Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method

Spectral discretization methods are well established methods for the computation of characteristic roots of time-delay systems. In this paper a method is presented for computing all characteristic roots in a given right half plane. In particular, a procedure for the automatic selection of the number of discretization points is described. This procedure is grounded in the connection between a spectral discretization and a rational approximation of exponential functions. First, a region that contains all desired characteristic roots is estimated. Second, the number of discretization points is selected in such a way that in this region the rational approximation of the exponential functions is accurate. Finally, the characteristic roots approximations, obtained from solving the discretized eigenvalue problem, are corrected up to the desired precision by a local method. The effectiveness and robustness of the procedure are illustrated with several examples and compared with DDE-BIFTOOL.

[1]  A. Bellen,et al.  Numerical methods for delay differential equations , 2003 .

[2]  Dimitri Breda,et al.  Solution operator approximations for characteristic roots of delay differential equations , 2006 .

[3]  Dirk Roose,et al.  Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL , 2002, TOMS.

[4]  Eric A. Butcher,et al.  On the Chebyshev spectral continuous time approximation for constant and periodic delay differential equations , 2011 .

[5]  Koen Verheyden.  Numerical bifurcation analysis of large-scale delay differential equations , 2007 .

[6]  Dimitri Breda,et al.  Computing the characteristic roots for delay differential equations , 2004 .

[7]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[8]  Wim Michiels,et al.  An Arnoldi like method for the delay eigenvalue problem , 2010 .

[9]  S. Lunel,et al.  Delay Equations. Functional-, Complex-, and Nonlinear Analysis , 1995 .

[10]  Tomás Vyhlídal,et al.  Mapping Based Algorithm for Large-Scale Computation of Quasi-Polynomial Zeros , 2009, IEEE Transactions on Automatic Control.

[11]  D. D. Perlmutter,et al.  Stability of time‐delay systems , 1972 .

[12]  Dimitri Breda,et al.  Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations , 2005, SIAM J. Sci. Comput..

[13]  G. Samaey,et al.  DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations , 2001 .

[14]  Dirk Roose,et al.  Numerical bifurcation analysis of delay differential equations , 2000 .

[15]  Dirk Roose,et al.  Efficient computation of characteristic roots of delay differential equations using LMS methods , 2008 .

[16]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[17]  Wim Michiels,et al.  A predictor-corrector type algorithm for the pseudospectral abscissa computation of time-delay systems , 2010, Autom..

[18]  Ing. Tomáš Vyhlídal.  Analysis and Synthesis of Time Delay System Spectrum , 2003 .

[19]  Wim Michiels,et al.  Stability and Stabilization of Time-Delay Systems (Advances in Design & Control) (Advances in Design and Control) , 2007 .

[20]  S. Niculescu,et al.  Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach , 2007 .

[21]  Dirk Roose,et al.  Efficient numerical stability analysis of delay equations: a spectral method , 2005 .

[22]  K. ENGELBORGHS,et al.  On Stability of LMS Methods and Characteristic Roots of Delay Differential Equations , 2002, SIAM J. Numer. Anal..

[23]  R. Vermiglio,et al.  TRACE-DDE: a Tool for Robust Analysis and Characteristic Equations for Delay Differential Equations , 2009 .

[24]  S. Niculescu.  Delay Effects on Stability: A Robust Control Approach , 2001 .