1. Introduction
With the continuous development of aerospace technology, pursuing better performance also requires higher robustness and reliability of aeroengines [
1]. In addition, the safety of aeroengine operation can be significantly assured by the contribution of fault diagnosis systems in detecting, isolating, and estimating faults [
2,
3]. Meanwhile, considering that uncertainty is an essential feature of complex systems [
4], it can be classified as stochastic and epistemic uncertainty [
5]. According to [
6], the control system stochastic uncertainty and aeroengine epistemic uncertainty form the multi-source uncertainty affecting aeroengine operation. However, the traditional design process of the fault diagnosis system is based on a deterministic model and ignores the widespread uncertainty effects [
2]. With the development of UQ, fault diagnosis of aeroengine control systems under multi-source uncertainty is gradually becoming a research hotspot, and fruitful results have been achieved in sensor fault diagnosis [
6] and estimation [
7]. But more research on actuator fault estimation is needed. Therefore, to improve the performance and effectiveness of the diagnostic system to ensure the safety of aeroengine operation, it is of great importance to consider multi-source uncertainty while designing the actuator fault estimation system.
Due to the higher interpretability and reliability of model-based methods, fruitful research results of actuator fault estimation are achieved by designing appropriate filters or observers [
8,
9]. The LKF is the most classical state-estimation method for linear dynamic systems, and Simon et al. designed a Kalman filter bank for the fault diagnosis of the aeroengine control system in 2003 [
10,
11]. It is widely known that the health management of aeroengines is mainly achieved by treating the unknown health parameters input as state and estimating them with the corresponding ASKF [
12,
13]. The actuator fault can also be treated as unknown input and is estimated in the same way as the health parameters. However, as the number of states increases, the computational effort of the ASKF increases geometrically [
14]. To reduce computation ford while guaranteeing state estimation accuracy, TSKF, composed of two sub-filters, is proposed by introducing U-V transformations [
14]. By extending the sub-filter of the TSKF from LKF to the extended Kalman filter [
15] or unscented Kalman filter [
16], the enhanced TSKF can realize more accurate fault estimation in a more extensive range. However, the above methods need to consider uncertainties sufficiently, and more advanced fault estimation methods are required for more complex environments [
17,
18]. Under non-Gaussian noise conditions, Lu et al. combined the particle swarm algorithm with LKF to propose the Kalman particle filter for state estimation and used it for aeroengine health management [
19]. With the development of UQ, Combastel first proposed the ZKF and achieved optimal estimation of the state motion boundary [
20]. And in 2021, Zhang et al. used ZKF for fault interval estimation [
21]. However, the ZKF-based estimation is an interval estimation method and cannot characterize the distribution of states. PCE, the most popular way for UQ today, can adequately characterize the statistical properties of the state [
22] and is increasingly used by scholars for state estimation [
23,
24], fault diagnosis [
6] and performance monitoring [
25].
In summary, research on fault estimation of aeroengine actuators under multi-source uncertainty needs to be revised, especially considering both stochastic processes and systems. On the foundation of PCE-based UQ of the discrete stochastic model, HeKF is proposed in this paper and realizes optimal filtering under multi-source uncertainty. Further, the TSHeKF is proposed and achieves optimal actuator fault estimation under multi-source uncertainty. Moreover, by adding estimation of the unknown state deviation, the HFO is established on the basis of the TSHeKF; it provides fault estimation with better stability and reliability and is less conservative.
This paper is organized as follows. The mathematical description of aeroengine under multi-source uncertainty is presented in
Section 2. PCE-based discrete stochastic model quantification and the further optimal filtering under multi-source uncertainty are described in
Section 3. The TSHeKF is proposed in
Section 4, and the HFO is designed and realized conservativeness-reduced fault estimation. Numerical and MC simulations are presented in
Section 5. Some brief conclusions are shown in
Section 6.
Notation: In this article,
represents the set of real numbers, and
and
denote real vectors and real matrices of appropriate dimensions shown in superscript, respectively.
and
represent the null matrix and identity square matrix, respectively; dimensions are also shown in the corresponding superscript. The bounded basic function set is represented by
, and
represents the
space. For a given real vector
, its square is written as
and is defined as follows. Using
to represent the sum of each element, for a given matrix
,
has the following expression.
2. Mathematical Description of the Problem
From [
6], the multi-source uncertainty faced by aeroengines consists of the stochastic uncertainty of control system and the epistemic uncertainty of aeroengine. Based on uncertainty modeling of one type or a batch of aeroengines [
26], the linear dynamical system considers the actuator fault with independent probabilistic parameters, as shown in Equation (2):
for
with state functions
; similarly, input functions
and output function
.
,
and
are continuous independent disturbance functions, which represent system noise, measurement noise and disturbance of actuator, respectively.
is the expectation of
and
is the deterministic function which characterize the intensity of actuator stochastic uncertainty.
,
,
, and
denote the control matrices of appropriate dimensions with probabilistic system parameters which satisfy the controllability and the observability.
is the additive actuator fault vector;
and
are fault-drive matrices with constant parameters.
and
are constant drive matrices; especially when the disturbance of each state or output is independent,
and
become diagonal matrices. Use
to represent
,
, and
, which is a continuous zero-mean random process with independent increments. According to [
27],
can only be expressed as
, where
is the Brownian motion. Using
as the sampling time, the linear discrete stochastic dynamical system with actuator fault is shown in Equation (3):
where
; similarly,
and
obey the mutually independent multi-source Gaussian distribution.
is used to substitute
; the superscript
represents the discretization of the system matrix, which is obtained by Equation (4), where
is the expansion order.
Temporarily ignoring the parameter uncertainty of the system matrix in Equation (3), the actuator failure can be optimally estimated based on either the ASKF [
13] or TSKF [
15] by treating
as an augmented state. Moreover, the LKF, which is the basis for the ASKF and TSKF, needs to be replaced by optimal filtering considering multi-source uncertainty. However, since the above model is for a type or a batch of aeroengines rather than for a specific engine, the fault estimation based on Equation (3) is also optimal for a range of aeroengines, and robust for an individual one. In order to improve the accuracy of fault estimation for an individual aeroengine, the adaptiveness of the fault estimation needs to be enhanced. The linear discrete stochastic system for one specific aeroengine is shown in Equation (5):
where the overscore indicates the expectation of the stochastic matrix below, and
represents the differences of the following control matrix between the aimed aeroengine and the expectation of the corresponding type or batch of aeroengines, which can also be seen as the uncertainty of the discrete model. It is obvious that both the state function and output function are affected. In order to improve the adaptiveness of the state estimation, the effect of model uncertainty on the state and output needs to be quantified.
and
are used to represent the bias of state and output caused by model uncertainty, respectively. By treating
as unknown input, the effect of model uncertainty on the state can be estimated by the TSKF optimally. Moreover, since
is additive to the output and does not affect the state, it is impossible to estimate
by considering it as the augmented state. Considering that the optimal output estimation of an individual aeroengine is unachievable, the optimal output estimation for a range of aeroengines is the best available option. Therefore, the conservativeness-reduced actuator fault estimation system for individual aeroengine is designed optimally based on Equation (6).
To sum up, when considering multi-source uncertainty, TSKF, as the classical fault-estimation method when treating unknown inputs, is no longer feasible. TSKF, therefore, needs to be adapted from the optimal filtering under multi-source uncertainty to obtain optimal actuator fault estimation. However, directly using the optimal state estimation and output prediction for a batch of aeroengines would result in an unnecessary loss of accuracy by completely ignoring the differences between individuals. Because of the existence of and , the optimal output prediction for individual aeroengine does not exist. Therefore, by combining the adaptive state estimation for individual aeroengine with the optimal output estimation for the corresponding range of aeroengines, the accuracy of actuator fault estimation is maximized under multi-source uncertainty. With the optimal estimation of , the conservativeness-reduced estimation of is achieved based on the proposed HFO, which is the main contribution of this paper.
6. Conclusions
This paper presents an innovative HFO for handling aeroengine actuator fault estimation under multi-source uncertainty consisting of system noise, measurement noise, input noise, and probabilistic system parameters. On the foundation of the PCE-based UQ of the discrete stochastic model, the optimal filtering under multi-source uncertainty is realized. Additionally, the TSHeKF-based fault estimation is also proposed by replacing the basic filter of the TSKF with HeKF, realizing optimal filtering for the linear stochastic system under multi-source uncertainty, and leading to the robust estimation for individuals. Since the bias in output function is relatively small and not estimable, by adding an estimate of state function bias, the HFO provides the best possible fault estimation for individuals and significantly reduces the conservativeness. Numerical simulations demonstrate the high accuracy of the HFO-based state estimation, output prediction, and unknown deviation estimation. Meanwhile, the effectiveness of the HFO in handling single and multiple faults has also been proven. Furthermore, MC simulations demonstrate that, compared with estimations based on the TSKF and TSHeKF, the proposed HFO has higher accuracy and better adaptiveness with guaranteed robustness when considering multi-source uncertainty. This enormously improves the fault diagnosis performance of the engine control system in complex environments and better ensures the safety and reliability of the aeroengine.