×

Critical delays and polynomial eigenvalue problems. (English) Zbl 1166.65040

The author proposes a method for parameterizing the critical delays, \(h_k\), which give the boundary of the stability region of the delay-differential equation \(x'(t)=A_0x(t)+\sum _{k=1}^m A_kx(t-h_k)\) for \(t>0\), with \(x(t)=\varphi (t)\) for \(t\in [-h_m,0]\), where the \(A_k\) are real \(n\times n\) matrices. A key step in the method requires the solution of a quadratic eigenvalue problem for \(n^2\times n^2\) matrices. In the case in which the delays are integer multiples of the smallest delay, the author discusses the relationship of this work to earlier work of J. Chen, G. Gu and C. N. Nett [Syst. Control Lett. 26, No. 2, 107–117 (1995; Zbl 0877.93117)].

MSC:

65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
34K20 Stability theory of functional-differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
93C23 Control/observation systems governed by functional-differential equations
65L07 Numerical investigation of stability of solutions to ordinary differential equations
93B35 Sensitivity (robustness)

Citations:

Zbl 0877.93117

Software:

SOAR

References:

[1] Bai, Z.; Su, Y., SOAR: A second-order Arnoldi method for the solution of the quadratic eigenvalue problem, SIAM J. Matrix Anal. Appl., 26, 3, 640-659 (2005) · Zbl 1080.65024
[2] Bélair, J.; Campbell, S. A., Stability and bifurcations of equilibria in a multiple-delayed differential equation, SIAM J. Appl. Math., 54, 5, 1402-1424 (1994) · Zbl 0809.34077
[3] Beretta, E.; Kuang, Y., Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33, 5, 1144-1165 (2002) · Zbl 1013.92034
[4] Breda, D.; Maset, S.; Vermiglio, R., Computing the characteristic roots for delay differential equations, IMA J. Numer. Anal., 24, 1-19 (2004) · Zbl 1054.65079
[5] Cahlon, B.; Schmidt, D., Stability criteria for certain third-order delay differential equations, J. Comput. Appl. Math., 188, 2, 319-335 (2006) · Zbl 1094.34051
[6] Chen, J.; Gu, G.; Nett, C. N., A new method for computing delay margins for stability of linear delay systems, Syst. Control Lett., 26, 2, 107-117 (1995) · Zbl 0877.93117
[7] Cooke, K. L.; Grossman, Z., Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86, 592-627 (1982) · Zbl 0492.34064
[8] Ergenc, A. F.; Olgac, N.; Fazelina, H., Extended Kronecker summation for cluster treatment of LTI systems with multiple delays, SIAM J. Control Optimization, 46, 1, 143-155 (2007) · Zbl 1136.93029
[9] Fridman, E., Stability of systems with uncertain delays: A new “complete” Lyapunov-Krasovskii functional, IEEE Trans. Automat. Control, 51, 5, 885-890 (2006) · Zbl 1366.93547
[10] Fu, P.; Niculescu, S.-I.; Chen, J., Stability of linear neutral time-delay systems: Exact conditions via matrix pencil solutions, IEEE Trans. Autom. Control, 51, 6, 1063-1069 (2006) · Zbl 1366.34091
[11] Gu, K.; Kharitonov, V.; Chen, J., Stability of time-delay systems, (Control Engineering (2003), Birkhäuser: Birkhäuser Boston, MA) · Zbl 1039.34067
[12] Gu, K.; Niculescu, S.-I.; Chen, J., On stability crossing curves for general systems with two delays, J. Math. Anal. Appl., 311, 1, 231-253 (2005) · Zbl 1087.34052
[13] Hale, J., Dynamics and delays, Delay differential equations and dynamical systems, (Proc. Conf., Claremont/CA (USA) 1990. Proc. Conf., Claremont/CA (USA) 1990, Lect. Notes Math., vol. 1475 (1991)), 16-30 · Zbl 0735.34051
[14] Hale, J.; Huang, W., Global geometry of the stable regions for two delay differential equations, J. Math. Anal. Appl., 178, 2, 344-362 (1993) · Zbl 0787.34062
[15] Hale, J.; Lunel, S. M.V., Introduction to Functional Differential Equations (1993), Springer-Verlag · Zbl 0787.34002
[16] Hertz, D.; Jury, E.; Zeheb, E., Simplified analytic stability test for systems with commensurate time delays, IEE Proc., Part D, 131, 52-56 (1984) · Zbl 0535.93054
[17] Jarlebring, E., A quadratic eigenproblem in the analysis of a time delay system, Proc. Appl. Math. Mech., 6, 63-66 (2006)
[18] Knopse, C. R.; Roozbehani, M., Stability of linear systems with interval time delays excluding zero, IEEE Trans. Autom. Control, 51, 8, 1271-1288 (2006) · Zbl 1366.34099
[19] Louisell, J., A matrix method for determining the imaginary axis eigenvalues of a delay system, IEEE Trans. Autom. Control, 46, 12, 2008-2012 (2001) · Zbl 1007.34078
[20] Mackey, S.; Mackey, N.; Mehl, C.; Mehrmann, V., Structured polynomial eigenvalue problems: Good vibrations from good linearizations, SIAM J. Matrix Anal. Appl., 28, 1029-1051 (2006) · Zbl 1132.65028
[21] Mackey, S.; Mackey, N.; Mehl, C.; Mehrmann, V., Vector spaces of linearizations for matrix polynomials, SIAM J. Matrix Anal. Appl., 28, 971-1004 (2006) · Zbl 1132.65027
[22] Malakhovski, E.; Mirkin, L., On stability of second-order quasi-polynomials with a single delay, Automatica, 42, 6, 1041-1047 (2006) · Zbl 1135.93025
[23] Niculescu, S.-I., Delay Effects on Stability. A Robust Control Approach (2001), Springer-Verlag: Springer-Verlag London · Zbl 0997.93001
[24] Niculescu, S.-I., On delay-dependent stability under model transformations of some neutral linear systems, Int. J. Control, 74, 6, 609-617 (2001) · Zbl 1047.34088
[25] Niculescu, S.-I.; Fu, P.; Chen, J., On the stability of linear delay-differential algebraic systems: Exact conditions via matrix pencil solutions, (Proceedings of the 45th IEEE Conference on Decision and Control (2006))
[26] Nussbaum, R. D., Differential-delay equations with two time lags, Mem. Am. Math. Soc., 205, 62 (1978) · Zbl 0406.34059
[27] Z. Rekasius, A stability test for systems with delays, Proc. of Joint Autom. Contr. Conf San Francisco, 1980; Z. Rekasius, A stability test for systems with delays, Proc. of Joint Autom. Contr. Conf San Francisco, 1980 · Zbl 0429.93017
[28] Ruhe, A., Rational Krylov: A practical algorithm for large sparse nonsymmetric matrix pencils, SIAM J. Sci. Comput., 19, 5, 1535-1551 (1998) · Zbl 0914.65036
[29] Saad, Y., Variations on Arnoldi’s method for computing eigenelements of large unsymmetric matrices, Linear Algebra Appl., 34, 269-295 (1980) · Zbl 0456.65017
[30] Sipahi, R.; Olgac, N., Complete stability robustness of third-order LTI multiple time-delay systems, Automatica, 41, 8, 1413-1422 (2005) · Zbl 1086.93049
[31] Sleijpen, G. L.; Booten, A. G.; Fokkema, D. R.; der Vorst, H. A.V., Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36, 3, 595-633 (1996) · Zbl 0861.65035
[32] Thowsen, A., The Routh-Hurwitz method for stability determination of linear differential-difference systems, Int. J. Control, 33, 991-995 (1981) · Zbl 0474.93051
[33] Tisseur, F.; Meerbergen, K., The quadratic eigenvalue problem, SIAM Rev., 43, 2, 235-286 (2001) · Zbl 0985.65028
[34] Verheyden, K.; Luzyanina, T.; Roose, D., Efficient computation of characteristic roots of delay differential equations using LMS methods, J. Comput. Appl. Math. (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.