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Research Papers

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

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

Department of Mechanical Engineering,
Ordnance Factory Estate,
e-mail: vcprakash@iith.ac.in

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

John McPhee

Systems Design Engineering,
University of Waterloo,

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

## Abstract

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.

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## Figures

Fig. 1

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

Fig. 2

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

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

Fig. 4

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

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.

Fig. 6

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

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.

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.

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.

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