Online gradient descent method has been widely applied for parameter learning in neuro-fuzzy systems. The success of the application relies on the convergence of the learning procedure. However, there barely have been convergence analyses on the online learning procedure for neuro-fuzzy systems. In this paper, an online gradient learning algorithm with adaptive learning rate is proposed to identify the parameters of the neuro-fuzzy systems representing the Mamdani fuzzy model with Gaussian fuzzy sets. We take the reciprocals of the variances of the Gaussian membership functions, rather than the variances themselves, as independent variables when computing the gradient with respect to the variance parameters. Subsequently, oscillation of the gradient value in the learning process can be avoided. Furthermore, some convergence results for this online learning scheme are studied. Finally, three numerical examples are provided to illustrate the performance of the proposed algorithm.

在线梯度下降方法已广泛应用于神经模糊系统中的参数学习。应用程序的成功依赖于学习过程的收敛。然而，几乎没有关于神经模糊系统的在线学习过程的收敛分析。在本文中，提出了一种具有自适应学习率的在线梯度学习算法来识别代表具有高斯模糊集的 Mamdani 模糊模型的神经模糊系统的参数。在计算关于方差参数的梯度时，我们将高斯隶属函数的方差的倒数而不是方差本身作为自变量。随后，可以避免学习过程中梯度值的振荡。此外，研究了该在线学习方案的一些收敛结果。最后，提供了三个数值例子来说明所提出算法的性能。