By using the radial basis function neural network (RBF NN), this paper presents tracking control problem of the nonlinear systems with the input time delay, in which the unknown continuous functions may be nonlinearly parameterized. Different from the existing results which deal with the nonlinearly parameterized functions by using the separation principle, in this paper, the nonlinearly parameterized functions are lumped into the continuous functions, and then, the neural networks (NNs) are applied to approximate them. Moreover, through a state transformation the system can be easily transformed into a system without the input time delay. Finally, based on the minimal learning parameters (MLP) algorithm and the adaptive backstepping dynamic surface control (DSC) technique, a new adaptive NN backstepping control scheme is developed, and only two parameters need to be adjusted online in the controller design procedure. Thus, the proposed control method cannot only overcome the problem of “explosion of complexity” inherently existing in traditional backstepping design methods, but also reduce the computational burden greatly. It is proven that the proposed design method is able to guarantee that all the signals in the closed-loop system are bounded and the tracking error can converge to a small neighborhood of the origin with an appropriate choice of design parameters. Finally, three examples are used to show the effectiveness of the proposed approach.

通过使用径向基函数神经网络（RBF NN），提出了具有输入时滞的非线性系统的跟踪控制问题，其中未知连续函数可以被非线性参数化。与使用分离原理处理非线性参数化函数的现有结果不同，本文将非线性参数化函数集中到连续函数中，然后应用神经网络对其进行逼近。此外，通过状态转换，可以轻松地将系统转换为没有输入时间延迟的系统。最后，基于最小学习参数（MLP）算法和自适应反步动态表面控制（DSC）技术，开发了一种新的自适应NN反步控制方案，在控制器设计过程中，仅需要在线调整两个参数。因此，提出的控制方法不仅克服了传统反推设计方法固有的“复杂性爆炸”问题，而且大大减轻了计算量。事实证明，所提出的设计方法能够保证在闭环系统所有的信号都是有界和跟踪误差可收敛到原点的一个小邻区的设计参数的适当选择。最后，通过三个例子说明了该方法的有效性。提出的控制方法不仅克服了传统反推设计方法固有的“复杂性爆炸”问题，而且大大减轻了计算量。事实证明，所提出的设计方法能够保证在闭环系统所有的信号都是有界和跟踪误差可收敛到原点的一个小邻区的设计参数的适当选择。最后，通过三个例子说明了该方法的有效性。提出的控制方法不仅克服了传统反推设计方法固有的“复杂性爆炸”问题，而且大大减轻了计算量。事实证明，所提出的设计方法能够保证在闭环系统所有的信号都是有界和跟踪误差可收敛到原点的一个小邻区的设计参数的适当选择。最后，通过三个例子说明了该方法的有效性。