Chaotic system synchronization with an unknown master model using a hybrid HOD active control approach q Shengzhi Du a,b, * , Barend J. van Wyk b , Guoyuan Qi b , Chunling Tu b a Department of EAD, ICT Faculty, Tshwane University of Technology, Pretoria 0001, South Africa b French South Africa Technical Institute of Electronics (F’SATIE), Tshwane University of Technology, Pretoria 0001, South Africa article info Article history: Accepted 25 March 2009 abstract In this paper, a hybrid method using active control and a High Order Differentiator (HOD) methodology is proposed to synchronize chaotic systems. Compared to some traditional active control methods, this new method can synchronize chaotic systems where only out- put states of the master system are available, i.e. the system is considered a black box. The HOD is used to estimate the derivative information of the master system followed by an active control methodology relying on HOD information. The Qi hyperchaotic system is used to verify the performance of this hybrid method. The proposed method is also com- pared to some traditional methods. Experimental results show that the proposed method has high synchronization precision and speed and is robust against uncertainties in the master system. The circus implements of the proposed synchronizing scheme are included in this paper. The simulation results show the feasibility of the proposed scheme. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Synchronization, aiming at making two or more chaotic systems achieve equal or approximative states, is valuable both in theory and in practice [1]. Many chaotic system synchronization methods that have been proposed recently include linear [2,3] and nonlinear [4–7] methods to control and synchronize various chaos systems [8–11]. Since active control strategies using the Lyapunov function in their design exhibit good stability, they draw much atten- tion [4,11–14]. Unfortunately, most results on active control are based on the assumption that the model for the master cha- otic system is known, i.e. a white or grey box model [11,13]. In this paper this is not the case since a black box model for the master system is assumed. By combining active control strategies and a High Order Differentiator (HOD) methodology [3,15], a synchronizing method allowing the master chaotic system to be unknown, is proposed. The HOD is used to estimate the derivative information of the master system which is necessary to implement an active control strategy. Related active con- trol methods where the master system is assumed known are compared to the black box method proposed in this paper. For numerical experiments, the Qi hyperchaotic system [16] is used as the master system. Four cases of master-slave sys- tem pairs are considered to test the performance of this hybrid method including (a) identical structure and parameters, (b) identical structure and different parameters, (c) different structures, and (d) uncertainty present in the master system. Each case is divided into two subcases to compare traditional methods with the one proposed, one assuming the master system as a white box and the other one assuming the master system as a black box. To verify the practical feasibility of the proposed methodology an experiment was performed where the master and slave systems are both Qi hyperchaoic systems, but with different parameters. Some corrections to the original circuit in the 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.101 q This work is supported by Research Funding of TUT, grants from the National Scientific Foundation of China (10772135, 60774088), the Scientific Foundation of Tianjin City, PR China (07JCYBJC05800), the Scientific and Technological Research Foundation of Ministry of Education, PR China (207005). * Corresponding author. Address: Department of EAD, ICT Faculty, Tshwane University of Technology, Pretoria 0001, South Africa. E-mail address: dushengzhi@gmail.com (S. Du). Chaos, Solitons and Fractals 42 (2009) 1900–1913 Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos