Summary
An overview of eigenvalue based tools for the stability analysis of linear periodic systems with delays is presented. It is assumed that both the system matrices and the delays are periodically varying. First the situation is considered where the time-variation of the periodic terms is fast compared to the system’s dynamics. Then averaging techniques are used to relate the stability properties of the time-varying system with these of a time-invariant one, which opens the possibility to use frequency domain tools. As a special characteristic the averaged system exhibits distributed delays if the delays in the original system are time-varying. Both analytic and numerical tools for the stability analysis of the averaged system are discussed. Special attention is paid to the characterization of situations where a variation of a delay has a stabilizing effect. Second, the assumption underlying the averaging approach is dropped. It is described how exact stability information of the original, periodic system can be directly computed. The two approaches are briefly compared with respect to generality, applicability and computational efficiency. Finally the results are illustrated by means of two examples from mechanical engineering. The first example concerns a model of a variable speed rotating cutting tool. Based on the developed theory and using the described computational tools, both a theoretical explanation and a quantitative analysis are provided of the beneficial effect of a variation of the machine speed on enhancing stability properties, which was reported in the literature. The second example concerns the stability analysis of an elastic column, subjected to a periodic force.
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References
Bai Z., Demmel J., Dongarra J., Ruhe A., van der Vorst, H. Eds. (2000) Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. Software, Environments, and Tools, vol. 11, SIAM, Philadelphia
Bellen A., Zennaro M. (2003) Numerical methods for delay differential equations. Oxford University Press, London
Breda D. (2004) Numerical computation of characteristic roots for delay differential equations. PhD thesis, Department of Mathematics and Computer Science, University of Udine
Breda D., Maset S. Vermiglio R. (2004) Computing the characteristic roots for delay differential equations. IMA J. Numerical Analysis 24(1):1–19.
Butcher E.A., Ma H., Bueler E., Averina V., Szabo, Z. (2004) Stability of linear time-periodic delay-differential equations via Chebyshev polynomials. International J. Numerical Methods in Engineering 59:895–922
Cooke K. L. Grossman Z. (1982) Discrete delay, distributed delay and stability switches. J. Mathematical Analysis and Applications 86:592–627
Doedel E.J., Champneys A.R., Fairgrieve T.J., Kuznetsov Y.A., Sandstede B., Wang X.-J. (1998) AUTO97: Continuation and bifurcation software for ordinary differential equations. Technical report, Dept. of Computer Science, Concordia University
Doedel E.J., and Kernévez J.P. (1986) AUTO: Software for continuation and bifurcation problems in ordinary differential equations. Applied Mathematics Report, California Institute of Technology, Pasadena, U.S.A.
Engelborghs K., Luzyanina T., in’t Hout K., Roose, D. (2000) Collocation methods for the computation of periodic solutions of delay differential equations SIAM J. Sci. Comput. 22:1593–1609
Engelborghs K., Luzyanina T., Roose D. (2002) Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Transactions on Mathematical Software 28(1):1–21
Engelborghs K., Luzyanina T., Samaey G. (2001) DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equation. T.W. Report 330, Department of Computer Science, K.U. Leuven
Engelborghs K. and Roose D. (2002) On stability of LMS methods and characteristic roots of delay differential equations. SIAM J. Numerical Analysis 40(2):629–650
Fox L., Parker I. B. (1972) Chebyshev polynomials in numerical analysis. Oxford mathematical handbooks, Oxford University Press, London
Goubet-Bartholoméüs A., Dambrine M., Richard J.-P. (1997) Stability of perturbed systems with time-varying delays. Systems & Control Letters 31:155–163
Gu K., Kharitonov V.L., Chen, J. (2003) Stability of time-delay systems. Birkhauser, Boston
Hale J. K., Verduyn Lunel S. M. (1993) Introduction to Functional Differential Equations. Applied Math. Sciences, vol. 99, Springer Verlag, Berlin Heidelberg New York
Halmos P. (1966) Measure Theory. The university series in higher mathematics, Van Nostrand Princeton
Insperger T., Stepan B. (2002) Semi-discretization method for delayed systems. International J. for Numerical Methods in Engineering 55(5):3–18
Jayaram S., Kapoor S.G., DeVor, R.E. (2000) Analytical stability analysis of variable spindle speed machines. J. of Manufacturing and Engineering 122:391–397
Kharitonov, V.L. and Niculescu, S.-I.: On the stability of linear systems with uncertain delay. IEEE Trans. Automat. Contr. 48 (2003) 127–133
Lehman B., Bentsman J., Verduyn Lunel S., Verriest, E.I. (1994) Vibrational Control of Nonlinear Time Lag Systems with Bounded Delay: Averaging Theory, Stabilizability, and Transient Behavior. IEEE Transactions on Automatic Control 39:898–912
Louisell J. (1992) Growth estimates and asymptotic stability for a class of differential-delay equation having time-varying delay. J. Mathematical Analysis and Applications 164:453–479
Luzyanina T., Engelborghs K., Roose, D. (2003) Computing stability of differential equations with bounded distributed delays. Numerical Algorithms 34(1):41–66
Luzyanina T., Roose, D. (2004) Equations with distributed delays: bifurcation analysis using computational tools for discrete delay equations. Functional Differential Equations 11:87–92
Luzyanina T., Roose D., Engelborghs K. (2004) Numerical stability analysis of steady state solutions of integral equations with distributed delays. Applied Numerical Mathematics 50:75–92
Michiels W., Engelborghs K., Roose D., Dochain, D. (2002) Sensitivity to in-finitesimal delays in neutral equations. SIAM Journal of Control and Optimization 40(4):1134–1158
Michiels W., Niculescu S.-I., Moreau, L. (2004) Using delays and time-varying gains to improve the output feedback stabilizability of linear systems: a comparison. IMA Journal of Mathematical Control and Information 21(4):393–418
Michiels W., Van Assche V., Niculescu, S.-I. (2005) Stabilization of time-delay systems with a controlled, time-varying delay and applications. IEEE Transactions on Automatic Control 50(4):493–504
Moreau L., Aeyels D. (1999) Trajectory-based global and semi-global stability results. In Modern Applied Mathematics Techniques in Circuits, Systems and Control, N.E. Mastorakis, Ed., pp.71–76, World Scientific and Engineering Society Press 71–76
Moreau L., Aeyels, D. (2000) Practical stability and stabilization. IEEE Transactions on Automatic Control 45(8):1554–1558
Moreau L., Michiels W., Aeyels D., Roose D. (2002) Robustness of nonlinear delay equations w.r.t. bounded input perturbations: a trajectory based approach. Math. Control Signals Systems 15(4):316–335
Niculescu S.-I. (2001) Delay effects on stability: A robust control approach. LNCIS, vol. 269, Springer, Berlin Heidelberg New York
Niculescu S.-I., Gu K., Abdallah, C.T. (2003) Some remarks on the delay stabilizing effect in SISO systems. In Proc. 2003 American Control Conference, Denver, Colorado
Niculescu S.-I., de Souza C. E., Dion J.-M., Dugard, L. (1998) Robust exponential stability of uncertain systems with time-varying delays. IEEE Transactions on Automatic Control 43:743–748
Richard J.-P. (2003) Time-delay systems: an overview of recent advances and open problems. Automatica 39(10):1667–1694
Roesch O., Roth H., Niculescu S.-I. (2005) Remote coontrol of mechatronic systems over communication networks. In Proc. 2005 IEEE Int. Conf. Mechatronics & Automation, Niagara Falls, Canada
Rudin W. (1966) Real and complex analysis. McGraw Hill, New York
Sanders J.A., Verhulst F. (1985) Averaging Methods in Nonlinear Dynamical Systems. Applied Mathematical Sciences, vol. 59, Springer Verlag, Berlin Heidelberg New York
Sexton J., Stone B. (1978) The stability of machining with continuously varying spindle speed. Ann. CIRP 27:321–326
Sridhar R., Hohn R.E., Long G.W. (1968) A general formultion of the milling process equation. ASME J. Engineering for Industry 90:317–324
Tarbouriech S., Abdallah C.T., Chiasson, J.N. eds. (2005) Advances in Communication Control Networks. LNCIS, vol. 308, Springer Verlag, Berlin Heidelberg New York
Trefethen L.N. (2000) Spectral methods in Matlab. Software, Environments, and Tools, vol. 10, SIAM, Philadelphia
Van Assche V., Ganguli A., Michiels W. (2004) Practical stability analysis of systems with delay and periodic coefficients. In Proc. 5th IFAC Workshop on Time-Delay Systems, Leuven, Belgium
Verheyden K., Lust, K. (2005) A Newton-Picard collocation method for periodic solutions of delay differential equations. BIT 45:605–625
Verheyden K., Lust K., Roose D. (2005) Computation and stability analysis of solutions of periodic delay differential algebraic equations. In Proc. of ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Long Beach, USA
Verheyden K., Luzyanina T., Roose D. (2004) Efficient computation of characteristic roots of delay differential equations using LMS methods. T.W. Report 383, Department of Computer Science, K.U. Leuven
Verheyden K., Roose D. (2004) Efficient numerical stability analysis of delay equations: a spectral method. In Proc. 5th IFAC Workshop on Time-Delay Systems, Leuven, Belgium
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Michiels, W., Verheyden, K., Niculescu, SI. (2007). Mathematical and Computational Tools for the Stability Analysis of Time-Varying Delay Systems and Applications in Mechanical Engineering. In: Chiasson, J., Loiseau, J.J. (eds) Applications of Time Delay Systems. Lecture Notes in Control and Information Sciences, vol 352. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-49556-7_13
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