0
Research Papers

Galerkin Approximations for Retarded Delay Differential Equations With State-Dependent Delays

[+] Author and Article Information
C. P. Vyasarayani

Department of Mechanical Engineering,
Indian Institute of Technology Hyderabad,
Ordnance Factory Estate,
Andhra Pradesh 502205, India
e-mail: vcprakash@iith.ac.in

Adarsh Gupta

Indian Oil Corporation Limited,
Western Region Pipelines,
Jaipur, Rajasthan 303901, India

John McPhee

Systems Design Engineering,
University of Waterloo,
Waterloo, ON N2L 3G1, Canada

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 15, 2012; final manuscript received July 12, 2013; published online August 23, 2013. Assoc. Editor: Sean Brennan.

J. Dyn. Sys., Meas., Control 135(6), 061006 (Aug 23, 2013) (6 pages) Paper No: DS-12-1015; doi: 10.1115/1.4025152 History: Received January 15, 2012; Revised July 12, 2013

In this work, approximations for state dependent delay differential equations (DDEs) are developed using Galerkin's approach. The DDE is converted into an equivalent partial differential equation (PDE) with a moving boundary, where the length of the domain dependents on the solution of the PDE. The PDE is further reduced into a finite number of ordinary differential equations (ODEs) using Galerkin's approach with time dependent basis functions. The nonlinear boundary condition is represented by a Lagrange multiplier, whose expression is derived in closed form. We also demonstrate the validity of the developed method by comparing the numerical solution of the ODEs to that of the original DDE.

FIGURES IN THIS ARTICLE
<>
Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Fig. 1

Representation of a DDE with constant delay as a boundary control problem of advection equation (PDE)

Grahic Jump Location
Fig. 2

Representation of a DDE with state dependent delay as a boundary control problem of advection equation (PDE) with a moving boundary

Grahic Jump Location
Fig. 3

Time response of z(t) of Eq. (24) obtained from direct numerical integration as compared to a0(t) obtained from Galerkin approximation with (a) N = 3 and (b) N = 5

Grahic Jump Location
Fig. 4

Maximum value of absolute error max|a0(t)-z(t)| for increasing N

Grahic Jump Location
Fig. 5

(a) Time response of z(t) of Eq. (25) obtained from direct numerical integration as compared to a0(t) obtained from Galerkin approximation with N = 6. (b) Error between approximate and actual numerical solution.

Grahic Jump Location
Fig. 6

Maximum value of absolute error |a0(t)-z(t)| for frequencies 2≤ω≤10 for N = 6

Grahic Jump Location
Fig. 7

(a) Plant output z(t) of Eq. (25) for ω = 8 as compared to a0(t) obtained from the observer with N = 6. (b) Tracking error between plant output and observed output.

Grahic Jump Location
Fig. 8

(a) Time response of z(t) of Eq. (26) obtained from direct numerical integration as compared to a0(t) obtained from Galerkin approximation with N = 8. (b) Error between approximate and actual numerical solution. The system parameters are b = 0.5, c = 0.1, k = 2.5, and ω = 2.

Grahic Jump Location
Fig. 9

(a) Time response of z(t) of Eq. (27) obtained from direct numerical integration as compared to a0(t) obtained from Galerkin approximation with N = 11. (b) Error between approximate and actual numerical solution. The system parameters are k = 2, b = 0.5, c = 0.1, ω = 2, F = 2, and Ω = 3.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In