## 3 Bearing State Space Model Establishment and Fault Identification Process
### Fault Detection and Identification Process of Bearing  
Under the influence of internal and external factors, the normal working rolling bearing will generate excitation force, thus promoting the system's vibration. Therefore, the system's vibration mode includes both natural vibration and fault vibration. The natural vibration is almost unaffected by the working state, because it is only related to the vibration transmission path, processing links, materials and other factors. Fault vibration can indicate the working state of bearing. By analyzing the vibration characteristics, we can judge whether the bearing has faults, such as wear fault, corrosion fault, fracture fault and indentation fault [26]. The failure caused by the incipient weak local damage of the bearing will produce vibration and shock. However, due to the short duration of shock, energy divergence, wide frequency range, and small vibration amplitude, small faults are often submerged in background noise and are difficult to find and extract [27; 28]. This article uses fiber Bragg grating resonant sensors to monitor bearing. According to the collected data, noise reduction, time domain, power spectrum and improved autocorrelation envelope spectrum analyses are performed in sequence to realize the detection and diagnosis of bearing faults. The process of bearing fault detection and identification is shown in Figure 1.  
In this article, the fiber Bragg grating resonant sensor is used to monitor the bearing. The fiber Bragg grating resonant sensor uses generalized resonance as the monitoring principle and fiber Bragg grating as the monitoring method, which can realize the monitoring of the incipient weak fault signals of the bearing. The specific process of fiber Bragg grating resonant sensor monitoring bearing is as follows: when a partial failure occurs on the surface of a certain component of the bearing, it will collide with the surface of other components during the loaded operation. This produces a concentrated shock pulse force. The sensor absorbs shock energy to generate a generalized resonance wave, and releases energy at the sensor's own high-frequency natural frequency. After processing the high-frequency resonance wave, a vibration waveform that eliminates low-frequency vibration interference but is rich in fault information can be obtained. The vibration signal in this process is measured by the fiber Bragg grating. The fiber Bragg grating converts the vibration signal into the axial strain of the fiber Bragg grating, and then uses the high-speed fiber Bragg grating demodulator to obtain the center wavelength of the fiber Bragg grating. By monitoring the wavelength dynamics of the fiber Bragg grating change, the bearing vibration can be monitored [29; 30]. The process of picking up weak faults is shown in Figure 2.  
Due to the harsh working environment of the bearing, the vibration signal collected by the sensor contains a lot of noise. In order to accurately realize the detection and identification of the faulty bearing's signal, it is necessary to reduce the noise of the vibration signal collected by the sensor. This article establishes the AR model, determines the AR model order through the FPE criterion, and then determines the AR model parameters through the RLS algorithm, and finally uses the Kalman filter to achieve noise reduction. The main process is shown in Figure 3.  
Figure 1: Bearing fault detection and identification process. The detection process of the bearing faulty signal is as follows: first, the time-domain analysis is performed on the signal after noise reduction. If the time-domain characteristic curve has attenuated oscillation, it indicates that there is a generalized resonance wave. Then, analyze the power spectrum of the signal after noise reduction, and the extracted high frequency components is close to the natural frequency of the fiber Bragg grating resonant sensor, which further shows that the bearing and the fiber Bragg grating resonant sensor have a generalized resonance phenomenon.