# Hybrid Fault Diagnosis of Bearings: Adaptive Fuzzy Orthonormal-ARX Robust Feedback Observer
## 1 Introduction
The feedback linearization observer is a powerful technique for signal estimation. The main idea of this approach is to algebraically transform the non-linear system dynamic parameters into a linearized system so that the feedback observation algorithm can be applied. This observer is based on the dynamics of the system's behavior. Thus, it works perfectly if all parameters are known. Apart from the stability and reliability of this observation technique, it suffers from a lack of robustness. To address this issue, the variable structure technique and sliding mode procedure are used to improve the fault estimation performance in certain and uncertain conditions. Despite the advantages of this technique, it suffers from system modeling complexities and poor estimation accuracy in uncertain conditions [9]. Hybrid-based fault diagnosis techniques are considered to be a reliable, stable, and robust techniques. These techniques use various algorithms in parallel to develop the achievement of the fault detection and fault classification algorithm [6; 7]. In this work, a hybrid-based observer that uses a combination of the fuzzy orthonormal-ARX technique, a fuzzy multi-structure feedback observer, and a machine learning-based classification called support vector machine (SVM) is introduced for the diagnosis of bearing faults of various severities. System modeling is the main argument in designing model-based techniques for fault diagnosis [12; 13; 14]. However, numerous procedures have been used to model REBs and can be classified into two main groups: mathematical-based system modeling such as five-degrees-of-freedom mathematical modeling of vibration signals [13], and system identification techniques such as ARX-Laguerre vibration and bearing acoustic emission signal modeling [10; 12]. Mathematical-based bearing modeling is reliable and stable but is limited in uncertain and unknown conditions [11; 15]. To decrease these limitations,an identification technique is advised in the literature [12; 16]. The ARX-Laguerre technique is robust and stable, but has some limitations when applied to non-linear and non-stationary signal modeling [10; 12; 17]. To address these problems, a fuzzy orthonormal-ARX technique for vibration and acoustic emission bearing signals is prescribed in this work. Once the rotary machinery bearing is modeled based on a system identification technique (the fuzzy orthonormal-ARX method), and the rotary machinery bearing signals are estimated based on the fuzzy extended-state observer (the fuzzy orthonormal-ARX fuzzy multi-structure feedback observer), a decision regarding the bearing conditions can be made. In this work, a machine learning-based classification technique called a support vector machine (SVM) is employed to implement the decision-making procedure for bearing fault diagnosis to complete the proposed technique for diagnosis the fault [18; 19; 20]. In this work, a hybrid-based approach is developed for fault diagnosis in REBs. First, fuzzy orthonormal-ARX is used to model the vibration and acoustic emission (AE) for bearing signals. Next, to estimate the signals, a feedback observer (FO) is developed for two kinds of signals. Then, to solve the challenge of robustness in feedback observer, the multi-structure technique is represented to develop a multi-structure feedback observer (SFO). Also, the fuzzy technique is used to improve the accuracy of the multi-structure feedback observer. Finally, an adaptive technique is used to optimize the multi-structure parameter, which is important to improving robustness and accuracy. The fuzzy orthonormal-ARX adaptive fuzzy logic-structure feedback observer (proposed method) is used for accurate signal estimation. After generating the energy residual signal, the SVM is used for fault classification.