Network Structure Identification Based on Measured Output Data Using Koopman Operators

This paper considers the identification problem of network structures of interconnected dynamical systems using measured output data. In particular, we propose an identification method based on the measured output data of each node in the network whose dynamic is unknown. The proposed identification method consists of three steps: we first consider the outputs of the nodes to be all the states of the dynamics of the nodes, and the unmeasurable hidden states to be dynamical inputs with unknown dynamics. In the second step, we define the dynamical inputs as new variables and identify the dynamics of the network system with measured output data using Koopman operators. Finally, we extract the network structure from the identified dynamics as the information transmitted via the network. We show that the identified coupling functions, which represent the network structures, are actually projections of the dynamical inputs onto the space spanned by some observable functions. Numerical examples illustrate the validity of the obtained results

A synchronization condition for coupled nonlinear systems with time-delay: A frequency domain approach

This paper considers the synchronization problem for nonlinear systems with time-delay couplings. We assume that the error dynamics can be rewritten as a feedback connection of a linear delay system with multiple inputs and outputs and nonlinear elements which are decentralized and satisfy a sector condition. Then, we derive a synchronization condition for time-delay coupled systems by applying the multivariable circle criterion. Unlike the conventional synchronization criteria, the derived criterion is based on a frequency-domain stability condition and avoids the use of the Lyapunov Krasovskii approach. As a result, the condition based on the circle criterion does not contain the conservativeness caused by the LyapunovKrasovskii approach. The effectiveness of the proposed criterion is shown by examples of coupled Chua systems with delay coupling

Delay-Independent Synchronization and Network Topology of Systems with Transmission Delay Couplings

We investigate the relationship between graph topology and delay-independent synchronization occuring in networks of the identical nonlinear systems with transmission delays. In this paper, we show that if networks contain a cycle subgraph of an odd number of nodes, partial synchronization corresponding to the equitable graph partition with the fewest cells occurs for a sufficiently large coupling strength regardless of the length of time-delay. The validity of the obtained results are examined through numerical simulations of Hindmarsh-Rose neuron systems networks

Synchronization in networks of chaotic systems with time-delay coupling

In this paper, we consider synchronization of N identical nonlinear systems unidirectionally or bidirectionally coupled with time delay. First we show, using the small-gain theorem, that trajectories of coupled strictly semi-passive systems converge to a bounded region. Next, we consider the network structure under which the synchronization error dynamics has a trivial solution at zero and derive a necessary condition for synchronization with respect to the network structure. Using these facts, we then derive sufficient conditions for synchronization of the systems in terms of linear matrix inequalities via the Lyapunov-Krasovskii functional approach. The obtained results are illustrated on networks of Lorentz systems with coupling delay

Robust Anticipating Synchronization of Nonlinear Systems

This paper considers the state prediction of chaotic behaviour using a master-slave synchronization scheme. We have already derived a sufficient condition for perfect state prediction of the master system via a time-delayed output signal of the slave system. In this paper, we consider the robust convergency of the prediction error of the anticipating synchronization of nonlinear systems with model uncertainties. The effectiveness of the obtained result is illustrated by a simulation example

Prediction of chaotic behavior

This paper considers the prediction of chaotic behavior using a master-slave synchronization scheme. Based on the stability theory for retarded systems using a Lyapunov-Krasovskii functional, we derive a sufficient condition for perfect state prediction of the master system via a time-delayed output signal of the slave system. The obtained result is based on the delay-dependent stability of time-delay systems. In addition, we derive an upper bound of the admissible time delay by using linear matrix inequality techniques. Finally, we show the effectiveness of the proposed predictor by two numerical examples

State prediction based on synchronization and an application

This paper proposes a controller design method for nonlinear systems with time-delay. The method consists of a state feedback and a state predictor based on synchronization of coupled systems. We consider the robust convergency of the prediction error under perturbation and model uncertainties from a practical viewpoint. Then combining the state predictor with a static feedback, we propose a predictorbased control of nonlinear systems with time-delay in the input. The effectiveness of the proposed method is illustrated by a numerical simulation