# Hybrid Fault Diagnosis of Bearings: Adaptive Fuzzy Orthonormal-ARX Robust Feedback Observer
## 1 Introduction  
Energy is essential for the survival and strength of modern industrial civilization. Generally, the sources of energy are fossil fuels such as oil and coal. Use of these fuels has significant impacts on the environment, one of the main challenges being greenhouse gas (GHG) emissions, with carbon dioxide (CO2) as their major component [1]. Regularly, 29 Gigatons (Gt) of CO2 are emitted every year, of which, about 7 Gt can be absorbed naturally [2]. The remainder enters the atmosphere, causing issues such as global warming and pollution. In recent years, the rate of industrialization has sharply increased, causing an increase in the rate of GHG emissions. About 50% of the energy produced by various techniques is consumed by heavy industries. The most widely used electro-mechanical equipment, consuming about 80% of the industrial electricity total, is various types of motors. Based on the energy efficiency trends seen in the energy audits of the various industries, optimization of power consumption, condition monitoring and fault diagnosis, and power quality improvements are important factors in energy efficiency. There is also serious attention onthe disturbing effects of systems anomalies on energy efficiency and thus the environment since conservation of the environment's quality and energy efficiency are major challenges in most countries around the world [2; 3]. Most research and industry initiatives over the last decade have resulted in a significant decrease in the share of energy losses due to faulty industrial systems. However, in some fields, such as induction motors, reliable techniques must still be designed. Induction motors have been used in several applications. This electro-mechanical system has non-linear dynamic behaviors and non-linearities, which can lead to various challenges in fault detection and diagnosis. Mechanical and electrical malfunctions are the two main categories of induction motor faults. Various types of induction motor anomalies have been defined and are categorized as: (a) mechanical faults (79%) and (b) electrical faults (21%). The mechanical anomalies can be categorized into two main groups: (a) bearing faults (69%) and rotor failures (10%) [4]. The associated task complexity and non-linear parameters in REBs have led to numerous anomalies detection and classification difficulties. Four types of faults found in REBs such as inner faults, outer faults, ball faults, and cage faults [5]. Several methods have been advised for anomaly detection and identification in REBs. These techniques can be divided into four principal divisions: model-based techniques, signal-based approaches, data-driven algorithms, and hybrid-based procedures [6; 7]. The model-based fault diagnosis techniques are reliable and robust algorithms and have been used in various applications. Observation-based algorithms are the main model-based techniques used for bearing fault diagnosis [8]. Despite the advantages of observation-based approaches, these techniques have some limitations in the presence of uncertain and unknown conditions [8; 9]. Non-linear-based observation techniques (e.g., sliding mode observer, feedback linearization observer) and linear-based observation algorithms (e.g., proportional-integral (PI) observer) are the main procedures used to develop observation to estimate the signals [10]. The PI observer is a linear and easy to implement technique for system estimation but have limited robustness and accuracy [10; 11; 12]. The sliding mode observer is a non-linear and high-gain observer that can improve a system's dynamic and reduce the estimator error infinite time. This technique is robust and reliable but is prone to chattering phenomenon and limited estimation accuracy. To minimize the chattering phenomenon, the higher-order super-twisting sliding mode observer has been recommended in the literature. This technique suffers from a somewhat reduced estimation accuracy [8]. To minimize the effect of the chattering phenomenon, a feedback linearization observer was developed.