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A control perspective on synchronization and the Takens-Aeyels-Sauer Reconstruction Theorem
Abstract
A method, based on ideas from control theory, is described for the synchronization of discrete time transmitter /receiver dynamics. Conceptually, the methodology consists of constructing observer-receiver dynamics that exploit at each time instant the drive signal and past values of the drive signal. In this way, the method can be viewed as a dynamic reconstruction mechanism. PACS numbers: 02.10.Jf 02.90.+p 05.45.+b 47.52.+j 89.90.+n 1 Introduction Following Pecora and Caroll [14] 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 [14] 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) where x 1 (Delta) and x 2 (Delta) are vectors of dimension m ...
... This method of modulating and demodulating relies on the knowledge of the dynamics used to generate the carrier. This form of synchronization, in its recursive format, is equivalent to the classical problem of observer design familiar from systems theory [Nijmeijer & Mareels, 1997;Angeli et al., 1995;Huijberts & Nijmeijer, 2001;Huijberts et al., 1999;Nijmeijer, 2001]. ...
... This is a departure from the classical synchronization problem, where normally the synchronization is only an asymptotic requirement. Nevertheless, finite time synchronization (a variant of a dead beat observer problem) is of course of interest and has been studied, see also [Angeli et al., 1995] and [Huijberts et al., 1999;James, 1994;Moraal & Grizzle, 1995]. Within the communication context we suggest that finite time synchronization is essential. ...
... This is based on the fact that in the trajectory recovery problem a real trajectory, generically a unique real trajectory, is the target. (The generic solvability of this problem for sufficiently large T is a consequence of the Takens-Aeyels embedding theorem, for a discussion see [Huijberts et al., 1999].) ...
Prediction, smoothing, filtering and synchronization or observer design given finitely many measurements and a given (possibly nonlinear) dynamical map are discussed from a computational complexity point of view. All these problems are particular instances of finding a zero of an appropriately defined function. The recognition of this fact enables one to approach these questions from a computational complexity point of view. For polynomial maps the computational complexity of a global Newton algorithm adapted to identify the finite trajectory of the dynamical system's state over the desired window scales in a polynomial manner with the condition number (an invariant for the problem at hand) and the degree of the polynomials required to describe the models. The computational complexity analysis allows one to identify the most efficient manner to approach synchronization (prediction, smoothing, filtering) problems. Moreover differences between adaptive and nonadaptive formulations are revealed based on the condition number of the associated zero finding problem. The advocated formulation, with the associated global Newton algorithm has good robustness properties with respect to measurement errors and model errors for both adaptive and nonadaptive problems. These aspects are illustrated through a simulation study based around the Hénon map.
... From a control perspective, the problem of synchronization can be regarded as an observer problem, cf. [13] for continuous-time and [5] for discrete-time systems. This work focuses on a design of nonlinear observers for discrete-time systems (transmitters) of the form ...
... For w = ǫ, system (5.24) is identical to the one presented in [1]. An observer design via EOF for w = ǫ was already considered in [5]. ...
... In this paper, we will, among others, consider the example from [Badola et al., 1992]. The present paper extends the paper [Huijberts et al., 1999a], and is an expanded version of the paper [Huijberts et al., 1999b]. ...
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.
... 3) The n-time delayed value of u, u n− is restituted in the last reconstructed dynamic. This approach could seem not at all original because in [5], [6], [10] the idea of a " delayed observer " was developed. The authors, used the (i − N )-times delayed values of the output y in order to observe the i-th dynamic. ...
Through out this paper a new secured data transmission based on a discrete time hyperchaotic cryptography is presented. First, this technique is implemented for a Rossler hyperchaotic generator. Then, the formal approach of this technic is developed for triangular systems. Some open questions and future investigations conclude this paper.
Synchronization of complex/chaotic systems is reviewed from a dynamical control perspective. It is shown that the notion of an observer is essential in the problem of how to achieve synchronization between two systems on the basis of partial state measurements of one of the systems. An overview of recent results on the design of synchronizing systems using observers is given. Examples are given to demonstrate the main results.
We study the synchronization problem in discrete-time via an extended Kalman filter (EKF). That is, synchronization is obtained of transmitter and receiver dynamics in case the receiver is given via an EKF that is driven by a noisy drive signal from a noisy transmitter dynamics. The convergence of the filter dynamics towards the transmitter dynamics is rigorously shown using recent results in extended Kalman filtering. Two extensive simulation examples show that the filter is indeed suitable for synchronization of (noisy) chaotic transmitter dynamics. An application to private communication is also given.
SUMMARYA set membership method for right inversion of nonlinear systems from data is proposed in the paper. Both the cases where the system to invert is known or unknown and therefore identified from data are addressed. The method does not require the invertibility of the regression function describing the system and ensures tight bounds on the inversion error. In the case of unknown system, the method allows the derivation of a robust right-inverse, guaranteeing the inversion error bound for all the systems belonging to the uncertainty set which can be defined from the available prior and experimental information. Based on such a set membership inversion, two methods for robust control of nonlinear systems from data are introduced: nonlinear feed-forward control (NFFC) and nonlinear internal model control (NIMC). Both the design methods ensure robust stability and bounded tracking errors for all the systems belonging to the involved uncertainty set. Two applicative examples of robust control from data are presented: NFFC control of semi-active suspension systems and NIMC control of vehicle lateral dynamics.Copyright © 2013 John Wiley & Sons, Ltd.
Synchronization of complex/chaotic systems is reviewed from a dynamical control perspective. It is shown that the notion of
an observer is essential in the problem of how to achieve synchronization between two systems on the basis of partial state
measurements of one of the systems. An overview of recent results on the design of synchronizing systems using observers is
given. Examples are given to demonstrate the main results.
A systematic coupling procedure is introduced for synchronizing arbitrary chaotic dynamical systems. This coupling exploits the existing contraction properties of the flow and suppresses divergence only along those directions in state space, where the underlying flow is not contracting. In this way, systems can be synchronized using a minimum of transmitted information for guaranteed high-quality synchronization. Applications in combination with sporadic driving and in partitioned state spaces are numerically illustrated.
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.
A chaotic system is self-synchronizing if it can be decomposed into subsystems; for some synchronizing chaotic systems, the ability to synchronize is robust, as in the Lorenz system that is decomposable into two separate response subsystems that will each synchronize to the drive system when started from any initial condition. Attention is here given to the synchronizing properties of the Lorenz system as an analog circuit, and the use of the Lorenz circuit for communications applications.
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.
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.
Negativity of the conditional Lyapunov exponents is a necessary and in many cases a sufficient condition for the occurrence of synchronization between a chaotic drive and the response subsystem. Treating one of the variables of coupled logistic maps as the drive, our numerical simulations show that the negativity of the conditional Lyapunov exponents does not always guarantee synchronization and, additionally, the domain of initial conditions for the drive variables needs to be specified in which case the synchronization occurs. Additionally, an example showing the pitfalls of the numerical study of synchronization when the conditional Lyapunov exponent is positive is presented.