Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations

Breda, Dimitri; Maset, Stefano; Vermiglio, Rossana

doi:10.1137/030601600

Cited by 229 publications

(234 citation statements)

References 9 publications

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“…This method was first proposed by Breda et al [12]. Later, Butcher and Bobrenkov [13] constructed an identical method approaching the problem from the framework of continuous-time approximation [26] and therefore called the technique Chebyshev spectral continuous-time approximation.…”

Section: Pseudospectral Collocation (Psc) Methodsmentioning

confidence: 99%

“…In Figure 1/A, these lines define the red surface which approximates the exact solution of (12)- (13). Note that, due to (12), the isocurves of the blue surface are lines with slope 1. In Figure 1/B the thin red lines define an approximation for the solution segment x t of (9) at each time instant t. These solution segment approximations are shown by thick red lines for time instants 0, t 1 and t 2 .…”

Section: Tau Approximationmentioning

confidence: 99%

“…Focusing on the stability of the original RFDE, some properties of the PsT method are analyzed through numerical case studies. Furthermore, the PsT method is compared with the pseudospectral collocation [12,13] (PsC) also called Chebyshev spectral continuous-time approximation, with the spectral Legendre tau (SLT) method [14] and with the spectral element (SE) method [15]. The comparison is based on the stability analysis of three linear autonomous RFDEs: the Hayes equation, an oscillator with two delays and an oscillator with distributed delay.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A pseudospectral tau approximation for time delay systems and its comparison with other weighted‐residual‐type methods

Lehotzky

Insperger

2016

Numerical Meth Engineering

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This paper presents a method which applies pseudospectral tau approximation for retarded functional differential equations (RFDEs). The goal is to construct a system of ordinary differential equations (ODEs), which provides a finite dimensional approximation of the original RFDE. The method can be used to determine approximate stability diagrams for RFDEs. Thorough numerical case studies show that the rightmost characteristic roots of the ODE approximation converge to the rightmost characteristic roots of the original RFDE. Application of the method to time-periodic RFDEs is also demonstrated, and the convergence of the stability boundaries is verified numerically. The method is compared with recently developed highly efficient numerical methods: the pseudospectral collocation (also called Chebyshev spectral continuous-time approximation), the spectral Legendre tau method and the spectral element method. The comparison is based on the stability analysis of three linear autonomous RFDEs. The efficiency of the methods is measured by the convergence rate of stability boundaries in the space of system parameters, by the convergence rate of the rightmost characteristic exponent and by the computation time of the stability charts.

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Section: Pseudospectral Collocation (Psc) Methodsmentioning

confidence: 99%

Section: Tau Approximationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A pseudospectral tau approximation for time delay systems and its comparison with other weighted‐residual‐type methods

Lehotzky

Insperger

2016

Numerical Meth Engineering

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“…In fact, the characteristic equation (4) is transcendental and the infinitely many characteristic roots cannot be computed analytically, rather a finite set of them can be numerically approximated. In the recent years the authors presented a family of numerical techniques focused on the discretization of the infinitesimal generator of the solution semigroup associated to (3) [2,4,5]. The discretization via pseudospectral differencing techniques [5] is based on n + 1 Chebyshev nodes on the delay interval [−τ, 0] and it leads to a matrix whose eigenvalues give approximations to the rightmost characteristic roots.…”

Section: Stability Chartsmentioning

confidence: 99%

“…Moreover, let us remark that the algorithm presented in this paper is used (together with the one of [5]) in "Trace-DDE" ( [3,13]), a Matlab graphic user interface devoted to the computation of characteristic roots and of stability charts of DDEs.…”

Section: The Overall Algorithmmentioning

confidence: 99%

An adaptive algorithm for efficient computation of level curves of surfaces

2009

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A new efficient algorithm for the computation of z = constant level curves of surfaces z = f (x, y) is proposed and tested on several examples. The set of z-level curves in a given rectangle of the (x, y)-plane is obtained by evaluating f on a first coarse square grid which is then adaptively refined by triangulation to eventually match a desired tolerance. Adaptivity leads to a considerable reduction in terms of evaluations of f with respect to uniform grid computation as in Matlab®'s contour. Furthermore, especially when the evaluation of f is computationally expensive, this reduction notably decreases the computational time. A comparison of performances is shown for two real-life applications such as the determination of stability charts and of ε−pseudospectra for linear time delay systems. The corresponding Matlab code is also discussed.

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Stability analysis and stabilization of delayed reduced-order model of large electric power system

Kishor

Haarla

Purwar

2016

Int. Trans. Electr. Energ. Syst.

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SUMMARYThis paper discusses stability analysis using spectral discretization of time-delayed electric power system. The stability assessment is discussed in terms of influence of time delays on inter-area oscillation modes of the power system models: well-known 4-generator and 14-generator Southeast Australian power system. The three dominant inter-area modes available in full-order power system are retained in its equivalent representation, i.e., reduced-order model, obtained with the input-output variables of the generator that participates actively to low-frequency oscillation modes. The pseudospectrum analysis suggests the most dominant modes to be sensitive against time delay. The rightmost characteristic roots and the robust spectral abscissa are computed over a wide range of time delays in the input-output variables. The approach applied in study computes all the characteristic roots in the right half s-plane with the number of discretization points selected in such a way that rational approximation of the exponential functions remains accurate in this region. The effect of time delays on the computation of spectral abscissa is also investigated. Further, reduced-order time-delayed model is also used to design the wide-area controller by optimizing (minimizing) the robust spectral abscissa.

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Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations

Cited by 229 publications

References 9 publications

A pseudospectral tau approximation for time delay systems and its comparison with other weighted‐residual‐type methods

A pseudospectral tau approximation for time delay systems and its comparison with other weighted‐residual‐type methods

An adaptive algorithm for efficient computation of level curves of surfaces

Stability analysis and stabilization of delayed reduced-order model of large electric power system

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