Advertisement

Synchronization for Chaotic System Through an Observer Using the Immersion and Invariance (I&I) Approach

  • Rafael Martínez-GuerraEmail author
  • Claudia Alejandra Pérez-Pinacho
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

In this chapter, the named master–slave configuration, where nonlinear systems observers and chaos synchronization are used together. The key idea is to design observers to accomplish chaos synchronization, where the slave is actually an observer coupled to the master through its corresponding output. This chapter aims at the master–slave synchronization by applying the Immersion and Invariance (I&I) method to solve the chaos synchronization problem for a kind of simple chaotic systems. To this end, a class of feedback-linearized chaotic systems are characterized. Afterwards, the I&I method is applied to propose the corresponding observer, or slave system, for such systems. This observer, which has some robust properties, allows asymptotic estimation of the underlying dynamics of the master system. Notably, the I& I approach has been successfully applied to control, identify and observe a wide range of nonlinear systems. The seminal ideas of the I& I approach and its application can be found in Astolfi and Ortega (IEEE Trans. Autom. Control 48(4):590–606, 2003, [1], X. Liu (IEEE Trans. Autom. Control 55(9):2209–2214, 2010, [2]). The chapter is organized as follows. In Sect. 4.1, the problem statement is established. In Sect. 4.2, the observer is proposed to solve the synchronization problem, by applying the I& I method. Section 4.3 shows the results of some numerical comparisons with other well-known observers. The conclusions are given in Sect. 4.4.

References

  1. 1.
    Alessandro Astolfi and Romeo Ortega. Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear system. Automatic Control, IEEE Transactions on, 48(4):590–606, 2003.Google Scholar
  2. 2.
    Xiangbing Liu, Romeo Ortega, Hongye Su, and Jian Chu. Immersion and invariance adaptive control of nonlinearly parameterized nonlinear systems. Automatic Control, IEEE Transaction on 55(9):2209–2214, 2010.Google Scholar
  3. 3.
    Martínez-Guerra, Rafael, and Christopher Diego Cruz-Ancona. Algorithms of Estimation for Nonlinear Systems: A Differential and Algebraic Viewpoint. Springer, 2017.Google Scholar
  4. 4.
    AN Atassi and HK Khalil. Separation results for the stabilization of nonlinear systems using different high-gain observer designs. Systems & Control Letters, 39(3): 183–191, 2000.MathSciNetCrossRefGoogle Scholar
  5. 5.
    J Heagy and WL Ditto. Dynamics of a two-frequency parametrically driven duffing oscillator. Journal of Nonlinear Science, 1(4):423–455, 1991.ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Rafael Martínez-Guerra
    • 1
    Email author
  • Claudia Alejandra Pérez-Pinacho
    • 1
  1. 1.Automatic ControlCINVESTAV-IPNMexico CityMexico

Personalised recommendations