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On the Codimension of the Singularity at the Origin for Networked Delay Systems Read chapter Read first chapter

2016 | OriginalPaper | Chapter

3. Model Reduction for Norm Approximation: An Application to Large-Scale Time-Delay Systems

Authors : Igor Pontes Duff, Pierre Vuillemin, Charles Poussot-Vassal, Corentin Briat, Cédric Seren

Published in: Delays and Networked Control Systems

Publisher: Springer International Publishing

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Abstract

The computation of \(\mathscr {H}_2\) and \(\mathscr {H}_{2,\varOmega }\) norms for LTI Time-Delay Systems (TDS) are important challenging problems for which several solutions have been provided in the literature. Several of these approaches, however, cannot be applied to systems of large dimension because of the inherent poor scalability of the methods, e.g., LMIs or Lyapunov-based approaches. When it comes to the computation of frequency-limited norms, the problem tends to be even more difficult. In this chapter, a computationally feasible solution using \(\mathscr {H}_2\) model reduction for TDS, based on the ideas provided in [3], is proposed. It is notably demonstrates on several examples that the proposed method is suitable for performing both accurate model reduction and norm estimation for large-scale TDS.

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previous chapter Stability and Stabilization for Continuous-Time Difference Equations with Distributed Delay
next chapter General Formula for Event-Based Stabilization of Nonlinear Systems with Delays in the State
Literature
1.
A.-C. Antoulas, Approximation of Large-Scale Dynamical Systems (SIAM, 2005)
2.
A.-C. Antoulas, An overview of model reduction methods and a new result, in IEEE Conference on Decision and Control (CDC) (2009)
3.
C. Beattie, S. Gugercin, Realization-independent \({\cal H}_2\)-approximation, in IEEE Conference on Decision and Control (CDC) (2012)
4.
G. Flagg, C. Beattie, S. Gugercin, Convergence of the iterative rational Krylov algorithm. Syst. Control Lett. 61(6), 688–691 (2012)MathSciNetCrossRefMATH
5.
K. Gu, V.L. Kharitonov, J. Chen, Stability of Time-Delay Systems (Birkhäuser, 2003)
6.
S. Gugercin, A.C. Antoulas, C. Beattie, \({\cal H}_2\) model reduction for large-scale linear dynamical systems. SIAM J. Matrix Anal. Appl. 30(2), 609–638 (2008)
7.
S. Gugercin, C. Beattie, A.-C. Antoulas, Rational Krylov methods for optimal \({\cal H}_2\) model reduction. ICAM Technical report (2006)
8.
Y. Halevi, Frequency weighted model reduction via optimal projection. IEEE Trans. Autom. Control 37(10), 1537–1542 (1992)
9.
A. Ionita, A. Antoulas, Data-driven parametrized model reduction in the Loewner framework. SIAM J. Sci. Comput. 36(3), A984–A1007 (2014)MathSciNetCrossRefMATH
10.
E. Jarlebring, K. Meerbergen, W. Michiels, Computing a partial Schur factorization of nonlinear eigenvalue problems using the infinite Arnoldi method. SIAM J. Matrix Anal. Appl. 35(2), 411–436 (2014)MathSciNetCrossRefMATH
11.
E. Jarlebring, J. Vanbiervliet, W. Michiels, Characterizing and computing the norm of time-delay systems by solving the delay Lyapunov equation. IEEE Trans. Autom. Control 56(4), 814–825 (2011)MathSciNetCrossRefMATH
12.
F. Leibfritz, W. Lipinski, Description of the benchmark examples in compleib 1.0. Technical report. Department Mathematics, University of Trier, Germany (2003)
13.
C. Magruder, C. Beattie, S. Gugercin, Rational Krylov methods for optimal \({\cal L}_2\) model reduction, in IEEE Conference on Decision and Control (CDC) (2010)
14.
A.-J. Mayo, A.-C. Antoulas, A framework for the solution of the generalized realization problem. Linear Algebra Appl. 425(2), 634–662 (2007)
15.
S.I. Niculescu, Delay Effects on Stability. A Robust Control Approach (Springer, Heidelberg, 2001)
16.
I. Pontes Duff, P. Vuillemin, C. Poussot-Vassal, C. Seren, C. Briat, Large-scale time delay systems stability analysis by model approximation techniques (submitted)
17.
C. Poussot-Vassal, P. Vuillemin, Introduction to MORE: a MOdel REduction toolbox, in IEEE International Conference on Control Applications (ICCA) (2012)
18.
J.-P. Richard, Time-delay systems: an overview of some recent advances and open problems. Automatica 39(10), 1667–1694 (2003)
19.
J. Rommes, N. Martins, Efficient computation of multivariable transfer function dominant poles using subspace acceleration. IEEE Trans. Power Syst. 21(4), 1471–1483 (2006)CrossRef
20.
J.T. Spanos, M.H. Milman, D.L. Mingori, A new algorithm for \(\fancyscript {L}_2\) optimal model reduction. Automatica 28(5), 897–909 (1992)
21.
R. Van Beeumen, K. Meerbergen, W. Michiels, A rational Krylov method based on hermite interpolation for nonlinear eigenvalue problems. SIAM J. Sci. Comput. 35(1), A327–A350 (2013)MathSciNetCrossRefMATH
22.
P. Van Dooren, K.-A. Gallivan, P.-A. Absil, \({\cal H}_2\)-optimal model reduction of MIMO systems. Appl. Math. Lett. 21(12), 1267–1273 (2008)
23.
P. Vuillemin, C. Poussot-Vassal, D. Alazard, Poles residues descent algorithm for optimal frequency-limited \({\cal H}_2\) model approximation, in European Control Conference (ECC) (2014)
24.
P. Vuillemin, C. Poussot-Vassal, D. Alazard, Spectral expression for the frequency-limited \({\cal H}_2\)-norm of LTI dynamical systems with high order poles, in European Control Conference (ECC) (2014)
25.
P. Vuillemin, C. Poussot-Vassal, D. Alazard, Two upper bounds on the \({\cal H}_{\infty }\)-norm of LTI dynamical systems, in IFAC World Congress (2014)
26.
D.A. Wilson, Optimum solution of model-reduction problem. Proc. Inst. Electr. Eng. 117(6), 1161–1165 (1970)
27.
P. Zitek, T. Vyhlídal, Rating the significance of infinite chains of poles in time-delay systems, in IFAC World Congress (2011)
Metadata
Title
Model Reduction for Norm Approximation: An Application to Large-Scale Time-Delay Systems
Authors
Igor Pontes Duff
Pierre Vuillemin
Charles Poussot-Vassal
Corentin Briat
Cédric Seren
Copyright Year
2016
Book
Delays and Networked Control Systems

Print ISBN: 978-3-319-32371-8

Electronic ISBN: 978-3-319-32372-5

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-32372-5_3