# Nonlinear Extended-state ARX-Laguerre PI Observer Fault Diagnosis of Bearings
## 1 Introduction
The higher order sliding mode observer (HOSMO) was presented to solve the chattering phenomenon in the sliding mode observer [25]. This observer works based on the system's dynamic behavior, and thus the observer performance can be significant if most system dynamic parameters are known. To address the challenges of proportional integral observer (PIO) and SMO, the sliding mode extended-state ARX-Laguerre PI observer technique is introduced in this research. This technique is designed with the following steps. In the first step, the ARX-Laguerre technique is used for the system (induction motor) modeling. In the second step, the first observer (ARX-Laguerre PI observer) is designed for system and fault estimation. Apart from the numerous positive attributes of the ARX-Laguerre PI observer, this technique has an issue of fault estimation accuracy. To improve fault estimation based on the ARX-Laguerre PI observer, in the next step, an extended-state ARX-Laguerre PI observer is used. Based on this method, the performance of fault estimation increased sharply. The residual generation plays a vital role in the observer-based fault diagnosis. To generate a robust residual signal, in the fourth step, the sliding mode technique is utilized. This method is robust and effective for motor fault diagnosis at variable speeds. To process the threshold value, in the fifth step, a robust sliding mode technique is used. After processing the robust residual signal and the threshold value, in the sixth step, the proposed extended-state ARX-Laguerre PI observer generated two different scenarios for fault diagnosis: (a) variable crack sizes and (b) variable motor speeds. This paper presents three different problems and the solutions to solve these problems. The problems and solutions are briefly described as follows:  
**Problem 1:** System (induction motor) modeling is a significant issue, especially to the design model-reference technique for FDD. **Solution 1:** The robust system estimation technique based on the ARX-Laguerre technique is employed for system estimation (please see Section 4.1). **Problem 2:** Fault detection based on the proposed method in the presence of unknown parameters. **Solution 2:** After designing the proposed method, robust sliding mode residual generation is introduced to find the robust residual signal and generate it. Based on the comparison between the band of uncertainty and threshold value, the abnormal condition can be detected, as seen in Section 4.3. **Problem 3:** Fault identification and estimation are the third issue of this research. **Solution 3:** For fault estimation and identification, the proposed extended-state observer is used to improve the performance of the proposed ARX-Laguerre PIO and reduce the effect of uncertainty (Section 4.4). The rest of this paper is organized as follows: Section 2 gives the dataset. Section 3 presents the problem statements and the proposed method objectives. Section 4 introduces the proposed nonlinear extended-state ARX-Laguerre PI observer method and the effectiveness of the proposed method for fault detection and diagnosis. Section 5 shows the results and discussions, and Section 6 concludes this paper.