# Bearing Fault Identification Using Machine Learning and Adaptive Cascade Fault Observer
## 1 Introduction
The (machine/deep) learning techniques can be used for fault feature extraction and including convolutional neural networks [20; 21; 22; 23; 24; 25; 26] to autonomously extract the features, generative adversarial networks [27] to generate the new signals that resemble the original ones, different types of autoencoders [28; 29] for latent coding for signal reconstruction; generation; compression; anomaly detection, and deep neural network (DNN) [30] to increase the performance of classification accuracy in high-dimensional and uncertain input data. The second scenario is the modern control-based algorithm for fault feature identification. The observation-based approach is one of the powerful techniques in the family of modern control algorithms and can be classified into two main categories: linear and nonlinear observers. The modern control procedures using linear observers have been used in real industries. The main issues of these techniques are robustness and reliability. To address these issues, two different scenarios have been defined by researchers: nonlinear-based observers that have the challenge of complexities and hybrid approaches. Various kinds of hybrid approaches have been reported in [31; 32; 33; 34]. The first step to develop a hybrid observer is a function approximation [35]. Although function approximation using a mathematical approach is reliable, it has two important problems: high complexity and low accuracy in uncertain conditions [36; 37; 38]. System identification techniques are the next scenario for function approximation. Several system identification techniques have been used for function approximation such as auto-regressive with external inputs (ARX), ARX-Laguerre, and intelligent-based ARX-Laguerre techniques [35; 36; 39]. To estimate the different classes of signals using observers, diverse methods have been introduced such as sliding mode, feedback linearization, backstepping, fuzzy, and proportional-integral (PI) observers [40; 41; 42]. High reliability and robustness are the main characteristics of sliding mode observer, but the most important negative characteristics of this technique are high-frequency fluctuation and complexity [41; 43]. The second scenario to estimate the unknown signal is the feedback linearization observer. A lack of robustness and complexity are the main negative characteristics of this technique [44]. To address the issues of complexity of implementation for sliding mode and feedback linearization observers, the PI observer was developed. Implementing this technique is simple, but the main drawbacks of this technique are estimation accuracy and resistance, especially when the signal is non-stationary. The ARX-Laguerre PI observer technique was recommended in [36; 39; 45] to improve the accuracy estimation. The extended technique has been recommended to solve the challenge of robustness in the ARX-Laguerre PI observer [39]. The ARX-Laguerre procedure does not provide a favorable result when dealing with the complex, non-stationary, and nonlinear faults that occur in rotating machinery. In this work, this issue is addressed by proposing a fuzzy orthonormal regressive technique. After approximating the normal signal of bearing using the fuzzy orthonormal regressive, the adaptive cascade observer is developed in four steps. First, the linear observation technique using a proportional-integral (PI) observer with the fuzzy orthonormal regressive signal approximation is developed. In the second step, to increase the power of uncertainties rejection in the PI observer, the structure procedure is used serially. Next, the fuzzy like observer is selected to increase the accuracy of structure PI observer. Moreover, the adaptive technique is used to develop the reliability of the cascade (fuzzy-structure PI) observer. Therefore, in this work, an adaptive cascade observer is recommended for highly accurate signal estimation.