Optimal Backstepping Control for Genesio–Tesi chaotic system Using Genetic Algorithm Mohammad Reza Modabbernia † , Ali Reza Sahab ‡ , Masoud Taleb Ziabari + And Seyed Amin Sadjadi Alamdari † Department of Electronics, Chamran Community Collage of Technology, Rasht , Iran. ‡ Faculty of Engineering, Electrical Group, Islamic Azad University, Lahijan Branch, Iran. + Faculty of Engineering, Mechatronic Group, Islamic Azad University, Qazvin Branch, Iran. Faculty of Engineering, Mechatronic Group, Islamic Azad University, Qazvin Branch, Iran. Email: m_modabbernia@afr.ac.ir, Ali.reza.sahab@gmail.com, m.t.ziabari@gmail.com, asadjadi@gmail.com Abstract– This paper has presented chaos synchronization in the Genesio-Tesi using the backstepping approach. Backstepping approach consists of parameters which accept positive values. The parameters are usually chosen optionally. The system response is different for each value. It is necessary to select proper parameters to obtain a good response because the improper selection of the parameters leads to inappropriate responses or even to instability of system. Genetic algorithm can select appropriate and optimal values for the parameters. GA by minimizing the fitness function can find the optimal values for the parameters. This selected fitness function is for minimizing the least square error. Fitness function forces the system error to decline to zero rapidly that causes the system to have a short and optimal setting time. Fitness function also makes an optimal controller and causes overshoot to reach its minimum value. This hybrid makes an optimal backstepping controller. 1. Introduction Chaos is a very interesting nonlinear phenomenon and has applications in many areas. One of the important problems in chaotic systems is synchronization. A robust adaptive PID controller for chaotic systems had been presented in [1]. Linear feedback for controlling chaos and Routh–Hurwitz criteria based on stability analysis has been done in [2]. Chaos suppression of Genesio system is achieved to use adaptive feedback linearization-based controller in [3]. Exponential Synchronization in the Genesio Tesi via a novel feedback control has been presented in [4]. Considerable effort has been also done to design control systems using feedback linearization and backstepping design technique for deterministic as well as uncertain chaotic systems [5-10]. Synchronization in the Genesio Tesi via Backstepping Approach has been presented in [11]. Backstepping design based on synchronization of two Genesio chaotic systems is proposed in [12]. Genetic algorithms(GAs) have been extensively applied to the off-line design of controllers [13]. Until now for controlling synchronization, Genesio-Tesi chaos has been used. In many of these methods, the designed controller has high overshoot and the system is reached stability after a long time. In some other controllers, the system is reached stability in appropriate time, but the system error is too much for a while. In the backstepping controllers, the parameters of controller are chosen arbitrarily. In these controllers if we change some of these values, the system will be led to instability and it won’t respond well. In these controllers, the system’s behavior may not be good and they may have high overshoot. In some other controllers, the system has mush setting time and they may have oscillation behavior and show bad errors. The paper is organized as follows: Section 2 describes Genesio-Tesi chaotic system. In section 3, a backstepping controller for synchronization is designed. Section 4 describes GA and algorithm used here. In section 5 a backstepping controller is designed for step tracking. Section 6 provides the conclusion. 2. Genesio-Tesi Chaotic System Genesio–Tesi chaotic system can be represented by following set of nonlinear differential equations [14]: ku dx cx bx ax x x x x x + + − − − = = = 2 1 1 2 3 3 3 2 2 1 (1) where 2 1 , x x and 3 x are state variables, and b a , and c are positive real constants satisfying c ab < . For instance, the system is chaotic for the parameters 2 . 1 = a , 92 . 2 = b and 6 = c . Here, b a , and c are linear parameters and d is nonlinear parameter which is taken as one without loss of generality. Constant scalar k is assumed to be known and u is the control input to the model. The initial condition for the states is taken as T 0.2] [0.1;-0.2; x(0) = . Before controlling (u=0) the nonlinear Genesio-Tesi given by equation (1) exhibits varieties of dynamical behavior including chaotic motion - displayed in Figure 1. 2010 International Symposium on Nonlinear Theory and its Applications NOLTA2010, Krakow, Poland, September 5-8, 2010 - 450 -