Synchronization of Complex Dynamical Networks with Nonidentical Nodes and Derivative Coupling via Distributed Adaptive Control

Adaptive synchronization control is proposed for a new complex dynamical network model with nonidentical nodes and nonderivative andderivative couplings.Thedistributed adaptive learning laws of periodically time-varying and constant parameters and distributed adaptive control are designed.The newmethod which can obtain the synchronization error of closed-loop complex network system is asymptotic convergence in the sense of square error norm. What is more, the coupling matrix is not assumed to be symmetric or irreducible. Finally, a simulation example shows the feasibility and effectiveness of the approach.


Introduction
Complex dynamical networks have become a focus issue and have attracted increasing attention in many fields.The structure of many real systems in nature can be modeled by complex networks, such as biological neural networks, ecosystems, food webs, and World Wide Web [1,2].It has been demonstrated that many cooperative behavior mechanisms and complex phenomena in nature and society have close relationships with network synchronization [3].Thus, the synchronization of complex systems has been widely studied, and many control schemes, such as adaptive control [4,5], sliding mode control [6], and state observer-based control [7], have been focused on this topic.However, many existing works on synchronization are based on the networks with same nodes.Since almost all complex dynamical networks in engineering have different nodes, it is unrealistic to assume that all network nodes are the same.For example, in a multirobot system, the robots can have distinct structures or have different parameters such as masses and inertias.With much more complicated behaviors, the synchronization schemes for complex networks with identical nodes cannot be directly applied to the dynamical networks with nonidentical nodes.Therefore, it is strongly necessary to conduct further investigation of new synchronization schemes for a complex dynamical network with nonidentical nodes.
Recently, some papers have studied the synchronization problem with nonidentical nodes.A pinning control scheme was developed to realize the cluster synchronization of community networks with nonidentical nodes [8].Moreover, other weaker forms of synchronization, namely, output synchronization and bounded synchronization, were proposed in [9,10].It is so difficult for a complex network with nonidentical nodes to achieve asymptotic synchronization that little work has been reported to deal with the problem.By constructing a common Lyapunov function for all the nodes, a synchronization criterion for a nonidentical nodes system was given in [11].Some local robust controllers were designed for the network with strictly different nodes to achieve generalized synchronization in [12].In [13], the asymptotic synchronization for complex dynamical networks with nonidentical nodes was studied.Based on solving a number of lower dimensional matrix inequalities and scalar inequalities, global synchronization criteria were given.Except the propriety of nonidentical nodes, the time-varying parameter and unknown system parameters should be considered meanwhile.Reference [14] introduced the adaptive synchronization of complex network systems with unknown time-varying nonlinear coupling.Although it considered nonlinearly parameterized complex dynamical networks with unknown time-varying parameters, the nonidentical nodes were not considered yet.
On the other hand, the aforementioned papers only considered nonderivative coupling.Taking into account the complexity of the network, such as population ecology [15] and robots in contact with rigid environments [16], there always exist delayed coupling [17], derivative coupling, and so on.In [18], the authors proposed a general complex dynamical network with non-derivative and derivative coupling.Under the designed adaptive controllers, the network system can asymptotically synchronize to a given trajectory.The paper was limited to the inner coupling matrix of derivative being identity matrix.A complex dynamic network model with derivative coupling and nonidentical nodes was studied in [19].By designing the adaptive strategy with time derivative terms, the authors have dealt with the known derivative coupling successfully.However, they did not consider nonlinear parameterized complex network with time-varying coupling strength and the unknown derivative coupling strength.
In recent years, the adaptive learning control has been proposed to deal with the unknown constant or time-varying parameters of the system in order to obtain control input [20,21].Under the designed adaptive learning control, the convergence of tracking error can be proved by using the composite energy function.Meanwhile, the sliding mode control (SMC) strategy has been successfully applied to a class of systems with nonlinearities and uncertainties [22][23][24].The advantage of using SMC method is the concepts of sliding mode surface design and equivalent control.In this paper, under the adaptive control strategy and inequality technique, we solve the proposed problem successfully.For the nonlinearities function, further work can focus on SMC method.
Motivated by the above discussion, the synchronization problem via new distributed adaptive control is proposed for nonlinear parametric complex dynamical networks with derivative coupling and nonidentical nodes.The main difficulty is how to deal with the unknown derivative coupling.Using the adaptive control and skill of inequality zoom, we find a way to overcome this challenge.By designing the distributed adaptive laws, the effect of derivative coupling can be eliminated.Based on a Lyapunov-Krasovskii-like composite energy function (CEF) [25,26], the adaptive asymptotical synchronization is obtained for nonlinearly parameterized complex dynamical networks with nonidentical nodes and derivative coupling.The theoretical analysis shows that the designed effective distributed adaptive strategy guarantees that the system asymptotically stable in the  2  norm sense and all closed-loop signals are bounded.And it is worth mentioning that the coupling configuration matrix may be arbitrary matrix with appropriate dimensions [27].
The rest of the paper is organized as follows.The problem formulation and preliminaries are given in Section 2. In Section 3, adaptive synchronization scheme of the complex dynamical networks is presented.In Section 4, a simulation example is provided to illustrate the effectiveness of the proposed controller.Finally, conclusion is given in Section 5.

Problem Statement and Preliminaries
A complex dynamical network consisting of  nonidentical nodes is described as ẋ  () =   (,   ()) where Remark 1.In this paper, we assume that the strength of matrix Γ is unknown, which is more general than the inner coupling matrices assumed to be diagonal matrix in [18].As the authors in papers [19] and [21] only consider the known coupling matrix strength, it should be pointed out that the unknown derivative coupling consists of more information than the dynamical model in [19,21].The coupling configuration matrices  and  need not be symmetric or irreducible.Thus, the results can be suitable to more complex dynamical networks.
Let () ∈   be a solution of the dynamics of the isolated node to which all   () are expected to synchronize.
Since the row sum of coupling matrix is zero, we have the following dynamical error equation: Our objective is to design adaptive control laws   () such that the synchronization errors converge to zero in the norm sense.
In order to derive the main results, some assumptions and lemmas are introduced as follows.
Here ℎ(⋅, ⋅) is a known nonnegative continuous function with constant upper bounded , and (⋅) is an unknown nonnegative continuous function.
Remark 3.According to separation principle [28], Assumption 2 can be easily satisfied.Under the assumption, we can "separate" the parameters from the nonlinear function.For the unknown parameters, we can solve the problem using adaptive method.
Assumption 4 (Lipschitz condition).For system (1), there exist positive constants   > 0,  = 1, . . .,  such that Assumption 5.In network (1),   () is an unknown timevarying function with a known period ; thus, (  ()) is also a periodic function with the same period .Suppose where   () is an unknown continuous function with a known period  and   is an unknown constant parameter.
Assumption 7. Assume that the state and the state derivative of system (1) are measurable.
Remark 8.This assumption is necessary to design controller and adaptive laws.Assumption 7 seems to be restrictive.The observer for state derivative will be considered in the near future.

Adaptive Synchronization of the Complex Dynamical Networks
In this section, adaptive controller is designed to make the states of (1) synchronize to the trajectory (), and some sufficient conditions are given to ensure the asymptotic stability of the synchronization process.
For network (1), we design the controller   () ∈   as follows: Remark 11.The main design difficulty is how to deal with the unknown derivative coupling and Γ.In this paper, we use the known upper bound to magnify the inner coupling matrix.Thus, the controller is related to every element of the synchronization error vector.Under the new controller, we can deal with the derivative terms easily.
The constant parameter distributed update laws are designed as follows: The time-varying distributed periodic adaptive learning law is designed as where   ,   ,   are positive constants and  0 () is a continuous and strictly increasing nonnegative function which satisfies  0 (0) = 0,  0 () =   , which ensure φ () is continuous when  = ,  = 1, 2, . . ., .
The following theorem will give a sufficient condition of asymptotical synchronization for the controlled network (3).
Theorem 12.Under Assumptions 2-6, the control law (10) with the distributed adaptive learning laws (12)-( 14) guarantees that the controlled network (4) achieves asymptotic synchronization in the  2  norm sense; that is to say, Moreover, all closed-loop signals are bounded.
Proof.Define a Lyapunov-Krasovskii function as follows: for where and  is a sufficiently large positive constant which will be determined later.
It should be noted that we define from the distributed adaptive law ( 14), and we get Firstly, the finiteness property of () for the period [0, ) is given.Consider (4) and the proposed distributed control laws ( 12)-( 14), it can be seen that the right-hand side of ( 4) is continuous with respect to all arguments.According to the existence theorem of differential equation, (4) has unique solution in the interval [0,  1 ) ⊂ [0,), with 0 <  1 ≤ .This can guarantee the boundedness of () over [0,  1 ).Therefore, we need only focus on the interval [ 1 , ).

Mathematical Problems in Engineering
We can choose the appropriate  such that According to (31), we have Since   () is continuous and periodic, the boundedness can be obtained.The boundedness of   () leads to the boundedness of V().For ( 1 ) is bounded, the finiteness of () is obvious, for all  ∈ [0, ).
Next, we will prove the asymptotical convergence of ().

Conclusion
In this paper, a new distributed adaptive learning control method is applied to the synchronization of complex dynamical networks with nonidentical nodes, nonlinear nonderivative coupling, and derivative coupling.The coupling matrix is not assumed to be symmetric or irreducible.By combining inequality techniques and the parameter separation, introducing the composite energy function, the convergence of the tracking error and the boundedness of the system signals are derived.Simulation results demonstrate the effectiveness of the proposed control method.Future effort is needed to design the observer for state derivative.