Abstract
It is a challenge problem to stably control the well-known Lorenz system with uncertain parameters because of its nonlinearity and singularity. In this paper, by combining Zhang dynamics (ZD) and gradient-algorithm, a novel Zhang-Gradient (ZG) controller is designed and developed for solving the controlling problem of the uncertainty Lorenz system. In order to improve computing efficiency, the stochastic parallel gradient descent algorithm is also introduced to perform an incremental adjustment of the unknown parameters for the uncertain Lorenz system. The presented theoretical analysis in this paper shows that our such presented method could conquer the possible singularity which is a difficult problem in typical backstepping controller design. The computer simulation results exhibit that, the controlled system can be stable globally and the tracking error converges to zero asymptotically, which are further demonstrate the effectiveness and feasibility of our presented ZG controller.
Highlights
The interesting nonlinear phenomenon chaos is widely encountered in practical engineering systems
Inspired by [22], in this paper, Zhang dynamics (ZD) is firstly applied to construct three error functions related to the three Lorenz differential equations by the Lyapunov stability theorem, and gradientbased dynamics (GD) is exploited to obtain values of the controller and the system parameters in the form of u, p1, p2, and p3, respectively
The Stochastic Parallel Gradient Descent (SPGD) performs an incremental adjustment of control parameter {γj} using a realtime estimation for the gradient {Jj = ∂J /∂γj} with the replacement of J = δJ δγj in real applications [33], where δ is defined as a perturbation
Summary
The interesting nonlinear phenomenon chaos is widely encountered in practical engineering systems. I) Since the Lorenz system is a nonlinear dynamical system, the methods using linear state feedback cannot guarantee the global stability for the system in some situations [17]. Inspired by [22], in this paper, ZD is firstly applied to construct three error functions related to the three Lorenz differential equations by the Lyapunov stability theorem, and GD is exploited to obtain values of the controller and the system parameters in the form of u, p1, p2, and p3, respectively. I) Based on the Zhang dynamics and gradient-algorithm, a novel Zhang-Gradient controller is designed and developed for the uncertain well-known Lorenz chaotic system. Ii) To lower the computation complexity, a SPGD design procedure is presented to get the estimated values of the Lorenz system. Iii) Computer simulation results via an illustrative example are presented and analyzed to show the effectiveness of the presented ZG controller with the SPGD computation method
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