## 1 Introduction
To estimate the original signals, diverse algorithms have been suggested in recent years. The observation-based technique is one of the powerful mechanisms for signal estimation [25]. Artificial Intelligence (AI)-based observers and modern control-based observers are two main groups for signal estimation [19; 20]. AI-based observers, such as the fuzzy logic observer [20], the neural network observer [21], and the neuro-fuzzy observer [22], have the same challenge (namely, reliability). Modern control-based observers (including linear-based observers and nonlinear-based observers) have been used for signal estimations [18; 19; 32]. The sliding mode observer is one of the most robust and reliable estimators for fault diagnosis. The application of the sliding mode observer for fault diagnosis was presented in [18]. Despite its stability, reliability, and robustness, the sliding mode observer suffers from complexity and the chattering phenomenon. The next nonlinear-based observer is the feedback linearization approach [33]. However, this approach solves the issue of the chattering phenomenon; robustness and reliability are two important drawbacks. Thus, the main challenge of nonlinear modern control-based observers (such as the sliding mode observer and the feedback linearization observer) is complexity. To reduce complexity, modern linear control-based observers, such as the proportional integral (PI) observer [22] and the proportional multi-integral (PMI) observer [34], have been suggested. Although PI observers and PMI observers have good performance in terms of accuracy and complexity, they have some problems related to robustness about uncertainties [34]. To address the robustness issue, a combination of PMI observer and sliding mode approach was used in [34]. The main limitation of this algorithm is to increase the amount of chattering and signal fluctuations in the presence of variation in torque load and motor speed. To fault and state condition classifications, diverse classification algorithms such as decision trees [35], nearest neighbor classifiers [36], support vector machines [37], and ensemble classifiers have been introduced. In the paper, the hybrid-based fault diagnosis approach is suggested for anomaly identification of the bearing. The proposed hybrid-based approach has three main steps: (i) preprocessing and feature extraction step, (ii) signal modeling and estimation step, and (iii) classification step. Based on the above, the feature extraction technique is considered as the first stage. In this research, the main stage for designing the hybrid technique for rolling element bearing crack identification is to compute residual signals, which are calculated based on the difference between original signals and estimated signals. Modeling is also the main critical challenge in modern control-based observers for signal estimation. Subsequently, the first step in estimating a signal using observers is signal modeling to approximate the state-space function of the signal. Therefore, in the paper, we apply machine learning-based regression (MBR) to the ARX with uncertainty input (ARXU) technique for signal approximation, which is called \"AMRXU\". After signal modeling, the original signals are estimated before computing the residual signals. To do this, first, the PMI observer is recommended. After that, to address the robustness limitation in the PMI, the variable structure (VS)-Lyapunov is adopted. Apart from reducing the complexity and increasing the accuracy and robustness of this technique, the flexibility about uncertainties in various torque loads, motor speeds, or crack sizes can be a weakness. To overcome this issue, we use the self-tuning hybrid-based observer which is a combination of a self-tuning AI-based technique and modern control-based approach to improve the positive points, which is called \"AVSPMI\". Next, the residual signals are computed using the difference between the original signals and the estimated ones.