In this article we present sufficient conditions for parametric and state estimation using **adaptive** **observers** which cover high- gain designs—cf. [35], [51] 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 [47]–[49]); 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 [50]); 3) the conditions we set for our **adaptive** **observers** in- tersect (and generalize in certain ways) with high-gain designs for nonlinear **systems**, similar to those in [62], [63] 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 **chaotic** **systems**); 5) the condition in terms of PE covers cases consid- ered for instance in [7], [8], [12], [47]–[49], [55]–[58]. Our re- sults are for a (structurally) similar class of **systems** as that con- sidered in [12] except that in the latter **systems** are assumed to be partially linear; another fundamental difference is that in [12] **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. [64].

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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

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ﬁrst found by Mackey and Glass []. For **chaotic** **systems** with time-delay and distur- bance, several works have proposed the problem for various **chaotic** **systems** in the lit- erature [–]. In [], an **adaptive** control law was derived and applied to achieve the state lag-**synchronization** of two nonidentical time-delayed **chaotic** **systems** with unknown parameters. In [], an output coupling and feedback scheme were proposed to achieve the **robust** **synchronization** of noise-perturbed **chaotic** **systems** with multiple time-delays. An impulse control was proposed by Qian and Cao [] to synchronize two nonidentical **chaotic** **systems** 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 **chaotic** **synchronization** with time-varying delay is an important topic.

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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 **chaotic** **systems** is built in [16], the **adaptive** modified function projective **synchronization** of multiple time-delayed **chaotic** Rossler **systems** is discussed in [17], the hybrid **synchronization** of L¨u hyperchaotic system with disturbances is investigated in [18], and the **robust** **synchronization** of perturbed Chen’s fractional-order **chaotic** **systems** is studied in [19]. However, these results are just about one kind of **chaotic** **systems**, which limits their applied scope. Hence how to design the impulsive strategy to realize the **synchronization** suitable for more **chaotic** **systems** activates our research.

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Chaos **synchronization** (CS) has been widely investigated based on the results of [1]. 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. [19] dis- cussed **robust** **synchronization** of uncertain complex networks by using impulsive con- trol. Yu and Cao [20] addressed the **synchronization** of **chaotic** **systems** with time delay. The **synchronization** and anti-**synchronization** via active control approach on fractional **chaotic** ﬁnancial system were studied by Huang et al. in [21]. On the basis of control the- ory, a lot of methods, such as **adaptive** control [22], sliding control [23, 24], pinning control [25], intermittent control [26, 27], have been developed for CS.

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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 **robust** **synchronization** scheme to guarantee the precise mimicry of **chaotic** **systems** 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 [15]-[17]. 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.

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engineering, economics, neuroscience and many more [7,8,10,11,13,33,37]. In search of the stable, fast and **robust** **synchronization** 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 [32]. 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** [38].

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In this paper, the lag projective (anti-)**synchronization** problems for a kind of master-slave **chaotic** **systems** 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.

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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.

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Isochronal function projective **synchronization** between **chaotic** and time-delayed **chaotic** **systems** 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 **chaotic** **systems**. 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.

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Remark 2 This example shows a diﬀerence between fractional and integer order calculus. And the results in [32] are still right because their proofs are valid as long as we replace Lemma 5 in [32] 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).

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Furthermore, time delays are also introduced to fractional differential **chaotic** **systems** [22, 23]. In 2007, Deng et al. [24] 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. [25] 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. [26] 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.

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presented the **adaptive** **synchronization** of fractional Lorenz **systems** using a re- duced number of control signals and parameters [13]. Kajbaf et al. used sliding mode controller to obtain **chaotic** **systems** [14]. Wang et al. proposed a new feedback **synchronization** criterion based on the largest Lyapunov exponent [15]. 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 [16] 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. [17], 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.

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This paper proposes a new fuzzy **adaptive** exponential **synchronization** controller for uncertain time-delayed **chaotic** **systems** 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.

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Chaos is an important phenomenon, happens vastly in both natural and man-made **systems**. Lorenz [1] faced to the first **chaotic** attractor in 1963. In continue, a lot of researches were achieved on **chaotic** **systems** [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 [28], backstepping [28] and **adaptive** control [28] are three different methods for **synchronization** of T system. Active control [28] and backstepping [28] methods are selected when system parameters are known, and **adaptive** control [28] 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.

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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 **chaotic** **systems** from different initial conditions was introduced in the seminal work in [1]. After that, several **synchronization** schemes were introduced in [50,58–66]. Besides, the practical applications of **chaotic** **synchronization** 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 [3]. Therefore, the chaos generator model from Equation (3) in Generalized Hamiltonian form, is given by

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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].

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