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This work deals with chaos synchronisation. One of the commonly used methods is the master-slave configuration, where the response of the chaotic receiver (slave) system must converge to the chaotic trajectory of the transmitter (master). The major problem with this configuration is the delay that occurs during the transmission of the drive signal from the master to the slave, which can degrade the synchronisation. The objective of this paper is to propose a solution to the synchronisation problem of nonlinear fractional-order chaotic systems in the presence of a transmission delay at the output and unknown disturbance. The proposed approach consists of combining a fractional high-order sliding mode observer (FHOSMO) and a predictor arranged in a cascade to compensate for the delayed transmission signal from the transmitter to the receiver. The observer permits to estimate the delayed states and the delayed total disturbance (uncertainties and external disturbances) in finite time. These estimates are injected into the predictor to obtain the estimated states at the current time. The finite time convergence of the proposed method is established. The numerical simulations illustrate the theoretical results and the performance of the proposed method. The proposed synchronisation method is compared with other synchronisation methods using the chattering free sliding mode approach such as the fractional first-order sliding mode observer using the continuous function instead of the discontinuous sign function and the fractional second-order sliding mode observer. In addition, a conventional first-order sliding mode observer with the sign function is also considered.

High-order sliding mode-based synchronisation of fractional-order chaotic systems subject to output delay and unknown disturbance