Chaotification of nonlinear discrete systems via immersion and invariance

This paper is concerned with the chaotification of nonlinear discrete systems. A novel method based on (system) immersion and (manifold) invariance (I&I) is introduced to chaotify nonlinear discrete systems. Its basic idea is to immerse an ideal system which holds chaotic properties and may be a lower dimension into the plant system, and then control trajectories of the plant system to converge toward the invariant manifold where the ideal system is immersed. For a class of linearizable systems, we present the immersion and the control law such that these systems can be chaotified through I&I design. An illustrative example with simulation is presented to validate the proposed chaotification scheme.

[1]  Alessandro Astolfi,et al.  Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems , 2001, IEEE Trans. Autom. Control..

[2]  Jun-Guo Lu,et al.  Generating chaos via decentralized linear state feedback and a class of nonlinear functions , 2005 .

[3]  Guanrong Chen,et al.  Feedback anticontrol of discrete chaos , 1998 .

[4]  Guanrong Chen,et al.  Chaotifying a stable LTI system by tiny feedback control , 2000 .

[5]  Riccardo Marino,et al.  Nonlinear control design: geometric, adaptive and robust , 1995 .

[6]  Liang Chen,et al.  Controlling chaos in Internet congestion control model , 2004 .

[7]  Alessandro Astolfi,et al.  A constructive solution for stabilization via immersion and invariance: The cart and pendulum system , 2008, Autom..

[8]  Alessandro Astolfi,et al.  Nonlinear and adaptive control with applications , 2008 .

[9]  A. Isidori.  Nonlinear Control Systems , 1985 .

[10]  Guanrong Chen,et al.  Making a dynamical system chaotic: feedback control of Lyapunov exponents for discrete-time dynamical systems , 1997 .

[11]  Zhengzhi Han,et al.  Controlling and synchronizing chaotic Genesio system via nonlinear feedback control , 2003 .

[12]  Robin J. Evans,et al.  Control of chaos: Methods and applications in engineering, , 2005, Annu. Rev. Control..

[13]  Guanrong Chen,et al.  Chaotifying a stable map via smooth small‐amplitude high‐frequency feedback control , 2000, Int. J. Circuit Theory Appl..

[14]  L. Chua,et al.  Generalized synchronization of chaos via linear transformations , 1999 .

[15]  Xiang Li,et al.  Generating chaos by an Elman network , 2001 .

[16]  Qigui Yang,et al.  A simple time-delay feedback anticontrol method made rigorous. , 2004, Chaos.

[17]  Guanrong Chen,et al.  Chaotification via arbitrarily Small Feedback Controls: Theory, Method, and Applications , 2000, Int. J. Bifurc. Chaos.

[18]  M. Hénon,et al.  A two-dimensional mapping with a strange attractor , 1976 .

[19]  Wei Zhang,et al.  Stabilization of center systems via immersion and invariance , 2009 .

[20]  Y. Soh,et al.  The stabilization and synchronization of Chua's oscillators via impulsive control , 2001 .

[21]  Alessandro Astolfi,et al.  Immersion and invariance adaptive control of linear multivariable systems , 2003, Syst. Control. Lett..

[22]  Guanrong Chen,et al.  FEEDBACK CONTROL OF LYAPUNOV EXPONENTS FOR DISCRETE-TIME DYNAMICAL SYSTEMS , 1996 .