General Dynamical Networks Research Articles

Asymptotic stability of general linear dynamical networks under distributed relative-state feedback control

This paper investigates general linear dynamical networks (GLDNs), distributed relative-state feedback control, and pinning control. For symmetric GLDNs under distributed relative-state feedback control, some necessary and sufficient conditions for asymptotic stability are proposed. While for the asymmetric case, some sufficient conditions are derived. If the obtained stability conditions are not satisfied, one can design some pinning controllers to asymptotically stabilize the GLDNs. Compared with the existing results, the considered dynamical network model is more general, and the obtained theoretical results are novel.

Verification of Neural-Network Control Systems by Integrating Taylor Models and Zonotopes

We study the verification problem for closed-loop dynamical systems with neural-network controllers (NNCS). This problem is commonly reduced to computing the set of reachable states. When considering dynamical systems and neural networks in isolation, there exist precise approaches for that task based on set representations respectively called Taylor models and zonotopes. However, the combination of these approaches to NNCS is non-trivial because, when converting between the set representations, dependency information gets lost in each control cycle and the accumulated approximation error quickly renders the result useless. We present an algorithm to chain approaches based on Taylor models and zonotopes, yielding a precise reachability algorithm for NNCS. Because the algorithm only acts at the interface of the isolated approaches, it is applicable to general dynamical systems and neural networks and can benefit from future advances in these areas. Our implementation delivers state-of-the-art performance and is the first to successfully analyze all benchmark problems of an annual reachability competition for NNCS.

Explicit Characterization of Performance of a Class of Networked Linear Control Systems

We quantify performance of a class of interconnected linear dynamical networks with arbitrary nodal dynamics. It is shown that the steady-state variance of the output, which is equivalent to the squared $\mathcal {H}_2$ norm of the network, as a performance measure, can be expressed explicitly as a summation of a rational function over the Laplacian eigenvalues of the underlying graph of the network. Our results generalize the existing works on $\mathcal {H}_2$ -norm characterization for the first- and second-order consensus networks to general consensus-type linear dynamical networks with arbitrary nodal dynamics. We characterize several fundamental limits and scaling laws for the performance of this class of networks. To show the practical usefulness of our results, we discuss how to design decentralized observer-based output feedback mechanisms and how to analyze the performance of composite networks. Numerous examples support our theoretical contributions.

Topology Reconstruction of Dynamical Networks via Constrained Lyapunov Equations

The network structure (or topology) of a dynamical network is often unavailable or uncertain. Hence, we consider the problem of network reconstruction. Network reconstruction aims at inferring the topology of a dynamical network using measurements obtained from the network. In this technical note we define the notion of solvability of the network reconstruction problem. Subsequently, we provide necessary and sufficient conditions under which the network reconstruction problem is solvable. Finally, using constrained Lyapunov equations, we establish novel network reconstruction algorithms, applicable to general dynamical networks. We also provide specialized algorithms for specific network dynamics, such as the well known consensus and adjacency dynamics.

Successive lag synchronization on complex dynamical networks via delay-dependent impulsive control

This paper studies the successive lag synchronization (SLS) on complex dynamical networks by using impulsive control method. For the purpose of achieving SLS of a general dynamical network, a novel impulsive control which depends on synchronization delay is designed. Furthermore, we also discuss the SLS of an impulsive dynamical network with communication delay. Based on stability theory of differential equations, several sufficient conditions for guaranteeing the stability of SLS are derived. Finally, for verifying the obtained theoretical results, three numerical examples are performed.

Asymptotically Local Synchronization of an Interdependent Network with Two Sub-Networks and Delays

In this article, the asymptotically local synchronization for a general interdependent dynamical network with two sub-networks and delays is investigated. The coupling function under consideration in this network is continuous and on behalf of both the intra-network and inter-network. Based on the Master Stability Function method, Linearization method and Lyapunov–Krasovskii Stability Theory combing with the Delay Differential Inequality technique, some elegant strategies are established which guarantee the asymptotically exponential local synchronization for each node in the network. Finally, several numerical simulations show the effectiveness and feasibility of the proposed schemes.

Exponential cluster synchronization of hybrid-coupled impulsive delayed dynamical networks: average impulsive interval approach

In this paper, the problem of exponential cluster synchronization of general hybrid-coupled impulsive dynamical networks with internal delay and delayed coupling is investigated. A more general delayed coupling term including different transmission delay and self-feedback delay is taken into account. By using average impulsive interval approach and the analysis technique, some novel globally exponential cluster synchronization criteria are derived analytically. It is noted that the internal delay, transmission delay and self-feedback delay are all time-varying and can be different from each other, the inner connecting matrices are not demanded to be diagonal and the coupling matrices are not restricted to be symmetric or irreducible, more consistent with the realistic networks. Particularly, it is shown that the derived cluster synchronization criteria are simultaneously applicable for studying general delayed dynamical networks with synchronizing impulses or desynchronizing impulses. Numerical examples are finally given to illustrate the effectiveness of the obtained theoretical results.

Erratum to: Robust adaptive synchronization of general dynamical networks with multiple delays and uncertainties

Erratum to: Robust adaptive synchronization of general dynamical networks with multiple delays and uncertainties

Robust adaptive synchronization of general dynamical networks with multiple delays and uncertainties

In this article, a general complex dynamical network which contains multiple delays and uncertainties is introduced, which contains time-varying coupling delays, time-varying node delay, and uncertainties of both the inner- and outer-coupling matrices. A robust adaptive synchronization scheme for these general complex networks with multiple delays and uncertainties is established and raised by employing the robust adaptive control principle and the Lyapunov stability theory. We choose some suitable adaptive synchronization controllers to ensure the robust synchronization of this dynamical network. The numerical simulations of the time-delay Lorenz chaotic system as local dynamical node are provided to observe and verify the viability and productivity of the theoretical research in this paper. Compared to the achievement of previous research, the research in this paper seems quite comprehensive and universal.

Synchronization between two general complex networks with time-delay by adaptive periodically intermittent pinning control

This letter investigates the problem of synchronization between two general delayed dynamical complex networks. Based on the Lyapunov stability theory and the adaptive periodical pinning intermittent control technique, some criteria are obtained for the synchronization between two general dynamical complex networks. Moreover, two typical BA scale complex networks with time-delayed Chua oscillator are chosen to illustrate the effectiveness of the theoretical results.

Bounded synchronisation of a time-varying dynamical network with nonidentical nodes

This paper investigates the global bounded synchronisation problem of complex dynamical networks of coupled nonidentical nodes with general time-varying topology through the Lyapunov function and graph theory. Several sufficient conditions in form of scalar inequalities are established so that the global bounded synchronisation of the general dynamical network can be evaluated by the stability of a linear time-varying system and the boundedness of a nonlinear function, both associated with the dynamics of nonidentical nodes. These analytical results are simple yet generic, without assuming the symmetry coupling configuration matrix or calculating their eigenvalues. They can be used to explore synchronisation issues of various complex networks. Numerical simulations show their effectiveness.

Restrictions and stability of time-delayed dynamical networks

In this paper we present a criteria for the global stability of general time-delayed dynamical networks. We show that under our criteria a network's stability is invariant with respect to the removal of time delays and the addition of single type time delays. As modifying a network's delays can have a destabilizing effect on the system's dynamics, this introduces a new and stronger form of global stability, which we call intrinsic stability. To carry out this analysis we introduce a family of graph transformations that can be used to maintain or modify the spectral radius of a graph. We then introduce the notion of an implicit delay and show that by removing a network's implicit delays the result is a lower dimensional system, which we term a network restriction. We demonstrate that such restrictions can be used to obtain improved estimates of a network's global stability. The effectiveness of our approach is illustrated by applications to various classes of Cohen–Grossberg neural networks.

Global Bounded Synchronization of General Dynamical Networks With Nonidentical Nodes

This note addresses the problem of synchronization for general dynamical networks with nonidentical nodes. The coupling strength, outer coupling configuration and inner connection in such networks are all time varying. Neither an equilibrium for each node nor a synchronization manifold is assumed to exist. An estimate of the convergence domain for a general class of time-varying nonlinear systems is given. By introducing the average dynamics of all nodes and based on this estimate, a criterion of global synchronization in the sense of boundedness of the maximum state deviation between nodes is developed. An explicit bound of the maximum state deviation between nodes is obtained by the maximum difference between each node dynamics and the average dynamics. The proposed criterion is an extension of several related synchronization criteria for the case of identical nodes to the case of nonidentical nodes.

A general fractional-order dynamical network: Synchronization behavior and state tuning

A general fractional-order dynamical network model for synchronization behavior is proposed. Different from previous integer-order dynamical networks, the model is made up of coupled units described by fractional differential equations, where the connections between individual units are nondiffusive and nonlinear. We show that the synchronous behavior of such a network cannot only occur, but also be dramatically different from the behavior of its constituent units. In particular, we find that simple behavior can emerge as synchronized dynamics although the isolated units evolve chaotically. Conversely, individually simple units can display chaotic attractors when the network synchronizes. We also present an easily checked criterion for synchronization depending only on the eigenvalues distribution of a decomposition matrix and the fractional orders. The analytic results are complemented with numerical simulations for two networks whose nodes are governed by fractional-order Lorenz dynamics and fractional-order Rössler dynamics, respectively.

Stability of dynamical networks with non-identical nodes: A multiple [formula omitted]-Lyapunov function method

Global asymptotic stability of general dynamical networks with non-identical nodes is studied by introducing multiple V -Lyapunov functions. In such a network, the coupling strength, inner coupling matrix and outer coupling matrix are all allowed to be state-dependent and nonlinear. A stability criterion is proposed in terms of matrix norms and eigenvalues of some lower-dimensional matrices. Based on this criterion, an optimization problem is formed whose solution can be used to test global asymptotic stability. We also study the problem of how to achieve global asymptotic stability by design of controllers under the scheme of multiple V -Lyapunov functions. The control action is regarded as a re-shape of outer coupling topology and the associated controllers are designed. In particular, a method of adding or removing a certain number of links is proposed. Stability analysis of coupled non-identical Lorenz systems and a design example are also given to illustrate the proposed method.

Lyapunov formulation of ISS cyclic-small-gain in continuous-time dynamical networks

This paper provides a Lyapunov formulation of the cyclic-small-gain theorem for general dynamical networks (large-scale systems) composed of continuous-time input-to-state stable (ISS) subsystems. ISS-Lyapunov functions for continuous-time dynamical networks satisfying cyclic-small-gain conditions are constructed from the ISS-Lyapunov functions of the subsystems.

Stabilization of complex dynamical networks with noise disturbance under performance constraint

Under given performance constraint, this paper studies the stabilization problem of general dynamical network subject to noise disturbance. The newly presented dynamical network model includes both intrinsic disturbance of single node and communication noise over the network connections, which appear typically in a network environment. Single controller is pinned into one of the nodes for the exponential stabilization of dynamical network, and the prescribed performance constraint is satisfied. The reason why only one controller is valid for stabilization of dynamical network is the full utilization of network’s local connections. One important feature of this paper is the introduction of the performance constraint concept into the stabilization of complex dynamical network with intrinsic and communication noises. The derived criteria are expressed in terms of linear matrix inequalities (LMIs), which are easy to be verified by resorting to recently developed algorithm. Numerical example is utilized to illustrate the effectiveness of the proposed results.

Identifying topology of general dynamical networks from noisy data

The reverse engineering problem of identifying the topology of dynamical networks withtime varying intersystem coupling delays from noisy observation data is treated in thispaper. A three-stage approach is adopted to solve this identification problem: first,feedback control is applied to the network and it is driven to its equilibrium state; then, theinformation obtained from the observed steady state responses is explored and theidentification problem is converted into a mixed integer optimization problem; finally, aninference algorithm is applied to obtain all elements of the connectivity matrix.Several dynamical network examples are provided for illustration. Identificationresults show that the proposed method is robust to noise and computationallyefficient.

Global Synchronization of Dynamical Networks with Non-identical Nodes: a Multiple V-Lyapunov Function Method

The global synchronization problem for general dynamical networks with non-identical nodes is studied by introducing multiple V-Lyapunov functions. In such a network, the coupling strength, inner coupling matrix and outer coupling matrix are all allowed to be state-dependent and nonlinear. A global synchronization criterion is proposed in terms of matrix norms and eigenvalues of some lower-dimensional matrices. We also study the problem of how to achieve global synchronization by design of controllers under the scheme of multiple V-Lyapunov functions. The control action is regarded as a re-shape of outer coupling topology and the associated controllers are designed. In particular, a method of global synchronization by adding or removing a certain number of links is proposed.

Dynamical networks: Continuous time and general discrete time models

Dynamical networks are characterized by 1) their topology (structure of the graph of interactions among the elements of a network); 2) the interactions between the elements of the network; 3) the intrinsic (local) dynamics of the elements of the network. A general approach to studying the commulative effect of all these three factors on the evolution of networks of a very general type has been developed in [1]. Besides, in this paper there were obtained sufficient conditions for a global stability (generalized strong synchronization) of networks with an arbitrary topology and the dynamics which is a composition (action of one after another) of a local dynamics of the elements of a network and of the interactions between these elements. Here we extend the results of [1] on global stability (generalized strong synchronization) to the case of a general dynamics in discrete time dynamical networks and to general dynamical networks with continuous time.