This paper focus on three inherent demerits of adaptive feedback RBFNN control with lattice distribution of hidden nodes: 1) The approximation area of adaptive RBFNN is difficult to be obtained in priori; 2) Only partial persistence of excitation (PE) can be guaranteed; 3) The number of hidden nodes is the exponential growth with the increase of the dimension of the input vectors and the polynomial growth with the increase of the number of the hidden nodes in each channel which is huge especially for the high dimension of inputs of the RBFNN. Adaptive feedforward RBFNN control with lattice distribution of hidden node can improve solve the demerits 1) but just improve demerits 2) and 3) slightly. This paper proposes an adaptive feedforward RBFNN control strategy with an optimal distribution of hidden nodes. It solves the demerits 2) and 3) that the standard PE can be guaranteed and the number of hidden nodes is linear increase with the complexity of the desired state trajectory rather than the exponential growth with the increase of the dimension of the input vectors. In addition, we articulate that PID is the special case of adaptive feedforward RBFNN control for the set points tracking problem and we named the controller is enhanced PID. It is very easy tuning our algorithm which just more complex than PID slightly and the tuning experience of PID can be easily transferred to our scheme. In the case of the controller implemented by digital equipment, the control performance can equal or even better than it in model-based schemes such as computed torque control and feedforward nonlinear control after enough time to learn. Simulations results demonstrate the excellent performance of our scheme. The paper is a significant extension of deterministic learning theory.

中文翻译：

---

增强型 PID：具有最佳隐藏节点分布的机器人机械手的自适应前馈 RBF 神经网络控制

本文重点讨论了具有隐藏节点格分布的自适应反馈 RBFNN 控制的三个固有缺点：1）自适应 RBFNN 的近似区域难以先验获得；2) 只能保证部分励磁持久性（PE）；3）隐藏节点的数量随着输入向量维数的增加呈指数增长，随着每个通道中隐藏节点数量的增加呈多项式增长，特别是对于输入的高维数是巨大的RBF神经网络。具有隐藏节点格分布的自适应前馈RBFNN控制可以改善解决缺点1）但只是稍微改善缺点2）和3）。本文提出了一种具有最优隐藏节点分布的自适应前馈 RBFNN 控制策略。它解决了缺点2）和3），可以保证标准PE，隐藏节点的数量随着期望状态轨迹的复杂度线性增加，而不是随着输入向量维数的增加呈指数增长。此外，我们明确指出 PID 是针对设定点跟踪问题的自适应前馈 RBFNN 控制的特例，我们将控制器命名为增强型 PID。调整我们的算法非常容易，只是比 PID 稍微复杂一点，PID 的调整经验可以很容易地转移到我们的方案中。在控制器由数字设备实现的情况下，经过足够的学习时间后，控制性能可以与基于模型的方案（例如计算转矩控制和前馈非线性控制）相当甚至更好。仿真结果证明了我们方案的优异性能。该论文是确定性学习理论的重要延伸。