Comparison between pinning control of different chaotic complex dynamical networks

The stabilization properties of various typical complex dynamical networks composed of chaotic nodes are theoretically investigated and numerically simulated in detail. Some local stability properties of such pinned networks are derived and the valid stability regions are estimated based on eigenvalue analysis. Numerical simulations of such networks are given to explain why significantly less local controllers are needed by the specifically pinning scheme, which pins the most highly connected nodes in scale-free networks, than that required by the randomly pinning scheme. Also, it is explained why there is no significant difference between the two schemes for controlling random-graph networks and small-world networks.

Cluster consensus of second-order multi-agent systems via pinning control

This paper investigates the cluster consensus problem for second-order multi-agent systems by applying the pinning control method to a small collection of the agents. Consensus is attained independently for different agent clusters according to the community structure generated by the group partition of the underlying graph and sufficient conditions for both cluster and general consensus are obtained by using results from algebraic graph theory and the LaSalle Invariance Principle. Finally, some simple simulations are presented to illustrate the technique.

Disturbance rejection and H_∞ pinning control of linear complex dynamical networks

This paper concerns the disturbance rejection problem of a linear complex dynamical network subject to external disturbances. A dynamical network is said to be robust to disturbance, if the H∞ norm of its transfer function matrix from the disturbance to the performance variable is satisfactorily small. It is shown that the disturbance rejection problem of a dynamical network can be solved by analysing the H∞ control problem of a set of independent systems whose dimensions are equal to that of a single node. A counter-intuitive result is that the disturbance rejection level of the whole network with a diffusive coupling will never be better than that of an isolated node. To improve this, local feedback injections are applied to a small fraction of the nodes in the network. Some criteria for possible performance improvement are derived in terms of linear matrix inequalities. It is further demonstrated via a simulation example that one can indeed improve the disturbance rejection level of the network by pinning the nodes with higher degrees than pinning those with lower degrees.

Cost and effect of pinning control for network synchronization

The problem of pinning control for the synchronization of complex dynamical networks is discussed in this paper. A cost function of the controlled network is defined by the feedback gain and the coupling strength of the network. An interesting result is that a lower cost is achieved by using the control scheme of pinning nodes with smaller degrees. Some strict mathematical analyses are presented for achieving a lower cost in the synchronization of different star-shaped networks. Numerical simulations on some non-regular complex networks generated by the Barabasi-Albert model and various star-shaped networks are performed for verification and illustration.

Pinning control of a generalized complex dynamical network model

This paper investigates the local and global synchronization of a generalized complex dynamical network model with constant and delayed coupling. Without assuming symmetry of the couplings, we proved that a single controller can pin the generalized complex network to a homogenous solution. Some previous synchronization results are generalized. In this paper, we first discuss how to pin an array of delayed neural networks to the synchronous solution by adding only one controller. Next, by using the Lyapunov functional method, some sufficient conditions are derived for the local and global synchronization of the coupled systems. The obtained results are expressed in terms of LMIs, which can be efficiently checked by the Matlab LMI toolbox. Finally, an example is given to illustrate the theoretical results.

SYNCHRONIZATION OF SINGULAR MARKOVIAN JUMPING NEUTRAL COMPLEX DYNAMICAL NETWORKS WITH TIME-VARYING DELAYS VIA PINNING CONTROL

This article discusses the synchronization problem of singular neutral complex dynamical networks(SNCDN)with distributed delay and Markovian jump parameters via pinning control.Pinning control strategies are designed to make the singular neutral complex networks synchronized.Some delay-dependent synchronization criteria are derived in the form of linear matrix inequalities based on a modified Lyapunov-Krasovskii functional approach.By applying the Lyapunov stability theory,Jensen's inequality,Schur complement,and linear matrix inequality technique,some new delay-dependent conditions are derived to guarantee the stability of the system.Finally,numerical examples are presented to illustrate the effectiveness of the obtained results.

Exponential synchronization of coupled memristive neural networks via pinning control

This paper is concerned with the exponential synchronization problem of coupled memristive neural networks. In contrast to general neural networks, memristive neural networks exhibit state-dependent switching behaviors due to the physical properties of memristors. Under a mild topology condition, it is proved that a small fraction of controlled sub- systems can efficiently synchronize the coupled systems. The pinned subsystems are identified via a search algorithm. Moreover, the information exchange network needs not to be undirected or strongly connected. Finally, two numerical simulations are performed to verify the usefulness and effectiveness of our results.

Novel pinning control strategies for synchronisation of complex networks with nonlinear coupling dynamics

This paper considers the global stability of controlling an uncertain complex network to a homogeneous trajectory of the uncoupled system by a local pinning control strategy. Several sufficient conditions are derived to guarantee the network synchronisation by investigating the relationship among pinning synchronisation, network topology, and coupling strength. Also, some fundamental and yet challenging problems in the pinning control of complex networks are discussed： （1） what nodes should be selected as pinned candidates？ （2） How many nodes are needed to be pinned for a fixed coupling strength？ Furthermore, an adaptive pinning control scheme is developed. In order to achieve synchronisation of an uncertain complex network, the adaptive tuning strategy of either the coupling strength or the control gain is utilised. As an illustrative example, a network with the Lorenz system as node self-dynamics is simulated to verify the efficacy of theoretical results.

Pinning control of complex Lur’e networks

This paper addresses the control problem of a class of complex dynamical networks with each node being a Lur＇e system whose nonlinearity satisfies a sector condition, by applying local feedback injections to a small fraction of the nodes. The pinning control problem is reformulated in the framework of the absolute stability theory. It is shown that the global stability of the controlled network can be reduced to the test of a set of linear matrix inequalities, which in turn guarantee the absolute stability of the corresponding Lur＇e systems whose dimensions are the same as that of a single node. A circle-type criterion in the frequency domain is further presented for checking the stability of the controlled network graphically. Finally, a network of Chua＇s oscillators is provided as a simulation example to illustrate the effectiveness of the theoretical results.

Cluster synchronization in community network with nonidentical nodes via intermittent pinning control

In this paper, cluster synchronization in community network with nonidentical nodes is investigated. By combining intermittency with a pinning control scheme, some effective controllers are designed. In the control scheme, only one node in each community is controlled and coupling weights of a spanning tree in each community are enhanced. Based on the Lyapunov function method and mathematical analysis technique, two results for achieving cluster synchronization are obtained. Noticeably, by introducing an adaptive strategy, some universal adaptive intermittent pinning controllers are designed for different networks. Finally, two numerical simulations are performed to verify the correctness of the derived results.

Observer-Based Intra-Cluster Lag Consensus of Multi-Agent Systems via Intermittent Adaptive Pinning Control

The intra-cluster lag consensus means that the agents in the same cluster can achieve lag consensus asymptotically while the agents in different clusters can achieve different consensus.In this paper,the authors investigate the observer-based intra-cluster lag consensus problems of multi-agent systems(MASs)with general linear dynamics and nonlinear dynamics via intermittent adaptive pinning control.The interaction network is considered to be weakly connected,i.e.,it is not necessary to be strongly connected,in-degree balanced or contain a directed spanning tree.To realise the intra-cluster lag consensus,a class of observers is designed to estimate the states of followers.Then a class of observer-based intermittent adaptive pinning control protocols is proposed according to the difference that the agents receive information source.Moreover,the pinning gains are designed to be intermittent adaptive and have an exponential convergence rate,which will effectively reduce communication costs,avoid the pinning gains being larger than those needed in real applications and guarantee the pinning gains quickly converge to steady value.Correspondingly,some sufficient consensus criteria are derived and rigorous proofs are given based on matrix theory and Lyapunov stability theory.Finally,the effectiveness for the proposed intermittent adaptive pinning control strategy is validated by a numerical simulation.

Pinning-controlled synchronization of complex networks with bounded or unbounded synchronized regions

This paper studies pinning-controlled synchronization of complex networks with bounded or unbounded synchronized regions. To study a state-feedback pinning-controlled network with N nodes, it first converts the controlled network to an extended network of N＋1 nodes without controls. It is shown that the controlled synchronizability of the given network is determined by the real part of the smallest nonzero eigenvalue of the coupling matrix of its extended network when the synchronized region is unbounded; but it is determined by the ratio of the real parts of the largest and the smallest nonzero eigenvalues of the coupling matrix when the synchronized region is bounded. Both theoretical analysis and numerical simulation show that the portion of controlled nodes has no critical values when the synchronized region is unbounded, but it has a critical value when the synchronized region is bounded. In the former case, therefore, it is possible to control the network to achieve synchronization by pinning only one node. In the latter case, the network can achieve controlled synchronization only when the portion of controlled nodes is larger than the critical value.

Pinning Control and Controllability of Complex Dynamical Networks

In this article, the notion of pinning control for directed networks of dynamical systems is introduced, where the nodes could be either single-input single-output （SISO） or multi-input multi-output （MIMO） dynamical systems, and could be non-identical and nonlinear in general but will be specified to be identical linear time-invariant （LTI） systems here in the study of network controllability. Both state and structural controllability problems will be discussed, illustrating how the network topology, node-system dynamics, external control inputs and inner dynamical interactions altogether affect the controllability of a general complex network of LTI systems, with necessary and sufficient conditions presented for both SISO and MIMO settings. To that end, the controllability of a special temporally switching directed network of linear time-varying （LTV） node systems will be addressed, leaving some more general networks and challenging issues to the end for research outlook.

Adaptive Synchronization of Fractional Order Complex-Variable Dynamical Networks via Pinning Control

In this paper, the synchronization of fractional order complex-variable dynamical networks is studied using an adaptive pinning control strategy based on close center degree. Some effective criteria for global synchronization of fractional order complex-variable dynamical networks are derived based on the Lyapunov stability theory. From the theoretical analysis, one concludes that under appropriate conditions, the complex-variable dynamical networks can realize the global synchronization by using the proper adaptive pinning control method. Meanwhile, we succeed in solving the problem about how much coupling strength should be applied to ensure the synchronization of the fraetionla order complex networks. Therefore, compared with the existing results, the synchronization method in this paper is more general and convenient. This result extends the synchronization condition of the real-variable dynamical networks to the complex-valued field, which makes our research more praetical. Finally, two simulation examples show that the derived theoretical results are valid and the proposed adaptive pinning method is effective.

Distributed Secondary Control Strategy Based on Q-learning and Pinning Control for Droop-controlled Microgrids

A distributed secondary control(DSC) strategy that combines Q-learning and pinning control is originally proposed to achieve a fully optimal DSC for droop-controlled microgrids(MGs). It takes advantages of cross-fusion of the two algorithms to realize the high efficiency and self-adaptive control in MGs. It has the following advantages. Firstly, it adopts the advantages of reinforcement learning in autonomous learning control and intelligent decision-making, driving the action value of pinning control for feedback adaptive correction. Secondly, only a small part of points selected as pinned points needs to be controlled and pre-learned, hence the actual control problem is transformed into a synchronous tracking problem and the installation number of controllers is further reduced.Thirdly, the pinning matrix can be modified to adapt to plugand-play operation under the distributed control architecture.Finally, the effectiveness and versatility of the proposed strategy are demonstrated with a typical droop-controlled MG model.

Controlled synchronization of complex network with different kinds of nodes

In this paper, a new dynamical network model is introduced, in which the nodes of the network are different. It is shown that by the designed controllers, the state of the network can exponentially synchronize onto a homogeneous stationary state. Some criteria are derived and some examples are presented. The numerical simulations coincide with theoretical analysis.

Pinning consensus analysis of multi-agent networks with arbitrary topology

In this paper the pinning consensus of multi-agent networks with arbitrary topology is investigated. Based on the properties of M-matrix, some criteria of pinning consensus are established for the continuous multi-agent network and the results show that the pinning consensus of the dynamical system depends on the smallest real part of the eigenvalue of the matrix which is composed of the Laplacian matrix of the multi-agent network and the pinning control gains. Meanwhile, the relevant work for the discrete-time system is studied and the corresponding criterion is also obtained. Particularly, the fundamental problem of pinning consensus, that is, what kind of node should be pinned, is investigated and the positive answers to this question are presented. Finally, the correctness of our theoretical findings is demonstrated by some numerical simulated examples.

Pinning synchronization of time-varying delay coupled complex networks with time-varying delayed dynamical nodes

This paper deals with the pinning synchronization of nonlinearly coupled complex networks with time-varying coupling delays and time-varying delays in the dynamical nodes.We control a part of the nodes of the complex networks by using adaptive feedback controllers and adjusting the time-varying coupling strengths.Based on the Lyapunov-Krasovskii stability theory for functional differential equations and a linear matrix inequality（LMI）,some sufficient conditions for the synchronization are derived.A numerical simulation example is also provided to verify the correctness and the effectiveness of the proposed scheme.

Controllability robustness of complex networks

This article presents an overview on the state-of-the-art development in complex network controllability and its robustness against malicious attacks and random failures.Specifically,it first reviews the concepts of network pinning control and controllability,and then discusses the network controllability robustness against destructive attacks by means of node-and/or edge-removal.The related issue of network connectivity robustness is also discussed.To that end,it furthermore provides an brief overview on the recent development of a machine-learning approach for predicting optimal network controllability robustness,which may shed some lights on the understanding of optimal network structures for various design considerations.

Trajectory Control of Scale-Free Dynamical Networks with Exogenous Disturbances

In this paper, the trajectory control of multi-agent dynamical systems with exogenous disturbances is studied. Suppose multiple agents composing of a scale-free network topology, the performance of rejecting disturbances for the low degree node and high degree node is analyzed. Firstly, the consensus of multi-agent systems without disturbances is studied by designing a pinning control strategy on a part of agents, where this pinning control can bring multiple agents＇ states to an expected consensus track. Then, the influence of the disturbances is considered by developing disturbance observers, and disturbance observers based control （DOBC） are developed for disturbances generated by an exogenous system to estimate the disturbances. Asymptotical consensus of the multi-agent systems with disturbances under the composite controller can be achieved for scale-free network topology. Finally, by analyzing examples of multi-agent systems with scale-free network topology and exogenous disturbances, the verities of the results are proved. Under the DOBC with the designed parameters, the trajectory convergence of multi-agent systems is researched by pinning two class of the nodes. We have found that it has more stronger robustness to exogenous disturbances for the high degree node pinned than that of the low degree node pinned.