# Fault Parameter Estimation Using Adaptive Fuzzy Fading Kalman Filter
## 1 Introduction  
System faults can be classified into three types: (a) actuator faults, (b) sensor faults, and (c) components faults. The actuator faults and sensor faults are commonly modeled as additive perturbations of the system dynamics. The component faults, which are associated with the change of the dynamics, can be modeled as a multiplicative quantity to the state variables. A general fault model for the linear system is given as follows [1]:  
\[x(t+1)=(A+\Delta A_{c})x(t)+(B+\Delta B_{c})u(t)+E_{1}(t)n_{1}(t)+Bf_{a}(t) \tag{1}\]  
\[y(t)=(C+\Delta C_{c})x(t)+(D+\Delta D_{c})u(t)+E_{2}(t)n_{2}(t)+f_{s}(t) \tag{2}\]  
where \(f_{a}(t)\) and \(f_{s}(t)\) are actuator faults and sensor faults, \(\Delta A_{c}\), \(\Delta B_{c}\), \(\Delta C_{c}\), and \(\Delta D_{c}\) are component faults, \(n_{1}\) and \(n_{2}\) are model uncertainties, \(E_{1}\) and \(E_{2}\) are weighting matrices. The time characteristics of system faults are classified into three types: (a) abrupt, (b) incipient, and (c) intermittent (Figure 1) [2]. Abrupt faults are caused by hardware damages, while incipient faults are caused by a slow parametric change, such as the effect of aging. Intermittent faults are considered as the most difficult fault category to detect because they appear and disappear repeatedly. In general, fault detection and diagnosis (FDD) methods are categorized into model-based methods and data-based methods [3]. A large number of works in FDD have been developed using various approaches including observers, (joint) Kalman filters, recursive least square (RLS) algorithms, principal component analysis (PCA), expert systems, etc. PCA and expert systems are a data-based method that utilizes a large amount of historical data. The data is then transformed and presented as _a priori_ knowledge for diagnosing the system. The approach with observers and Kalman filters is a typical model-based method that utilizes the difference (i.e., residual) between the measurement output and the estimated output. When no fault occurs, the mean of the residual will be close to zero. Conversely, faults may result in the mean of the residual deviated from zero. The approach with RLS algorithms and joint Kalman filters can be seen as a model-based method by which system parameters are directly estimated through the input-output relationship. Meanwhile, FDD methods have to consider cooperation with fault-tolerant control (FTC). FTC will more effectively satisfy the given control objectives if FDD provides appropriate information which is needed for reconfiguration of the control structure. Typical information used in FTC includes fault parameters and fault models. For example, FTC using an adaptive control strategy requires the exact value of fault parameters [3], for which an accurate parameter estimation technique is essential. The estimation of fault parameters can be reformulated as the state estimation problem by augmenting the parameters as additional states, which is known as the augmented Kalman filter or joint Kalman filter. The joint Kalman filter has been studied for the problem of estimating the states and the parameters simultaneously. The joint Kalman filter was first proposed in [4; 5; 6]. However, the conventional approach cannot estimate the parameters that are repeatedly changing due to system faults. As Kalman filters depend on the accuracy of _a priori_ knowledge about the system model, any unknown information may seriously degrade the estimation performance. When a conventional joint Kalman filter is adopted to estimate the fault parameter of a dynamic system, incomplete information of the model will result in a serious bias or divergence on the estimation. To overcome this problem, researchers have studied adaptive Kalman filter techniques [7; 8], which can be classified into two types: (a) the innovation-based adaptive estimation (IAE) and (b) the multiple-model-based adaptive estimation (MMAE). In this work, IAE-based adaptive Kalman filters are examined.