Summary
In the recent years the authors developed numerical schemes to detect the stability properties of different classes of systems involving delayed terms. The base of all methods is the use of pseudospectral differentiation techniques in order to get numerical approximations of the relevant characteristic eigenvalues. This chapter is aimed to present the freely available Matlab package TRACE-DDE devoted to the computation of characteristic roots and stability charts of linear autonomous systems of delay differential equations with discrete and distributed delays and to resume the main features of the underlying pseudospectral approach.
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Keywords
- Functional Differential Equation
- Delay Differential Equation
- Characteristic Root
- Spectral Accuracy
- Stability Chart
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References
Altintas, Y., Engin, S., Budak, E.: Analytical stability prediction and design of variable pitch cutters. ASME J. Manuf. Sci. E.-T. 121, 173–178 (1999)
Breda, D.: The infinitesimal generator approach for the computation of characteristic roots for delay differential equations using BDF methods. Technical Report RR17/2002, Department of Mathematics and Computer Science, University of Udine (2002)
Breda, D.: Solution operator approximation for characteristic roots of delay differential equations. Appl. Numer. Math. 56(3-4), 305–317 (2006)
Breda, D., Iannelli, M., Maset, S., Vermiglio, R.: Stability analysis of the Gurtin-MacCamy model. J. Numer. Anal. 46(2), 980–995 (2008)
Breda, D., Maset, S., Vermiglio, R.: Computing the characteristic roots for delay differential equations. IMA J. Numer. Anal. 24(1), 1–19 (2004)
Breda, D., Maset, S., Vermiglio, R.: Efficient computation of stability charts for linear time delay systems. In: Proc. ASME-IDETC/CIE 2005, Long Beach, USA (2005)
Breda, D., Maset, S., Vermiglio, R.: An adaptive algorithm for efficient computation of level curves of surfaces. To appear in Numer Algorithms (2009) (preprint)
Breda, D., Maset, S., Vermiglio, R.: Pseudospectral differencing methods for characteristic roots of delay differential equations. SIAM J. Sci. Comput. 27(2), 482–495 (2005)
Breda, D., Maset, S., Vermiglio, R.: Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions. Appl. Numer. Math. 56(3-4), 318–331 (2006)
Breda, D., Maset, S., Vermiglio, R.: Numerical approximation of characteristic values of partial retarded functional differential equations. Numer. Math. (2009), doi:10.1007/s00211-009-0233-7
Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.O.: Delay Equations - Functional, Complex and Nonlinear Analysis. In: AMS series, vol. 110. Springer, New York (1995)
Engelborghs, K., Luzyanina, T., Roose, D.: Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM T. Math. Software 28(1), 1–21 (2002)
Engelborghs, K., Roose, D.: On stability of LMS methods and characteristic roots of delay differential equations. SIAM J. Numer. Anal. 40(2), 629–650 (2002)
Fattouh, A., Sename, O., Dion, J.M.: h ∞ controller and observer design for linear systems with point and distributed delays. In: Proc. 2nd IFAC workshop on Linear Time Delay Systems, Ancona, Italy (2000)
Hale, J.K., Verduyn Lunel, S.M.: Introduction to functional differential equations. AMS series, vol. 99. Springer, New York (1993)
Insperger, T., Stépán, G.: Updated semi-discretization method for periodic delay-differential equations with discrete delay. Int. J. Numer. Meth. Eng. 61, 117–141 (2004)
Luzyanina, T., Engelborghs, K., Roose, D.: Computing stability of differential equations with bounded distributed delays. Numer. Algorithms 34(1), 41–66 (2003)
Michiels, W., Niculescu, S.I.: Stability and stabilization of time-delay systems. An eigenvalue based approach. In: Advances in Design and Control, vol. 12. SIAM, Philadelphia (2007)
Niculescu, S.-I.: Delay Effects on Stability: A Robust Control Approach. In: TLNCIS Monograph, vol. 269. Springer, London (2001)
Olgac, N., Sipahi, R.: An exact method for the stability analysis of time delayed LTI systems. IEEE T. Automat. Contr. 47(5), 793–797 (2002)
Richard, J.-P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39, 1667–1694 (2003)
Sechi, D.: Sviluppo di interfaccia grafica per lo studio della stabilità di sistemi differenziali con ritardo (in italian). Master’s thesis, University of Udine (2005)
Stépán, G.: Retarded dynamical systems. Longman, Harlow (1989)
Trefethen, L.N.: Spectral methods in MATLAB. Software - Environment - Tools series. SIAM, Philadelphia (2000)
Vyhlídal, T., Zítek, P.: Mapping the spectrum of a retarded time-delay system utilizing root distribution features. In: Proc. 6th IFAC workshop on Linear Time Delay Systems, L’Aquila, Italy (2006)
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Breda, D., Maset, S., Vermiglio, R. (2009). TRACE-DDE: a Tool for Robust Analysis and Characteristic Equations for Delay Differential Equations. In: Loiseau, J.J., Michiels, W., Niculescu, SI., Sipahi, R. (eds) Topics in Time Delay Systems. Lecture Notes in Control and Information Sciences, vol 388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02897-7_13
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