Stability of Interconnected Systems under Structural Perturbation: Decomposition-Aggregation Approach
In this paper, the decomposition-aggregation method
is used to carry out connective stability criteria for general linear
composite system via aggregation. The large scale system is
decomposed into a number of subsystems. By associating directed
graphs with dynamic systems in an essential way, we define the
relation between system structure and stability in the sense of
Lyapunov. The stability criteria is then associated with the stability
and system matrices of subsystems as well as those interconnected
terms among subsystems using the concepts of vector differential
inequalities and vector Lyapunov functions. Then, we show that the
stability of each subsystem and stability of the aggregate model
imply connective stability of the overall system. An example is
reported, showing the efficiency of the proposed technique.
[1] M. Araki, "Application of M-matrix to the stability problems of
composite system" Journal of mathematical analysis and application, No.
52, pp. 309-321, 1975.
[2] F. N. Bailey "The application of Lyapunov-s second method to
interconnected systems" SIAM Journal of control, No. 3, pp. 443-462,
1966.
[3] R. Bellman, "Vector Lyapunov functions-, SIAM Journal of control,
vol.1, pp. 32-34, 1962.
[4] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrisnan "Linear matrix
inequalities in system and control theory" Philadelphia, PA: SIAM,
1994.
[5] F. H. Clark, Yu. S. Ledyaev "Nonsmooth analysis and control theory"
New-York: Springer-Verlag, 1998.
[6] R. Cogill, S. Lall "Control design for topology independent stability of
interconnected systems".
[7] 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.
[8] W. M Haddad, V. Chellaboina "Impulsive and Hybrid dynamical
systems" Princeton University Press, Princeton NJ 2006.
[9] R. A. Horn and C. R. Johnson "Topics in matrix analysis" Cambridge,
UK: Cambridge University Press, 1991.
[10] 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.
[11] M. Jamshidi "Large-scale systems: Modeling, control, and fuzzy logic"
Prentice Hall, Inc., New-Jersey, 1997.
[12] M. Kidouche and A. Charef "A constructive methodology of Lyapunov
functions of composite systems with nonlinear interconnection term"
International Journal of Robotics and Automation, vol.21, No.1, pp. 19-
24, 2006.
[13] M. Kidouche "The variable gradient method for generating Lyapunov
functions of nonlinear composite system" 11th IEEE International
Conference on Methods and Models in Automation and Robotics,
Miedzyzdroje, Poland,pp.233-237, Aug. 2005.
[14] G. S. Ladde, D. D. Siljak "Multiplex Control Systems: Stochastic
stability and dynamic reliability" International Journal of Control, N0.
38, pp. 514 - 524, 1983.
[15] V. Lakshmikantham, et al. "Vector Lyapunov functions and stability
analysis of nonlinear systems" Dordrecht, The Netherlands: Kluwer
Academic Publishers, 1991.
[16] V. M. Matrasov "On theory of stability of motion" Prikladnia
Matematika I Mekhanika, vol.26, pp. 992-1000, 1962.
[17] V. M. Matrasov "Nonlinear Control theory and Applications" Fizmatlit,
Moscow 2000.
[18] 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.
[19] A. N. Michel, B. Hu "Stability analysis of interconnected hybrid
dynamical systems" in Proc. IFAC symposium on large scale system,
1998.
[20] A. N. Michel "Qualitative analysis of dynamical systems, 2nd ed. New-
York: Marcel Dekker, 2001.
[21] A. Papachristodoulou and S. Prajma "On the construction of Lyapunov
functions using the sum of squares decomposition" Proc. of the 41st
IEEE Conference on Decision and Control, Las Vegas, Nevada, USA,
pp. 3482-3486, Dec. 2002.
[22] D. Reinhard "Graph theory"2nd Ed. New-York: Springer Verlag, 2000.
[23] S. Ruan "Connective stability of competitive equilibrium" Journal of
mathematical analysis and applications, vol. 160, pp. 480-484, 1991.
[24] D. D. Siljak "Decentralized control of complex systems" Academic
Press, New-York, 1991.
[25] D. D. Siljak "Connective stability of discontinuous interconnected
systems via parameter dependent Lyapunov functions" in Proc. of the
American control conference Arlington, VA, pp. 4189-4196, 2001.
[26] X. L. Tan, M. Ikeda "Decentralized stabilization for expanding
construction of large scale systems" IEEE Transactions on Automatic
Control NO. 35, pp. 644 - 651, 1990.
[27] H. C. Tang, D. D. Siljak "A learning scheme for dynamic neural
networks: Equilibrium manifold and connective stability" Neural
Networks, N0. 8, pp. 853 - 864, 1995.
[1] M. Araki, "Application of M-matrix to the stability problems of
composite system" Journal of mathematical analysis and application, No.
52, pp. 309-321, 1975.
[2] F. N. Bailey "The application of Lyapunov-s second method to
interconnected systems" SIAM Journal of control, No. 3, pp. 443-462,
1966.
[3] R. Bellman, "Vector Lyapunov functions-, SIAM Journal of control,
vol.1, pp. 32-34, 1962.
[4] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrisnan "Linear matrix
inequalities in system and control theory" Philadelphia, PA: SIAM,
1994.
[5] F. H. Clark, Yu. S. Ledyaev "Nonsmooth analysis and control theory"
New-York: Springer-Verlag, 1998.
[6] R. Cogill, S. Lall "Control design for topology independent stability of
interconnected systems".
[7] 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.
[8] W. M Haddad, V. Chellaboina "Impulsive and Hybrid dynamical
systems" Princeton University Press, Princeton NJ 2006.
[9] R. A. Horn and C. R. Johnson "Topics in matrix analysis" Cambridge,
UK: Cambridge University Press, 1991.
[10] 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.
[11] M. Jamshidi "Large-scale systems: Modeling, control, and fuzzy logic"
Prentice Hall, Inc., New-Jersey, 1997.
[12] M. Kidouche and A. Charef "A constructive methodology of Lyapunov
functions of composite systems with nonlinear interconnection term"
International Journal of Robotics and Automation, vol.21, No.1, pp. 19-
24, 2006.
[13] M. Kidouche "The variable gradient method for generating Lyapunov
functions of nonlinear composite system" 11th IEEE International
Conference on Methods and Models in Automation and Robotics,
Miedzyzdroje, Poland,pp.233-237, Aug. 2005.
[14] G. S. Ladde, D. D. Siljak "Multiplex Control Systems: Stochastic
stability and dynamic reliability" International Journal of Control, N0.
38, pp. 514 - 524, 1983.
[15] V. Lakshmikantham, et al. "Vector Lyapunov functions and stability
analysis of nonlinear systems" Dordrecht, The Netherlands: Kluwer
Academic Publishers, 1991.
[16] V. M. Matrasov "On theory of stability of motion" Prikladnia
Matematika I Mekhanika, vol.26, pp. 992-1000, 1962.
[17] V. M. Matrasov "Nonlinear Control theory and Applications" Fizmatlit,
Moscow 2000.
[18] 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.
[19] A. N. Michel, B. Hu "Stability analysis of interconnected hybrid
dynamical systems" in Proc. IFAC symposium on large scale system,
1998.
[20] A. N. Michel "Qualitative analysis of dynamical systems, 2nd ed. New-
York: Marcel Dekker, 2001.
[21] A. Papachristodoulou and S. Prajma "On the construction of Lyapunov
functions using the sum of squares decomposition" Proc. of the 41st
IEEE Conference on Decision and Control, Las Vegas, Nevada, USA,
pp. 3482-3486, Dec. 2002.
[22] D. Reinhard "Graph theory"2nd Ed. New-York: Springer Verlag, 2000.
[23] S. Ruan "Connective stability of competitive equilibrium" Journal of
mathematical analysis and applications, vol. 160, pp. 480-484, 1991.
[24] D. D. Siljak "Decentralized control of complex systems" Academic
Press, New-York, 1991.
[25] D. D. Siljak "Connective stability of discontinuous interconnected
systems via parameter dependent Lyapunov functions" in Proc. of the
American control conference Arlington, VA, pp. 4189-4196, 2001.
[26] X. L. Tan, M. Ikeda "Decentralized stabilization for expanding
construction of large scale systems" IEEE Transactions on Automatic
Control NO. 35, pp. 644 - 651, 1990.
[27] H. C. Tang, D. D. Siljak "A learning scheme for dynamic neural
networks: Equilibrium manifold and connective stability" Neural
Networks, N0. 8, pp. 853 - 864, 1995.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:54136", author = "M. Kidouche and H. Habbi and M. Zelmat", title = "Stability of Interconnected Systems under Structural Perturbation: Decomposition-Aggregation Approach", abstract = "In this paper, the decomposition-aggregation method
is used to carry out connective stability criteria for general linear
composite system via aggregation. The large scale system is
decomposed into a number of subsystems. By associating directed
graphs with dynamic systems in an essential way, we define the
relation between system structure and stability in the sense of
Lyapunov. The stability criteria is then associated with the stability
and system matrices of subsystems as well as those interconnected
terms among subsystems using the concepts of vector differential
inequalities and vector Lyapunov functions. Then, we show that the
stability of each subsystem and stability of the aggregate model
imply connective stability of the overall system. An example is
reported, showing the efficiency of the proposed technique.", keywords = "Composite system, Connective stability, Lyapunovfunctions.", volume = "2", number = "1", pages = "13-5", }
