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On stability crossing curves for general systems with two delays. (English) Zbl 1087.34052

Authors’ abstract: For the general linear scalar time-delay systems of arbitrary order with two delays, this article provides a detailed study on the stability crossing curves consisting of all the delays such that the characteristic quasipolynomial has at least one imaginary zero. The crossing set, consisting of all the frequencies corresponding to all the points in the stability crossing curves, are expressed in terms of simple inequality constraints and can be easily identified from the gain response curves of the coefficient transfer functions of the delay terms. This crossing set forms a finite number of intervals of finite length. The corresponding stability crossing curves form a series of smooth curves except at the points corresponding to multiple zeros and a number of other degenerate cases. These curves may be closed curves, open ended curves, and spiral-like curves oriented horizontally, vertically, or diagonally. The category of curves are determined by which constraints are violated at the two ends of the corresponding intervals of the crossing set. The directions in which the zeros cross the imaginary axis are explicitly expressed. An algorithm may be devised to calculate the maximum delay deviation without changing the number of right half-plane zeros of the characteristic quasipolynomial (and preservation of stability as a special case).

MSC:

34K20 Stability theory of functional-differential equations
34K06 Linear functional-differential equations
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[1] Bellman, R. E.; Cooke, K. L., Differential-Difference Equations (1963), Academic Press: Academic Press New York
[2] Beretta, E.; Kuang, Y., Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33, 1144-1165 (2002) · Zbl 1013.92034
[3] Bélair, J.; Campbell, S. A., Stability and bifurcations of equilibria in a multiple-delayed differential equation, SIAM J. Appl. Math., 54, 1402-1424 (1994) · Zbl 0809.34077
[4] Bruce, J. W.; Giblin, P. J., Curves and Singularities (1992), Cambridge Univ. Press · Zbl 0770.53002
[5] Chen, J.; Latchman, H. A., Frequency sweeping tests for stability independent of delay, IEEE Trans. Automat. Control, 40, 1640-1645 (1995) · Zbl 0834.93044
[6] Chen, J.; Gu, G.; Nett, C. N., A new method for computing delay margins for stability of linear delay systems, Systems Control Lett., 26, 101-117 (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] Cooke, K. L.; van den Driessche, P., On zeroes of some transcendental equations, Funkcial. Ekvac., 29, 77-90 (1986) · Zbl 0603.34069
[9] Datko, R., A procedure for determination of the exponential stability of certain differential-difference equations, Quart. Appl. Math., 36, 279-292 (1978) · Zbl 0405.34051
[10] Diekmann, O.; van Gils, S. A.; Verduyn-Lunel, S. M.; Walther, H.-O., Delay Equations, Functional-, Complex and Nonlinear Analysis, Appl. Math. Sci., vol. 110 (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0826.34002
[11] El’Sgol’ts, L. E.; Norkin, S. B., Introduction to the Theory and Application of Differential Equations with Deviating Arguments (1973), Academic Press: Academic Press New York · Zbl 0287.34073
[12] Górecki, H.; Fuksa, S.; Grabowski, P.; Korytowski, A., Analysis and Synthesis of Time-Delay Systems (1989), Polish Scientific Publishers: Polish Scientific Publishers Warszawa · Zbl 0695.93002
[13] Gu, K.; Kharitnov, V. L.; Chen, J., Stability of Time-Delay Systems (2003), Birkhäuser: Birkhäuser Boston
[14] K. Gu, S.-I. Niculescu, J. Chen, On calculating the maximum radius of delay deviation, in: Sixteenth International Symposium on Mathematical Theory of Networks and Systems, Leuven, Belgium, July 5-9, 2004; K. Gu, S.-I. Niculescu, J. Chen, On calculating the maximum radius of delay deviation, in: Sixteenth International Symposium on Mathematical Theory of Networks and Systems, Leuven, Belgium, July 5-9, 2004
[15] Guggenheimer, H. W., Differential Geometry (1977), Dover: Dover New York · Zbl 0116.13402
[16] Hale, J. K.; Huang, W., Global geometry of the stable regions for two delay differential equations, J. Math. Anal. Appl., 178, 344-362 (1993) · Zbl 0787.34062
[17] Hale, J. K.; Infante, E. F.; Tsen, F.-S. P., Stability in linear delay equations, J. Math. Anal. Appl., 105, 535-555 (1985) · Zbl 0569.34061
[18] Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations, Appl. Math. Sci., vol. 99 (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0787.34002
[19] Kolmanovskii, V. B.; Nosov, V. R., Stability of Functional Differential Equations, Math. Sci. Engrg., vol. 180 (1986), Academic Press: Academic Press New York · Zbl 0593.34070
[20] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press Boston · Zbl 0777.34002
[21] Levinson, N.; Redheffer, R. M., Complex Variables (1970), Holden-Day: Holden-Day Baltimore · Zbl 0201.40202
[22] Niculescu, S. I., Delay Effects on Stability—A Robust Control Approach, Lecture Notes in Comput. Sci., vol. 269 (2001), Springer-Verlag: Springer-Verlag Heidelberg
[23] Niculescu, S. I., On delay robustness of a simple control algorithm in high-speed networks, Automatica, 38, 885-889 (2002) · Zbl 1010.93077
[24] Nussbaum, R., Differential delay equations with two time lags, Mem. Amer. Math. Soc., 205 (1978) · Zbl 0406.34059
[25] Olgac, N.; Sipahi, R., An exact method for the stability analysis of time-delayed LTI systems, IEEE Trans. Automat. Control, 47, 793-797 (2002) · Zbl 1364.93576
[26] Ruan, S.; Wei, J., On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynam. Contin. Discrete Impuls. Systems, 10, 863-874 (2003) · Zbl 1068.34072
[27] Z.V. Rekasius, A stability test for systems with delays, Proc. of Joint Automatic Control Conf., Paper No. TP9-A, 1980; Z.V. Rekasius, A stability test for systems with delays, Proc. of Joint Automatic Control Conf., Paper No. TP9-A, 1980 · Zbl 0429.93017
[28] Stépán, G., Retarded Dynamical Systems: Stability and Characteristic Function (1989), Wiley: Wiley New York · Zbl 0686.34044
[29] Stépán, G., Delay-differential equation models for machine tool chatter, (Moon, F. C., Dynamics and Chaos in Manufacturing Process (1998), Wiley: Wiley New York), 165-192
[30] R. Sipahi, N. Olgac, Stability analysis of multiple time delay systems using the direct method, Paper #41495, ASME IMECE, Washington, DC, November 2003; R. Sipahi, N. Olgac, Stability analysis of multiple time delay systems using the direct method, Paper #41495, ASME IMECE, Washington, DC, November 2003 · Zbl 1100.93033
[31] R. Sipahi, N. Olgac, A novel stability study on multiple time-delay systems (MTDS) using the root clustering paradigm, in: American Control Conference, Boston, June 30-July 2, 2004; R. Sipahi, N. Olgac, A novel stability study on multiple time-delay systems (MTDS) using the root clustering paradigm, in: American Control Conference, Boston, June 30-July 2, 2004
[32] Walton, K.; Marshall, J. E., Direct method for TDS stability analysis, IEE Proc. Pt. D, 134, 101-107 (1987) · Zbl 0636.93066
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