Nonlinear Dynamics

, Volume 79, Issue 4, pp 2589–2608 | Cite as

Neural learning control of pure-feedback nonlinear systems

Original Paper


This paper is concerned with the problem of learning control for a class of pure-feedback systems with unknown non-affine terms in dynamical environments. The implicit function theorem and the mean value theorem are firstly used to transform the closed-loop system into a semi-affine form. By combining the quadratic-type Lyapunov function and the appropriate inequality technology, a concise adaptive neural control scheme is developed to simplify the system stability analysis and guarantee the convergence of the tracking error in a finite time. After the stable control design, we decompose the closed-loop system into a series of linear time-varying perturbed subsystems with the help of the linear state transformation. Using a recursive design, the partial persistent excitation (PE) condition for the radial basis function neural network is satisfied during tracking control to a recurrent reference trajectory. Under the PE condition, accurate approximations of the implicit desired control dynamics are recursively achieved in a local region along recurrent orbits of closed-loop signals. Subsequently, a neural learning control method which effectively utilizes the learned knowledge without re-adapting to the unknown system dynamics is proposed to achieve the closed-loop stability and improved control performance. Simulation studies are performed to demonstrate that the proposed learning control scheme not only can approximate accurately the implicit desired control dynamics, but also can reuse the learned knowledge to achieve the better control performance with the faster tracking convergence rate and the smaller tracking error.


Adaptive neural control  Dynamic learning Pure-feedback systems  Persistent excitation Exponential stability 



This work was partially supported by the National Natural Science Foundation of China (Nos. 61473121, 61374119, 61104108 and 61004065), the National Science Fund for Distinguished Young Scholars (No. 61225014), the Science and Technology New Star of Zhujiang, and the Fundamental Research Funds for the Central Universities.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.School of Automation Science and Engineering, Center for Control and OptimizationSouth China University of TechnologyGuangzhouChina

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