Physics Letters A 374 (2010) 2835–2840 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Feedback control design for Rössler and Chen chaotic systems anti-synchronization S. Hammami a,c,∗ , M. Benrejeb a , M. Feki b , P. Borne c a UR LARA Automatique, Ecole Nationale d’Ingénieurs de Tunis, BP 37, Tunis Le Belvédère, 1002, Tunisia b UR MECA, Ecole Nationale d’Ingénieurs de Sfax, BP 3038, Sfax, Tunisia c LAGIS, Ecole Centrale de Lille, BP 48, 59651 Villeneuve d’Ascq Cedex, France article info abstract Article history: Received 6 February 2010 Received in revised form 7 April 2010 Accepted 4 May 2010 Available online 7 May 2010 Communicated by A.R. Bishop Keywords: Chaotic systems Anti-synchronization Aggregation techniques Arrow form matrix Feedback stabilization The proposed anti-synchronization conditions of coupled chaotic systems are based, in this Letter, on the use of aggregation techniques for the stability study of the error dynamics. The schemes are, successfully, applied to coupled Rössler and Chen chaotic systems making the instantaneous characteristic matrix under the arrow form. Numerical simulations are performed to illustrate the efficiency of the proposed approach.  2010 Elsevier B.V. All rights reserved. 1. Introduction After the pioneering work on chaotic systems of Pecora and Carroll [1], the concept of synchronization has been extended to the generalized synchronization [2–4], the phase synchroniza- tion [2], the lag synchronization [5] and the anti-phase synchro- nization [6,7]. Anti-synchronization is a phenomenon that the state vectors of the synchronized systems have the same amplitude but opposite signs as those of the driving system. Therefore, the sum of two signals is expected to converge to zero when either anti- synchronization or anti-phase synchronization appears. Recently, several stability methods have been applied to anti-synchronize chaotic systems [8–12]. To lead stability study, one can start with a system description (respectively stability methods), then choose an adapted analysis method (respectively system description). Then, the problem of Lyapunov functions construction and the problem of determining the largest stability domain, for example, can be transformed by a suitable system description choice [13–16]. Several approaches are considered in the literature using de- composition of a large scale system into subsystems, or of the * Corresponding author at: UR LARA Automatique, Ecole Nationale d’Ingénieurs de Tunis, BP 37, Tunis Le Belvédère, 1002, Tunisia. E-mail addresses: sonia.hammami@enit.rnu.tn (S. Hammami), mohamed.benrejeb@enit.rnu.tn (M. Benrejeb), moez.feki@enig.rnu.tn (M. Feki), pierre.borne@ec-lille.fr (P. Borne). vector Lyapunov function or the vector norm taking into account specific theoretical or physical properties of the process. Based on aggregation techniques and on the use of the Borne– Gentina practical stability criterion applied to continuous systems (Appendix A) [17,18], this Letter sets out to establish a new output feedback stabilizing approach for nonlinear continuous hierarchical systems. The proposed approach, which constitutes an extension of the used state feedback stabilizing approach formulated in our previous work [4], is carried out through the determination of con- trol laws, guaranteeing the asymptotic stabilisability property by making the matrix description of the controlled system under the arrow form [16–19]. This arrow form representation was used in previous works on asymptotic and global stability of nonlinear systems (Appendix B) [16] and, recently, on chaotic systems synchronization [4]. It ap- pears very suitable for two-level hierarchical systems with many nonlinearities and can be extended without any difficulty to mul- tilevel hierarchical system description. Indeed, the main purpose in this work is to design an adaptive output feedback controller guaranteeing the asymptotic stability then the anti-synchronization of nonlinear continuous error of two identical and different chaotic systems. The Letter is organized as follows. After a brief description of the studied systems and the definition of the anti-synchronization concept, Section 2 investigates the design of the proposed anti- synchronous output feedback controller. The case of two identical Rössler systems is considered in Section 3. Section 4 deals, finally, 0375-9601/$ – see front matter  2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.05.008