Modified projective synchronization of unknown chaotic dissipative gyroscope systems via Gaussian radial basis adaptive variable structure control

This paper proposes the modified projective synchronization method for unknown chaotic dissipative gyroscope systems via Gaussian radial basis adaptive variable structure control. Because of the nonlinear terms of the dissipative gyroscope system, the system exhibits chaotic motions. As chaotic signals are usually broadband and noise-like, synchronized chaotic systems can be used as cipher generators for secure communication. Obviously the importance of obtaining these objectives is specified when the dynamics of the gyroscope system are unknown. In this paper, using the neural variable structure control technique, control laws are established, which guarantees the modified projective synchronization of an unknown chaotic dissipative gyroscope system. Switching surfaces are adopted to ensure the stability of the error dynamics in variable structure control. In the neural variable structure control, Gaussian radial basis functions are utilized online to estimate the system dynamic functions. Also, the adaptation laws of the online estimators are derived in the sense of Lyapunov function. Thus, the unknown chaotic gyroscope system can be guaranteed to be asymptotically stable. Also, the synchronization objectives have been achieved. The proposed method allows us to arbitrarily adjust the desired scaling by controlling the slave system. It is not necessary to calculate the Lyapunov exponents and the eigenvalues of the Jacobian matrix, which makes it simple and convenient. Also, it is a systematic procedure for modified projective synchronization of chaotic systems and it can be applied to a variety of chaotic systems no matter whether it contains external excitation or not. The designed control system is robust versus model uncertainty. Numerical simulations are presented to verify the proposed synchronization method.