A new Tool for Input to State Stabilization of Delayed System via FDE Dissipativity

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Published: Jun 29, 2009
DOI: https://doi.org/10.37936/ecti-eec.201082.172110
Keywords:
Function Differential Equation (FDE) FDE-Dissipativity Input to State Stability (ISS) Feedback Connection

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Adha Imam Cahyadi
Department of Electrical Engineering, Diploma Program Engineering Faculty, Gadjah Mada University
Yoshio Yamamoto
Department of Precision Engineering School of Engineering, Tokai University

Abstract

In this paper, we derive a new tool for Input to State Stabilization (ISS) of a delayed systems. The considered system is in a very general form of Functional Differential Equation (FDE). Firstly, a notion of FDE Dissipativity that is applicable to a class of delayed nonlinear systems is introduced and explored. It will be shown that FDE-Dissipativity will imply Input to State Stability (ISS) for time delay systems. Moreover it has equivalent characteristics with the non-delayed nonlinear systems, for instance, feedback interconnection of two FDE-Dissipative systems is also FDE-Dissipative and it is possible to reconstruct loop-transformation with properties similar to the non-delayed passive system. Based on this, we construct a Lyapunov-Krasovskii like functional as the main tool for the IS-Stabilization of the FDE systems. Finally, a numerical studies are presented in order to verify the e®ectiveness of the proposed tool.

Article Details

How to Cite
Cahyadi, A. I., & Yamamoto, Y. (2009). A new Tool for Input to State Stabilization of Delayed System via FDE Dissipativity. ECTI Transactions on Electrical Engineering, Electronics, and Communications, 8(2), 239–245. https://doi.org/10.37936/ecti-eec.201082.172110
Issue
Vol. 8 No. 2 (2010): Special Session from ISMAC 2009
Section
Research Article

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References

[1] T. A. Burton, "Voltera Integral and Differential Equations," Mathematics in Science and Engineering, Elsevier, second edition edition, vol.202, 2005.

[2] T. A. Burton, "Stability theory for functional differential equations," Transactions of American Math Society, vol. 255, pp.263-275, 1979.

[3] R. D. Driver, "Existence and stability solutions of a delay differential system," Arch. Rational Mech. Analysis, vol.10, pp.401-426, 1962.

[4] S. Gan, "Dissipativity of µ-methods for Nonlinear Volterra Delay-Integro-Differential Equations," Journal of Computational and Applied Mathematics, vol.206, pp.898-907, 2007.

[5] J. Hale, Theory of Functional Differential Equations, Springer Publications, 1977.

[6] C.M. Huang and G.N. Chen, "Dissipativity of runge-kutta methods for dynamical systems with delay," IMA Journal of Numerical Analysis, vol.20, pp.153-166, 2000.

[7] H. Hulsen, T. Trtiper and S. Fatikow, "Control System for the automatic Handling of biological Cells with mobile Microrobots," Proceedings of the 2004 American Control Conference, 2004.

[8] I. G. Polushin, A. Tayebi and H. J. Marquez, "Control Schemes for Stable Teleoperation with Communication delay based on IOS Small Gain Theorem," Automatica, vol.42, pp.905-915, 2006.

[9] M. Jankovic, "Control lyapunov-razumikhin functions and robust stabilization of time delay systems," IEEE Transactions on Automatic Control, vol.46, pp.1048-1060, 2001.

[10] X. Jiao, T. Shen, Y. Sun, and K. Tamura, "Krasovskii functional, razumikhin function and backstepping," In 2004 International Conference on Control, Automation, Robotics and Vision, pp. 1200-1205. IEEE, December 2004.

[11] H. K. Khalil, Nonlinear Systems,Prentice Hall, 2002.

[12] N.N. Krasovskii, Stability of Motion. Stanford University Press, 1963.

[13] F. Mazenc and P.A. Bliman, "Backstepping design for time-delay nonlinear systems," IEEE Transactions on Automatic Control, vol.51, pp.149-154, 2006.

[14] S.K. Nguang, "Robust stabilization of a class of time-delay nonlinear systems," IEEE Transactions on Automatic Control, vol.45, pp 756-762, 2000.

[15] R. Ortega, A. Loria and R. Kelly, "A Semiglobally Stable Output Feedback PI2D Regulator," IEEE Transactions on Automatic Control for Robot Manipulators, vol.40, pp.1432-1436, 1995.

[16] P. Pepe and Z. P. Jiang, "A lyapunov-krasovskii methodology for iss of time delay systems," In Proceedings of the IEEE Conference on Decision and Control, pp. 57825787, December 2005.

[17] I. G. Polushin, A. Tayebi and H. J. Marquez, "Control schemes for stable teleoperation with communication delay based on ios small gain theorem," Automatica, vol.42, pp.905-915, 2006.

[18] Andrew R. Teel, "Connections between razumikhin-type theorems and iss nonlinear gain theorem," IEEE Transactions on Automatic Control, vol.43, pp.960-964, 1998.

[19] H.J Tian, "Numerical and analytic dissipativity of the µ-method for delay differential equation with bounded variable lag," Journal of Biffurcation and Chaos, vol.14, pp.1839-1845, 2004.

[20] L.Wen and S. Li, "Dissipativity of volterra functional differential equations," J. Math. Anal. Appl., vol.324, pp.696-706, 2006.