Mathematical Problems in Engineering

Volume 2010, Article ID 591639, 24 pages

http://dx.doi.org/10.1155/2010/591639

## Robust Filtering for State and Fault Estimation of Linear Stochastic Systems with Unknown Disturbance

^{1}Electrical Engineering Department of ESSTT, Research Unit C3S, Tunis University, 5 Avenue Taha Hussein, BP 56, 1008 Tunis, Tunisia^{2}Electrical Engineering Department, Research Laboratory CRAN (CNRS UMR 7039), Nancy University, 2 Avenue de la forêt de Haye, 54516 Vandoeuvre-les-Nancy Cedex, France

Received 16 February 2010; Revised 24 August 2010; Accepted 27 September 2010

Academic Editor: J. Rodellar

Copyright © 2010 Fayçal Ben Hmida et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a new robust filter structure to solve the simultaneous state and fault estimation problem of linear stochastic discrete-time systems with unknown disturbance. The method is based on the assumption that the fault and the unknown disturbance affect both the system state and the output, and no prior knowledge about their dynamical evolution is available. By making use of an optimal three-stage Kalman filtering method, an augmented fault and unknown disturbance models, an augmented robust three-stage Kalman filter (ARThSKF) is developed. The unbiasedness conditions and minimum-variance property of the proposed filter are provided. An illustrative example is given to apply this filter and to compare it with the existing literature results.

#### 1. Introduction

The joint fault and state estimation for linear stochastic systems with unknown disturbance is concerned in this paper. The most important aim is to obtain an unbiased robust estimation of the fault and the state despite the presence of the unknown disturbance. This can be useful to solve a fault detection and isolation (FDI) problem [1–4] or a fault tolerant control (FTC) problem [5].

State estimation for stochastic linear systems with unknown inputs has gained the interest of many researchers during the last decades. In this context, this problem has been extensively studied using the Kalman filtering approach, see, for example, [1, 6–23]. When the model of the unknown input is available, it is possible to obtain an optimal estimation by using the Augmented State Kalman Filter (ASKF). To reduce computation costs of the ASKF, Friedland [7] has introduced the Two-Stage Kalman Filter (TSKF). His approach consists of decoupling the ASKF into the state subfilter and unknown-input subfilter. Friedland’s filter is only optimal for constant bias. Many authors have extended the Friedland’s idea to treat the stochastic bias, for example, [6, 10, 13–19]. In the same context, Hsieh and Chen [14] have generalized Friedland’s filter by destroying the bias noise effect to obtain the Optimal Two-Stage Kalman Filter (OTSKF). Chen and Hsieh [15] proposed a generalization of the OTSKF to get the Optimal MultiStage Kalman Filter (OMSKF). Recently, Kim et al. [17] have developed an adaptive version of TSKF noted ATSKF (Adaptive Two-Stage Kalman Filter) and they have analysed the stability of this filter in [18].

On the other hand, when the unknown input model is not available, the unbiased minimum variance (UMV) state estimations are insensitive with the unknown inputs. Kitanidis [20] has developed a Kalman filter with unknown inputs by minimizing the trace of the state error covariance matrix under an algebraic constraint. Darouach and Zasadzinski [21] have used a parameterizing technique as an extension of the Kitanidis’s results to derive an UMV estimator. Hsieh [13] has developed a robust filter in two-stage noted RTSKF (Robust Two-Stage Kalman Filter) equivalent to Kitanidis’s filter. Next, the same author [11] has proposed an extension of the RTSKF (named ERTSKF) to solve the addressed general unknown-input filtering problem. To obtain ERTSKF, the author has introduced a new constrained relationship to have an equivalent structure to the optimal unbiased minimum-variance filter (OUMVF) presented in [12]. Gillijns and Moor [22] have treated the problem of estimating the state in the presence of unknown inputs which affect the system model. They developed a recursive filter which is optimal in the sense of minimum-variance. This filter has been extended by the same authors [23] for joint input and state estimation to linear discrete-time systems with direct feedthrough where the state and the unknown input estimation are interconnected. This filter is called recursive three-step filter (RTSF) and is limited to direct feedthrough matrix with full rank. Recently, Hsieh [8] has extended the RTSF [23] noted ERTSF, where he solved a general case when the direct feedthrough matrix has an arbitrary rank.

Model-based fault detection and isolation (FDI) problem for linear stochastic discrete-time systems with unknown disturbance is several studied. In [2, 3], the optimal filtering and robust fault diagnosis problem has been studied for stochastic systems with unknown disturbance. An optimal observer is proposed, which can produce disturbances decoupled state estimation with minimum-variance for linear time-varying systems with both noise and unknown disturbance. Recently, the unknown input filtering idea [8] is extended by [1] to solve the previously problem. Indeed, Ben Hmida et al. [1] present a new recursive filter to joint fault and state estimation of linear time-varying stochastic discrete-time systems in the presence of unknown disturbance. The method is based on the assumption that no prior knowledge about the dynamical evolution of the fault and the unknown disturbance is available. Moreover, it considers an arbitrary direct feedthrough matrix of the fault. However, it may in certain cases suffer from poor quality fault estimation.

The main objective of this paper is to develop a robust filter structure, that can solve the problem of simultaneously estimating the state and the fault in presence of the unknown disturbance. If the fault and the unknown disturbance affect the system state, we develop the robust three-stage Kalman filter (RThSKF) on two steps. Firstly, we make three-stage* U*-*V* transformations in order to decouple the covariance matrix on the augmented state Kalman Filter (ASKF) thus, we obtain an optimal structure named optimal three-stage Kalman filter (OThSKF). Then, we use a modification in measurement update equations of the OThSKF in order to obtain an unbiased fault and state estimation. On the other hand, when the fault and the unknown disturbance affect both the state and the measurement equations, we propose an augmented robust three-stage Kalman filter (ARThSKF) to overcome this problem. This latter is obtained by a direct application of the RThSKF on the augmented fault and unknown disturbance models. The performances of the resulting filter are established in the sense of the unbiased minimum-variance estimation.

This paper is organized as follows. Section 2 states the problem of interest. In Section 3 we design the OThSKF. A robust three-stage Kalman filter (RThSKF) is developed in Section 4. In Section 5, the augmented robust three-stage Kalman filter is derived. Finally, an illustrative example of the proposed filter is presented.

#### 2. Statement of the Problem

The problem consists of designing a filter that gives a robust state and fault estimation for linear stochastic systems in the presence of unknown disturbance. This problem is described by the bloc diagram in Figure 1.