×

zbMATH — the first resource for mathematics

On positivity of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate point delays. (English) Zbl 1117.93034
Summary: This paper deals with the positivity properties of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate constant point delays. The main idea behind the investigation is that its main body is performed based on the construction of the whole state-space trajectory solution without using as usual equivalence or similarity transformations on the matrix of dynamics in order to split the state-trajectory solution into two parts, one being typically associated with a nilpotent matrix. In that way, the whole state trajectory solution contains impulsive terms associated with the initial conditions and inputs. Some extensions concerning positivity aspects are given for a special canonical form which separates the dynamics associated with the nilpotent matrix obtained from an equivalence transformation on the singular matrix of the dynamic system.

MSC:
93C05 Linear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
93C70 Time-scale analysis and singular perturbations in control/observation systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. De la Sen, S. Alonso, Adaptive control of time-invariant systems with discrete delays subject to multiestimation, Discrete Dynamics in Nature and Society, Art. No. 41973, doi:10.1155/DDNS/2006/41973.
[2] M. De la Sen, Stabilization criteria for continuous-time linear time-invariant systems with constant lags, Discrete Dynamics in Nature and Society, Art. No. 87062, doi:10.1155/DDNS/2006/87062. · Zbl 1099.93040
[3] de la Sen, M., Stability of impulsive time-varying systems and compactness of the operators mapping the input space into the state and output spaces, Journal of mathematical analysis and applications, 321, 2, 621-650, (2006) · Zbl 1111.93072
[4] de la Sen, M., Absolute stability of feedback systems independent of the internal point delays, IEE Proceedings – control theory and applications, 152, 5, 567-574, (2005)
[5] Marchenko, V.M.; Poddubnaya, O.N.; Zackiewicz, Z., On the observability of linear differential-algebraic systems with delays, IEEE transactions on automatic control, 51, 8, 1387-1392, (2006) · Zbl 1366.93079
[6] M. de la Sen, On positivity and stability of a class of time-delay systems, Nonlinear Analysis-Theory Methods & Applications, doi:10.1016/j.norwa.2006.03.002, in press. · Zbl 1141.93056
[7] Kaczorek, T., Realization problem for positive linear systems with time delay, Mathematical problems in engineering, 2005, 4, 455-463, (2005) · Zbl 1200.93062
[8] Yan, J.R.; Zhao, A.M.; Zhang, Q.X., Oscillation properties of nonlinear impulsive delay differential equations and applications to population models, Journal of mathematical analysis and applications, 322, 1, 359-370, (2006) · Zbl 1107.34056
[9] Liao, X.Y.; Zhu, H.Y.; Chen, F.D., On asymptotic stability of delay-difference systems, Applied mathematics and computation, 176, 2, 775-784, (2006)
[10] Appleby, J.A.D.; Gyori, I.; Reynolds, D.W., On exact rates of decay of solutions of linear systems of Volterra equations with delay, Journal of mathematical analysis and applications, 320, 1, 56-77, (2006) · Zbl 1148.45003
[11] Dai, B.X.; Zhang, N., Stability and global attractivity for a class of nonlinear delay difference equations, Discrete dynamics in nature and society, 2005, 3, 227-234, (2005) · Zbl 1135.39002
[12] Jiang, W.; Shong, W.Z., Controllability of singular systems with control delay, Automatica, 37, 11, 1873-1877, (2001) · Zbl 1058.93012
[13] Richard, J.P., Time-delay systems: an overview of some recent advances and open problems, Automatica, 39, 10, 1667-1694, (2003) · Zbl 1145.93302
[14] L.H. Ding, W.X. Li, Stability and bifurcations of numerical discretization Nicholson blowflies equation with delay, Discrete Dynamics in Nature and Society, Art. No. 19413, doi:10.1155/DDNS/2006/19413.
[15] Yang, Z.D., Existence of explosive positive solutions of quasilinear elliptic equations, Applied mathematics and computation, 177, 2, 581-588, (2006) · Zbl 1254.35085
[16] Yin, H.H.; Yang, Z.D., Some new results on the existence of bounded positive entire solutions for quasilinear elliptic equations, Applied mathematics and computation, 177, 2, 606-613, (2006) · Zbl 1254.35094
[17] Yang, Z.L., Positive solutions of a second-order integral boundary value problem, Journal of mathematical analysis and applications, 321, 2, 751-765, (2006) · Zbl 1106.34014
[18] Li, X.P.; Liu, T.H.; Zhou, .J., The existence of positive solutions and bounded oscillation for even-order neutral differential equations with unstable type, Applied mathematics and computation, 176, 2, 632-641, (2006) · Zbl 1105.34042
[19] Z.G. Wang, L.S. Liu, Y.H. Wu, Multiple positive solutions of Sturm-Liouville equations with singularities, Discrete Dynamics in Nature and Society, Art. No. 32018 , doi:10.1155/DDNS/2006/32018.
[20] He, Z.M.; Jiang, X.M., Triple positive solutions of boundary value problems for p-Laplacian dynamic equations on time scales, Journal of mathematical analysis and applications, 321, 2, 911-920, (2006) · Zbl 1103.34012
[21] Afrouzi, G.A.; Khademioo, S., A numerical method to find positive solution of semilinear elliptic Dirichlet problems, Applied mathematics and computation, 174, 2, 1408-1415, (2006) · Zbl 1090.65125
[22] Huo, H.F.; Li, W.T., Permanence and global stability of positive solutions of nonautonomous discrete ratio-dependent predator – prey model, Discrete dynamics in nature and society, 2005, 2, 35-144, (2005) · Zbl 1111.39007
[23] Kaczorek, T., Positive 1D and 2D systems, Communications and control engineering series, (2002), Springer-Verlag London · Zbl 1005.68175
[24] Berman, A.; Plemmons, R.J., Nonnegative matrices in the mathematical sciences, Classics in applied mathematics, (1994), SIAM Philadelphia, PA · Zbl 0815.15016
[25] Farina, L.; Rinaldi, S., Positive linear systems. theory and applications, Series on pure and applied mathematics, (2000), John Wiley & Sons New York · Zbl 0988.93002
[26] Hinrichsen, D.; Manthey, W.; Helmke, U., Minimal partial realizations by descriptor systems, Linear algebra and its applications, 326, 1-3, 45-84, (2001) · Zbl 0986.93017
[27] Dao, H.I., Structured perturbations of Drazin inverse, Applied mathematics and computation, 158, 2, 419-432, (2004) · Zbl 1072.15005
[28] Castro-González, N.; Dopazo, E.; Robles, J., Formulas for the Drazin inverse of special block matrices, Applied mathematics and computation, 174, 1, 252-270, (2006) · Zbl 1097.15005
[29] Lei, T.G.; Wei, Y.M.; Woo, C.W., Condition numbers and structured perturbation of the W-weighted Drazin inverse, Applied mathematics and computation, 165, 1, 185-194, (2005) · Zbl 1077.15006
[30] H.D. Kurz, N. Salvadori, The dynamic Leontief model and the theory of endogenous growth, in: Twelfth International Conference on Input-Output Techniques, New York 18-22 May 1998 (Special Session 11).
[31] De la Sen, M., Asymptotic hyperstability under unstructured and structured modelling deviations from the linear behaviour, Non linear analysis-real world applications, 7, 2, 248-264, (2006) · Zbl 1093.93023
[32] De la Sen, M., Some conceptual links between dynamic physical systems and operator theory issues concerning energy balances and stability, Informatica, 16, 3, 395-406, (2005) · Zbl 1140.93473
[33] de la Sen, M., On the adaptive control of a class of SISO dynamic hybrid systems, Applied numerical mathematics, 56, 5, 618-647, (2006) · Zbl 1111.93040
[34] de la Sen, M., Robust stable pole-placement adaptive control of linear systems with multiestimation, Applied mathematics and computation, 172, 2, 1145-1174, (2006) · Zbl 1111.93029
[35] Saadaoui, H.; Manamanni, N.; Djemai, M.; Barbot, J.P.; Floquet, T., Exact differentiation and sliding mode observers for switched Lagrangian systems, Nonlinear analysis – theory methods & applications, 65, 5, 1050-1069, (2006) · Zbl 1134.93320
[36] Agrawal, A.K.; Xu, Z.; He, W.L., Ground motion pulse-based active control of a linear base-isolated benchmark building, Structural control & health monitoring, 13, 2-3, 792-808, (2006)
[37] Pozo, F.; Ikhouane, F.; Pujol, G.; Rodellar, J., Adaptive backstepping control of hysteric base-isolated structures, Journal of vibration and control, 12, 4, 373-394, (2006) · Zbl 1182.74171
[38] Haddad, W.M.; Chellaboina, V.; Nersesov, S.G., Hybrid nonnegative and compartmental dynamic systems, Mathematical problems in engineering, 8, 6, 493-515, (2002) · Zbl 1049.34062
[39] De la Sen, M., Sufficient conditions for lyapunov’s global stability of time-varying hybrid linear systems, International journal of control, 72, 2, 107-114, (1999) · Zbl 0953.93057
[40] M. De la Sen, About positivity of a class of hybrid dynamic linear systems, Applied Mathematics and Computation, doi:10.1016/j.amc.2006.11.182. · Zbl 1119.93038
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.