Abstract
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.
Highlights
The problem of the unidirectional synchronization of chaotic systems consists of finding an appropriate control law such that when this is applied to a system with coupled inputs called “slave” or “response,” such system follows the dynamics of an autonomous chaotic system called “master” or “drive” [1,2,3,4,5,6,7,8,9,10,11]
(6) with Lipschitz constant γ and the gain K and the matrices Λ, Q0, A0 = A − K, R = Λ, Q = γ2Λ−1 + Q0 are selected according to Lemma 4 in such a way that the matrix Riccati equation AT0 P + PA0 + PRP + Q = 0 has a unique positive definite solution P and the control law u = −Ke is applied to the slave system (7), the synchronization error e = ws −wm converges exponentially to zero
If the function f in equations (45) and (46) is locally Lipschitz on the attractor of the autonomous system (45) with Lipschitz constant γ and the gain K and the matrices Λ, Ω, Q0, A0 = A−K, R = Λ+Ω−1, Q = γ2Λ−1+Q0 are selected according to Lemma 4 in such a way that the matrix Riccati equation AT0 P1 + P1A0 + P1RP1 + Q = 0 has a unique positive definite solution P1 and the control law u = −Ke is applied to the slave system (46), the norm of the synchronization error ‖e‖ = ‖ws − wm‖ converges exponentially to a zone bounded by √Ψ/(αλmin(P1)) where α = λmin(P−11/2Q0P−11/2)
Summary
The problem of the unidirectional synchronization of chaotic systems consists of finding an appropriate control law such that when this is applied to a system with coupled inputs called “slave” or “response,” such system follows the dynamics of an autonomous chaotic system called “master” or “drive” [1,2,3,4,5,6,7,8,9,10,11]. Complexity into account the conditions on which this quadratic form is negative definite, they could guarantee the asymptotic convergence to zero This result was obtained by using only a unique control input. In all the aforementioned works, only the asymptotic convergence of the synchronization error can be guaranteed To overcome this situation and the limitations of Wang’s technique, in this paper, a new linear feedback controller based on a matrix Riccati equation is presented. This procedure requires knowing a priori Lipschitz constant of nonlinear function of the chaotic system on its attractor.
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