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Abstract

A method is described for the synchronization of nonlinear discrete time dynamics. The methodology consists of constructing observer-receiver dynamics that exploit at each time instant the drive signal and buffered past values of the drive signal. In this way, the method can be viewed as a dynamic reconstruction mechanism, in contrast to existing static inversion methods from the theory of dynamical systems. Keywords: Synchronization, observers, nonlinear discrete time systems 1 Introduction Following Pecora and Carroll [15] a huge interest in the synchronization of two coupled systems has arisen. This research is partly motivated by its possible use in secure communications, cf. [6]. Often, like in [15] a drive/response, or transmitter/receiver, viewpoint is assumed. In a discrete-time context, this typically allows for a description of the transmitter as a n-dimensional dynamical system x 1 (k+1) = f 1 (x 1 (k); x 2 (k)) (1) x 2 (k+1) = f 2 (x 1 (k); x 2 (k)) (2) This paper has...
... In some other papers, the output injection term is allowed to depend, besides the current output value, also on a finite number of its past values, reducing that way the restrictions * Corresponding author. Email: vkaparin@cc.ioc.ee on possibility to construct such transformations (Huijberts, 1999;Huijberts, Lilge, and Nijmeijer, 1999). A corollary of the results, obtained in Huijberts et al. (1999), is that when the number of past output values equals n − 1 (where n is denoted the state dimension), the system can always be transformed into the extended observer form, provided the system under consideration is strongly observable. ...
... Email: vkaparin@cc.ioc.ee on possibility to construct such transformations (Huijberts, 1999;Huijberts, Lilge, and Nijmeijer, 1999). A corollary of the results, obtained in Huijberts et al. (1999), is that when the number of past output values equals n − 1 (where n is denoted the state dimension), the system can always be transformed into the extended observer form, provided the system under consideration is strongly observable. ...
... Though very simple, the results of this paper have the disadvantages of not being intrinsic. Another point to mention is that our results assume (but so do those in Huijberts (1999) and Huijberts et al. (1999)) that the i/o equation, corresponding to the state equations, can be easily found from the state equations. Under observability assumption, one may always find the i/o equations, at least locally, using the state elimination algorithm. ...
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The paper focuses on the problem of transforming the discrete-time single-input single-output nonlinear state equations into the extended observer form, which, besides the input and output, also depends on a finite number of their past values. The simple necessary and sufficient conditions for the existence of the extended coordinate change and the output transformation, allowing to solve the problem, are formulated in terms of certain partial derivatives, related to the input–output equation, corresponding to the state equations. Moreover, a certain algorithm for transforming the state equations into the observer form is proposed.
... Theorem 1. The limit of Algorithm 1 is the observable space of system (5). ...
... where matrix K is chosen such that the eigenvalues of A + KC are inside the unit circle. It has been proven in [5] that any single-output observable discrete-time system without inputs can be taken into the extended observer form with buffer n − 1. The proof carries over to the input-dependent and multi-input multi-output case (MIMO). ...
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The paper studies the possibility of constructing observer©based residuals to detect faults in a nonlinear discrete©time system. The residuals are generated in such a manner that they detect one specific fault and are not affected by other faults and disturbances. Thus, a bank of residuals has been found to detect and isolate different faults in the system. An algebraic method called functions’ algebra is used to construct an algorithm which computes the residuals. The key fact in residual generation is that any discrete©time observable system can be taken into the extended observer form. This form is used to construct the observer to estimate the system states under the assumption that there are no faults in the system. The state estimates are then compared to the measured values of the states. An example is added to illustrate the theoretical results. In the example it is also demonstrated how to combine the fault detection with the plant reconfiguration step of fault tolerant control.
... In the second approach, the system equations are extended and one transforms this extended system into the classical observer form [12]- [16]. It has been also proved in the discrete-time case that when the number of past output values in the nonlinear term is large enough (but bounded by the dimension of state equations), then one can always transform the observable system into the extended form [17]. Of course, in many cases a smaller number of past outputs will be sufficient and there exist conditions which allow one to determine exactly how many of them are needed. ...
Conference Paper
The paper addresses the problem of transforming single-output continuous and discrete-time nonlinear dynamical systems into their respective extended observer forms. The output injection term in the extended observer form of degree I depends, besides on the output also on its first I derivatives (in the continuous-time case) or first I past values (in the discrete-time case). Intrinsic necessary and sufficient conditions are given for the existence of the state transformation that transforms the equations into the extended observer form, both for continuous- and discrete-time cases. The conditions can be directly checked from state equations and do not rely on the input-output equation as in the earlier papers.
... The paper [3] provides the conditions under which a given single-output discrete-time system may be transformed into the extended observer form by means of an extended coordinate change (i.e. a coordinate transformation that depends on the state of the system and a finite number of past output values) and an output transformation. A corollary of the results considered in [7] is that when the number of past output values equals n − 1 (where n is the dimension of the state space of the system under consideration), the system can be always transformed into the extended observer form, provided the system under consideration is strongly observable. The necessary and sufficient conditions for transformation of the input dependent system into the extended observer form were presented in [8], where certain partial derivatives related to the input-output equation have been computed, and the function, necessary for the output transformation, is easy to find from the conditions. ...
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The paper addresses the problem of transforming the discrete-time single-input single-output nonlinear state equations into the extended observer form, which, besides the input and output, also depends on a finite number of their past values. The simple necessary and sufficient conditions for the existence of the extended coordinate change and the output transformation, allowing to solve the problem, are formulated in terms of differential one-forms, associated with the input-output equation, corresponding to the state equations.
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