In this paper, a new linear feedback controller for synchronization of two identical chaotic systems in a master-slave configuration is presented. This controller requires knowing a priori Lipschitz constant of the nonlinear function of the chaotic system on its attractor. The controller development is based on an algebraic Riccati equation. If the gain matrix and the matrices of Riccati equation are selected in such a way that a unique positive definite solution is obtained for this equation, then, with respect to previous works, a stronger result can be guaranteed here: the exponential convergence to zero of the synchronization error. Additionally, the nonideal case is also studied, that is, when unmodeled dynamics and/or disturbances are present in both master system and slave system. On this new condition, the synchronization error does not converge to zero anymore. However, it is still possible to guarantee the exponential convergence to a bounded zone. Numerical simulation confirms the satisfactory performance of the suggested approach.
It is considered a controller design in order to stabilize a control system. The technique that is used for designing the controller includes a linear regulator and an asymptotical estimator which form the controller. The linear regulator is designed by state estimators feedback for symmetrical systems minimizing a quadratic performance index. Computing the gain matrix of optimal feedback is solved by Riccati's equation, whilst the gain estimator matrix is computed by making use of symmetrical systems properties. The symmetrical systems properties allow us to find the solution for Riccati's dual equation without needing to solve two equations, one for the feedback and other for the estimator, using the controllability and observability matrices can be found a symmetrical and invertible matrix which can be used for computing the Riccati's dual solution. Having the regulator and the estimator gain matrices, we proceed to compute the transfer function of the controller based on estimator. Then, we use this method to stabilize the magnetic levitation system.
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