Adaptive Control for Modified Projective Synchronization-Based Approach for Estimating All Parameters of a Class of Uncertain Systems: Case of Modified Colpitts Oscillators

Abstract

A method of estimation of all parameters of a class of nonlinear uncertain dynamical systems is considered, based on the modified projective synchronization (MPS). The case of modified Colpitts oscillators is investigated. Through a suitable transformation of the dynamical system, sufficient conditions for achieving synchronization are derived based on Lyapunov stability theory. Global stability and asymptotic robust synchronization of the considered system are investigated. The proposed approach offers a systematic design procedure for robust adaptive synchronization of a large class of chaotic systems. The combined effect of both an additive white Gaussian noise (AWGN) and an artificial perturbation is numerically investigated. Results of numerical simulations confirm the effectiveness of the proposed control strategy.

1. Introduction

Synchronization of chaotic systems and their potential applications in wide areas of physics and engineering sciences is currently a field of great interest ([1, 2] and references cited therein). The first idea of synchronizing two identical chaotic systems with different initial conditions is introduced by Pecora and Carroll [3] and the method is realized in electronic circuits. Synchronization techniques have been improved in recent years, and many different methods are applied theoretically and experimentally to synchronize the chaotic systems which include back stepping design technique [4], projective synchronization (PS) [5], modified projective synchronization (MPS) [6, 7], generalized synchronization [8], adaptive modified projective synchronization [9], lag synchronization [10], anticipating synchronization [11], phase synchronization [12], and their combinations [13]. Synchronization may involve several systems without a prescribed hierarchy (bidirectional) as it is the case in synchronization of networks of systems [14, 15], often happening naturally, for instance, in certain biological systems. Another intensive area of research to emphasize within bidirectional synchronization is the study of the consensus paradigm (see an excellent text in [16]). Amongst all kinds of chaos synchronization, MPS is the state-of-the-art of synchronization schemes. MPS means that the master and slave systems could be synchronized up to a constant scaling matrix. Recently, various control methods which include adaptive control [17, 18] and active control [7, 19, 20] have been introduced. Most of the works done on MPS of chaotic systems have used active control method since it is easy to design a control input and to deal with equations including scaling functions. The controller based on the active control method is complex and contains various variables, so it may not be suitable for real practical purpose. In fact, it is obvious that practical controllers should have simple structures. Besides, the projective synchronization (PS) has been used in the research on secure communication [21] because of the unpredictability of the scaling factor which may be a useful element.

Adaptive control technique is used when the system parameters are unknown. In adaptive method, control law and a parameter update rule for unknown parameters are designed in such a way that the chaotic response system is controlled by the chaotic drive system. Most of the studies in synchronization involve two identical/nonidentical systems under the hypotheses that all the parameters of the master and slave systems are known a priori. A controller is constructed with the known parameters and systems are free from external perturbations. But in practical situations the uncertainties like parameter mismatch and external disturbances may destroy the synchronization and even break it. So it is necessary to design an adaptive controller and parameter update law for the control and synchronization of chaotic systems consisting of unknown parameters to get rid of internal and external noises. In the presence of model uncertainties and external disturbances, an appropriate adaptive control scheme is applied to stabilize a group of chaotic systems. In [22] Salarieh and Shahrokhi proposed an adaptive synchronization of two different chaotic systems with time-varying unknown parameters. An adaptive synchronization between two different hyperchaotic systems was developed by Wu et al. [23] while Li et al. developed a complete (anti-) synchronization of chaotic systems with fully uncertain parameters by adaptive control [24]. In [25] Wang et al. present an adaptive control and synchronization for chaotic systems with parametric uncertainties. The same control strategy is used by Mossa et al. for the antisynchronization of two identical and different hyperchaotic systems with uncertain parameters [26].

From the literature survey, it is seen that, with the development of nonlinear control theory, nowadays adaptive projective synchronization method has become very much effective to control and synchronize the chaotic and hyperchaotic systems with uncertain parameters and external disturbances. Recently, many authors have studied the adaptive synchronization for the chaotic systems. In [27] Shen et al. demonstrated that two chaotic Colpitts circuits can be properly synchronized with employment of adaptive controllers while the circuit parameters and the channel are time varying. An adaptive projective synchronization between different chaotic systems with parametric uncertainties and external disturbances was presented by Mayank et al. [28] while Jia et al. develop a generalized projective synchronization of a class of chaotic (hyperchaotic) systems with uncertain parameters [29].

Most of the adaptive control scheme is based on the dynamic parameter estimation. In their book entitled Stable Adaptive Systems [30], Narendra and Annaswamy show that the estimation is feasible when the parameters of the chaotic system can be written in the matrix form. This approach was used in 1991 by Mossayebi et al. to present an adaptive estimation and synchronization of chaotic systems [31]. The authors enlarge the parameter estimation concept extended to well-known chaotic systems in the literature. Nowadays, many other works are carried out to show that this topic is not new in nonlinear science but there is not generalized method planned for this issue. The problem of estimating the unknown parameters using adaptive control has been extensively investigated in the literature for linear and nonlinear systems. For instance, in [32], Fotsin and Daafouz analyzed the adaptive synchronization of uncertain chaotic Colpitts oscillators based on parameters identification. Based on Lyapunov stabilization theory, Huang et al. [33] proposed an adaptive controller with parameters identification for synchronizing a class of chaotic systems with unknown parameters. In the work [34], synchronization-based estimation of all parameters of chaotic systems from time series is presented by Huang. But we notice that the underlying assumption in those papers is that the chaotic systems used as benchmark examples to investigate synchronizations are in a form that provides an easy identification of these parameters. The techniques mentioned in the above articles are not suitable for direct estimation of all system parameters of the circuits in which certain coefficients are arguments of some other nonlinear functions (jerk family, modified Colpitts, etc.). An adequate estimation of the MCO parameters then appears inescapable because of its many advantages that it offers in the various chaos applications. To the best of our knowledge there is no study in the synchronization between two identical chaotic oscillators having the topology described above with estimation of all its parameters. Besides the topology of the modified Colpitts oscillator presents a complex nonlinear term in exponential form and one of our purposes consists in estimating its arguments.

Recently, there have been many efforts for the study of dynamical properties of this oscillator introduced by Ababei and Marculescu in [35] where it was used in the qualitative numerical transmission of information. The particular feature of this oscillator is the real possibility to control chaos using a single resistor, without varying any parameter of the intrinsic Colpitts oscillator, which offers the possibility of an electronic analog or digital control on the system dynamics. Most previous studies in the literature have predominantly concentrated on standard systems such as the Lorentz, the classic Colpitts oscillator, the Chua system, the Chen system, the Lu system, or the Rössler system either in the studies of their stability analysis and periodic oscillations or of their synchronization. It has been shown that the MCO can exhibit complicated dynamics with reference to the classical Colpitts oscillator linked to its nonlinearity topology which is a great advantage in telecommunication. In general, the security of chaos-based communication systems is dependent on the complexity degree of master’s dynamics, carrying signal as well as the encryption scheme used [21]. There are few studies in the synchronization of the MCO [36, 37], though these systems are widely encountered in practice, in particular in communication [38].

In this paper, we first transform the original system equations of MCO by rigorous mathematical theory and secondly we will study modified synchronization of uncertain MCO which is presented based on Lyapunov stability theory.

The organization of this paper is as follows. In Section 2 we first give a brief presentation of the model and its chaotic behavior is introduced. In Section 3, we present the theory of transformation of the MCO. Problem formulation is relaxed in Section 4. Main results presenting the synchronization behavior of two unknown MCO with artificial perturbation are studied in Section 5. Numerical simulations are given in Section 6. Finally, conclusions are presented in Section 7.

2. The Modified Colpitts Oscillator

The simplest configuration of the MCO is shown in Figure 1(a) [36]. This circuit uses a bipolar junction transistor (BJT) as the gain element and a resonant network consisting of an inductor (𝐿) and a pair of capacitors (𝐶₁ and 𝐶₂). The resistor 𝑅_𝑑 is the additive element compared to the classical system. The nonlinear element is the BJT (here the NPN type) for which the simplified model shown in Figure 1(b) consists of nonlinear voltage-controlled resistance 𝑅_𝐸 and a linear current-controlled current source 𝐼_𝐸 [39]. Parasite capacitances of the BJT can be neglected in the frequency range of oscillation since their effect would only be a frequency shift. We assume 𝛼_𝐹=1, where 𝛼_𝐹 is the common-base forward short-circuiting current gain. This corresponds to neglecting the base current. The V–I characteristic of the nonlinear resistor 𝑅_𝐸 (which corresponds normally to the emitter-base diode) and its approximate expression are defined as usual by 𝐼_𝐸=𝑓𝑉_BE=𝐼_𝑆exp𝑒𝑉_BE𝑘_𝐵𝑇−1≈𝐼_𝑆exp𝑒𝑉_BE𝑘_𝐵𝑇,(1) where 𝐼_𝑆 is the saturation current of the emitter-base diode, 𝑒 the elementary charge, 𝑘_𝐵 the Boltzmann constant, and 𝑇 the absolute temperature. The thermal voltage at room temperature is given by 𝑉_𝑇=𝐾_𝐵𝑇/𝑒≈26 mV. Here we set 𝐼_𝑆=10⁻¹¹ A. The base-emitter voltage drop 𝑉_BE is given by 𝑉_BE=−𝑢₂−𝑅_𝑑𝑖_𝐿+𝑖_𝐵,(2) where 𝑖_𝐵 is the base current. Moreover, the relations between the currents are 𝑖_𝐿=𝑖₁+𝑖_𝑐;𝑖_𝐿=𝑖₂+𝑖₃−𝑖_𝐵.(3) Since we have 𝑖_𝐿≫𝑖_𝐵, the simplified state equations for the schematic in Figure 1(a) are the following: 𝑑𝑢₁=𝑖𝑑𝑡𝑐^𝐿₁−𝐼_𝐸𝑐₁,𝑑𝑢₂=𝑖𝑑𝑡𝑐^𝐿₂−𝑢₂𝑅_𝑒𝑐₂−𝑉_CC𝑅_𝑒𝑐₂,𝑑𝑖_𝐿=𝑉𝑑𝑡𝐿^CC−𝑅_𝐶+𝑅_𝑑𝐿𝑖_𝐿−𝑢₁𝐿−𝑢₂𝐿.(4) We now introduce a set of dimensionless state variables (𝑥, 𝑦, and 𝑧) where we normalize voltages currents and time according to the following relations: 𝑢₁=𝑉_CC𝑥,𝜃=𝐿𝑐₁𝑢,𝑡=𝜃𝜏,₂=𝑉_CC𝑦,𝑖_𝐿=𝑉_CC𝐿/𝑐₁𝑧.(5) Using (2), (3), and (5), the system of (4) can be rewritten in the following form: 𝑑𝑥𝑑𝜏=𝑧−𝑎₂exp−𝑎𝑧−𝑏𝑦,𝑑𝑦𝑑𝜏=−𝑏₀−𝑏₀𝑦+𝑧,𝑑𝑧𝑑𝜏=1−𝑥−𝑦−𝑐₁₁𝑧(6) with 𝑏₀=𝜃/𝑅_𝑒𝑐₂, 𝑏=𝜃𝐼_𝑆/𝑐₂𝑉_CC, 𝑐₁₁=𝜃(𝑅_𝐶+𝑅_𝑑)/𝐿, 𝑎=(𝑉_CC/𝑉_𝑇)(𝑅_𝑑/𝐿/𝑐₁), and 𝑎₂=𝐼_𝑆𝜃/𝑐₁𝑉_CC.

(a)

(b)

Figure 1 

Circuit model: (a) Schematic of the Colpitts oscillator. (b) BJT model in common base configuration.

Let us select the parameters: 𝐿=98.5 μH, 𝑅_𝐶=35Ω, 𝑅_𝐸=400Ω, 𝑉_CC=𝑉_EE=5V, 𝐶₁=𝐶₂=54 nF and 𝑅_𝑑=0.5Ω which yields 𝑎=2.25, 𝑏=192.3, 𝑏₀=0.106𝑐₁₁=0.934, and 𝑎₂=8.51⋅10⁻¹¹; the system shows a chaotic behavior characterized by a maximal Lyapunov exponent 𝜆_max=0.06 which confirms the occurrence of chaotic oscillations.

3. Transformation Analysis of MCO

Consider the dynamics system of MCO given by 𝑑𝑥₁𝑑𝜏=𝑥₃−𝑎₁exp−𝑎𝑥₃−𝑏𝑥₂,𝑑𝑥₂𝑑𝜏=−𝑏₀−𝑏₀𝑥₂+𝑥₃,𝑑𝑥₃𝑑𝜏=1−𝑥₁−𝑥₂−𝑐₁₁𝑥₃.(7) In order to quantify the parameters inside the exponential term argument, let us introduce the coordinates change in state and output-space: 𝑧₁,𝑧₂,𝑧₃=𝑥₁,𝑒^−𝑏𝑥2,𝑒^−𝑎𝑥3.(8) Derivatives in the considered space are given by 𝑥̇₁,𝑥̇₂,𝑥̇₃=𝑧̇₁1,−𝑏𝑧̇₂𝑧₂1,−𝑎𝑧̇₃𝑧₃.(9) In these new coordinates, the system (7) takes the form 𝑧̇₁=−𝜎₁ln𝑧₃−𝜎₂𝑧₂𝑧₃,𝑧̇₂=𝜎₃𝑧₂−𝜎₄𝑧₂ln𝑧₂+𝜎₆𝑧₂ln𝑧₃,𝑧̇₃=𝜎₀𝑧₁−1𝑧₃−𝜎₇𝑧₃ln𝑧₂−𝜎₅𝑧₃ln𝑧₃.(10) with 𝜎₁=1/𝑎, 𝜎₂=𝑎₁, 𝜎₃=𝑏𝑏₀, 𝜎₄=𝑏₀, 𝜎₅=𝑐₁₁, 𝜎₀=𝑎, 𝜎₆=𝑏𝑎⁻¹, and 𝜎₇=𝜎₆⁻¹. The system (10) presents five parameters as system (7) which confirms the linearity of the transformation.

3.1. Dissipation and Existence of Attractors

Preliminary insights concerning the existence of attractive sets [40] that might coexist in the system could be gained by evaluating the volume contraction/expansion rate (Γ=𝑉⁻¹𝑑𝑉/𝑑𝜏) of the oscillator modeled by (10) at any given point (𝑧₁,𝑧₂,𝑧₃)^𝑇 of the space. The following expression can be derived: 𝜕𝑧̇Γ=¹𝜕𝑧₁+𝜕𝑧̇₂𝜕𝑧₂+𝜕𝑧̇₃𝜕𝑧₃=𝜎₃−𝜎₄−𝜎₅−𝜎₀+𝜎₀𝑧₁−ln𝑧^𝜎4+𝜎7₂𝑧^{𝜎5−𝜎6}₃=𝜎₀𝑧₁𝑧−ln^𝜎4+𝜎72𝑧^𝜎5₃𝛿𝑧^𝜎6₃,(11) where ln𝛿=𝜎₃−𝜎₄−𝜎₅−𝜎₀ and 𝑧̇=(𝑧̇₁,𝑧̇₂,𝑧̇₃).

Consider the fact that the system (10) is dissipative which is expressed as follows: div(𝑧)̇=𝜎₀𝑧₁𝑧−ln^𝜎4+𝜎72𝑧^𝜎5₃𝛿𝑧^𝜎6₃<0.(12) This condition implies that the solutions of the new system are bounded as 𝜏→∞. We may rewrite condition (12) as follows:𝑧₁<1𝜎₀𝑧ln^𝜎4+𝜎72𝑧^𝜎5₃𝛿𝑧^𝜎6₃.(13a) We notice that any initial volume element Γ₀=Γ(𝜏=0) will be continuously contracted by the flow. In other words, each volume element containing the system trajectory shrinks to zero as time elapses. Consequently, all system orbits will be confined to a specific bounded subset of zero volume in phase space and the asymptotic motion converges to an attractor.

It is easily shown that the system (7) has one equilibrium point 𝐸=(𝑥₁₀;𝑥₂₀;𝑥₃₀) such that 𝐸=(1.02;−0.10;0.09). The characteristic equation of the Jacobean matrix of the system (7) about the equilibrium point is 𝛽₀𝜆³+𝑏₀+𝑐₁₁𝜆²+1+𝑏₀𝑐₁₁+𝑚₂𝜆+𝑚₁+𝑏₀𝑚₂=0,(13b) where 𝑚₁=𝑎₁𝑏exp(−𝑎𝑥₃₀−𝑏𝑥₂₀); 𝑚₂=1+𝑎𝑎₁exp(−𝑎𝑥₃₀−𝑏𝑥₂₀). Let 𝛽₀=1, 𝛽₁=𝑏₀+𝑐₁₁, 𝛽₂=1+𝑏₀𝑐₁₁+𝑚₂, and 𝛽₃=𝑚₁+𝑏₀𝑚₂. We obtain the following conditions 𝛽₁>0, 𝛽₂>0, and  𝛽₁𝛽₂<𝛽₃. According to Routh-Hurwitz criteria the equilibrium point is unstable. The singularity is saddle point. The transformation (8) leads to the following equilibrium point in the new space: 𝑧₁₀,𝑧₂₀,𝑧₃₀=𝑥₁₀,𝑒^{−𝑏𝑥20},𝑒^{−𝑎𝑥30}.(13c)Then 𝐸=(1.02;4046.67;0.48) and div(𝑧(̇𝑧₁₀,𝑧₂₀,𝑧₃₀))=2.72>0 which confirms the divergence of trajectories at the equilibrium point. In fact, an infinitesimal deviation of the initial conditions will eventually result in the divergence of nearby starting orbits. After a while, the system initially unstable becomes dissipative and stays unchanged with respect to its dynamical variables which strongly justify the synchronization process of MCO in the space considered.

3.2. Model Description

Let us consider a general class of chaotic systems described by the following differential equation: 𝑧̇=𝜙(𝑧(𝜏),𝜏)𝜃₁+𝜀(̃𝑧(𝜏),𝜏)𝜃₂+^𝑝_𝑖=1𝜃_𝑖Ω_𝑖𝜀(𝑧(𝜏),𝜏),𝑝=3,(14) where 𝑧∈𝑅^𝑛 denotes the state vectors, 𝜙∶𝑅^𝑛→𝑅^𝑛 are continuous function matrices, and 𝜀,̃𝜀∶𝑅^𝑛→𝑅^𝑛 are a continuous nonlinear function, where 𝜀̃ denotes the product part of the state variables and 𝜀 the made up function in which the arguments are the system variables and 𝜃₁, 𝜃₂, and 𝜃_𝑖 are parameters vectors such that 𝜃₁,𝜃₂,𝜃_𝑖∈𝑅𝑛_×𝑛 and Ω_𝑖∈𝑅^𝑛×𝑛, respectively. Note that many chaotic and hyperchaotic systems such as jerk family system, two-cell quantum-CNN, and modified Bloch equations with feedback field could be described by system (14). Expression (14) is an interesting form because it describes the whole complex chaotic system and makes its dynamics behavior analysis easy such as stability and bifurcations.

Assumption 1. The states of the chaotic system described by (14) are bounded, and the nonlinear function with two variables 𝑓(𝑎^′𝑣₃,𝑏^′𝑣₂) is locally Lipschitz; that is, there exists a positive constant 𝑘_𝑓 such that 𝑓𝑎^′𝑣₃,𝑏^′𝑣₂−𝑓𝑎^′𝑣₃̃,𝑏^′𝑣₂̃≤𝑎^′𝑘_𝑓𝑣₃̃−𝑣₃+𝑏^′𝑘_𝑓𝑣₂̃−𝑣₂,(15) where 𝑣=(𝑣₁,𝑣₃,𝑣₂), 𝑎^′  and  𝑏^′ are positive constants.
The relation (15) can be put in the following form: 𝑓𝑎^′𝑣₃,𝑏^′𝑣₂−𝑓𝑎^′𝑣₃̃,𝑏^′𝑣₂̃≤𝑀𝑎^′,𝑏^′𝑒_⟂,(16) where 𝑒_⟂=(‖𝑣₁̃−𝑣₁‖,‖𝑣₂̃−𝑣₂‖,‖𝑣₃̃−𝑣₃‖)^𝑇, 𝑀(𝑎^′,𝑏^′)=^0000𝜀′0_00𝜀′′ with 𝜀^′=𝑏^′𝑘_𝑓 and 𝜀^′′=𝑎^′𝑘_𝑓.
Let us consider two 3-dimensional chaotic systems which can be represented in a more generic form as follows: ̇(𝑥(𝑡)),̇𝑦(𝑡),𝑥=𝑓,𝑡(17)𝑦=𝑓,𝑡(18) where 𝑥∈𝑅³ and 𝑦∈𝑅³ are 3-dimensional state vectors of the system and 𝑓∶𝑅³→𝑅³ is analytic.
We assume that the asymptotic convergence of (17) and (18) is satisfied.
In the new space, the coordinates can be written in the following form: ̇(𝑧(𝑡)),̇(𝑧(𝑡)),𝑧=𝑓,𝑡(19)𝑧=𝑓,𝑡(20)𝑧(𝑡)=𝜓(𝑥(𝑡)), and 𝑧(𝑡)=𝜓(𝑦(𝑡)) where 𝜓(∗) is a transformation function.

Proposition 2. If the master system (17) and slave system (18) asymptotically synchronize, then the systems (19) and (20) which were derived from the change of space also asymptotically synchronize, such that lim_𝑡→∞‖𝑒(𝑡)‖=0⇒lim_𝑡→∞‖𝑒^′(𝑡)‖=0, where 𝑒(𝑡)=𝑦(𝑡)−𝑥(𝑡) and 𝑒^′(𝑡)=𝑧(𝑡)−𝑧(𝑡).

Proof. Let us write 𝑓(𝑧(𝑡),𝑡) as the sum of two functions: 𝑓(𝑧(𝑡),𝑡)=𝜙(𝑧(𝑡),𝑡)+𝜀(𝑧(𝑡),𝑡),(21) where 𝜙(𝑧(𝑡),𝑡) is the linear part of 𝑓(𝑧(𝑡),𝑡) and can take the form: 𝜙(𝑧(𝑡),𝑡)=𝐴𝑧(𝑡)=𝐴𝜓(𝑥(𝑡)),(22) where 𝐴 is a full rank constant matrix and all eigenvalues of 𝐴 have negative real parts. The simplest configuration of matrix 𝐴 is described by 𝐴=diag(𝜆₁,𝜆₂,…,𝜆_𝑛). Synchronization error between system (19) and system (20) is defined as 𝑒^′(𝑡)=𝑧(𝑡)(𝑡)𝑦(𝑡)(𝑥(𝑡))(𝑧(𝑡(𝑧(𝑡𝜓𝑦(𝑡)(𝑥(𝑡))(𝑧(𝑡(𝑧(𝑡−𝑧=𝐴𝜓,𝑡−𝐴𝜓,𝑡+𝜀))−𝜀))=𝐴,𝑡−𝜓,𝑡+𝜀))−𝜀))≤𝐴𝑀^′𝑦(𝑡)−𝑥(𝑡)+𝑀𝜓𝑦(𝑡),𝑡−𝜓(𝑥(𝑡),𝑡)=𝐴𝑀^′‖𝑒(𝑡)‖+𝑀𝑀^′‖𝑒(𝑡=()𝑀)‖𝐴+𝑀^′‖𝑒(𝑡)‖=𝑀^∗𝑒_⟂(23) with 𝑒_⟂=𝑀^′‖𝑒(𝑡)‖ and 𝑀^∗=(𝐴+𝑀)𝑀^′ being chosen such that all eigenvalues of the matrix 𝑀^∗ have negative real parts; the dynamic errors are asymptotically stable. This completes the proof.

4. Problem Formulation

An illustration of the modified projective synchronization is now presented. Taking into account the synchronization between two chaotic systems, take the drive system as follows: 𝑧̇_𝑚=𝜙𝑧_𝑚,𝜏𝜃₁+𝜀̃𝑧_𝑚,𝜏𝜃₂+^𝑝_𝑖=1𝜃_𝑖Ω_𝑖𝜀𝑧_𝑚,𝜏+𝜉𝑧_𝑚,𝜏,𝑝=3.(24) The response system is 𝑠̇=𝜙(𝑠,𝜏)𝜃_1𝑟+𝜀(̃𝑠,𝜏)𝜃_2𝑟+^𝑚_𝑖=1𝜃_𝑖𝑟Ω_𝑖𝑟𝜀(𝑠,𝜏)+𝑢𝑧_𝑚,𝑠,𝜏+𝑢^′𝑧_𝑚,𝑠,𝜏,(25) where 𝑠=[𝑠₁,𝑠₂,…,𝑠_𝑘]^𝑇∈𝑅^𝑘, (𝑢;𝑢^′)∈𝐶^ℓ[𝑅⁺×𝑅^𝑛×𝑅^𝑘,𝑅^𝑘,𝑅^𝑛] and ℓ≥1, 𝜉(𝑧_𝑚,𝜏) is the external disturbances bounded in magnitude ‖𝜉(𝑧_𝑚,𝜏)‖≤𝜌^′, 𝑢(𝑧_𝑚,𝑠,𝜏) is the controller attached in the response system to be determined later, and 𝑢^′(𝑧_𝑚,𝑠,𝜏) is the adaptive controller used for synchronizing the reference model with perturbation. If there exists a function 𝛼=diag𝛼₁,𝛼₂,…,𝛼_𝑛satisfyinglim_𝜏→∞‖𝑒(𝜏)‖=lim_𝜏→∞𝛼𝑠(𝜏)−𝑧_𝑚(𝜏)=0∀𝑧_𝑚𝜏₀,𝑠𝜏₀∈𝑅^𝑛,(26) then the systems (24) and (25) achieve modified projective synchronization, where 𝑒=(𝑒₁,𝑒₂,…,𝑒_𝑛)^𝑇and we call 𝛼 a “scaling factor.”

Remark 3. Chaos synchronization schemes such as complete synchronization and antisynchronization are special cases of modified projective synchronization when 𝛼=1 and 𝛼=−1, respectively.
From the definition of error signal (26), the error dynamics is 𝑒(̇𝜏)=𝛼𝜙(𝑠,𝜏)𝜃_1𝑟+𝛼𝜀(̃𝑠,𝜏)𝜃_2𝑟+𝛼^𝑝_𝑖=1𝜃_𝑖𝑟Ω_𝑖𝑟𝜀(𝑠,𝜏)+𝛼𝑢𝑧_𝑚−𝜙𝑧,𝑠,𝜏_𝑚,𝜏𝜃₁−𝜀̃𝑧_𝑚,𝜏𝜃₂+𝛼𝑢^′𝑧_𝑚,𝑠,𝜏−^𝑚_𝑖=1𝜃_𝑖Ω_𝑖𝜀𝑧_𝑚,𝜏−𝜉𝑧_𝑚,𝜏.(27) Let us choose the controllers in the following form: 1𝑢=𝛼𝜙𝑧_𝑚,𝜏𝜃_1𝑟−𝛼𝜙(𝑠,𝜏)𝜃_1𝑟−𝛼𝜀(̃𝑠,𝜏)𝜃_2𝑟+𝜀̃𝑧_𝑚,𝜏−𝛼^𝑝_𝑖=1𝜃_𝑖𝑟Ω_𝑖𝑟𝜀()+𝑠,𝜏_𝑝𝑖=1𝜃_𝑖𝑟Ω_𝑖𝑟𝜀𝑧_𝑚,𝜏−Λ𝑒,(28) where Λ=diag(Λ₁,Λ₂,…,Λ_𝑛), Λ_𝑖>0(𝑖=1,2,…,𝑛) are constants. One advantage of this type of controller is that it can be easily constructed through time-varying resistors, capacitors, or operational amplifier and their combinations, or using a digital signal processor together with the appropriate converters. Consequently, we get 𝑒(̇𝜏)=𝜙𝑧_𝑚,𝜏−𝛼𝜙(𝑠,𝜏)𝜃_1𝑟−𝜃₁+𝛼𝜀(̃𝑠,𝜏)−𝜀̃𝑧_𝑚,𝜏𝜃_2𝑟−𝜃₂+^𝑝_𝑖=1𝜃_𝑖𝑟−𝜃_𝑖Ω𝑒_𝑖𝛼𝜀(𝑠,𝜏)−𝜀𝑧_𝑚𝑧,𝜏−Λ𝑒−𝜉_𝑚,𝜏.(29) The parameter update rule is 𝜃̇_1𝑟=−𝜙𝑧_𝑚,𝜏−𝛼𝜙(𝑠,𝜏)^𝑇𝑒−Υ_𝛼𝜃_1𝑟,𝜃̇_2𝑟=−𝛼𝜀(̃𝑠,𝜏)−𝜀̃𝑧_𝑚,𝜏^𝑇𝑒−Υ_𝛽𝜃_2𝑟,̇𝜃_𝑖𝑟=−𝛼𝜀(𝑠,𝜏)−𝜀𝑧_𝑚,𝜏^𝑇Ω𝑒_𝑖−Υ_𝜆𝜃_𝑖𝑟,(30a) where 𝜃_1𝑟=𝜃_1𝑟−𝜃₁, 𝜃_2𝑟=𝜃_2𝑟−𝜃₂, 𝜃_𝑖𝑟=𝜃_𝑖𝑟−𝜃_𝑖, Ω𝑒_𝑖=Ω_𝑖𝑟−Ω_𝑖, and Υ_𝑗=diag(Υ_1𝑗,Υ_2𝑗,…,Υ_𝑝𝑗), 𝑗=(𝛼,𝛽,𝜆), are constant positive matrices.

Theorem 4 (Barbalat’s lemma [41]). Let one consider an uniform continuous function 𝜋∶[0,∞)→𝑅 such that ∀𝜏∈[0,∞): lim_𝜏→∞^𝜏₀𝜋(𝑡)𝑑𝑡𝑒𝑥𝑖𝑠𝑡𝑠𝑎𝑛𝑑𝑖𝑠𝑓𝑖𝑛𝑖𝑡𝑒;𝑡ℎ𝑒𝑛lim_𝜏→∞𝜋(𝜏)=0.(30b)

Theorem 5. For the given scaling factor 𝛼, the modified projective synchronization between drive system (24) and response system (25) will occur under the control law (28) and the parameter update rule (30b).

Proof. Construct dynamical Lyapunov function as follows: 𝑉̇=12𝑒^𝑇1𝑒+2²_𝑣=1𝜃^𝑇_𝑣𝑟𝜃_𝑣𝑟+12^𝑝_𝑖=1𝜃^𝑇_𝑖𝑟𝜃_𝑖𝑟.(31) Then the time derivative of Lyapunov function 𝑉 along the trajectory of error system (29) is 𝑉̇=𝑒^𝑇𝑒̇+²_𝑣=1𝜃^𝑇_𝑣𝑟𝜃̇_𝑣𝑟+^𝑝_𝑖=1𝜃^𝑇_𝑖𝑟𝜃̇_𝑖𝑟=𝑒^𝑇𝜙𝑧_𝑚,𝜏−𝛼𝜙(𝑠,𝜏)𝜃_1𝑟−𝜃₁+𝑒^𝑇𝛼𝜀(̃𝑠,𝜏)−𝜀̃𝑧_𝑚,𝜏𝜃_2𝑟−𝜃₂−𝑒^𝑇𝑚_𝑖=1𝜃_𝑖𝑟−𝜃_𝑖Ω𝑒_𝑖𝛼𝜀(𝑠,𝜏)−𝜀𝑧_𝑚,𝜏−𝑒^𝑇Λ𝑒−𝜃^𝑇_1𝑟𝜙𝑧_𝑚,𝜏−𝛼𝜙(𝑠,𝜏)^𝑇𝑒−𝜃^𝑇_1𝑟Υ_𝛼𝜃_1𝑟−𝜃^𝑇_2𝑟𝛼𝜀(̃𝑠,𝜏)−𝜀̃𝑧_𝑚,𝜏^𝑇𝑒−𝜃^𝑇_2𝑟Υ_𝛽𝜃_2𝑟−^𝑝_𝑖=1𝜃^𝑇_𝑖𝑟Ω𝑒_𝑖𝛼𝜀(𝑠,𝜏)−𝜀𝑧_𝑚,𝜏^𝑇−^𝑝_𝑖=1𝜃^𝑇_𝑖𝑟Υ_𝜆𝜃_𝑖𝑟−𝑒^𝑇𝜉𝑧_𝑚,𝜏=−𝑒^𝑇Λ𝑒−𝜃^𝑇_1𝑟Υ_𝛼𝜃_1𝑟−𝜃^𝑇_2𝑟Υ_𝛽𝜃_2𝑟−^𝑝_𝑖=1𝜃^𝑇_𝑖𝑟Υ_𝜆𝜃_𝑖𝑟−𝑒^𝑇𝜉𝑧_𝑚,𝜏.(32) Since the systems (24) and (25) are contained in an attractor, we can suppose that the whole error system (27) can be in the domain Ω={𝑒∈𝑅^𝑛,‖𝑒‖≤𝑟₀,𝑟₀>0}. Set ‖𝜉(𝑧_𝑚,𝜏)‖≤𝜌^′ and consider a bounded positive function Λ^′(𝜏) such that 𝜉(𝑧_𝑚,𝜏)=−Λ^′(𝜏)𝑒. It is obvious to note that ‖Λ^′(𝑡)𝑒‖≤𝑟₀𝜌^′. Then, (32) may be written as follows: 𝑉̇≤−𝑒^𝑇Λ𝑒−𝜃^𝑇_1𝑟Υ_𝛼𝜃_1𝑟−𝜃^𝑇_2𝑟Υ_𝛽𝜃_2𝑟−^𝑝_𝑖=1𝜃^𝑇_𝑖𝑟Υ_𝜆𝜃_𝑖𝑟+𝑒^𝑇Λ^′(𝜏)𝑒≤−𝑒^𝑇Λ−Λ^′(𝜏)𝑒−𝜃^𝑇_1𝑟Υ_𝛼𝜃_1𝑟−𝜃^𝑇_2𝑟Υ_𝛽𝜃_2𝑟−^𝑝_𝑖=1𝜃^𝑇_𝑖𝑟Υ_𝜆𝜃_𝑖𝑟<0.(33) We can see that 𝑉 is positive definite if Λ−Λ^′(𝜏)≤0 and 𝑉̇ is negative definite; according to Lyapunov stability theory, the error vectors 𝑒_𝜎(𝜎=1,2,…,𝑛) asymptotically tend to zero, leading to the modified projective synchronization between drive (24) and response system (25). Therefore, the proof is completed.

5. Main Results

The equations of MCO in the new space can be expressed as in equations (24), with the following parameters: 𝑧_𝑚=𝑧₁𝑧₂𝑧₃,Ω₁=00−10−𝑧₂000−𝑧₃,Ω₂=00000𝑧₂0−𝑧₃0,𝜙(𝑧(𝜏))=0,𝜏𝑧₂−𝑧₃,𝜀(𝑧(𝜏),𝜏)=ln𝑧₁ln𝑧₂ln𝑧₃,𝜀(̃𝑧(𝜏),𝜏)=−𝑧₂𝑧₃0𝑧₁𝑧₃,𝜃₂=𝜎₂0𝜎₀,𝜃₁=0𝜎₃𝜎₀,𝜃₁=𝜎₁𝜎₄𝜎₇,𝜃₂=0𝜎₆𝜎₅.(34) For the sake of clarity and the matter to handle easily the calculations, expression (34) can be put in the particular form which will enable us to underline modified projective synchronization. Let us consider the following MCO master with artificial perturbation and slave systems: 𝑧̇₁(𝜏)=−𝜎₁ln𝑧₃(𝜏)−𝜎₂𝑧₂(𝜏)𝑧₃(𝜏)−𝑘𝑧₁(𝜏),𝑧̇₂(𝜏)=𝜎₃𝑧₂(𝜏)−𝜎₄𝑧₂(𝜏)ln𝑧₂(𝜏)+𝜎₆𝑧₂(𝜏)ln𝑧₃(𝜏),𝑧̇₃(𝜏)=𝜎₀𝑧₁(𝜏)−1𝑧₃(𝜏)−𝜎₇𝑧₃(𝜏)ln𝑧₂(𝜏)−𝜎₅𝑧₃(𝜏)ln𝑧₃(𝜏),𝑠̇(35)₁(𝜏)=−𝜎^′₁ln𝑠₃(𝜏)−𝜎^′₂𝑠₂(𝜏)𝑠₃(𝜏)+𝑢₁+𝑢^′₁𝑠₂̇(𝜏)=𝜎^′₃𝑠₂(𝜏)−𝜎^′₄𝑠₂(𝜏)ln𝑠₂(𝜏)+𝜎^′₆𝑠₂(𝑡)ln𝑠₃(𝜏)+𝑢₂+𝑢^′₂,𝑠₃̇(𝜏)=𝜎^′₀𝑠₁(𝜏)−1𝑠₃(𝜏)−𝜎^′₇𝑠₃(𝜏)ln𝑠₂(𝜏)−𝜎^′₅𝑠₃(𝜏)ln𝑠₃(𝜏)+𝑢₃+𝑢^′₃,(36) where 𝜎₁,𝜎₂,𝜎₃,𝜎₄, and 𝜎₅ are unknown parameters of the master system and 𝜎^′₁,𝜎^′₂,𝜎^′₃,𝜎^′₄, and 𝜎^′₅ are parameters of the slave system which need to be estimated. Assuming that the character of (35) could be altered by adding an artificial perturbation 𝑘𝑧₁(|𝑘|≤𝑘_𝑚isapertubationcoefficient), the adaptive control input was added into the first equation of the driven model.

By assumption, the master system operates in the chaotic regime; hence, all master signals are bounded. Furthermore, let us temporarily assume that the trajectory of the slave system in closed loop, that is, 𝑠₁(𝜏), is bounded for all 𝜏 (this will be relaxed and proved at the end). Then, there exists 𝛽 such that sup_𝜏≥0𝑠₁(𝜏)≤𝛽.(37) The error state variables are defined as follows: 𝑒₁(𝜏)=𝛼₁𝑠₁(𝜏)−𝑧₁(𝜏),𝑒₂(𝜏)=𝛼₂𝑠₂(𝜏)−𝑧₂(𝜏),𝑒₃(𝜏)=𝛼₃𝑠₃(𝜏)−𝑧₃(𝜏).(38) From these equations, we have the following error dynamics: 𝑒₁̇=𝜎₁ln𝑧₃(𝜏)+𝜎₂𝑧₂(𝜏)𝑧₃(𝜏)−𝛼₁𝜎^′₁ln𝑠₃−𝛼₁𝜎^′₂𝑠₂𝑠₃+𝑘𝑧₁(𝜏)+𝛼₁𝑢₁+𝛼₁𝑢^′₁,𝑒₂̇=−𝜎₃𝑧₂(𝜏)−𝜎₄𝑧₂(𝜏)ln𝑧₂(𝜏)+𝜎₆𝑧₂(𝜏)ln𝑧₃(𝜏)+𝛼₂𝜎^′₃𝑠₂−𝛼₂𝜎^′₄𝑠₂ln𝑠₂+𝜎^′𝛼₂𝑠₂ln𝑠₃+𝛼₂𝑢₂+𝛼₂𝑢^′₂,𝑒₃̇=−𝜎₀𝑧₃(𝜏)+𝑧₁(𝜏)𝑧₃(𝜏)+𝜎₇𝑧₃(𝜏)ln𝑧₂(𝜏)+𝜎₅𝑧₃(𝜏)ln𝑧₃(𝜏)+𝛼₃𝜎^′₀𝑠₃+𝑠₁𝑠₃−𝛼₃𝜎^′₇𝑠₃ln𝑠₂+𝛼₃𝜎₅𝑠₃ln𝑠₃+𝛼₃𝑢₃+𝛼₃𝑢^′₃.(39) The following update laws for system (39) are designed: 𝑢₁=1𝛼₁−𝛼₁𝜎^′₁ln𝑠₃−𝛼₁𝜎^′₂𝑠₂𝑠₃+𝜎^′₁ln𝑧₃(𝜏)−𝜎^′₂𝑧₂(𝜏)𝑧₃(𝜏)−𝑘₁𝑒₁,𝑢₂=1𝛼₂−𝜎^′₃𝑧₂(𝜏)+𝜎^′₄𝑧₂(𝜏)ln𝑧₂(𝜏)−𝜎^′₃𝑧₂(𝜏)ln𝑧₃(𝜏)+𝛼₂𝜎^′₃𝑠₂−𝛼₂𝜎^′₄𝑠₂ln𝑠₂−𝛼₂𝑠₂ln𝑠₃−𝑘₂𝑒₂,𝑢₃=1𝛼₃−𝜎^′₀𝑧₁(𝜏)+1𝑧₃(𝜏)+𝜎^′₇𝑧₃(𝜏)ln𝑧₂(𝜏)+𝜎^′₅𝑧₃(𝜏)ln𝑧₃(𝜏)−𝛼₃𝜎^′₀𝑠₁+1𝑠₃+𝛼₃𝜎^′₆𝑠₃ln𝑠₂+𝛼₃𝜎^′₅𝑠₃ln𝑠₃−𝑘₃𝑒₃,(40) where 𝑘_𝑖(𝑖=1,2,3) are positive constants.

Substituting the control input (40) into (39) gives 𝑒₁̇=𝜎₁−𝜎^′₁ln𝑧₃(𝜏)+𝜎₂−𝜎^′₂𝑧₂(𝜏)𝑧₃(𝜏)−𝑘₁𝑒₁−𝑘𝑧₁(𝜏)+𝛼₁𝑢^′₁,𝑒₂̇=−𝜎₃−𝜎^′₃𝑧₂(𝜏)+𝜎₄−𝜎^′₄𝑧₂(𝜏)ln𝑧₂(𝜏)+𝜎₆−𝜎^′₆𝑧₂(𝜏)ln𝑧₃(𝜏)−𝑘₂𝑒₂+𝛼₂𝑢^′₂,𝑒₃̇=−𝜎₀−𝜎^′₀𝑧₃(𝜏)−𝑧₁(𝜏)𝑧₃(𝜏)+𝜎₇−𝜎^′₇𝑧₃(𝜏)ln𝑧₂(𝜏)−𝜎₅−𝜎^′₅𝑧₃(𝜏)ln𝑧₃(𝜏)−𝑘₃𝑒₃+𝛼₃𝑢^′₃.(41) The modified projective synchronization between master and slave systems will occur by the control law (40) and the following update rules for five unknown parameters 𝜎^′₁,𝜎^′₂,𝜎^′₃,𝜎^′₄, and 𝜎^′₅: ̇𝜎^′₁=𝑒₁ln𝑧₃(𝜏)−𝜎^′₁−𝜎₁,̇𝜎^′₂=𝑒₁𝑧₂(𝜏)𝑧₃(𝜏−𝜏)−𝜎^′₂−𝜎₂,̇𝜎^′₃=−𝑧₂(𝜏)𝑒₂−𝜎^′₃−𝜎₃,̇𝜎^′₄=𝑧₂(𝜏)𝑒₂ln𝑧₃(𝜏)−𝜎^′₄−𝜎₄,̇𝜎^′₅=𝑧₃(𝜏)𝑒₃ln𝑧₃(𝜏)−𝜎^′₅−𝜎₅.(42)

Theorem 6. For any nonzero scaling factors, the slave system (36) can synchronize the master system (35) if 𝑢^′₁=−(𝑔/𝛼₁)𝑒₁ and 𝑢^′₂=0, 𝑢^′₃=0, where g is an estimated feedback gain which is updated according to the following adaption algorithm: 𝑑𝑔𝑑𝜏=𝜎𝑒²₁,𝑤ℎ𝑒𝑟𝑒𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝜎>0.(43)

Proof. The error system is 𝑒₁̇=𝜎₁−𝜎^′₁ln𝑧₃(𝜏)+𝜎₂−𝜎^′₂𝑧₂(𝜏)𝑧₃(𝜏)−𝑘₁𝑒₁−𝑔𝑒₁−𝑘𝑧₁(𝜏),𝑒₂̇=−𝜎₃−𝜎^′₃𝑧₂(𝜏)+𝜎₄−𝜎^′₄𝑧₂(𝜏)ln𝑧₂(𝜏)+𝜎₆−𝜎^′₆𝑧₂(𝜏)ln𝑧₃(𝜏)−𝑘₂𝑒₂,𝑒₃̇=−𝜎₀−𝜎^′₀𝑧₃(𝜏)−𝑧₁(𝜏)𝑧₃(𝜏)+𝜎₇−𝜎^′₇𝑧₃(𝜏)ln𝑧₂(𝜏)−𝜎₅−𝜎^′₅𝑧₃(𝜏)ln𝑧₃(𝜏)−𝑘₃𝑒₃,𝑔̇=𝜌𝑒²₁.(44) Define the Lyapunov function 1𝑉=2𝑒^𝑇𝑒+𝜎₁−𝜎^′₁²+𝜎₂−𝜎^′₂²+𝜎₃−𝜎^′₃²+𝜎₄−𝜎^′₄²+𝜎₅−𝜎^′₅²−1𝜌𝑔−𝑔₁²,(45) where 𝑔₁>0 is an initial value of 𝑔 and 𝑒=(𝑒₁,𝑒₂,𝑒₃). The time derivative of the Lyapunov function along the trajectory of the error (44) is 𝑑𝑉𝑑𝑡=𝑒^𝑇𝑒̇+𝜎̇₁𝜎₁−𝜎^′₁+𝜎̇₂𝜎₂−𝜎^′₂+𝜎̇₃𝜎₃−𝜎^′₃+𝜎̇₄𝜎₄−𝜎^′₄+𝜎̇₅𝜎₅−𝜎^′₅−1𝜌𝑔̇𝑔−𝑔₁=−𝑘₁𝑒²₁−𝑘₂𝑒²₂−𝑘₂𝑒²₃−𝑘𝑒₁𝑧₁−𝑒₁𝜎₁−𝜎^′₁ln𝑧₃(𝜏)−𝜎₂−𝜎^′₂𝑧₂(𝜏)𝑒₁𝑧₃(𝜏)+𝜎₃−𝜎^′₃𝑧₂(𝜏)𝑒₂𝜎₄−𝜎^′₄−𝑧₂(𝜏)𝑒₂ln𝑧₂(𝜏)+𝜎₅−𝜎^′₅𝑧₃(𝜏)𝑒₃ln𝑧₃(𝜏)=−𝑔₁𝑒²₁−𝑘₁𝑒²₁−𝑘₂𝑒²₂−𝑘₃𝑒³₃−𝑘𝑒₁𝑧₁=−𝑔₁+𝑘₁𝑒₁+𝑘𝑧₁2𝑔₁+𝑘₁²−𝑘²𝑧₁²𝑔₁+𝑘₁−𝑘₂𝑒²₂−𝑘₃𝑒³₃.(46) If we choose 𝑔(0)=𝑔₁>0 and under the constraints that the constant 𝑘 could be neglected because 𝑘₁ is large enough, (46) can be rewritten in the following form: 𝑑𝑉𝑔𝑑𝑡≤−₁+𝑘₁𝑒²₁−𝑘₂𝑒²₂−𝑘₃𝑒³₃𝑒=−𝑒¹₂𝑒₃𝑔₁+𝑘₁000𝑘₂000𝑘₃𝑒₁𝑒₂𝑒₃=−𝑒^𝑇𝑃𝑒.(47) Since the Lyapunov function 𝑉 is positive definite and its derivative 𝑉̇ is negative semidefinite, we cannot immediately obtain that the origin of error system (27) is asymptotically stable. In fact, as 𝑉̇≤0, then 𝑒₁,𝑒₂,𝑒₃,(𝜎₁−𝜎^′₁),(𝜎₂−𝜎^′₂),(𝜎₃−𝜎^′₃),(𝜎₄−𝜎^′₄),(𝜎₅−𝜎^′₅),(𝑔−𝑔₁)∈𝐿_∞. From the error system (44), we have 𝑒₁̇,𝑒₂̇,𝑒₃̇,𝑔̇∈𝐿_∞. Since 𝑃 is a positive-definite matrix, we have ^𝜏₀𝜆_min‖𝑒‖²𝑑𝜏≤^𝜏₀−𝑒^𝑇𝑃𝑒𝑑𝜏≤^𝜏₀−̇(0)(𝜏)(0),𝑉𝑑𝜏=𝑉−𝑉≤𝑉(48) where 𝜆_min(𝑃) is the minimum eigenvalue of positive-definite matrix 𝑃. Thus 𝑒₁̇,𝑒₂̇,𝑒₃̇∈𝐿₂. According to Barbalat’s lemma presented in Theorem 4, we have 𝑒₁(𝜏),𝑒₂(𝜏),𝑒₃(𝜏)→0 as 𝜏→∞. Therefore, the response system (36) synchronizes the drive system (35) in the sense of modified projective synchronization. This completes the proof.

It is left to show that the trajectories of the slave system are bounded under the feedback. We invoke the following.(1)Since the systems are assumed to operate in chaotic mode without feedback, their trajectories converge to compact invariant set. Let 𝑟>0 and let the closed ball 𝐵_𝑟 strictly contain such a compact set (since the chaotic trajectories are bounded, we assume that they are contained in 𝐵_𝑟); let ∞>𝑇^∗≥0 be the smallest number such that 𝑒(𝜏)∈𝐵_𝑟 for all 𝜏≥𝑇^∗. The previous development shows that 𝑉 is positive definite Lyapunov function with a negative definite derivative for any values of the states contained in a compact set. Hence, there always exist control gains such that the system under feedback is forward complete.(2)It may be shown as in [42] that the system under control is forward complete; that is, if there exists a set of initial conditions and gains such that, together, they generate solutions that tend to infinity, these solutions may unboundedly grow only in infinite time. From this, it follows that, for each 𝛿>0 and 𝑇>0, there exists 𝑀(𝛿,𝑇) such that |𝑒(0)|≤𝛿⇒|𝑒(𝜏)|≤𝑀(𝛿,𝑇), for all 𝜏∈[0,𝑇], where 𝑀 is in general a nondecreasing function of its arguments. We assume that 𝑇 is the synchronization time. Since, by assumption, the system operates in open loop for all 𝜏∈[0,𝑇] for any 𝑇>0,  |𝑒(𝜏)|≤max{𝑀(𝛿,𝑇),𝑟} for all 𝜏∈[0,𝑇] for any 𝑇>0. That is, the solutions are bounded. Note that the bound max{𝑀(𝛿,𝑇),𝑟} is independent of the gains, and, for 𝑇≥𝑇^∗, we can safely assume that max{𝑀(𝛿,𝑇),𝑟}=𝑟 and 𝛽 depends only on 𝑟; hence, point (2) of the proposition follows. If 𝑇 and 𝛿 are given and max{𝑀(𝛿,𝑇),𝑟}=𝑟, then 𝛽 depends on 𝑇 and 𝛿 and so does 𝑘₁; hence, point (1) of the proposition follows. In either case, 𝑘₁ does not depend on the initial conditions 𝑒(0), nor is |𝑒(0)|≤𝑟 required.

6. Simulations Investigations

The calculations of all equations were carried out using the fourth-order Runge-Kutta algorithm. For this numerical simulation, we assume the initial conditions: 𝑧₁(0),𝑧₂(0),𝑧₃(0)=0.5,5×10⁻¹⁰,10⁻⁴𝑠₁(0),𝑠₂(0),𝑠₃(0),𝑔(0)=3×10⁻¹⁵,10⁻⁷,3×10⁻³,0.3(49) and control gains (𝑘₁,𝑘₂,𝑘₃)=(1,1,1). As a test of verification of strategy, let us take (𝛼₁,𝛼₂,𝛼₃)=(0.1,0.1,0.1); hence the error system has the initial values (𝑒₁(0),𝑒₂(0),𝑒₃(0))=(−0.500,9.5×10⁻⁹,−10⁻⁴). Let us choose 𝜌=1 and 𝑘_𝑚=0.05. For the specified values deduced from the circuit parameters, the system (34) exhibits a chaotic behavior where the phase diagram and time histories are provided in Figure 2. Furthermore, a kind of channel distortion by an AWGN was considered and added to (25). Figure 3 displays the time history of the trajectories 𝑧₁(𝜏) and 𝑧₃(𝜏) of the master system (35) in the presence of both artificial perturbation and AWGN when no control is applied. In addition, these graphs confirm the linearity of the transformation adopted and also justify the benefits of using the type of control scheme presented in this work. As it is expected in Figure 4 the divergence of the flows (10) confirms that the system (10) is dissipative (div(𝑧(̇𝜏))<0). The robustness of adaptive synchronization was validated with both artificial perturbation and AWGN added to the controlled system. Figure 5 presents the phase portrait of system (10) in the presence of artificial perturbation and AGWN. Figure 6 shows that the estimations 𝜎^′₁,𝜎^′₂,𝜎^′₃,𝜎^′₄, and 𝜎^′₅ of unknown parameters converge to 𝜎₁,𝜎₂,𝜎₃,𝜎₄, and 𝜎₅, respectively which permits us easily to find the constants inside exponential (𝑎 and 𝑏) and other parameters of the system by using the following relations: 𝑎=1/𝜎₁,  𝑏=𝜎₃/𝜎₄,  𝑏₀=𝜎₄,  𝑐₁₁=𝜎₅, and 𝑎₁=𝜎₂. Figure 7 shows that the system is well controlled in the presence of the artificial perturbation. Figure 8 shows the time evolution of the errors 𝑒₁(𝜏),𝑒₂(𝜏), and 𝑒₃(𝜏). The dynamics of the synchronization error is defined as 𝑒_𝑞(𝜏)=𝑒²₁(𝜏)+𝑒²₂(𝜏)+𝑒²₃(𝜏), whose time evolution is shown in Figure 9. One can observe that the convergence of the systems parameters and the synchronization errors are obtained with a good accuracy despite the perturbation, which is consistent with our expectations.

(a)

(b)

Figure 2 

(a) Time histories of MCO, (b) Phase portraits (versus 𝑥, 𝑦 versus 𝑥, 𝑦 versus 𝑧).

(a)

(b)

Figure 3 

Time histories of MCO when no control is applied ((a) 𝑍1 versus time 𝑧, (b) 𝑍3 versus time).

Figure 4 

Divergence of the flow of MCO in the new space.

Figure 5 

Phase portrait of MCO in the new space in the presence of artificial pertubation.

(a)

(b)

(c)

(d)

(e)

Figure 6 

Estimated values for unknown parameters.

Figure 7 

Time evolution of adaptive gain.

(a)

(b)

(c)

Figure 8 

Synchronization errors: (a) 𝑒1(𝜏), (b) 𝑒2(𝜏), (c) 𝑒3(𝜏).

Figure 9 

Time evolution of the dynamic error.

7. Conclusion

In this paper, based on Lyapunov stability theory, theory of changing space of variables, and Barbalat’s lemma we achieved lag synchronization of two modified Colpitts oscillators. This control strategy of the MCO with uncertainties including the coefficients of nonlinear terms was obtained via adaptive control. It appears that it is possible to introduce a specific controller to attenuate any artificial perturbation on the system. The final remark is that the proposed scheme is applicable to various other dynamical systems to efficiently estimate unknown parameters which could be arguments of some other nonlinear functions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.