been exploited to design a controller which places a subset of eigenvalues at desired locations. Stability is guaranteed by using a new algorithm developed in this paper, which is based on the Nyquist plot. The approach is validated through numerical examples.

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Figures

Grahic Jump Location
Fig. 1

Eigenspectrum of the system in the Example 1

Grahic Jump Location
Fig. 2

Schematic of a two degrees-of-freedom (2DOF) mass-spring-damper system

Grahic Jump Location
Fig. 3

Plot of p(jω)/(1+jω)4 for the system in Example 2

Grahic Jump Location
Fig. 4

Eigenspectrum of the closed-loop system in Example 2

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