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Discretized Lyapunov-Krasovskii functional for coupled differential-difference equations with multiple delay channels. (English) Zbl 1191.93120
Summary: Many practical systems have a large number of state variables, but only a few components involve time delays. These components are often scalar or low dimensional, and typically only one delay is present in each such component. A special form of coupled differential-difference equations with one delay per channel proposed recently is well suited to formulate such systems. This article extends the discretized Lyapunov-Krasovskii functional method to this class of systems. In addition to generality, this formulation also drastically reduces the computational cost for a typical system, and therefore is appropriate to use even for time-delay systems of retarded type. The discretized formulation is also simpler than the previous formulation for systems with multiple delays.

93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
34K20 Stability theory of functional-differential equations
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
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[1] Banks, H.T.; Burns, J.A., Hereditary control problems: numerical methods based on averaging approximations, SIAM journal on control and optimization, 16, 169-208, (1978) · Zbl 0379.49025
[2] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Philadelphia · Zbl 0816.93004
[3] Brayton, R., Nonlinear oscillations in a distributed network, Quarterly of applied mathematics, 24, 289-301, (1976) · Zbl 0166.35102
[4] Carvalho, L.A.V., On quadratic Liapunov functionals for linear difference equations, Linear algebra and its applications, 240, 41-64, (1996) · Zbl 0851.39006
[5] Cebotarev, N.G.; Meiman, N.N., The Routh-Hurwitz problem for polynomials and for entire functions, Trudy matematicheskogo instituta imeni V. A. steklova., 26, 1-332, (1949), (in Russian)
[6] Cruz, M.A.; Hale, J.K., Stability of functional differential equations of neutral type, Journal of differential equations, 7, 334-355, (1970) · Zbl 0191.38901
[7] Datko, R., Extending a theorem of A. M. Liapunov to Hilbert space, Journal of mathematical analysis and applications, 32, 610-616, (1970) · Zbl 0211.16802
[8] Datko, R., Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM journal on mathematical analysis, 3, 428-445, (1972) · Zbl 0241.34071
[9] Datko, R., Liapunov functionals for certain linear delay-differential equations in a Hilbert space, Journal of mathematical analysis and applications, 76, 37-57, (1980) · Zbl 0482.34055
[10] Delfour, M.C., The linear quadratic optimal control for hereditary differential systems: theory and numerical solution, Applied mathematics and optimization, 3, 2-3, 101-162, (1976) · Zbl 0404.49010
[11] Doyle, J. C., Wall, J., & Stein, G. (1982). Performance and robustness analysis for structured uncertainty. In 20th IEEE conference on decision and control(pp. 629-636).
[12] Fridman, E., Stability of linear descriptor systems with delay: A Lyapunov-based approach, Journal of mathematical analysis and applications, 273, 14-44, (2002) · Zbl 1032.34069
[13] Gouaisbaut, F., & Peaucelle, D. (2006a). Delay-dependent robust stability of time delay systems. In IFAC-ROCOND, Toulouse, France, 5-7 July, 2006. · Zbl 1293.93589
[14] Gouaisbaut, F., & Peaucelle, D. (2006b). Delay-dependent stability analysis of linear time delay systems. In The sixth IFAC workshop on time-delay systems, Aquila, Italy, 10-12 July.
[15] Gu, K., A further refinement of discretized Lyapunov functional method for the stability of time-delay systems, International journal of control, 74, 10, 967-976, (2001) · Zbl 1015.93053
[16] Gu, K., Refined discretized Lyapunov functional method for systems with multiple delays, International journal of robust and nonlinear control, 13, 11, 1017-1033, (2003) · Zbl 1039.93032
[17] Gu, K. (2010). Stability problem of systems with multiple delay channels. Automatica, in press (doi:10.1016/j.automatica.2010.01.028). · Zbl 1193.93157
[18] Gu, K.; Liu, Y., Lyapunov-krasovskii functional for uniform stability of coupled differential-functional equations, Automatica, 45, 3, 798-804, (2009) · Zbl 1168.93384
[19] Gu, K.; Niculescu, S.-I., Stability analysis of time-delay systems: A Lyapunov approach, (), 139-170 · Zbl 1217.93069
[20] Halanay, A.; Raˇsvan, V., Approximation of delays by ordinary differential equations, (), 155-198
[21] Hale, J.; Huang, W., Variation of constants for hybrid systems of functional differential equations, Proceedings of the royal society of Edinburgh, 125A, 1-12, (1993) · Zbl 0830.34055
[22] Hale, J.K.; Verduyn Lunel, S.M., Introduction to functional differential equations, (1993), Springer-Verlag New York · Zbl 0787.34002
[23] Han, Q.-L., Improved stability criteria and controller design for linear neutral systems, Automatica, 45, 8, 1948-1952, (2009) · Zbl 1185.93102
[24] Han, Q., & Yu, X. (2002). A discretized Lyapunov functional approach to stability of linear delay-differential systems of neutral type. In Proc. of the 15th IFAC world congress on automatic control, Barcelona, Spain, July 21-26, 2002.
[25] Kabakov, I.P., On steam pressure control, Inzhenernii sbornik, 2, 27-60, (1946), (in Russian)
[26] Kappel, F.; Schappacher, W., Autonomous nonlinear functional differential equations and averaging approximations, Nonlinear analysis: theory methods applications, 2, 391-422, (1978) · Zbl 0402.34056
[27] Kharitonov, V.L.; Zhabko, A.P., Lyapunov-krasovskii approach to the robust stability analysis of time-delay systems, Automatica, 39, 15-20, (2003) · Zbl 1014.93031
[28] Kolmanovskii, V.; Myshkis, A., Introduction to the theory and applications of functional differential equations, (1999), Kluwer Dordrecht, The Netherlands · Zbl 0917.34001
[29] Krasovskii, N.N., Stability of motion, (1963), Stanford University Press · Zbl 0109.06001
[30] Krasovskii, N.N., The approximation of a problem of analytic design of controls in a system with time lags, Journal of applied mathematics and mechanics, 28, 876-885, (1964) · Zbl 0143.32201
[31] Li, H., & Gu, K. (2008). Discretized Lyapunov-Krasovskii functional for systems with multiple delay channels. In 2008 ASME dynamic systems and control conference, Ann Arbor, MI, October 20-22, 2008.
[32] Martinez-Amores, P., Periodic solutions of coupled systems of differential and difference equations, Annali di matematica pura ed applicata, 121, 1, 171-186, (1979) · Zbl 0419.34069
[33] Niculescu, S.-I., ()
[34] Ochoa, G., & Kharitonov, V. L. (2005). Lyapunov matrices for neutral type of time delay systems. In 2nd int. conf. on electrical & electronics engineering and 11th conf. on electrical engineering, Mexico City, September 7-9, 2005.
[35] Peet, M., Papachristodoulou, A., & Lall, A. (2006). On positive forms and the stability of linear time-delay systems. In 45th conference on decision and control, San Diego, CA, Dec. 13-15. · Zbl 1187.34101
[36] Pepe, P., On the asymptotic stability of coupled delay differential and continuous time difference equations, Automatica, 41, 1, 107-112, (2005) · Zbl 1155.93373
[37] Pepe, P.; Jiang, Z.-P.; Fridman, E., A new Lyapunov-krasovskii methodology for coupled delay differential and difference equations, International journal of control, 81, 1, 107-115, (2007) · Zbl 1194.39004
[38] Pepe, P.; Verriest, E.I., On the stability of coupled delay differential and continuous time difference equations, IEEE transactions on automatic control, 48, 8, 1422-1427, (2003) · Zbl 1364.34104
[39] Raˇsvan, V., Absolute stability of a class of control processes described by functional differential equations of neutral type, () · Zbl 0266.93044
[40] Raˇsvan, V. (1998). Dynamical systems with lossless propagation and neutral functional differential equations. In Proc. MTNS 98, 1998 (pp. 527-531). Padoue, Italy.
[41] Raˇsvan, V. (2006). Functional differential equations of lossless propagation and almost linear behavior, Plenary Lecture. In 6th IFAC workshop on time-delay systems, L’Aquila, Italy.
[42] Raˇsvan, V.; Niculescu, S.-I., Oscillations in lossless propagation models: A Liapunov-krasovskii approach, IMA journal of mathematical control and information, 19, 157-172, (2002) · Zbl 1020.93010
[43] Repin, Yu.M., On the approximate replacement of systems with lags by ordinary differential equations, Journal of applied mathematics and mechanics, 29, 254-264, (1965) · Zbl 0143.10702
[44] Ross, D.W., Controller design for time lag systems with a quadratic criterion, IEEE transactions on automatic control, 16, 664-672, (1971)
[45] Ross, D.W.; Flugge-Lotz, I., An optimal control problem for systems with differential difference equation dynamics, SIAM journal on control and optimization, 7, 609-623, (1969) · Zbl 0186.48601
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