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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...

Content uploaded by Henri Huijberts

Author content

All content in this area was uploaded by Henri Huijberts on Sep 06, 2013

Content may be subject to copyright.

... 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. ...

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). ...

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. ...

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. ...

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.

The paper addresses the problem of transforming discrete‐time multi‐input multi‐output nonlinear state equations into the extended observer form, which, besides the inputs and outputs, also depends on a finite number of their past values. Necessary and sufficient conditions for the existence of the extended coordinate transformation are formulated in terms of differential one‐forms, associated with the input‐output equations, corresponding to the state equations. The difference between the single‐input single‐output and multi‐input multi‐output cases is described. The applicability of the conditions is illustrated by an example.

The paper addresses the problem of transforming 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. Necessary and sufficient conditions for the existence of both the extended coordinate and output transformations, solving the problem, are formulated in terms of differential one-forms, associated with the input-output equation, corresponding to the state equations. An algorithm for transformation of state equations into the extended observer form is proposed and illustrated by an example. Moreover, the considered approach is compared with the method of dynamic observer error linearisation, which likewise is intended to enlarge the class of systems transformable into an observer form.

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. The method is illustrated on a few simulation examples consisting of coupled chaotic logistic equations. Also, a discrete-time message reconstruction scheme is simulated using the extended observer mechanism.

The synchronization problem for complex discrete-time systems is
revisited from a control perspective and it is argued that the problem
may be viewed as an observer problem. It is shown that for several
classes of systems a solution for the synchronization (observer) problem
exists. Also, by allowing past measurements a dynamic mechanism for
state reconstruction is provided

The synchronization problem for complex discrete--time systems is revisited from a control perspective and it is argued that the problem may be viewed as an observer problem. It is shown that for several classes of systems a solution for the synchronization (observer) problem exists. Also, by allowing past measurements a dynamic mechanism for state reconstruction is provided. 2 CHAPTER 1. OBSERVERS AND SYNCHRONIZATION 1.1 Introduction Since the work of Pecora and Carroll [18], a huge interest in (chaos) synchronization has arisen. Among others, this is illustrated by the appearance of a number of special issues of journals devoted to the subject, cf. [29, 28, 30]. One clear motivation for this widespread interest lies in the fact that Pecora and Carroll indicated that chaos synchronization might be useful in communications. Although by now this claim is not fully justified yet, several interesting applications of (chaos) synchronization are envisioned. Synchronization as it was intr...

this paper, we consider the design of observers for discrete-time nonlinear systems by means of so called (extended) observer forms. Loosely speaking, a system in observer form is a linear observable (continuous-time or discrete-time) system that is interconnected with an output-dependent nonlinearity. Observers for this kind of systems may be built by building a classical linear Luenberger observer for the linear system, and adding the output-dependent nonlinearity to this observer. Thus, observer design for systems in observer form is relatively easy. By the same token, also observer design for systems that may be transformed into a system in observer form by means of a coordinate transformation and an output transformation is relatively easy. Observer design for systems in observer form was first studied, in the continuous-time setting, in [10],[11] (see also [15]). In these papers, conditions were given under which a nonlinear continuous-time system may be transformed into a system in observer form by means of a coordinate transformation and an output transformation. Basically, these conditions were given in terms of the integrability of certain codistributions. Later on, the observer design for discrete-time systems in observer form was studied (see [1],[12],[13] and the references therein), and conditions were given under which a nonlinear discrete-time system may be transformed into a system in observer form by means of a coordinate transformation and an output transformation. These conditions came down to the question whether certain functions could be factorized in a certain way. For single-output systems, conditions under which this factorization is indeed possible were given when only output transformations are allowed. (In fact, [13] also claims to give cond...

This paper focuses on the design of nonlinear observers for discrete-time systems by means of a so called extended nonlinear observer canonical form which is computed via a nonlinear observability canonical form. In contrast to other approaches by Brodmann 1994 and Lin and Byrnes 1995 using a two-step-transformation, past measurements of the system output are used. This allows to extend the class of systems for which an observer can be designed and leads to several observers with different characteristics. An application to a pendulum on a cart shows the efficiency of the design method.

This paper focuses on the design of non-linear observers for discrete-time systems by means of a transformation into non-linear observer canonical form based on recent results presented by Brodmann and by Lin and Byrnes. In contrast to their approaches, past measurements of the system output are used. This allows extension of the class of systems for which an observer can be designed and leads to several observers with different characteristics. Simulations show the efficiency of the design method and the differences between possible observers. In this paper, the observer design is limited to multi-input and single-output systems.

A dynamical system consists of a smooth vectorfield defined on a differentiable manifold, and a smooth mapping from the manifold to the real numbers. The vectorfield represents the dynamics of a physical system. The mapping stands for a measuring device by which experimental information on the dynamics is made available. The information itself is modeled as a sampled version of the image of the state trajectory under the smooth mapping. In this paper the observability of this set-up is discussed from the viewpoint of genericity. First the observability property is expressed in terms of transversality conditions. Then the theory of transversal intersection is called upon to yield the desired results. It is shown that almost any measuring device will combine with a given physical system to form an observable dynamical system, if (2n plus 1) samples are taken and not fewer, where n is the dimension of the manifold.

We demonstrate the successful control of a periodic orbit associated with two unstable manifolds in a system comprised of two coupled diode resonators. It is shown that due to symmetries generic to spatially extended systems a one-parameter control is not possible. A novel method of determining the local Liapunov exponents utilizing orthogonal control as well as geometric information is presented.

In this paper the observability of autonomous discrete time systems is studied from a purely differential geometric point of view. As with continuous time systems, this approach leads to a local canonical form for an observable system. A proposal for the generalization of an invariant subspace is made.

We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.

This paper presents necessary and sufficient conditions under which a discrete-time autonomous system with outputs is locally state equivalent to an observable linear system or a system in the nonlinear observer form (Krener and Isidori, 1983). In particular, an open problem raised in Lee and Nam (1991), namely the observer linearization problem, is solved for a nonlinear system which may not be invertible (i.e., the mapping f may not be a local diffeomorphism). As a consequence, the nonlinear observer design problem is solved by means of exact linearization techniques.

Mathematical formulations of the embedding methods commonly used for the reconstruction of attractors from data series are discussed. Embedding theorems, based on previous work by H. Whitney and F. Takens, are established for compact subsetsA of Euclidean space Rk. Ifn is an integer larger than twice the box-counting dimension ofA, then almost every map fromR
k
toR
n
, in the sense of prevalence, is one-to-one onA, and moreover is an embedding on smooth manifolds contained withinA. IfA is a chaotic attractor of a typical dynamical system, then the same is true for almost everydelay-coordinate map fromR
k
toR
n
. These results are extended in two other directions. Similar results are proved in the more general case of reconstructions which use moving averages of delay coordinates. Second, information is given on the self-intersection set that exists whenn is less than or equal to twice the box-counting dimension ofA.

In [1] we have shown that almost all dynamical systems are observable with respect to an almost arbitrary sample program consisting of 2n + 1 samples (n is the dimension of the differentiable manifold supporting the dynamical system). In this paper we construct a dynamical system which is unobservable with respect to any sample program consisting of 1n samples. Small perturbations of the dynamics do not destroy the non-observability. This shows that the results obtained in [1] are the best ones possible in general.