# Bearing Fault Identification Using Machine Learning and Adaptive Cascade Fault Observer
## 1 Introduction
After approximating the normal signal function using a fuzzy orthonormal regressive method and estimating the signals using an adaptive cascade observer, the residual signals are generated and faults can be classified. Moreover, the residual signals are calculated by the difference between various conditions of the original signals and estimated signals obtained using an adaptive cascade observer. Since the adaptive cascade observer is tuned for working in the normal condition, the estimated signal is generated by the proposed adaptive cascade observer with minimum error for normal states. However, in the abnormal condition, the accuracy of signal estimation is reduced. Therefore, the difference between original signal and estimated signal in the abnormal condition will increase. In addition, the residual signals can be used as high discriminative features for fault detection and diagnosis of bearing. These residual signals are used at the next step as the input for machine learning technique to perform fault identification and the values of the residual of signals for adaptive cascade observer. In this work, we employ a machine learning technique for classification of the faults using the support vector machines (SVM) algorithm [46] to complete the proposed hybrid adaptive cascade fault diagnosis method. The SVM is known as a robust machine learning algorithm that is insensitive to the curse of dimensionality problem [47]. One of the main advantages of the SVM is that it can be efficiently applied for classification of both linear and nonlinear-separable types of datasets which is possible due to the availability of different types of kernels, such as linear, polynomial, and radial basis function kernels [48]. However, despite its flexibility, several challenges in this classifier should be addressed to obtain the best possible performance on the target task. Specifically, the first challenge is the selection of the kernel function itself. This is an important step since it directly affects the performance of the classifier when applied to the specific dataset. The kernel selection dilemma can be resolved by applying the prior knowledge about the data being analyzed (i.e., whether it is linearly separable or not) or the kernel can be selected experimentally by trial and error. The second challenge, which is also closely related to the first one, is the selection of the hyperparameter values which is dependent on the kernel type. In this paper, the linear kernel was selected for the SVM classifier experimentally because it demonstrated the most accurate separation of the features belonging to different classes in the dataset used. The conventional grid search algorithm was applied to fine-tune its hyperparameter value (i.e., the maximum distance for the boundary). Figure 1 illustrates the block diagram of the adaptive cascade observer with the SVM method for fault diagnosis in rotary machinery. According to this figure, this method has three main blocks: signal approximation using fuzzy orthonormal regressive (FOR), signal estimation using a FOR-adaptive cascade observer (FOR-ACO), and fault decision using SVM. For signal approximation, in the first step, a regressive technique is proposed. To increase the strength of resistance to disturbances in the regressive technique, the orthonormal regressive is used in the next step. Besides, to increase the accuracy of the signal approximation, the fuzzy orthonormal regressive technique (FOR) is developed. After approximating the normal signal using the FOR technique, the FOR-PI observer (FOR-PIO) fault estimation is developed. This technique presents two important challenges: low robustness and  
high estimation error. The structure fault observer is proposed to modify the robustness of FOR-PIO. Moreover, the fuzzy logic algorithm is represented to reduce the signal estimation error in FOR-SPIO. Additionally, the adaptive technique is proposed to improve the reliability of the cascade observer.