Synchronisation of Nonlinear Systems by Bidirectional Coupling with Time-varying Delay

This paper considers the synchronization problem for coupled nonlinear systems with time-varying delay. In previous work, we have already derived a sufficient condition for synchronisation and boundedness of two identical strictly semi-passive systems coupled using state feedback with time-delay. This method, however, requires that all states of each system are coupled to the other systems and the coupling delay is constant. In this paper we extend the conditions to identical systems coupled using output feedback with time-varying delay where a bound on the length of the delay and an upper bound of the time-derivative is known. Firstly, we show, using the small-gain theorem, that the trajectories of coupled strictly semi-dissipative systems converge to a bounded region. Then we derive a sufficient condition for synchronisation of the systems coupled with time-varying delay by using a delay range dependent on the stability criterion.