Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 31100
The First Integral Approach in Stability Problem of Large Scale Nonlinear Dynamical Systems

Authors: M. Kidouche, H. Habbi, M. Zelmat, S. Grouni


In analyzing large scale nonlinear dynamical systems, it is often desirable to treat the overall system as a collection of interconnected subsystems. Solutions properties of the large scale system are then deduced from the solution properties of the individual subsystems and the nature of the interconnections. In this paper a new approach is proposed for the stability analysis of large scale systems, which is based upon the concept of vector Lyapunov functions and the decomposition methods. The present results make use of graph theoretic decomposition techniques in which the overall system is partitioned into a hierarchy of strongly connected components. We show then, that under very reasonable assumptions, the overall system is stable once the strongly connected subsystems are stables. Finally an example is given to illustrate the constructive methodology proposed.

Keywords: lyapunov stability, Comparison principle, First integral, Large scale system

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1267


[1] E. P. Akpan "Stability of large scale systems and the method of cone valued Lyapunov functions" Nonlinear analysis, theory, methods and applications, vol. 26, No. 10, pp.1613-1620, 1996.
[2] A. S. "The comparison method in asymptotic stability problem" Journal of applied mathematics and mechanics, vol. 70, pp. 865-875, 2006.
[3] F. N. Bailey "The application of Lyapunov-s second method to interconnected systems" SIAM Journal of control, No. 3, pp. 443-462, 1966.
[4] R. Bellman, "Vector Lyapunov functions-, SIAM Journal of control, vol.1, pp. 32-34, 1962.
[5] C. Berge, "Graphs and hypergraphs", Amesterdam, North Holland, 1973.
[6] A. Bhaya, "Diagonal stability in the large scale system approach- Proc. Of the 35th conference on decision and control, Kobe, Japan, pp. 293- 311, Dec. 1996.
[7] V. Chellaboina, W.M Haddad "A unification between partial stability and stability theory for time varying systems" IEEE Control systems Mag.22, pp. 66-75, 2002.
[8] R. Cogill, S. Lall "Control design for topology independent stability of interconnected systems" Proceedings of the American Control Conferences, Boston, MA, pp. 3717-3722, 2004.
[9] N. Deo "Graph theory with application to engineering and computer science, Englewood, NJ Prentice-Hall, 1974.
[10] S. Dubljevic "A new Lyapunov design approach for nonlinear systems based on a Zubov-s method" Automatica, vol. 38, pp. 1999-2007, 2002.
[11] L. Faubourg and J.B. Ponet "Control Lyapunov functions for homogeneous systems" ESAIM: Control Optim. Calculus Variations, Vol. 5, pp. 293-311, June 2000.
[12] R.D. Gai, S. Zhang "Stability of linear large scale composite systems" Proc. of American Control Conference, Baltimore, Maryland, pp. 2207- 2211, June 1994.
[13] B. L. Griffin, G. S. Ladde "Qualitative properties of stochastic iterative processes under random structural perturbations" Mathematics and Computers in simulation, pp. 181-200, 67/2004.
[14] W. M Haddad, V. Chellaboina "Large scale nonlinear dynamical systems: a vector dissipative systems approach" Proc. IEEE Conference on decision and Control, Hawai, pp. 5603-5608, Dec. 2005.
[15] T. Hu and Z. Lin "Properties of the composite quadratic Lyapunov functions" IEEE Trans. on Automatic Control, vol.49, No.7, pp. 1162- 1167, July 2004.
[16] M. Jamshidi "Large-scale systems: Modeling, control, and fuzzy logic" Prentice Hall, Inc., New-Jersey, 1997.
[17] J. Jian, X. Liao "Partial exponential stability of nonlinear time varying large scale systems" Nonlinear Analysis, Vol. 59, pp. 789-800, July 2004.
[18] P.S. Krasilnoikov "A generalized scheme for constructing Lyapunov functions from first integrals" Journal Appl. Maths. Mechs., Vol. 65, No.2, pp. 195-204; 2001.
[19] A. K. Kevorkian "Structural aspects of large dynamical systems" presented at sixth IFAC World Congress, Boston, MA , paper 19.3 ; Aug. 1975.
[20] M. Kidouche; H. Habbi "Stability of interconnected system under structural perturbation: decomposition aggregation approach" International Journal of Mathematical, Physical and Engineering Sciences, Vol.2, No.3, pp. 121-125, February 2008.
[21] M. Kidouche "A constructive methodology of Lyapunov function function of composite systems" International Journal of Robotics and Automation, vol.21, No.1, 2006.
[22] G. S. Ladde, D. D. Siljak "Multiplex Control Systems: Stochastic stability and dynamic reliability" International Journal of Control, N0. 38, pp. 514 - 524, 1983.
[23] V. Lakshmikantham, et al. "Vector Lyapunov functions and stability analysis of nonlinear systems" Dordrecht, The Netherlands: Kluwer Academic Publishers, 1991.
[24] V. M. Matrasov "On theory of stability of motion" Prikladnia Matematika I Mekhanika, vol.26, pp. 992-1000, 1962.
[25] V. M. Matrasov "Nonlinear Control theory and Applications" Fizmatlit, Moscow 2000.
[26] A. A. Martynyuk "Novel stability and instability conditions for interconnected systems with structural perturbations" Dynamics of continuous discrete and impulsive systems, N0. 7, pp. 307 - 324, 2000.
[27] A. N. Michel "Qualitative Analysis of Large Scale Dynamical Systems" Academic Press Inc. New-York, 1977.
[28] A.N. Michel et al." Lyapunov stability of interconnected subsystems: Decomposition into strongly connected components" IEEE Trans. on circuits and systems, vol . CAS-25, no.9, pp 799-809, Sept. 1978.
[29] D.D. Sijak "Dynamic Graph" Nonlinear analysis: Hybrid Systems, doi: 1016/j.nahs. pp 1-24, 2006.
[30] D.D. Siljak "Large scale systems: Stability and Structure" Amesterda,, North Holland, 1978.
[31] R.Tarjan « Depth-first search and linear graph algorithm » SIAM J. Computer vol. 1.pp 146-160 June 1972.