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

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


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.


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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

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