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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.

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... Accurate predictions depend on the forcing scheme and, most importantly, the available observations. Construction of the forcing term is referred to as the "observer problem" in control theory: Given a dynamical system, the "observer" refers to a forced system that assimilates limited data with the objective of synchronizing to the original system (Huijberts & Nijmeijer 2001). Although different forcing strategies have been developed (Pogromsky & Nijmeijer 1998;Mohan et al. 2017), they have a relatively minor influence on the success of synchronization in comparison to the amount of available observations of the true state. ...

Synchronization of turbulence in channel flow is investigated using continuous data assimilation. The flow is unknown within a region of the channel. Beyond this region the velocity field is provided, and is directly prescribed in the simulation, while the pressure is unknown throughout the entire domain. Synchronization takes place when the simulation recovers the full true state of the flow, or in other words when the missing region is accurately re-established, spontaneously. Successful synchronization depends on the orientation, location and size of the missing layer. For friction Reynolds numbers up to one thousand, wall-attached horizontal layers can synchronize as long as their thickness is less than approximately thirty wall units. When the horizontal layer is detached from the wall, the critical thickness increases with height and is proportional to the local wall-normal Taylor microscale. A flow-parallel, vertical layer that spans the height of the channel synchronizes when its spanwise width is on the order of the near-wall Taylor microscale, while the criterion for a crossflow vertical layer is set by the advection distance within a Lyapunov timescale. Finally, we demonstrate that synchronization is possible when only planar velocity data are available, rather than the full outer state, as long as the unknown region satisfies the condition for synchronization in one direction. These numerical experiments demonstrate the capacity of accurately reconstructing, or synchronizing, the missing scales of turbulence from observations, using continuous data assimilation.

... Besides filtering of unwanted signals the results we derive may also be relevant in synchronization and secure communication. Indeed deterministic observers have already found applicability in this field [3], [5], [11]. Using state augmentation, one obtains a combined model for the extended state z ′ = [x ′ , b ′ ]: ...

A result of Friedland for efficient filtering in the presence of a static bias is extended to the case where the bias signals are given by the response of persistent autonomous systems with random initial conditions. It is shown that the optimal filter decouples into a bias free and a bias error correction filter. We apply the results to filtering of a system with delay when the initial data is missing. Theses results have potential applications in secure communication, synchronization and networked control.

Synchronization of turbulence in channel flow is investigated using continuous data assimilation. The flow is unknown within a region of the channel. Beyond this region the velocity field is provided, and is directly prescribed in the simulation, while the pressure is unknown throughout the entire domain. Synchronization takes place when the simulation recovers the full true state of the flow, or in other words when the missing region is accurately re-established, spontaneously. Successful synchronization depends on the orientation, location and size of the missing layer. For friction Reynolds numbers up to one thousand, wall-attached horizontal layers can synchronize as long as their thickness is less than approximately thirty wall units. When the horizontal layer is detached from the wall, the critical thickness increases with height and is proportional to the local wall-normal Taylor microscale. A flow-parallel, vertical layer that spans the height of the channel synchronizes when its spanwise width is of the order of the near-wall Taylor microscale, while the criterion for a crossflow vertical layer is set by the advection distance within a Lyapunov time scale. Finally, we demonstrate that synchronization is possible when only planar velocity data are available, rather than the full outer state, as long as the unknown region satisfies the condition for synchronization in one direction. These numerical experiments demonstrate the capacity of accurately reconstructing, or synchronizing, the missing scales of turbulence from observations, using continuous data assimilation.

This paper investigates the cluster synchronization for network of linear systems via a generalized pinning control strategy which allows the network of each cluster to take relaxed topological structure. For the case with fixed topology, it is shown that a feasible feedback controller can be designed to achieve the given cluster synchronization pattern if the induced network topology of each cluster has a directed spanning tree and further compared to the couplings among different clusters, the couplings within the each cluster are sufficiently strong. An extra balanced condition is imposed on the network topology of each cluster to allow for the cluster synchronization under arbitrary switching network topologies. Such a balanced condition can be removed via the use of dwell-time technique. For all the cases, the lower bounds for such strengths of couplings within each cluster that secure the synchronization as well as cluster synchronization rate are explicitly specified. Finally, some illustrative examples are provided to demonstrate the effectiveness of the theoretical findings.

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.

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.

In this paper we investigate the problem of controlled synchronization as a regulator problem. In controlled synchronization one is given autonomous transmitter dynamics and controlled receiver dynamics. The question is to find a (output) feedback controller that achieves matching between transmitter and controlled receiver. Several variants of the problem where the standard solvability assumptions for the regulator problem are not met turn out to have a solution. Simulations on two standard synchronization examples are also included. Copyright © 2000 John Wiley & Sons, Ltd.

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 new method for designing asymptotic observers for a class of nonlinear systems is presented. The error between the state of the systems and the state of the observer in appropriate coordinates evolves linearly and can be made to decay aribtrarily exponentially fast.

Communication using chaotic systems is considered from a control
point of view It is shown that parameter identification methods may be
effective in building reconstruction mechanisms, even when a
synchronizing system is not available. Three worked examples show the
potentials of the proposed method

In the literature on dynamical systems analysis and the control of
systems with complex behavior, the topic of synchronization of the
response of systems has received considerable attention. This concept is
revisited in the light of the classical notion of observers from
(non)linear control theory,

This paper studies the problem of modeling and control of multiple
cooperative underactuated manipulators handling a rigid object. We
reveal holonomic property of such a system by presenting a smooth
feedback controller subject to two conditions: 1) there are not fewer
active joints than the degrees of freedom of the object; and 2) the
Jacobian matrix with respect to passive joints is not singular. This
controller is an extension of the PD plus gravity compensation scheme
and its asymptotic stability is guaranteed by the LaSalle theorem.
Furthermore, we develop a trajectory tracking controller that yields
asymptotic convergence of position errors and bounded interaction forces
simultaneously. The performance of the proposed controllers has been
investigated by simulations on two 6-DOF underactuated manipulators and
by experiments on the cooperative underactuated manipulator system
developed at CMU

The goal of this paper is twofold. It unifies and generalizes
existing results on input-output injection linearization of nonlinear
systems. The problem is solved as a realization problem since it is
based on the analysis of the structure of the input-output differential
equation. The necessary and sufficient conditions are derived from a
simplified and constructive procedure. For clarity, the paper is limited
to the case of single output systems. Exterior differential systems are
extensively used throughout the paper, giving constructive conditions

An observer for nonlinear systems is constructed under rather
general technical assumptions (the fact that some functions are globally
Lipschitz). This observer works either for autonomous systems or for
nonlinear systems that are observable for any input. A tentative
application to biological systems is described

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 ...

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...

Modern control design methods are based on the knowledge of all state variables of the considered system. Since a measurement of all states is in most cases not possible or too expensive, the use of observers is of great importance. Up to now, nonlinear observers have mainly been studied for continuous-time systems, however, discrete-time representations are of increasing interest. For a relatively small class of systems an observer design with linearizable error dynamics based on canonical forms is possible. This work gives an extension of the so called "Two-Step-Transformation" to nonlinear observer canonical form. This extension allows to enlarge the class of transformable systems considerably. Considering past measurements of the systems in- and output variables leads to the so called extended nonlinear observer canonical form which also allows to design an observer with linearizable error dynamics. The transformation into extended observer form includes several degrees of freedom which help to select the structure and the characteristics of the resulting observers. The extended observer form exists for every strongly locally observable system with one output. The transformation of a system with several outputs is subject to further conditions. Compared to the transformation into classical observer form, these conditions are noticeably less restrictive. The observers via extended observer form are compared to another design procedure, which can be found in the literature. For the latter, an alternative structure and an extension to systems with several outputs is presented in this work. The comparison of all considered observers includes the transient behaviour, robustness to noisy measurements, parameter sensitivity and the feasibility of the design procedure. One of the main tasks to use observers is the state feedback of dynamical systems. Since the separation principle which holds for all linear, timeinvariant systems does not hold in the nonlinear case, this work also focuses on the problem of nonlinear discrete-time observers for nonlinear state feedback. An experimental investigation of the closed loop dynamics was carried out for the stabilization of an inverted pendulum. The results show the general applicability to technical systems of all considered observers and furthermore significant differences between some observers in the closed loop were emphasized.

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.

A generic, sudden transition to chaos has been experimentally verified using electronic circuits. The particular system studied involves the near resonance of two coupled oscillators at 2 : 1 frequency ratio when the damping of the first oscillator becomes negative. We identified in the experiment all types of orbits described by theory. We also found that a theoretical, 1D limit map fits closely a map of the experimental attractor which, however, could be strongly disturbed by noise. In particular, we found noisy periodic orbits, in good agreement with noise theory.

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.

Synchronization of systems executing periodic and chaotic motions, both oscillatory and rotatory, is considered from the general standpoint in this review article. It is shown that the synchronization phenomenon goes hand in hand with the phenomenon of chaotization. The analogy of synchronization and phase transitions is shown. Several mechanical and physical examples are discussed.

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.

IntroductionSynchronization of identical systems Constructing pairs of synchronizing systemsTransversal instabilities and noiseSporadic drivingSpatially extended systemsSynchronization of nonidentical systems Generalized synchronization IGeneralized synchronization IINon-identical synchronization of identical systemsPhase synchronizationApplications and Conclusion Constructing pairs of synchronizing systems Generalized synchronization IGeneralized synchronization IINon-identical synchronization of identical systemsPhase synchronization

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.

Takens Embedding Theorem forms the basis of virtually all approaches to the analysis of time series generated by nonlinear deterministic dynamical systems. It typically allows us to reconstruct an unknown dynamical system that gives rise to a given observed scalar time series simply by constructing a new state space out of successive values of the time series. This provides the theoretical foundation for many popular techniques, including those for the measurement of fractal dimensions and Liapunov exponents, for the prediction of future behaviour, for noise reduction and signal separation, and most recently for control and targeting. Current versions of Takens Theorem assume that the underlying system is autonomous. Unfortunately this is not the case for many real systems; in the laboratory we often force an experimental system in order for it to exhibit interesting behaviour, whilst in the case of naturally occurring systems it is very rare for us to be able to isolate the system to ensure that there are no external influences. In this paper we therefore prove two versions of Takens Theorem relevant to forced systems: one applicable to the case where the forcing is unknown, and the other to the situation where we are able to determine independently the state of the forcing system (usually because we are responsible for the forcing ourselves). In a subsequent paper we shall show how to extend these results to give an analogue of Takens Theorem for randomly forced systems, leading to a new framework for the analysis of time series arising from nonlinear stochastic systems.

Observers can easily be constructed for those nonlinear systems which can be transformed into a linear system by change of state variables and output injection. Necessary and sufficient conditions for the existence of such a transformation are given.

Using Takens's theorem we show that any pair of uni-directionally coupled dynamical systems can be synchronised via almost all scalar observations of the driving system's state. Moreover, the values of the scalar observation at only a finite number of times are sufficient for synchronisation purposes.

We show how to perform targeting control using global models derived from data coming from possibly nonstationary dynamical systems where the varying parameters are known along with the system observations. We first identify a global model from the observations and the values of the accessible parameters. To carry out the control, we successively apply the usual targeting algorithm to the model system to get a perturbation of the accessible parameters for a given starting point and targeting point, then apply this perturbation to the true system to generate a new starting point. Because our model is approximate, we repeat the targeting and perturbation steps until the observed trajectory is near the target point.

We present a design strategy in terms of ordinary differential equations which creates chaotic attractors with an increasing number of positive Lyapunov exponents as the (finite) dimension of the system is increased. First, we introduce the most simple abstract equation containing only one nonlinearity. Second, we suggest a piecewise linear version of the abstract equation. Third, we propose a set of chemical reactions and demonstrate that the corresponding rate equations produce hyperchaotic behavior equivalent to the abstract system.

A method of synchronizing two chaotic systems by implementing an
extended Kalman filter (EKF) as part of the receiver is presented.
Communication is accomplished by parameter modulation in the transmitter
and parameter estimation in the receiver filter and tolerates nontrivial
parameter mismatches as well as additive noise. Several parameters may
be modulated and estimated concurrently, resulting in simultaneous
multiple access of a single channel

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On Selfsynchronization and Controlled Synchronization

- I I Blekhman
- A L Fradkov
- H Nijmeijer
- A Pogromsky
- Yu
- I.I. Blekhman

Communication by Chaotic Signals: The Inverse System Approach

- U Feldman
- M Hasler
- W Schwarz
- U. Feldman

Observer Based Robust Synchronization of Dynamical Systems

- A Pogromsky
- Yu
- H Nijmeijer
- A. Pogromsky