An I&I‐Based Observer to Solve the Output‐Feedback Synchronization Problem for a Class of Chaotic Systems

Abstract

A novel robust observer, intended to solve the output-feedback chaos synchronization problem for the Master/Slave Configuration, is proposed here. Assuming that the given Master system belongs to a specific class of feedback-linearized systems, our solution is based on the well-known Immersion and Invariance (I&I) method. The proposed observer is devoted to the asymptotic estimation of the Master system's underlying dynamics, and its effectiveness is illustrated via computer-based simulations that involve both the so-called Duffing's oscillator and the Genesio & Tesi system.

Transformation of Nonlinear MIMO Discrete‐Time Systems into the Extended Observer Form

“…such that in the new coordinates the state Eqs. (1) are in the extended observer form † , consisting of the following p blocks z [1] i,j = z i,j+1…”

“…, p may be substituted according to (1), such that the right-hand sides of Eqs. (9) depend only on and the † The extended observer form with one output and without inputs was considered earlier in [12].…”

“…For a system in the extended observer form (7), one may straightforwardly construct an observer aŝ z [1] i,…”

“…Note that [−1] 1,1 in (26) depends on y [1] 2 , which will result in unacceptable dependence of z [1] 1,1 on y [1] 2 . To prevent such a situation, one may introduce the y [1] 2 -independent function as̃[ −1] 1,1 ∶= [−1] 1,1 − y [1] 2 , and then use the forth equation of (26) to replace y [1] 2 by…”

“…In various control tasks, when the values of system states are required but their direct measurement is impossible or too expensive, observers prove to be very useful tools, estimating the values of the states from the knowledge of inputs and outputs . The desirable properties of any nonlinear observer are stability and linearity of error dynamics.…”