# Nonlinear Extended-state ARX-Laguerre PI Observer Fault Diagnosis of Bearings
## 1 Introduction  
Induction motors have been used in diverse industries, such as the machine tool and oil industries. The dynamic behavior of an induction motor is entirely nonlinear, which can cause various challenges in control and fault diagnosis. High-temperature environments, heavy-duty cycles, poor installation, overloading, and aging of components cause a diverse range of electrical and mechanical defects in motors. Diverse faults have been defined in induction motors, such as motor failures, air-gap faults, bearing and cage faults, and stator failures. The two main types of induction motor defects are mechanical and electrical failures. Various types of induction motor faults are mostly associated with mechanical defects (79%), such as bearing defects (69%) and rotor faults (10%). The other types of induction motor faults are electrical failures (21%), such as open circuits and short circuits in stator windings [1]. The most common defect in induction motors are mechanical failures. These faults are classified as bearing faults and rotor faults. Across industries, bearing defects are the most common critical effect increasing failure in induction motors [1]. The four main types of bearing defects are inner racewayfaults, outer raceway faults, ball failures, and cage faults [2; 3]. To analyze the condition of bearings, different types of condition monitoring techniques, such as acoustic emission (AE), stator current, shaft voltage, bearing circuit, and vibration analysis, have been reported in the literature [4]. Vibration and AE measurement techniques are the most widely used for monitoring the bearing conditions [4]. Since AE signals are strongly correlated with actual faults, this research utilizes acoustic emission sensors for data collection. Two main fault detection and diagnosis methods are hardware-based fault detection and diagnosis (FDD) and functional-based FDD [5]. Fault detection and identification based on the hardware method is a stable, reliable, and preventive maintenance technique, but it is also expensive. To address this issue, functional-based FDD has been presented [6; 7; 8; 9; 10; 11]. The four foremost types of functional-based fault detection and diagnosis are signal-based FDD [6; 7; 8; 9; 10; 11], knowledge-based FDD [12; 13], model-based FDD [14; 15; 16], and hybrid-based FDD [17; 18]. Several signal-based techniques, such as motor current signature analysis (MCSA), vibration analysis, the noise monitoring technique, and torque monitoring analysis, have been introduced. A common signal-based technique for FDD in the induction motor is the MCSA technique. This technique is more common for broken rotor bar (BRB) faults, air-gap eccentricity faults, and stator electric current faults. To diagnose faults in the mechanical parts of an induction motor, such as a bearing, vibration and acoustic emission analysis are more common [7]. The main drawbacks of signal-based FDD are reliability and robustness. Knowledge-based FDD has various advantages, but this technique needs massive quantities of data for training [19]. Model-based FDD has been considered a robust and reliable FDD technique. The main concept of model-based FDD is system modeling, which has been acknowledged by several researchers in the field [19]. The main techniques for model-reference FDD use output observers, system identification and parameter estimation, and the parity equation [20; 21]. Various researchers have used observational methods for FDD, such as a proportional integral (PI) observer [22], proportional multiple integral (PMI) observer [23] and sliding mode observer (SMO) [24]. The fault diagnosis in noisy conditions and state-estimation in the highly nonlinear systems are the main drawbacks in the PI and PMI observers. The sliding mode observer has been considered to solve the linear observer's drawback [24]. Apart from several advantages of the sliding mode observer, chattering phenomenon is the main drawback.