Evolving fuzzy systems (EFSs) are now well developed and widely used, thanks to their ability to self-adapt both their structures and parameters online. Since the concept was first introduced two decades ago, many different types of EFSs have been successfully implemented. However, there are only very few works considering the stability of the EFSs, and these studies were limited to certain types of membership functions with specifically predefined parameters, which largely increases the complexity of the learning process. At the same time, stability analysis is of paramount importance for control applications and provides the theoretical guarantees for the convergence of the learning algorithms. In this paper, we introduce the stability proof of a class of EFSs based on data clouds, which are grounded at the AnYa type fuzzy systems and the recently introduced empirical data analytics (EDA) methodological framework. By employing data clouds, the class of EFSs of AnYa type considered in this paper avoids the traditional way of defining membership functions for each input variable in an explicit manner and its learning process is entirely data driven. The stability of the considered EFS of AnYa type is proven through the Lyapunov theory, and the proof of stability shows that the average identification error converges to a small neighborhood of zero. Although, the stability proof presented in this paper is specially elaborated for the considered EFS, it is also applicable to general EFSs. The proposed method is illustrated with Box–Jenkins gas furnace problem, one nonlinear system identification problem, Mackey–Glass time series prediction problem, eight real-world benchmark regression problems as well as a high-frequency trading prediction problem. Compared with other EFSs, the numerical examples show that the considered EFS in this paper provides guaranteed stability as well as a better approximation accuracy.

中文翻译：

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基于数据云的演化模糊系统的稳定性

演化模糊系统 (EFS) 现在得到了很好的发展和广泛使用，这要归功于它们能够在线自适应其结构和参数。自 20 年前首次引入该概念以来，已成功实施了许多不同类型的 EFS。然而，考虑到 EFS 稳定性的工作很少，而且这些研究仅限于具有特定预定义参数的某些类型的隶属函数，这在很大程度上增加了学习过程的复杂性。同时，稳定性分析对于控制应用至关重要，为学习算法的收敛性提供了理论保证。在本文中，我们介绍了一类基于数据云的 EFS 的稳定性证明，以 AnYa 型模糊系统和最近引入的经验数据分析 (EDA) 方法论框架为基础。通过使用数据云，本文考虑的 AnYa 类型的 EFS 类避免了以显式方式为每个输入变量定义隶属函数的传统方式，其学习过程完全是数据驱动的。通过李雅普诺夫理论证明了所考虑的 AnYa 类型 EFS 的稳定性，稳定性证明表明平均识别误差收敛到零的小邻域。尽管本文中提出的稳定性证明是专门针对所考虑的 EFS 进行的，但它也适用于一般的 EFS。所提出的方法用 Box-Jenkins 煤气炉问题来说明，这是一个非线性系统识别问题，Mackey-Glass 时间序列预测问题、八个真实世界基准回归问题以及一个高频交易预测问题。与其他 EFS 相比，数值例子表明，本文所考虑的 EFS 提供了有保证的稳定性以及更好的逼近精度。