Adaptive observer-based synchronization of the nonlinear nonpassifiable systems

Abstract

Abstract-In this paper the relative degree limitation for adaptive observer-based synchronization schemes is overcome. The scheme is extended to nonpassifiable systems. Two synchronization methods are described and justified based on augmented error adaptive observer and high-order tuners. The solution is based on modern theory of nonlinear adaptive control, particularly on nonlinear observer structure and new classes of adaptation algorithms. Conditions of parametric convergence of the parameter estimation are established for the noiseless case. Robustness of the scheme to the bounded measurement error is established. The results are illustrated by example of application the proposed adaptive synchronization of chaotic Lorenz systems.

Adaptive Observers With Persistency of Excitation for Synchronization of Chaotic Systems

“…During the last 15 years or so, a number of methods to establish synchronization of chaotic systems have been proposed in more or less generality: for instance, focussed on the Lorenz system: [5], [10], [11] (one of the most popular chaotic oscillators) or covering relatively general classes of chaotic oscillators-cf. [6], [7], [12]- [16].…”

“…[17]- [19], for the second, distinct control approaches have been put to test-cf. [6], [7], [12]- [14]. Some papers rely on analytic study - [8], [20], [21] and others on numerical methods and validation in simulation-cf.…”

“…e.g., [20], [39]- [42]. Early work on observer-based synchronization includes [43]; more recent work, including parametric uncertainty and adaptation are the interesting papers [12], [44] of which we became aware after the original submission of this paper. In the first reference the authors present similar results to ours but under somewhat more restrictive assumptions: adaptive observers for partially linear systems, affine in the unmeasured variables and, under persistency of excitation conditions, conclude synchronization and parametric convergence with small errors.…”

Multi-parameter identification from scalar time series generated by a Malkus-Lorenz water wheel

“…Their design is well a) CHAOS 22, 013127 (2012) established in the control theory literature for the case of noise-free linear time invariant systems 9 and has recently been extended to cover linear time varying (LTV) systems with single output as well as multiple outputs. [9][10][11] Adaptive observers for nonlinear systems are also known. 12 The guarantee of global convergence comes, in general, at the cost of assumptions that are difficult to check (e.g., the persistent excitation assumption) and the applicability of a particular adaptive observers design is limited to specific structures of the estimation problem.…”

“…To estimate q, one would have to knowx 2 . In order to determine q from knowledge ofx 1 , a very different observer design was proposed by Andrievskii et al, 10 where the estimation problem for the Lorenz system with knownx 1 is treated as an estimation problem for linear time-varying systems. In this paper, we show how this latter type of observer can be used to simultaneously determine q and r (Sec.…”