×

Considering the attractor structure of chaotic maps for observer-based synchronization problems. (English) Zbl 1064.37026

Summary: The main purpose of this paper is to state some sufficient conditions for global synchronization of chaotic maps. The synchronization is viewed as a state reconstruction problem which is tackled by polytopic observers. Unlike most standard observers, polytopic observers can account for a special property of chaotic dynamics. Indeed, it is shown that many chaotic maps can be described in a so-called convexified form, involving a time-varying parameter which depends on the chaotic state vector. Such a form makes it possible to incorporate knowledge on the structure of the compact set wherein the parameter lies. This set depends implicitly on the structure of the chaotic attractor. It is proved that the conservatism of the polyquadratic stability conditions for the state reconstruction, stated in a companion paper, can be reduced when the corresponding linear matrix inequalities involve the vertices of the minimal convex hull of this set. Theoretical developments along with special emphasis on computational aspects are provided and illustrated in the context of adaptive synchronization.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37N35 Dynamical systems in control
93D15 Stabilization of systems by feedback
93C10 Nonlinear systems in control theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Devaney, R. L., An Introduction to Chaotic Dynamical Systems (1989), Addison-Wesley: Addison-Wesley Redwood City, CA · Zbl 0695.58002
[2] Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Phys. Rev. Lett., 64, 11, 1196-1199 (1990) · Zbl 0964.37501
[3] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys. Rev. Lett., 64, 821-824 (1990) · Zbl 0938.37019
[4] Pecora, L. M.; Carroll, T. L., Driving systems with chaotic signals, Phys. Rev. A, 44, 8, 2374-2383 (1991)
[5] Blondel, V. D.; Sontag, E. D.; Vidyasagar, M.; Willems, J. C., Open problems in mathematical systems and control theory, in: Communication and Control Engineering (1999), Springer Verlag · Zbl 0945.93005
[6] Blekhman, I. I.; Fradkov, A. L.; Nijmeijer, H.; Pogromsky, A. Y., On self-synchronization and controlled synchronization, Syst. Control Lett., 31, 5, 299-305 (1997) · Zbl 0901.93028
[7] H. Nijmeijer (Ed.), Control of chaos and synchronization, Special Issue of Syst. Control Lett. 31 (5) (1997) 259-322.; H. Nijmeijer (Ed.), Control of chaos and synchronization, Special Issue of Syst. Control Lett. 31 (5) (1997) 259-322. · Zbl 0901.93029
[8] M.P. Kennedy, M.J. Ogorzalek (Eds.), Special Issue. Chaos synchronization and control: theory and applications, IEEE Trans. Circuits Syst. I: Fundamental Theor. Appl. 40 (10) (1997) 853-1039.; M.P. Kennedy, M.J. Ogorzalek (Eds.), Special Issue. Chaos synchronization and control: theory and applications, IEEE Trans. Circuits Syst. I: Fundamental Theor. Appl. 40 (10) (1997) 853-1039.
[9] Control and synchronization of chaos, Special Issue of Int. J. Bifurcat. Chaos 10 (4) (2000).; Control and synchronization of chaos, Special Issue of Int. J. Bifurcat. Chaos 10 (4) (2000).
[10] I.I. Blekhman E. Mosekilde, A.L. Fradkov (Eds.), Special Issue of Chaos Synchronization and Control, vol. 58, Elsevier, 2002.; I.I. Blekhman E. Mosekilde, A.L. Fradkov (Eds.), Special Issue of Chaos Synchronization and Control, vol. 58, Elsevier, 2002.
[11] Nijmeijer, H.; Mareels, I. M.Y., An observer looks at synchronization, IEEE Trans. Circuits Syst. I: Fundamental Theor. Appl., 44, 882-890 (1997)
[12] Huijberts, H. J.C.; Lilge, T.; Nijmeijer, H., Nonlinear discrete-time synchronization via extended observers, Int. J. Bifurcat. Chaos, 11, 7, 1997-2006 (2001)
[13] Pogromsky, A.; Nijmeijer, H., Observer-based robust synchronization of dynamical systems, Int. J. Bifurcat. Chaos, 8, 11, 2243-2254 (1998) · Zbl 1140.93468
[14] Sira Ramirez, H.; Cruz Hernandez, C., Synchronization of chaotic systems: a generalized Hamiltonian approach, Int. J. Bifurcat. Chaos, 11, 5, 1381-1395 (2001) · Zbl 1206.37053
[15] Cruz, C.; Nijmeijer, H., Synchronization through filtering, Int. J. Bifurcat. Chaos, 110, 4, 763-775 (2000) · Zbl 1090.37570
[16] Millerioux, G.; Daafouz, J., Polytopic observer for global synchronization of systems with output measurable nonlinearities, Int. J. Bifurcat. Chaos, 13, 3, 703-712 (2003) · Zbl 1129.93510
[17] Daafouz, J.; Millerioux, G., Poly-quadratic stability and global chaos synchronization of discrete time hybrid systems, Special Issue of Math. Comput. Simul., 58, 295-307 (2002) · Zbl 0997.65139
[18] Becker, G.; Packard, A.; Philbrick, D.; Balas, G., Control of parametrically-dependent linear systems: a single quadratic Lyapunov approach, (Proceedings of the American Control Conference. Proceedings of the American Control Conference, San Fransisco (June 1993))
[19] L. El Ghaoui, S.-I. Niculescu (Eds.), Advances in Linear Matrix Inequality methods in control, in: SIAMs Advances in Design and Control, 2000.; L. El Ghaoui, S.-I. Niculescu (Eds.), Advances in Linear Matrix Inequality methods in control, in: SIAMs Advances in Design and Control, 2000.
[20] Gahinet, P.; Apkarian, P.; Chilali, M., Parameter-dependent Lyapunov functions for real parametric uncertainty, IEEE Trans. Automat. Control, 41, 3, 436-442 (1996) · Zbl 0854.93113
[21] Feron, E.; Apkarian, P.; Gahinet, P., Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions, IEEE Trans. Automat. Control, 41, 1041-1046 (1996) · Zbl 0857.93088
[22] Bliman, P.-A., Nonconservative LMI approach to robust stability for systems with uncertain scalar parameters, (Proceedings of the 41st IEEE Control and Decision Conference. Proceedings of the 41st IEEE Control and Decision Conference, Las Vegas, USA (December 2002))
[23] Daafouz, J.; Bernussou, J., Parameter dependent Lyapunov functions for discrete time systems with time-varying parametric uncertainties, Syst. Control Lett., 43, 355-359 (2001) · Zbl 0978.93070
[24] Graham, R. L., An efficient algorithm for determining the convex hull of a finite planar set, Inform. Process. Lett., 2, 1, 132-133 (1973) · Zbl 0236.68013
[25] Eddy, W. F., ACM Trans. Math. Soft, 3, 398 (1977)
[26] Preparata, F. P.; Shamos, M. I., Computational Geometry (October 1985), Springer-Verlag
[27] Allison, D. C.S.; Noga, M. T., Computing the three-dimensional convex hull, Comput. Phys. Commun., 103, 1, 74-82 (1997)
[28] Chatterjee, S., A note on finding extreme points in multivariate space, Comput. Stat. Data Anal., 10, 87-92 (1990) · Zbl 0825.62548
[29] Pardalos, P. M.; Li, Y.; Hager, W. W., Linear programming approaches to the convex hull problem in \(R^m\), Comput. Math. Appl., 29, 7, 23-29 (1995) · Zbl 0831.90085
[30] Du, C., An algorithm for automatic delaunay triangulation of arbitrary planar domains, Adv. Eng. Software, 27, 21-26 (1996)
[31] Wu, C. W.; Yang, T.; Chua, L. O., On adaptive synchronization and control of nonlinear dynamical systems, Int. J. Bifurcat. Chaos, 6, 3, 455-471 (1996) · Zbl 0875.93182
[32] Fradkov, A. L.; Markov, A. Y., Adaptive synchronization of chaotic systems based on speed-gradient method and passification, IEEE Trans. Circuits Syst. I: Fundamental Theor. Appl., 44, 10, 905-912 (1997)
[33] Dedieu, H.; Kennedy, M. P.; Hasler, M., Chaos shift keying: modulation and demodulation of a chaotic carrier using self-synchronizing Chua’s circuits, IEEE Trans. Circuits Syst. II: Analog Digital Signal Process., 40, 10, 634-642 (1993)
[34] Anstett, F.; Millerioux, G.; Bloch, G., Global adaptive synchronization based upon polytopic observers, (Proceedings of the IEEE International Symposium on Circuit and System, ISCAS’04. Proceedings of the IEEE International Symposium on Circuit and System, ISCAS’04, Vancouver, Canada (May 2004)) · Zbl 1064.37026
[35] I. Gumowski, C. Mira, Dynamique chaotique, Transformations ponctuelles, Transitions Ordre-Désdordre, Cépadues ed., 1980.; I. Gumowski, C. Mira, Dynamique chaotique, Transformations ponctuelles, Transitions Ordre-Désdordre, Cépadues ed., 1980. · Zbl 0442.93001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.