Two-stage exogenous Kalman filter for time-varying fault estimation of satellite attitude control system

Abstract

This paper addresses the study of observer-based two-stage extended Kalman filter (TSEKF) estimation problem for the satellite attitude control system (ACS) in the presence of unknown time-varying actuator faults. In the traditional TSEKF methods, the considered faults always refers to constant signal in the propagation of the filter estimation, even though time-varying faults are taken into account in simulation demonstrations. In order to promote the accuracy of the TSEKF algorithm, a nonlinear observer is designed to obtain the fault dynamics and the state estimation with consideration of the nonlinear nature of the satellite ACS, and its estimation results are treated as exogenous signals used for linearizing the nonlinear ACS model. Then based on the observed fault information and the linearized ACS model, the TSEKF estimator is designed to obtain the exogenous filtering scheme, which can simultaneously reconstruct the state and faults accurately. Finally, by using the so called two-stage exogenous Kalman filter (TSXKF), simulation results show that when a time-varying fault occurs, more ideal estimation results can be obtained than those of TSEKF and better dynamic performance can be achieved than that of nonlinear observers.

Introduction

The concept of two-stage Kalman filter (TSKF), also called the separate-bias estimation algorithm, is basically to estimate the states without biases and the unknown constant biases of a linear system separately, and to obtain the optimal state estimations using the coupling relationship between them.

Since Friedland [1] firstly derived the separate-bias estimation algorithm, many related researches have been conducted to find the optimal solution of TSKF. Alouani, Xia, and Rice [2] presented an algebraic constraint on the correlation between the state process and bias noises considering a white Gaussian bias noise correlated with the system noise. Keller and Darouach [3] developed an optimal solution of TSKF called OTSKF which can be used to obtain the optimal state estimate and the optimal random bias estimate, and applied OTSKF to dealing with unknown exogenous inputs [4]. Based on the OTSKF, Hsieh [5] proposed a robust two-stage Kalman filter (RTSKF) which was unaffected by the unknown inputs, and the estimation results were optimal and with high accuracy. Later, Khabbazi and Esfanjani [6] introduced the gain projection notion as a type of state constraint to RTSKF, which further enhanced the acccuracy.

Based on the above improvements, the TSKF algorithm is widely used for fault estimation nowadays, by treating faults as a kind of biases. In order to deal with unknown inputs in addition to faults, Ref. [7] derived the optimal three-stage Kalman filter by decoupling the covariance matrices of Augmented state Kalman filter(ASKF) using a three-stage U-V transformation. Through defining the control effectiveness factors to represent the multiplicative faults, Hajiyev [8] extended the fault estimation to the estimation of control distribution matrix elements, so that the TSKF could be applied to actuator/surface fault diagnosis and fault-tolerant control of F-16. Wang and Qi [9] introduced two adaptive factors to the bias-free state filter and the bias filter, respectively, and presented a fault diagnosis strategy for flight control systems based on the closed-loop subspace model identification algorithm and the adaptive TSKF.

However, the above mentioned methods are not effective for nonlinear systems. To deal with nonlinear systems, much dedication has been made to EKF-based linearization approach. Hilairet, Auger, and Berthelot [10] proposed a two-stage extended Kalman filter to estimate rotor flux and velocity of an induction motor. Ref. [11] presented a TSEKF algorithm for estimating both additive and multiplicative faults, and in Ref. [12], the proposed TSEKF-based fault estimation strategy was applied to filtering the telemetry data of the on-orbit satellite TS-1 of Harbin Institute of Technology to reconstruct the actuator and sensor faults successfully. Considering the uncertainty of the statisctic properties, in [13] Xiao et al. introduced the forgetting factor into the three-stage extended Kalman filter to adjust the filter’s parameters adaptively, and the two-dimensional radar tracking problem was well solved. But due to the error produced by the interruption of higher-order terms in the Taylor series expansion, EKF-based methods are not suitable for systems with strong nonlinearities.

To improve the accuracy of nonlinear filtering algorithm, researchers suggested the unscented Kalman filter(UKF)-based method [14], [15] and the cubature Kalman filter(CKF)-based method [16], [17]. Both of them could obtain more accurate estimations of the nonlinear state with large computing cost. Moreover, Johansen and Fossen [18] derived a nonlinear observer-based EKF algorithm, which was called the exogenous Kalman filter (XKF), to achieve global stability and local optimality simultaneously. Furthermore, based on the XKF theory, Ref. [19] designed a multiplicative XKF with globally exponentially stability for attitude estimation. After the observer estimates converge to the true values of the state, higher-order terms in the Taylor series expansion will be zeros and thereby the interruption won’t introduce errors into the correction of TSEKF estimates. The limitation of these XKF-based methods is that, faults/biases were not taken into account in the nonlinear system model while the observer estimates were not used to compensate the system model and thereby failed to provide a more accurate model for filtering.

Considering the above limitations, the objective of this paper is to design a nonlinear observer-based TSEKF algorithm based on the XKF ãtheory – two-stage exogenous Kalman filter (TSXKF), to estimate actuator faults and attitude information of a satellite ACS. Considering the existence of actuator faults in the nonlinear satellite ACS, observers in Ref. [18] are not suitable for the absence of fault models, and sliding mode observers are employed to designing the TSXKF algorithm due to the advantage of finite-time convergence [20], [21]. Meanwhile, owning to the unknown dynamic nature of the fault which is needed in our designing process, higher-order sliding mode observers seems the most appropriate selection [22], [23], [24]. To our best knowledge, no matter the unknown fault/bias was constant or not, the fault/bias prediction was usually equal to the fault estimate in the former step, i.e. ${\widehat{\mathit{b}}}_{k+1|k}={\widehat{\mathit{b}}}_{k|k}$ where ${\widehat{\mathit{b}}}_{k|k}$ denotes the estimate of the fault at the k-th step and ${\widehat{\mathit{b}}}_{k+1|k}$ represented the fault prediction at the (k+1)-th step. However, considering that the time dependency of faults can be distinguished as abrupt fault (stepwise), incipient fault (drift-like), intermittent fault and transient fault [25], [26], it is more general to define a time-varying model of actuator faults. Based on higher-order sliding mode observers, not only the fault dynamics can be known and taken into account for filtering, but also the observer estimates can be used to improving the system model linearization, which is the main improvement compared with the TSEKF in our previous work [11], [12]. In this paper, with the assumption of an unknown time-varying fault model covering that of a constant fault model, a more accurate model can be built to obtain better estimations. By employing higher-order sliding mode observers, the fault dynamics can be known. Then through linearizing the model by the Taylor series expansion at the observer estimates, higher-order terms of the Taylor series converge to zeros as the observations converge to the true value of states, and thereby the linearization error can be reduced in TSXKF.

This paper is organized as follows. Section 2 firstly describes the system model with actuator faults. Section 3.3 gives the design scheme of the nonlinear observer to obtain the fault dynamics, and applies the observations to improving the estimates of TSEKF, which can estimate state and faults simultaneously and accurately. In Section 4, by using the designed TSXKF, simulations under two simulation conditions are examined to demonstrate our scheme and conclusions are drawn in Section 5.