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]:
where
fa(
t) and
fs(
t) are actuator faults and sensor faults, Δ
Ac, Δ
Bc, Δ
Cc, and Δ
Dc are component faults,
n1 and
n2 are model uncertainties,
E1 and
E2 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. Many works on the IAE approach have focused on how to adapt the filter statistical information, which is the matrices
Q and
R, for improving the filter performance. In contrast, a relatively small number of researchers have paid attention to the adaptation of error covariance matrix
P, of which approach is known as the adaptive fading Kalman filter (AFKF). Note that AFKF can be significantly effective for the estimation of the fault parameters. Incomplete information about the dynamic equation caused by faults in the system can be compensated by increasing the magnitude of the error covariance matrix using a forgetting factor. Meanwhile, the convergence property during the estimation procedure becomes a critical issue when the estimation of fault parameter is combined with an FTC based on adaptive control. In AFKF, the convergence property is determined by the forgetting factor. A large value of forgetting factor leads to a sluggish estimation transition in the transient phase, while a small value of forgetting factor results in the poor estimation of the parameter change caused by the fault. Consequently, it is necessary to regulate the forgetting factor so as to improve both the estimation transition and its convergence time.
In order to improve the convergence property during the estimation of fault parameters, this paper presents a novel adaptive fuzzy fading Kalman filter in which the forgetting factor is determined by a fuzzy system. The use of the fuzzy system is an attractive approach for Kalman filtering, not only because of its simplicity but also its effectiveness. The benefit of a fuzzy system in Kalman filtering has been verified through various studies [
9,
10,
11].
The remaining part of the paper is organized as follows. In
Section 2, the existing works on the Kalman filter and joint Kalman filter are discussed. In
Section 3, the proposed adaptive fuzzy fading Kalman filter is explained. A set of numerical simulations are presented in
Section 4, to demonstrate the effectiveness of the proposed method, followed by concluding discussions in
Section 5.
3. Adaptive Fuzzy Fading Kalman Filter for Fault Parameter Estimation
The optimality of the Kalman filter can be ensured when
a priori knowledge of the process/measurement noise matrices,
Q/
R, and the state transition matrix
A are completely known. In practical applications, however, these matrices are often insufficiently known. It is known that incomplete information makes estimations biased or even diverged [
13,
14]. To overcome this problem, various adaptive techniques have been studied. As mentioned before, component faults can be modeled as the change in the value of the dynamic system parameter. See (1). Therefore, to estimate the fault parameter via a Kalman filter, an adaptation procedure is required to handle the incompleteness of
a priori information. In this paper, an adaptive fuzzy fading Kalman filter is proposed so that the convergence property during the estimation of fault parameter can be improved.
3.1. Existing Adaptive Fading Kalman Filter
The innovation covariance of the Kalman filter is given by
where
,
and
Rk are innovations, the predicted error covariance and the measurement noise covariance, respectively.
Ck is referred to as the calculated innovation covariance [
7,
8]. The innovation covariance shows the effect of any unaccounted errors, as it is directly involved in the computations of the innovation. When the exact dynamic equation of a nonlinear stochastic system is not available, the estimation error and predicted error covariance may increase due to the effect of incomplete information. Therefore, the change of innovation covariance can be used for an adaptive filter. The increased innovation covariance is estimated by
where,
M is window size.
is called as the estimated innovation covariance. The relationship between
Ck and
is defined as
. Then, the scalar value can be estimated by
where,
p is the dimension of
zk and
tr() denotes the matrix trace. The estimated innovation covariance shows the estimated filter-computed innovation covariance.
Generally, the convergence property worsened by insufficient
a priori information can be improved by introducing a forgetting factor to the predicted error covariance
Pk. Thus, the predicted error covariance must be increased to compensate for the effect of incomplete information on the dynamic system as follows:
where,
λk is the
forgetting factor which is given by:
This indicates that the ratio of the innovation covariance αk is mainly generated by λk. Therefore, αk is almost equal to λk (i.e., αk = λk). The predicted error covariance given as . This is referred to as “AFKF with rescaling-Pk.”
3.2. Design of Adaptive Fuzzy Fading Kalman Filter
The forgetting factor of AFKF is determined as the ratio of the calculated innovation covariance and the estimated innovation covariance in the k-step. If any component fault occurs abruptly, it causes a step-like change in the parameter value, and then the AFKF undergoes a significant uncertainty of the state transition matrix Ak instantaneously due to the incomplete model of Equation (5). In this case, AFKF will increase the forgetting factor to compensate for the effect of incomplete information. However, a large forgetting factor may result in the poor estimation performance, such as a longer estimation transition, in the transient phase of the filter, and eventually, the FTC based on the parameter estimation will be unstable. Therefore, it is necessary to regulate the forgetting factor to improve the estimation transition and its convergence time.
In this paper, in order to improve the convergence property of AFKF, a fading algorithm based on the fuzzy system is proposed. Since
αk is the random variable which is subject to uncertainties, it is necessary to use an approach that takes into account such uncertainties to derive relevant
λk. This can be achieved with a fuzzy logic approach with rules of the kind [
15,
16]:
where
property and
action, are the form of
, respectively. Further, χ, κ, Ο
i/Λ
i are input variable, output variable, and fuzzy set, respectively.
To implement the fuzzy fading algorithm,
k-step
αk and the change of
αk (i.e., Δ
αk =
αk −
αk−1) are used. The input variable and Δ
λk are defined as the output of fuzzy inference system (FIS). Δ
λk denotes the adjust term as follows:
Figure 2 depicts the algorithm of the proposed adaptive fuzzy fading Kalman filter (AFFKF). The fuzzy rule for the forgetting factor is defined in
Table 1. The FIS is implemented using three fuzzy sets for
αk: S = Small, M = Medium, B = Big, three fuzzy sets for the change of
αk: S = Small, M = Medium, B = Big, and four fuzzy sets for Δ
λk: Z = Zero, S = Small, M = Medium, B = Big. The membership functions, designed using a heuristic approach, are shown in
Figure 3. The proposed AFFKF is an extension of AFKF with the fuzzy inference system that determines the appropriate forgetting factor.