In this article we present sufficient conditions for parametric and state estimation using adaptiveobservers which cover high- gain designs—cf. ,  and others from the references cited above: 1) we lay sufficient general conditions in terms of persis- tency of excitation along trajectories for nonlinear systems (in contrast to –); 2) the class of nonlinear systems includes systems that are linear in the unknown variables but parametric uncertainty may appear anywhere in the model (in contrast to ); 3) the conditions we set for our adaptiveobservers in- tersect (and generalize in certain ways) with high-gain designs for nonlinear systems, similar to those in ,  however, in contrast to the former our method is not restricted to high-gain observers and, with respect to the latter, in this paper we cover the case of parametric uncertainty without controls; 4) the class of systems that we consider contains time-varying nonlineari- ties which may be regarded as neglected dynamics; in contrast with works relying on Lipschitz assumptions we allow for high order terms provided that the trajectories of the master system are bounded (which is not restrictive in the context of chaoticsystems); 5) the condition in terms of PE covers cases consid- ered for instance in , , , –, –. Our re- sults are for a (structurally) similar class of systems as that con- sidered in  except that in the latter systems are assumed to be partially linear; another fundamental difference is that in  synchronization is considered as making two respective outputs which are part of the state of the master and slave system con- verge to each other, as opposed to estimating the whole state of the system. The theoretical proofs that we present are original and rely on previous results for stability of parameterized linear systems—cf. .
Robustsynchronization of master slave chaoticsystems are considered in this work. First an approximate model of the error system is obtained using the ultra-local model concept. Then a Continuous Singular Ter- minal Sliding-Mode (CSTSM) Controller is designed for the purpose of synchronization. The proposed approach is output feedback-based and uses fixed-time higher order sliding-mode (HOSM) differentiator for state estimation. Numerical simulation and experimental results are given to show the effectiveness of the proposed technique.
Amin Ramezani (IEEE Member 2005) was born in Khuzestan, Iran, in 1979. He received the B.E. degree in electrical engineering from the Shahid Beheshti University of Tehran, Tehran, Iran, in 2001, and the M.sc in Control systems for Sharif of Technology, Tehran, Iran, in 2004 and Ph.D. degrees in electrical engineering from the University of Tehran, Tehran, Iran, in 2011. In 2012, he joined the Department of Electrical Engineering, Tarbiat Modares University, Tehran, Iran, as a Lecturer. He the chair of Advanced Control Lab and cooperates as the PC members of some conferences and special editor and reviewer of international journals in the field of control systems. His current research interests include Fault Tolerant Control Systems, Based Predictive Control Systems, Stochastic Control Systems, Hybrid Systems. Dr. Ramezani is a member of IEEE Iranian Community Instrument and Control Engineers since 2012. He has contributed in more
ﬁrst found by Mackey and Glass . For chaoticsystems with time-delay and distur- bance, several works have proposed the problem for various chaoticsystems in the lit- erature [–]. In , an adaptive control law was derived and applied to achieve the state lag-synchronization of two nonidentical time-delayed chaoticsystems with unknown parameters. In , an output coupling and feedback scheme were proposed to achieve the robustsynchronization of noise-perturbed chaoticsystems with multiple time-delays. An impulse control was proposed by Qian and Cao  to synchronize two nonidentical chaoticsystems with time-varying delay. Most of them are based on the fact that the time- delay is a constant, while, in real world applications, the time-delay is also varying over time. Hence the study of chaoticsynchronization with time-varying delay is an important topic.
Recently impulsive control theory and its application in chaos synchronization have become a research hotspot. For instance, based on impulsive control strategies, the reduced- order observer for the synchronization of generalized Lorenz chaoticsystems is built in , the adaptive modified function projective synchronization of multiple time-delayed chaotic Rossler systems is discussed in , the hybrid synchronization of L¨u hyperchaotic system with disturbances is investigated in , and the robustsynchronization of perturbed Chen’s fractional-order chaoticsystems is studied in . However, these results are just about one kind of chaoticsystems, which limits their applied scope. Hence how to design the impulsive strategy to realize the synchronization suitable for more chaoticsystems activates our research.
Chaos synchronization (CS) has been widely investigated based on the results of . A lot of works have been given on this theme because of its possible application in many ﬁelds such as communications, information processing [2–18]. For example, Liu et al.  dis- cussed robustsynchronization of uncertain complex networks by using impulsive con- trol. Yu and Cao  addressed the synchronization of chaoticsystems with time delay. The synchronization and anti-synchronization via active control approach on fractional chaotic ﬁnancial system were studied by Huang et al. in . On the basis of control the- ory, a lot of methods, such as adaptive control , sliding control [23, 24], pinning control , intermittent control [26, 27], have been developed for CS.
This feature of high dependence to any uncertainties motivates the use of them intentionally to create chaotic behaviour much more secure for cryptography. The problem is to design a sufficiently robustsynchronization scheme to guarantee the precise mimicry of chaoticsystems in a master-slave configuration. A good candidate to achieve this goal is the use of robust sliding mode observer design methods for the uncertain nonlinear dynamics. In essence, the use of variable structure techniques in the state re- construction of nonlinear systems has some advantages, like allowing the presence of matched uncertain elements in the model and convergence speed over other existing techniques like feedback linearization, extended linearization and traditional Lyapunov-based techniques -. Sliding mode observers (SMO) require the knowledge of a bounding function on the uncertainty but this will not be needed in our approach due to a built-in adaptation mechanism in the sliding mode filter.
engineering, economics, neuroscience and many more [7,8,10,11,13,33,37]. In search of the stable, fast and robustsynchronization methods, such as linear and nonlinear feedback controllers, active control, passive control, backstepping, sliding mode control and many more, have been developed [1–3, 12, 39, 41]. Similarly, several types of synchronization have been introduced [4, 14, 17–19, 24, 25, 40]. One of the most exciting synchronization types is the generalized synchronization (GS). This type of synchronization is character- ized by the existence of a functional relationship φ between the state Y (t) of the slave system and the state X(t) of the master system, so that Y (t) = φ(X(t)) after a transient time . Different types of synchronization, such as complete synchronization, anti- synchronization, projective synchronization and function projective synchronization, can be achieved from the generalized synchronization scheme depending on the choice of φ. A variation is represented by the inverse generalized synchronization (IGS), where the synchronization condition becomes X(t) = ϕ(Y (t)) after a transient time. However, since the type is still relatively new, IGS has been applied with successfully in continuous- time systems as well as discrete-time systems [27, 28]. Another interesting type of syn- chronization is the hybrid synchronization, where complete synchronization and anti- synchronization coexist. Faster and more secure digital communication could be achieved using the hybrid synchronization .
In this paper, the lag projective (anti-)synchronization problems for a kind of master-slave chaoticsystems by using the adaptive control method have been investigated. Based on the Lasalle invariance principle of diﬀerential equation and the idea of the bang-bang con- trol, an adaptive controller with simple updated laws has been proposed. Three numerical examples have shown that the obtained method is eﬀective.
We consider the coupling of two non-identical dynamical systems using an adaptive feed- back linearization controller to achieve partial synchronization between the two systems. In addition we consider the case where an additional feedback signal exists between the two systems, which leads to bidirectional coupling. We demonstrate the stability of the adaptive controller, and use the example of coupling a Chua system with a Lorenz system, both ex- hibiting chaotic motion, as an example of the coupling technique. A feedback linearization controller is used to show the difference between unidirectional and bidirectional coupling. We observe that the adaptive controller converges to the feedback linearization controller in the steady state for the Chua-Lorenz example. Finally we comment on how this type of partial synchronization technique can be applied to modeling systems of coupled nonlinear subsystems. We show how such modeling can be achieved where the dynamics of one system is known only via experimental time series measurements.
Isochronal function projective synchronization between chaotic and time-delayed chaoticsystems with unknown parameters is investigated in this article. Based on Lyapunov stability theory, adaptive controllers and parameter updating laws are designed to achieve the isochronal function projective synchronization between chaotic and time-delayed chaoticsystems. The scheme is applied to realize the synchronization between time-delayed Lorenz systems and time-delayed hyper- chaotic Chen systems, respectively. Numerical simulations are also presented to show the effectiveness of the proposed method.
Remark 2 This example shows a diﬀerence between fractional and integer order calculus. And the results in  are still right because their proofs are valid as long as we replace Lemma 5 in  by the following Lemma 4. In many closed-loop systems, it is diﬃcult to determine that all signals are bounded. With the following lemma, we can manage to do this. For example, the signals γ ˆ and d ˆ¯ in system (13).
Furthermore, time delays are also introduced to fractional differential chaoticsystems [22, 23]. In 2007, Deng et al.  studied the stability of n-dimensional linear fractional differential equation with time delays, and also considered the synchronization between the coupled Duffing oscillators with time delays by using the linear feedback control method and their theorem. Gu et al.  investigated the global synchronization for fractional-order multiple time-delayed memristor-based neural networks with the parameter uncertainty, and derived the synchronization conditions of fractional-order multiple time-delayed memristor-based neural networks with the parameter uncertainty. More Zhang et al.  studied the drive-response synchronization fractional-order memristive neural networks with switching jumps mismatch, and obtained some lag quasi-synchronization conditions by the Laplace transform and linear feedback control.
presented the adaptivesynchronization of fractional Lorenz systems using a re- duced number of control signals and parameters . Kajbaf et al. used sliding mode controller to obtain chaoticsystems . Wang et al. proposed a new feedback synchronization criterion based on the largest Lyapunov exponent . However, most synchronization criterions were obtained under ideal cir- cumstances. If parameters perturbation and external disturbance exist, this kind of criterions will take no effect. According to this practical problem, some solu- tions have been presented. For examples, Jiang et al. proposed a LMI criterion  for chaotic feedback synchronization. Although the simulations showed that it is robust to a random noise with zero mean, but no rigorous mathematical proof was provided and we can’t determine if their method is effective for other kinds of noise. In Ref. , parameters perturbation was involved in their scheme. The theoretical proof and numerical simulations were given in their work, but external disturbance didn’t receive attention, which made their method unila- teral.
This paper proposes a new fuzzy adaptive exponential synchronization controller for uncertain time-delayed chaoticsystems based on Takagi-Sugeno T-S fuzzy model. This synchronization controller is designed based on Lyapunov-Krasovskii stability theory, linear matrix inequality LMI, and Jesen’s inequality. An analytic expression of the controller with its adaptive laws of parameters is shown. The proposed controller can be obtained by solving the LMI problem. A numerical example for time-delayed Lorenz system is presented to demonstrate the validity of the proposed method.
Synchronization of chaos refers to a process in which two or more chaoticsystems (either identical or non-identical) adjust a given property of their motion to a common behaviour. Until 1990, it was considered impractical due to the well-known divergence of trajectories caused by sensitivity of chaoticsystems to initial conditions. But the pioneering work of Pecora and Car- roll on synchronization of two identical chaoticsystems  has led the topic to become an inter- esting area of research. It has been developed and studied comprehensively in the ensuing time. Little over the past two decades various methods for synchronization of chaoticsystems have been proposed such as linear and nonlinear feedback synchronization [2–4], adaptive feedback control , active control , optimal control , time delay feedback approach , sliding mode control , backstepping design method , tracking control  and so on. Due to fast growing interest in chaos synchronization variety of synchronization types and schemes have also been investigated such as complete synchronization , phase synchronization , anti- phase synchronization , lag synchronization , generalized synchronization , anti- synchronization , projective synchronization , function projective synchronization , hybrid synchronization .
Chaos is an important phenomenon, happens vastly in both natural and man-made systems. Lorenz  faced to the first chaotic attractor in 1963. In continue, a lot of researches were achieved on chaoticsystems [2-11]. A new 3D chaotic system (T-system) from the Lorenz system was derived in [12,13]. Over the last two decades, chaos control and synchronization have been absorbed increasingly attentions due to their wide applications in many fields [14–25]. Active control , backstepping  and adaptive control  are three different methods for synchronization of T system. Active control  and backstepping  methods are selected when system parameters are known, and adaptive control  method is applied when system parameters are unknown. GBM [26,27], a new method to optimize backstepping method, controls chaos in nonlinear systems better than backstepping design.
Chaos synchronization is an important problem in nonlinear science. During the last three decades, synchronization has received a great interest among various scientists [1–7]. The synchronization can be seen as the property shared by some objects to express a uniform rate of coexistence. For example, two harmonic oscillators can be synchronized if their periods are equal. However, for the case of chaotic oscillators, the concepts of frequency and phase are not well defined and, therefore, two chaotic oscillators can be synchronized if eventually, after a transitional time (a long or short time span), the oscillations coincide exactly at all times despite both oscillators started at different initial conditions. The idea of synchronizing two identical chaoticsystems from different initial conditions was introduced in the seminal work in . After that, several synchronization schemes were introduced in [50,58–66]. Besides, the practical applications of chaoticsynchronization has some limitations to accomplish identical synchronization. For example, parameter mismatch will probably destroy the manifold of a synchronization. To deal with this issue, generalized synchronization approaches were introduced [3,67]. In this manner, we perform the synchronization of two chaotic oscillators following the approach given in . Therefore, the chaos generator model from Equation (3) in Generalized Hamiltonian form, is given by
Synchronized activity and temporal correlation are crit- ical for encoding and exchanging information for neuronal information processing in the brain [2–4]. Synchronization approaches in neuronal systems are aimed at exploring the communication between neurons with the computing of coupling functions that resemble observed experimental electrical cell activity [6–10]. From the general synchro- nization point of view, synchronization approaches can be classified into two general groups [11, 12]: (i) natural coupling (self-synchronization) [13–21] and (ii) artificial coupling using state observers or feedback control approaches [22–34].
Synchronization of chaoticsystems has become an active research area because of its po- tential applications in diﬀerent industrial areas [–]. Communication security scheme is one of the hottest ﬁelds based on chaos synchronization. In this secure communication scheme, the message signals are injected to a chaotic carrier in the transmitter and then are masked or encrypted. The resulting masked signals are transmitted across a public channel to the receiver. To recover the message in the receiver, the synchronization be- tween the chaoticsystems at the transmitter and receiver ends is required. Since Pecora and Carroll  originally proposed the synchronization of the drive and response systems with diﬀerent initial states in , many synchronization techniques such as coupling control , adaptive control , feedback control , fuzzy control , observer-based control , etc. have been developed in the literature.