Nonlinear Dyn DOI 10.1007/s11071-017-3400-x ORIGINAL PAPER Fixed-time sliding mode controller design for synchronization of complex dynamical networks Alireza Khanzadeh · Mahdi Pourgholi Received: 30 July 2016 / Accepted: 6 February 2017 © Springer Science+Business Media Dordrecht 2017 Abstract This paper investigates the fixed-time syn- chronization of complex dynamical networks with non- identical nodes in the presence of bounded uncertain- ties and disturbances using sliding mode control tech- nique. Firstly, a novel sliding surface is introduced and fixed-time stability of the sliding mode dynam- ics is proven by benefiting from Gudermannian func- tion. Then, a novel sliding mode controller is proposed whereby fixed-time stability of the reaching mode is guaranteed. The outstanding feature of the proposed controller is that fixed convergence times of reaching and sliding modes are independent design parameters explicitly existing in the control law. This allows us not only to set the reaching time of reaching mode and settling time of sliding mode at any desired values in advance but also to adjust them independent of each other and in the most straightforward possible way. Finally, simulation results are reported in order to show the effectiveness of the proposed controller. Keywords Complex dynamical networks · Fixed-time stability · Sliding mode control · Sliding surface · Synchronization A. Khanzadeh · M. Pourgholi (B) Faculty of Electrical Engineering, Shahid Beheshti University, P.O. Box 1658953571, Tehran, Iran e-mail: m_pourgholi@sbu.ac.ir 1 Introduction In the last decades and especially in the recent years, considerable attention has been given to the complex networks. This can be attributed to its real-world and broad applications in different fields such as ecosys- tems, secure communication, biology systems, traffic and electrical power grids. Among different research areas of the complex networks, the majority of stud- ies have been devoted to the synchronization of the complex networks. The aim of synchronization is to force all the coupled nodes of the network to reach a common state as the network evolves. So far, dif- ferent types of synchronization of complex dynamical networks with different topologies and coupling char- acteristics have been achieved by employing different control techniques such as output-feedback control [1], state-feedback control [2–4], intermittent control [5], pinning control [6–8], adaptive control [9–11], impul- sive control [12–14] and sliding mode control [15–17]. These works guarantee the asymptotical stability of the synchronization process. This means that the synchro- nization is realized when time goes to infinity, whereas, in practical engineering situations, it is desirable that the synchronization of complex network be accom- plished within a finite time. Based on the benchmark work of Bhat and Bernstein in [18], finite-time synchronization in [19–33] has been obtained. Finite-time combination–combination syn- chronization of chaotic systems has been carried out in [19]. Sun et al. in [20] have designed a nonsingular 123