Geodesic
Consistency and Lyapunov stability of linear singular time delay systems: a geometric approach
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 11 (2018) no. 4, pp. 123-135 Cet article a éte moissonné depuis la source Math-Net.Ru

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When we consider the control design of practical systems (chemical engineering systems, lossless transmission lines, large-scale electric network control, aircraft attitude control, flexible arm control of robots, etc.), time-delay often appears in many situations. Singular time delayed systems are the dynamic systems described by a mixture of algebraic and differential equations with retarded argument. This paper investigates the geometric description of initial conditions that generate smooth solutions to such problems and the construction of the Lyapunov stability theory to bound the rates of decay of such solutions. The new delay dependent conditions for asymptotic stability for the class of systems under consideration were derived. Moreover, the result is expressed in terms of only systems matrices that naturally occur in the model, therefore avoiding the need to introduce algebraic transformations into the statement of the main theorem.
Keywords: singular time delayed systems, stability in the sense of Lyapunov, consistent initial conditions. 
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     title = {Consistency and {Lyapunov} stability of linear singular time delay systems: a~geometric approach},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {123--135},
     year = {2018},
     volume = {11},
     number = {4},
     language = {en},
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D. L. Debeljković; I. M. Buzurović; G. V. Simeunović. Consistency and Lyapunov stability of linear singular time delay systems: a geometric approach. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 11 (2018) no. 4, pp. 123-135. http://geodesic.mathdoc.fr/item/VYURU_2018_11_4_a8/

[1] Campbell S. L., Singular Systems of Differential Equations, Pitman, Marshfield, 1980 | MR | Zbl

[2] Müller P. C., “Stability of Linear Mechanical Systems with Holonomic Constraint”, Applied Mechanics Review, 46 (1993), 160–164

[3] Campbell S. L., Meyer C. D., Rose N. J., “Application of Drazin Inverse to Linear Systems of Differential Equations”, SIAM Journal on Applied Mathematics, 31 (1976), 411–425 | DOI | MR | Zbl

[4] Luenberger D. G., “Dynamic Equations in Descriptor Form”, IEEE Transactions on Automatic Control, 22:3 (1977), 312–321 | DOI | MR | Zbl

[5] Bajic V. B., Lyapunov's Direct Method in the Analysis of Singular Systems and Networks, Shades Technical Publications, Hillcrest, 1992

[6] Campbell S. L., Singular Systems of Differential Equations II, Pitman, Marshfield, 1982 | MR | Zbl

[7] Campbell S. L., “Consistent Initial Conditions for Singular and Nonlinear Systems”, Circuits, Systems and Signal Processing, 2 (1983), 45–55 | DOI | MR | Zbl

[8] Lewis F. L., “A Survey of Linear Singular Systems”, Circuits, Systems and Signal Processing, 5 (1986), 3–36 | DOI | MR | Zbl

[9] Lewis F. L., “Recent Work in Singular Systems”, Proceedings of Symposium on Electromagnetic Compatibility (Atlanta, 1987), 20–24

[10] Debeljkovic D. Lj., Milinkovic S. A., Jovanovic M. B., “Application of Singular Systems Theory in Chemical Engineering”, MAPRET Lecture Monograph, 12th International Congress of Chemical and Process Engineering (1996)

[11] Pjescic R. M., Chistyakov V., Debeljkovic D. Lj., On Dynamical Analysis of Particular Class of Linear Singular Time Delayed Systems: Stabilty and Robusteness, Faculty of Mechanical Engineering, Belgrade, 2008 (in Serbian)

[12] Circuits, Systems and Signal Processing, 5:1, Special Issue on Semi State Systems (1986)

[13] Circuits, Systems and Signal Processing, 8:3, Special Issue: Recent Advances in Singular Systems (1989)

[14] Zhu S., Zhang C., Cheng Z., Feng J., “Delay-Dependent Robust Stability Criteria for Two Classes of Uncertain Singular Time-Delay Systems”, IEEE Transactions on Automatic Control, 52:5 (2007), 880–885 | DOI | MR | Zbl

[15] Xu S., Lam J., Zou Y., “An Improved Characterization of Bounded Realness for Singular Delay Systems and Its Applications”, International Journal of Robust and Nonlinear Control, 18:3 (2008), 263–277 | DOI | MR | Zbl

[16] Sun X., Zhang Q., Yang C., Su Z., Shao Y., “An Improved Approach to Delay-Dependent Robust Stabilization for Uncertain Singular Time-Delay Systems”, International Journal of Automation and Computing, 7:2 (2010), 205–212 | DOI | MR

[17] Debeljkovic D. Lj., Stojanovic S. B., Jovanovic M. B., Milinkovic S. A., “Singular Time Delayed System Stabilty Theory: Approach in the Sense of Lyapunov”, Preprints of IFAC Workshop on Time Delay Systems, Leuven, 2004, 9886174, 19 pp. | MR

[18] Debeljkovic D. Lj., Stojanovic S. B., Jovanovic M. B., Milinkovic S. A., “Further Results on Singular Time Delayed System Stability”, Proceedings of IEEE ACC (Oregon, 2005), 8573603, 23 pp. | MR

[19] Debeljkovic D. Lj., Stojanovic S. B., Jovanovic M. B., Milinkovic S. A., “Further Results on Descriptor Time Delayed System Stability Theory in the Sense of Lyapunov: Pandolfi Based Approach”, The Fifth International Conference on Control and Automation (Budapest, 2005), 353–358 | MR

[20] Debeljkovic D.Lj., Stojanovic S. B., Jovanovic M. B., Milinkovic S. A., “Further Results on Descriptor Time Delayed System Stability Theory in the Sense of Lyapunov: Pandolfi Based Approach”, International Journal of Information and System Science, 2:1 (2006), 1–11 | MR | Zbl

[21] Debeljkovic D. Lj., Stojanovic S. B., Visnjic N. S., Milinkovic S. A., “Singular Time Delayed System Stability Theory in the Sense of Lyapunov: A Quite New Approach”, American Control Conference (N.Y., 2007), 21–29 | MR

[22] Debeljkovic D. Lj., Stojanovic S. B., Milinkovic S. A., Jacic A., Visnjic N., Pjescic M., “Stability in the Sense of Lyapunov of Generalized State Space Time Delayed Systems: A Geometric Approach”, International Journal of Information and System Science, 4:2 (2008), 278–300 | MR | Zbl

[23] Pandolfi L., “Controllability and Stabilization for Linear Systems of Algebraic and Differential Equations”, Journal of Optimization Theory and Applications, 30:4 (1980), 601–620 | DOI | MR | Zbl

[24] Owens D. H., Debeljkovic D. Lj., “Consistency and Lyapunov Stability of Linear Descriptor Systems: A Geometric Approach”, Journal on Mathematical Control and Information, 2 (1985), 139–151 | DOI | Zbl

[25] Lee T. N., Diant S., “Stability of Time-Delay Systems”, IEEE Transactions on Automatic Control, 26:4 (1981), 951–953 | DOI | MR | Zbl

[26] Xu S., Dooren P. V., Stefan R., Lam J., “Robust Stability and Stabilization for Singular Systems with State Delay and Parameter Uncertainty”, IEEE Transactions on Automatic Control, 47 (2002), 122–128 | MR

[27] Debeljkovic D. Lj., “Singular Control Systems”, Dynamics of Continuous, Discrete and Impulsive Systems, 11 (2004), 697–706 | MR

[28] Debeljkovic D. Lj., Time Delay Systems, Intechopen, Vienna, 2011

[29] Debeljkovic D. Lj., Stojanovic S. B., “Asymptotic Stability Analysis of Linear Time Delay Systems: Delay Dependent Approach”, Systems Structure and Control, 2008, 29–60 | MR

[30] Debeljkovic D. Lj., Nestorovic T., “Time Delay Systems. Stability of Linear Continuous Singular and Discrete Descriptor Systems over Infinite and Finite Time Interval”, Systems Structure and Control, 2011, 15–30

[31] Debeljkovic D. Lj., Nestorovic T., “Time Delay Systems. Stability of Linear Continuous Singular and Discrete Descriptor Time Delayed Systems”, Systems Structure and Control, 2011, 31–74

[32] Stojanovic S. B., Debeljkovic D. Lj., “Necessary and Sufficient Conditions for Delay Dependent Asymptotic Stability of Linear Continuous Large Scale Time Delay Autonomous Systems”, Asian Journal of Control, 7:4 (2005), 414–418 | DOI | MR

[33] Debeljkovic D. Lj., Stojanovic S. B., Milinkovic S. A., Jacic Lj. A., Visnjic N. S., Pjescic M. R., “Stability in the Sense of Lyapunov of Generalized State Space Time Delayed Systems: A Geometric Approach”, International Journal of Information and System Science, 4:2 (2008), 278–300 | MR | Zbl

[34] Debeljkovic D. Lj., Buzurovic I. M., “Lyapunov Stability of Linear Continuous Singular Systems: An Overview”, International Journal of Information and System Science, 7:2 (2011), 247–268 | MR | Zbl

[35] Debeljkovic D. Lj., Stojanovic S. B., Jovanovic A. M., “Finite Time Stability of Continuous Time Delay Systems: Lypunov – Like Approach with Jensens and Coppels Inequality”, Acta Polytechnica Hungarica, 10:7 (2013), 135–150