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

Authors: Fayçal Ben Hmida; Karim Khémiri; José Ragot; Moncef Gossa
Journal: Mathematical Problems in Engineering
 (2010)

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## 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.


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## Body

## 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-stageU-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. Section2 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 Figure1.Figure 1
State and fault estimator filter.The plantP represents the linear time-varying discrete stochastic system(2.1)xk+1=Akxk+Bkuk+Fkxfk+Ekxdk+wkx,(2.2)yk=Hkxk+Fkyfk+Ekydk+vk,
where xk∈Rn is the state vector, uk∈Rr is the known control input, fk∈Rp is the additive fault vector, dk∈Rq is the unknown disturbance and yk∈Rm is the observation vector. wkx and vk are uncorrelated white noises sequences of zero-mean and the covariance matrices are Qkx=ℰ[wkxwkT]≥0, and Rk=ℰ[vkvkT]>0, respectively, where ℰ[·] denotes the expectation operator. The matrices Ak, Bk, Fkx, Hk and Fky are known and have appropriate dimensions. We assume that (Ak,Hk) is observable, m≥p+q, rank(Hk+1Fkx)=rank(Fkx) and rank(Hk+1Ekx)=rank(Ekx). The initial state is uncorrelated with the white noises processes wkx and vk. The initial state x0 is a gaussian random variable with ℰ[x0]=x̂0 and ℰ[(x0-x̂0)(x0-x̂0)T]=P0x.Under system equations (2.1)-(2.2), we consider that the proposed filter has the following form: (2.3)x̂k+1/k+1=x¯k+1/k+1+V¯k+112fk+1/k+1+V¯k+113dk+1/k+1,f̂k+1/k+1=fk+1/k+1+V¯k+123dk+1/k+1,x¯k+1/k+1=x¯k+1/k+K¯k+1x(yk+1-Hk+1x¯k+1/k),fk+1/k+1=Kk+1f(yk+1-Hk+1x¯k+1/k),dk+1/k+1=Kk+1d(yk+1-Hk+1x¯k+1/k),x¯k+1/k=Akx̂k/k+Bkuk.The gain matricesK¯k+1x, Kk+1f, and Kk+1d and the correction parameters V¯k+112, V¯k+113, and V¯k+123 will be determined later.


## 3. Optimal Three-Stage Kalman Filter (OThSKF)

This section is devoted to the optimal three-stage Kalman filter design. We first recall the structure of the augmented state filter, then the UV transformations are defined which will be used later to decouple the augmented state Kalman filter equations into three subfilters.We should treatdk and fk as a random-walk processes with given wide-sense representation.Thus, the dynamics ofdk may be assumed as follows:(3.1)dk+1=dk+wkd.
The additive faults fk are generated by(3.2)fk+1=fk+wkf,
where, the noises wkf and wkd are zero-mean white noise sequences with the following covariances:(3.3)ℰ(wkfwℓfT)=Qkfδkℓ,ℰ(wkdwℓdT)=Qkdδkℓ,ℰ(wkxwℓfT)=Qkxfδkℓ,ℰ(wkxwℓdT)=Qkxdδkℓ,ℰ(wkfwℓdT)=Qkfdδkℓ.
The initial fault and unknown input satisfy the followings:(3.4)ℰ(f0)=f¯0,ℰ(d0)=d¯0,ℰ((f0-f¯0)(f0-f¯0)T)=P0f,ℰ((d0-d¯0)(d0-d¯0)T)=P0d,ℰ((x0-f¯0)(x0-f¯0)T)=P0xf,ℰ((x0-d¯0)(x0-d¯0)T)=P0xd,ℰ((f0-d¯0)(f0-d¯0)T)=P0fd.


### 3.1. Augmented State Kalman Filter (ASKF)

Treatingxk,fk and dk as the augmented system state, the ASKF is described by(3.5)xk+1/ka=Akaxk/ka+Bkauk,(3.6)Pk+1/ka=AkaPk/kaAkaT+Qka,(3.7)xk+1/k+1a=xk+1/ka+Kk+1a(yk+1-Hk+1axk+1/ka),(3.8)Kk+1a=Pk+1/kaHk+1aT(Hk+1aPk+1/kaHk+1aT+Rk+1)-1,(3.9)Pk+1/k+1a=(I-Kk+1aHk+1a)Pk+1/ka,
where (3.10)x(·)a=[x(·)f(·)d(·)],Aka=[AkFkxEkx0I000I],Bka=[Bk00],Hka=[HkFkyEky],P(·)a=[P(·)xP(·)xfP(·)xdP(·)fxP(·)fP(·)fdP(·)dxP(·)dfP(·)d],Qka=[QkxQkxfQkxdQkfxQkfQkfdQkdxQkdfQkd].
The filter model (3.5)–(3.9) may be used to produce the optimal state estimate. But, this filter has two main disadvantages: the increase of the computational cost with the augmentation of the state dimension and the rise of numerical problems during the implementation [13]. So, to solve these problems, we should use the three-stage Kalman filtering (ThSKF) technique.


### 3.2.U-V Transformations

According to [13, 14], the ThSKF is obtained by the application of a three-stage U-V transformations in order to decouple the ASKF covariance matrices, that is, Pk+1/ka and Pk+1/k+1a. The aim is to find matrices Uk+1 and Vk+1 such that
(3.11)Pk+1/ka=Uk+1P¯k+1/kaUk+1T,Pk+1/k+1a=Vk+1P¯k+1/k+1aVk+1T,
with P¯(·)a=diag{P¯(·)x,P¯(·)f,P¯(·)d}, where P¯(·)x,P¯(·)f and P¯(·)d denote the transformed covariance matrices.We define the structures of theUk+1 and Vk+1 matrices as follows: (3.12)Uk+1=[IUk+112Uk+1130IUk+12300I],Vk+1=[IVk+112Vk+1130IVk+12300I],Uk+1ij and Vk+1ij for i=1or2 and  j=2or3 are to be determined later.Using these transformations (3.12), the equations (3.5), (3.7), and (3.8) are transformed into  (3.13)xk+1/ka=Uk+1x¯k+1/ka,xk+1/k+1a=Vk+1x¯k+1/k+1a,Kk+1a=Vk+1K¯k+1a.
The inverse transformations of Uk+1 and Vk+1 (3.12) will have this form  (3.14)Uk+1-1=Ũk+1=[IŨk+112Ũk+1130IŨk+12300I],Vk+1-1=Ṽk+1=[IṼk+112Ṽk+1130IṼk+12300I].
Using these inverse transformations (3.14), we have
(3.15)x¯k+1/ka=Ũk+1xk+1/ka,P¯k+1/ka=Ũk+1Pk+1/kaŨk+1T,x¯k+1/k+1a=Ṽk+1xk+1/k+1a,K¯k+1a=Ṽk+1Kk+1a,P¯k+1/k+1a=Ṽk+1Pk+1/k+1aṼk+1T,
where (3.16)x¯(·)a=[x¯(·)f¯(·)d¯(·)],P¯(·)a=[P¯(·)x000P¯(·)f000P¯(·)d],K¯(·)a=[K¯(·)xK¯(·)fK¯(·)d].


### 3.3. Decoupling

If we use the two-step substitution method, the filter model (3.5)–(3.9) is transformed into(3.17)x¯k+1/ka=Ũk+1Πk+1x¯k/ka+Ũk+1Bkauk,(3.18)P¯k+1/ka=Ũk+1(Πk+1Pk/kaΠk+1T+Qka)Ũk+1,(3.19)x¯k+1/k+1a=Ṽk+1Πk+1x¯k+1/ka+K¯k+1a(yk+1-Sk+1x¯k+1/ka),(3.20)K¯k+1a=Ṽk+1Πk+1P¯k+1/kaSk+1T(Sk+1P¯k+1/kaSk+1T+Rk+1)-1,(3.21)P¯k+1/k+1a=(Ṽk+1Πk+1-K¯k+1aSk+1)P¯k+1/ka(Ṽk+1Πk+1)-1,
where  (3.22)Πk+1=AkaVk=[AkΠk+112Πk+1130IΠk+12300I],Sk+1=[Sk+11Sk+12Sk+13],
with
(3.23)Πk+112=AkVk12+Fkx,Πk+113=AkVk13+FkxVk23+Ekx,Πk+123=Vk23,(3.24a)Sk+11=Hk+1,(3.24b)Sk+12=Hk+1Uk+112+Fk+1y,(3.24c)Sk+13=Hk+1Uk+113+Fk+1yUk+123+Ek+1y.Now, we start by developing the equations (3.18) and (3.21), respectively.From (3.18), we obtain(3.25)P¯k+1/kx=AkP¯k/kxAkT+Q¯k1,P¯k+1/kf=P¯k/kf+Q¯k2,P¯k+1/kd=P¯k/kd+Qkd,
where
(3.26)Q¯k1=Qkx+Πk+112P¯k/kfΠk+112T-Uk+112P¯k+1/kfUk+112T+Πk+113P¯k/kdΠk+113T-Uk+113P¯k+1/kdUk+113T,Q¯k2=Qkf+Πk+123P¯k/kdΠk+123T-Uk+123P¯k+1/kdUk+123T,Uk+113=(Πk+113P¯k/kd+Qkxd)(P¯k+1/kd)-1,Uk+123=(Πk+123P¯k/kd+Qkfd)(P¯k+1/kd)-1,Uk+112=(Πk+112P¯k/kf+Πk+113P¯k/kdΠk+123T-Uk+113P¯k/kfUk+123T+Qkxf)(P¯k+1/kf)-1.
The development of (3.21), leads to(3.27)P¯k+1/k+1x=(I-K¯k+1xSk+11)P¯k+1/kx,(3.28)P¯k+1/k+1f=(I-K¯k+1fSk+12)P¯k+1/kf,(3.29)P¯k+1/k+1d=(I-K¯k+1dSk+13)P¯k+1/kd,(3.30)Vk+112=Uk+112-K¯k+1xSk+12,(3.31)Vk+113=Uk+113-Vk+112K¯k+1fSk+13-K¯k+1xSk+13,(3.32)Vk+123=Uk+123-K¯k+1fSk+13.
With reference to (3.17), (3.19) and (3.20), we obtain, respectively,(3.33)x¯k+1/k=Akx¯k/k+Bkuk+u¯k1,(3.34)f¯k+1/k=f¯k/k+u¯k2,(3.35)d¯k+1/k=d¯k/k,(3.36)x¯k+1/k+1=x¯k+1/k+K¯k+1x(yk+1-Sk+11x¯k+1/k),(3.37)f¯k+1/k+1=f¯k+1/k+K¯k+1f(yk+1-Sk+11x¯k+1/k-Sk+12f¯k+1/k),(3.38)d¯k+1/k+1=d¯k+1/k+K¯k+1d(yk+1-Sk+11x¯k+1/k-Sk+12f¯k+1/k-Sk+13d¯k+1/k),(3.39)K¯k+1x=P¯k+1/kxSk+11T(Sk+11P¯k+1/kxSk+11T+Rk+1)-1,(3.40)K¯k+1f=P¯k+1/kfSk+12T(Sk+12P¯k+1/kfSk+12T+Sk+11P¯k+1/kxSk+11T+Rk+1)-1,(3.41)K¯k+1d=P¯k+1/kdSk+13T(Sk+13P¯k+1/kdSk+13T+Sk+12P¯k+1/kfSk+12T+Sk+11P¯k+1/kxSk+11T+Rk+1)-1,
where(3.42)u¯k1=(Πk+112-Uk+112)f¯k/k+(Πk+113-Uk+113-Uk+112(Πk+123-Uk+123))d¯k/k,u¯k2=(Πk+123-Uk+123)d¯k/k.Finally, to correct the estimation of the state and the fault, we should follow these equations(3.43)x̂k+1/k+1=x¯k+1/k+1+Vk+112f¯k+1/k+1+Vk+113d¯k+1/k+1,P̂k+1/k+1x=P¯k+1/k+1x+Vk+112P¯k+1/k+1fVk+112T+Vk+113P¯k+1/k+1dVk+113T,f̂k+1/k+1=f¯k+1/k+1+Vk+123d¯k+1/k+1,P̂k+1/k+1f=P¯k+1/k+1f+Vk+123P¯k+1/k+1dVk+123T.
The OThSKF is optimal in the minimum mean square error (MMSE) sense. However, this filter loses its optimality, when the statistical properties of the models (3.1) and (3.2) are unknown or not perfectly known. So, it would be better to use a robust three-stage Kalman filter (RThKF) to get a good estimation of state and fault in the presence of unknown disturbance.


## 3.1. Augmented State Kalman Filter (ASKF)

Treatingxk,fk and dk as the augmented system state, the ASKF is described by(3.5)xk+1/ka=Akaxk/ka+Bkauk,(3.6)Pk+1/ka=AkaPk/kaAkaT+Qka,(3.7)xk+1/k+1a=xk+1/ka+Kk+1a(yk+1-Hk+1axk+1/ka),(3.8)Kk+1a=Pk+1/kaHk+1aT(Hk+1aPk+1/kaHk+1aT+Rk+1)-1,(3.9)Pk+1/k+1a=(I-Kk+1aHk+1a)Pk+1/ka,
where (3.10)x(·)a=[x(·)f(·)d(·)],Aka=[AkFkxEkx0I000I],Bka=[Bk00],Hka=[HkFkyEky],P(·)a=[P(·)xP(·)xfP(·)xdP(·)fxP(·)fP(·)fdP(·)dxP(·)dfP(·)d],Qka=[QkxQkxfQkxdQkfxQkfQkfdQkdxQkdfQkd].
The filter model (3.5)–(3.9) may be used to produce the optimal state estimate. But, this filter has two main disadvantages: the increase of the computational cost with the augmentation of the state dimension and the rise of numerical problems during the implementation [13]. So, to solve these problems, we should use the three-stage Kalman filtering (ThSKF) technique.


## 3.2.U-V Transformations

According to [13, 14], the ThSKF is obtained by the application of a three-stage U-V transformations in order to decouple the ASKF covariance matrices, that is, Pk+1/ka and Pk+1/k+1a. The aim is to find matrices Uk+1 and Vk+1 such that
(3.11)Pk+1/ka=Uk+1P¯k+1/kaUk+1T,Pk+1/k+1a=Vk+1P¯k+1/k+1aVk+1T,
with P¯(·)a=diag{P¯(·)x,P¯(·)f,P¯(·)d}, where P¯(·)x,P¯(·)f and P¯(·)d denote the transformed covariance matrices.We define the structures of theUk+1 and Vk+1 matrices as follows: (3.12)Uk+1=[IUk+112Uk+1130IUk+12300I],Vk+1=[IVk+112Vk+1130IVk+12300I],Uk+1ij and Vk+1ij for i=1or2 and  j=2or3 are to be determined later.Using these transformations (3.12), the equations (3.5), (3.7), and (3.8) are transformed into  (3.13)xk+1/ka=Uk+1x¯k+1/ka,xk+1/k+1a=Vk+1x¯k+1/k+1a,Kk+1a=Vk+1K¯k+1a.
The inverse transformations of Uk+1 and Vk+1 (3.12) will have this form  (3.14)Uk+1-1=Ũk+1=[IŨk+112Ũk+1130IŨk+12300I],Vk+1-1=Ṽk+1=[IṼk+112Ṽk+1130IṼk+12300I].
Using these inverse transformations (3.14), we have
(3.15)x¯k+1/ka=Ũk+1xk+1/ka,P¯k+1/ka=Ũk+1Pk+1/kaŨk+1T,x¯k+1/k+1a=Ṽk+1xk+1/k+1a,K¯k+1a=Ṽk+1Kk+1a,P¯k+1/k+1a=Ṽk+1Pk+1/k+1aṼk+1T,
where (3.16)x¯(·)a=[x¯(·)f¯(·)d¯(·)],P¯(·)a=[P¯(·)x000P¯(·)f000P¯(·)d],K¯(·)a=[K¯(·)xK¯(·)fK¯(·)d].


## 3.3. Decoupling

If we use the two-step substitution method, the filter model (3.5)–(3.9) is transformed into(3.17)x¯k+1/ka=Ũk+1Πk+1x¯k/ka+Ũk+1Bkauk,(3.18)P¯k+1/ka=Ũk+1(Πk+1Pk/kaΠk+1T+Qka)Ũk+1,(3.19)x¯k+1/k+1a=Ṽk+1Πk+1x¯k+1/ka+K¯k+1a(yk+1-Sk+1x¯k+1/ka),(3.20)K¯k+1a=Ṽk+1Πk+1P¯k+1/kaSk+1T(Sk+1P¯k+1/kaSk+1T+Rk+1)-1,(3.21)P¯k+1/k+1a=(Ṽk+1Πk+1-K¯k+1aSk+1)P¯k+1/ka(Ṽk+1Πk+1)-1,
where  (3.22)Πk+1=AkaVk=[AkΠk+112Πk+1130IΠk+12300I],Sk+1=[Sk+11Sk+12Sk+13],
with
(3.23)Πk+112=AkVk12+Fkx,Πk+113=AkVk13+FkxVk23+Ekx,Πk+123=Vk23,(3.24a)Sk+11=Hk+1,(3.24b)Sk+12=Hk+1Uk+112+Fk+1y,(3.24c)Sk+13=Hk+1Uk+113+Fk+1yUk+123+Ek+1y.Now, we start by developing the equations (3.18) and (3.21), respectively.From (3.18), we obtain(3.25)P¯k+1/kx=AkP¯k/kxAkT+Q¯k1,P¯k+1/kf=P¯k/kf+Q¯k2,P¯k+1/kd=P¯k/kd+Qkd,
where
(3.26)Q¯k1=Qkx+Πk+112P¯k/kfΠk+112T-Uk+112P¯k+1/kfUk+112T+Πk+113P¯k/kdΠk+113T-Uk+113P¯k+1/kdUk+113T,Q¯k2=Qkf+Πk+123P¯k/kdΠk+123T-Uk+123P¯k+1/kdUk+123T,Uk+113=(Πk+113P¯k/kd+Qkxd)(P¯k+1/kd)-1,Uk+123=(Πk+123P¯k/kd+Qkfd)(P¯k+1/kd)-1,Uk+112=(Πk+112P¯k/kf+Πk+113P¯k/kdΠk+123T-Uk+113P¯k/kfUk+123T+Qkxf)(P¯k+1/kf)-1.
The development of (3.21), leads to(3.27)P¯k+1/k+1x=(I-K¯k+1xSk+11)P¯k+1/kx,(3.28)P¯k+1/k+1f=(I-K¯k+1fSk+12)P¯k+1/kf,(3.29)P¯k+1/k+1d=(I-K¯k+1dSk+13)P¯k+1/kd,(3.30)Vk+112=Uk+112-K¯k+1xSk+12,(3.31)Vk+113=Uk+113-Vk+112K¯k+1fSk+13-K¯k+1xSk+13,(3.32)Vk+123=Uk+123-K¯k+1fSk+13.
With reference to (3.17), (3.19) and (3.20), we obtain, respectively,(3.33)x¯k+1/k=Akx¯k/k+Bkuk+u¯k1,(3.34)f¯k+1/k=f¯k/k+u¯k2,(3.35)d¯k+1/k=d¯k/k,(3.36)x¯k+1/k+1=x¯k+1/k+K¯k+1x(yk+1-Sk+11x¯k+1/k),(3.37)f¯k+1/k+1=f¯k+1/k+K¯k+1f(yk+1-Sk+11x¯k+1/k-Sk+12f¯k+1/k),(3.38)d¯k+1/k+1=d¯k+1/k+K¯k+1d(yk+1-Sk+11x¯k+1/k-Sk+12f¯k+1/k-Sk+13d¯k+1/k),(3.39)K¯k+1x=P¯k+1/kxSk+11T(Sk+11P¯k+1/kxSk+11T+Rk+1)-1,(3.40)K¯k+1f=P¯k+1/kfSk+12T(Sk+12P¯k+1/kfSk+12T+Sk+11P¯k+1/kxSk+11T+Rk+1)-1,(3.41)K¯k+1d=P¯k+1/kdSk+13T(Sk+13P¯k+1/kdSk+13T+Sk+12P¯k+1/kfSk+12T+Sk+11P¯k+1/kxSk+11T+Rk+1)-1,
where(3.42)u¯k1=(Πk+112-Uk+112)f¯k/k+(Πk+113-Uk+113-Uk+112(Πk+123-Uk+123))d¯k/k,u¯k2=(Πk+123-Uk+123)d¯k/k.Finally, to correct the estimation of the state and the fault, we should follow these equations(3.43)x̂k+1/k+1=x¯k+1/k+1+Vk+112f¯k+1/k+1+Vk+113d¯k+1/k+1,P̂k+1/k+1x=P¯k+1/k+1x+Vk+112P¯k+1/k+1fVk+112T+Vk+113P¯k+1/k+1dVk+113T,f̂k+1/k+1=f¯k+1/k+1+Vk+123d¯k+1/k+1,P̂k+1/k+1f=P¯k+1/k+1f+Vk+123P¯k+1/k+1dVk+123T.
The OThSKF is optimal in the minimum mean square error (MMSE) sense. However, this filter loses its optimality, when the statistical properties of the models (3.1) and (3.2) are unknown or not perfectly known. So, it would be better to use a robust three-stage Kalman filter (RThKF) to get a good estimation of state and fault in the presence of unknown disturbance.


## 4. Robust Three-Stage Kalman Filter (RThSKF)

In this section, we present a robust version of a filter to solve the joint fault and state estimation problem for linear stochastic system with unknown disturbance. We consider that the fault and the unknown disturbance only affect the state equation, that is, (Fky=Eky=0). This filter is obtained by modifying the measurement update equations of unknown disturbance subfilter and fault subfilter of the OThSKF.Equations (3.24b)-(3.24c) will be rewritten as follow:
(4.1)Sk+12=Hk+1Uk+112,Sk+13=Hk+1Uk+113.
By substituting (3.34) and (3.35) into (3.37) and (3.38) and combining (3.28) and (3.29) into (3.40) and (3.41), respectively, the measurement update equations of the unknown input subfilter and the fault subfilter are given as follow:(4.2)f¯k+1/k+1=(I-K¯k+1fSk+12)f¯k+1/k+K¯k+1f(yk+1-Sk+11x¯k+1/k),(4.3)K¯k+1f=P¯k+1/k+1fSk+12TCk+1-1,(4.4)d¯k+1/k+1=(I-K¯k+1dSk+13)d¯k/k+K¯k+1d(yk+1-Sk+11x¯k+1/k-Sk+12f¯k+1/k),(4.5)K¯k+1d=P¯k+1/k+1dSk+13T(Sk+12P¯k+1/kfSk+12T+Ck+1)-1,
where Ck+1=Hk+1P¯k+1/kxHk+1T+Rk+1.Firstly, to eliminate the two termsf¯k+1/k and d¯k/k, we will choose the matrix gains K¯k+1f and K¯k+1d that can satisfy the following algebraic constraints:(4.6)(I-K¯k+1fSk+12)=0,(4.7)(I-K¯k+1dSk+13)=0,(4.8)K¯k+1dSk+12=0.
In this case, (4.2) and (4.4) become
(4.9)f¯k+1/k+1=K¯k+1f(yk+1-Sk+11x¯k+1/k),d¯k+1/k+1=K¯k+1d(yk+1-Sk+11x¯k+1/k).
With substituting (4.3) and (4.5) into (4.6) and (4.7) and using (4.8), P¯k+1/k+1f and P¯k+1/k+1d can be rewritten as follows
(4.10)P¯k+1/k+1f=(Sk+12TCk+1-1Sk+12)+,P¯k+1/k+1d=(Sk+13TCk+1-1Sk+13)+,
where M+ denotes any one-condition generalized inverse of M, that is, MM+M=M.The gain matrixK¯k+1d is calculated in the assumption that the constraints (4.7) and (4.8) are satisfied:
(4.11)K¯k+1d=P¯k+1/k+1dSk+13TCk+1-1αk+1(I-Sk+12Sk+12+),
where αk+1 is an arbitrary matrix.Equations (3.33) and (3.34) are presented, respectively, as follows:
(4.12)x¯k+1/k=Akx̂k/k+Bkuk+ũk1,f¯k+1/k=f¯k/k+ũk2,
where(4.13)ũk1=(Fkx-Uk+112)f¯k/k+(Ekx-Uk+113+Uk+112Uk+123)d¯k/k,ũk2=(Πk+123-Uk+123)d¯k/k.
In order to return (4.12) robust against the fault and the unknown disturbance, we can choose ũk1=0 and ũk2=0.In this case, the new matricesUk+112, Uk+113, and Uk+123 can be written as follow:
(4.14)Uk+123=Πk+123=Vk23,Uk+112=Fkx,Uk+113=Ekx+FkxVk23.
Equation (4.12) becomes
(4.15)x¯k+1/k=Akx̂k/k+Bkuk,f¯k+1/k=f¯k/k.
Finally, an unbiased estimation of the state and the fault is obtained if and only if the constraints (4.6)–(4.8) are satisfied and(4.16)Vk+112=Fkx-K¯k+1xSk+12,Vk+123=-K¯k+1fHk+1Ekx,Vk+113=Ekx-K¯k+1xHk+1Ekx+Vk+112Vk+123.
Now, we summarize the robust three-stage Kalman filter (RThSKF) equations as follow: (4.17)x̂k+1/k+1=x¯k+1/k+1+Vk+112f¯k+1/k+1+Vk+113d¯k+1/k+1,P̂k+1/k+1x=P¯k+1/k+1x+Vk+112P¯k+1/k+1fVk+112T+Vk+113P¯k+1/k+1dVk+113T,f̂k+1/k+1=f¯k+1/k+1+Vk+123d¯k+1/k+1,P̂k+1/k+1f=P¯k+1/k+1f+Vk+123P¯k+1/k+1dVk+123T,
where x¯k+1/k+1 is given by(4.18)x¯k+1/k=Akx̂k/k+Bkuk,P¯k+1/kx=AkP̂k/kxAkT+Qkx,K¯k+1x=P¯k+1/kxHk+1TCk+1-1,x¯k+1/k+1=x¯k+1/k+K¯k+1x(yk+1-Hk+1x¯k+1/k),P¯k+1/k+1x=(I-K¯k+1xHk+1)P¯k+1/kx,
the fault f¯k+1/k+1 is given by(4.19)P¯k+1/k+1f=(Sk+12TCk+1-1Sk+12)+,K¯k+1f=P¯k+1/k+1fSk+12TCk+1-1,f¯k+1/k+1=K¯k+1f(yk+1-Hk+1x¯k+1/k),d¯k+1/k+1 is given by
(4.20)P¯k+1/k+1d=(Sk+13TCk+1-1Sk+13)+,K¯k+1d=P¯k+1/k+1dSk+13TCk+1-1αk+1(I-Sk+12(Sk+12)+),d¯k+1/k+1=K¯k+1d(yk+1-Hk+1x¯k+1/k).
The initial conditions are given as follow: x̂0/0=x¯0, P̂0/0x=P0x and V023.


## 5. Augmented Robust Three-Stage Kalman Filter (ARThSKF)

In this section, we consider that the fault and the unknown disturbance affect both the state and the measurement equations, that is, (Fky≠0 and Eky≠0). Using the same technique proposed by [9, 10], the measurement equation (2.2) can be rewritten as follows:
(5.1)yk=Hkxk+Fkyf̃k+Ekyd̃k+vk.
Hence, we augment the fault and the unknown disturbance as follow:
(5.2)dk→dka=[d̃kTdkT]T,fk→fka=[f̃kTfkT]T.
Thus, the system model (2.1) and (5.1) can be represented, respectively, by:(5.3)xk+1=Akxk+Bkuk+F¯kxfka+E¯kxdka+wkx,yk=Hkxk+F¯kyfka+E¯kydka+vk,
where(5.4)F¯kx=[0Fkx],F̅ky=[Fky0],E¯kx=[0Ekx],E¯ky=[Eky0].
Referring to (4.14) and (5.4), we define
(5.5)U¯k+112=F¯kx,U¯k+113=E¯kx+F¯kxV¯k23,U¯k+123=V¯k23.
Using (5.5), equations (3.24b)-(3.24c) can be written as follow:(5.6)S¯k+12=Hk+1F¯kx+F¯k+1y=[FkySk+12],S¯k+13=Hk+1E¯kx+E¯k+1y+S¯k+12V¯k23=[Ek+1y+Fk+1yVk23Hk+1Ek+1x+Sk+12Vk23]
The augmented fault and unknown disturbance RThKF (ARThSKF) is obtained by a direct application of the RThSKF with a minor modification
(5.7)x̂k+1/k+1=x¯k+1/k+1+V¯k+112fk+1/k+1+V¯k13dk+1/k+1,P̂k+1/k+1x=P¯k+1/k+1x+V¯k+112Pk+1/k+1fV¯k+112T+V¯k+113Pk+1/k+1dV¯k+113T,f̂k+1/k+1=fk+1/k+1+V¯k+123dk+1/k+1,P̂k+1/k+1f=Pk+1/k+1f+V¯k+123Pk+1/k+1dV¯k+123T,
where x¯k+1/k+1 is given by (4.18), fk+1/k+1 and dk+1/k+1 are given by(5.8)fk+1/k+1=Kk+1f(yk+1-Hk+1x¯k+1/k),Kk+1f=Pk+1/k+1fS¯k+12TCk+1-1,Pk+1/k+1f={S¯k+12TCk+1-1S¯k+12}+,dk+1/k+1=Kk+1d(yk+1-Hk+1x¯k+1/k),Kk+1d=Pk+1/k+1dS¯k+13TCk+1-1αk+1(I-S¯k+12S¯k+12+),Pk+1/k+1d={S¯k+13TCk+1-1S¯k+13}+,
where αk+1 is an arbitrary matrix.The unknown disturbance estimation errord̃k+1 is given by(5.9)d̃k+1=dk+1a-dk+1/k+1=(I-Kk+1dS¯k+13-Kk+1dS¯k+12Vk+123)dk+1a-Kk+1dS¯k+12fk+1a-Kk+1dek+1,
where ek+1=Hk+1(Akx̃k+wkx)+vk+1.The unbiasedness constraints to have an unbiased estimation of the unknown disturbance are given as follow:(5.10)I-Kk+1dS¯k+13=0,Kk+1dS¯k+12=0.
The fault estimation error f̃k+1 is presented as follows(5.11)f̃k+1=fk+1a-f̂k+1/k+1=fk+1a-fk+1/k+1-Vk+123dk+1/k+1=(I-Kk+1fS¯k+12)fk+1a-Kk+1f(S¯k+13-S¯k+12V¯k23)dk+1a-V¯k+123dk+1/k+1-Kk+1fek+1.
The estimator f̂k+1 is unbiased if and only if
(5.12)V¯k+123=-Kk+1f(S¯k+13-S¯k+12V¯k23),0=I-Kk+1fS¯k+12.
The state estimation error x̃k+1 has the following form:(5.13)x̃k+1=xk+1-x̂k+1/k+1=(I-K¯k+1x)(Akx̃k+wk)-K¯k+1xvk+1+(E¯kx-Kk+1xS¯k+12)fk+1a-V¯k+112fk+1/k+1+(E¯kx-K¯k+1xS¯k+13+K¯k+1xS¯k+12Vk23)dk+1a-V¯k+113dk+1/k+1.
The parameters V¯k+112 and V¯k+113 have the following relations:
(5.14)V¯k+112=F¯kx-K¯k+1xS¯k2,V¯k+113=E¯kx-K¯k+1x(S¯k+13-S¯k+12V¯k23)+V¯k+112V¯k+123.
If the gain matrices K¯k+1x, Kk+1f and Kk+1d are determined to satisfy the unbiasedness conditions (5.10) and (5.12), then the proposed filter ensure an unbiased minimum-variance estimation of state, fault and unknown disturbance. Indeed, the unknown disturbance, fault and state estimation errors are given, respectively, as follows:(5.15)d̃k+1=-Kk+1dek+1,f̃k+1=-Kk+1fek+1-V¯k+123Kk+1dek+1,x̃k+1=(I-K¯k+1x)(Akx̃k+wk)-K¯k+1xvk+1-V¯k+112Kk+1fek+1-V¯k+113Kk+1dek+1.


## 6. Illustrative Example

We consider the linearized model of a simplified longitudinal flight control system as follows:(6.1)xk+1=(Ak+ΔAk)xk+Bkuk+Fkafka+wkx,yk=(Hk+ΔHk)xk+Fksfks+vk,
where the state variables are: pitch angle δz, pitch rate ωz and normal velocity ηy, the control input uk is the elevator control signal. Fka and Fks are the matrices distribution of the actuator fault fka and sensor fault fks.The presented system equations can be rewritten as follow:(6.2)xk+1=Akxk+Bkuk+Fkxfk+Ekxdk+wkx,yk=Hkxk+Fkyfk+Ekydk+vk,
where Fkx and Fky are the distribution matrices of the fault vector in the state and measurement equations
(6.3)Fkx=[Fka0],Fky=[0Fks]
The terms Ekxdk and Ekydk represent the parameter perturbations in matrices Ak and Hk:(6.4)[ΔAkΔHk]=[EkxEky]ΔkM,dk=ΔkMxk,
where ΔkΔkT≤I.The numerical example given in [2, 3] is considered and slightly modified, where the parameters system are given as follows:  (6.5)Ak=[0.9944-0.1203-0.43020.00170.9902-0.074700.81870],Bk=[0.4252-0.00820.1813],Hk=I3×3,xk=[ηyωzδz]T,Qkx=diag{0.12,0.12,0.012},Rk=0.12I3×3.
We inject simultaneously two faults in the system, (6.6)[fkafks]=[2us(k-20)-2us(k-60)-us(k-30)+us(k-70)],
where us(k) is the unit-step function.In the simulation, the aerodynamic coefficients are perturbed;M=[-0.0200] and Δk=0.2sin(0.1k). In addition, we set uk=10, x0=[0-12]T, P0=0.1I3×3.We propose to apply the proposed filter (ARThSKF) to obtain a robust estimation of simultaneous actuator and sensor faults. The obtained results will be compared with existing filters in literature, in particular the ARTSKF [10].


### 6.1. Case  1

The matrices distribution of the fault and the unknown disturbance are taken as follow:(6.7)Ekx=[010]T,Eky=[100]T,Fka=[0.4252-0.00820.1813]T,Fks=[001]T.
In Figures 2 and 3, we focused on the presentation of the fault (fk) and the unknown disturbance (dk), respectively. We have plotted the actual value, the estimated values obtained by ARThSKF and ARTSKF.Actual faultfk and estimated fault f̂k/k (Case 1).
(a)(b)Figure 3
Actual unknown disturbancedk and its estimated d̂k/k (Case 1).The simulation results in Table1, show the average root mean square errors (RMSE) in the estimated states, fault and unknown disturbance. For example, the RMSE of the first component of state vector is calculated by (6.8)RMSE(x1,k)=1N∑k=1N(x1,k-x̂1,k/k)2.
From Figures 2, 3, and Table 1, the filtering performances of the ARThSKF appears to be much better that those of ARTSKF.Table 1
Performance of the ARThSKF and ARTSKF (Case1).
RMSEARThSKFARTSKFx1,k0.973110.0330x2,k0.14180.1418x3,k0.18270.3576fka1.17962.7494fks0.18870.3795dk0.22250.3035


### 6.2. Case  2

Here, we assume that the distribution matrices of the fault and the unknown disturbance are the followings:(6.9)Ekx=[000]T,Eky=[100]T,Fka=[0.4252-0.00820.1813]T,Fks=[001]T.
According to Table 2 and Figures 4 and 5, we note that the ARThSKF gives the best estimation of the first component of the state vector, the actuator fault and the unknown disturbance in comparison with the ARTSKF.Table 2
Performance of the ARThSKF and ARTSKF (Case2).
RMSEARThSKFARTSKFx1,k1.28935.3844x2,k0.14220.1418x3,k0.50280.5920fka2.59544.6809fks0.50870.5944dk1.28059.6895Actual faultfk and estimated fault f̂k/k (Case 2).
(a)(b)Figure 5
Actual unknown disturbancedk and its estimated d̂k/k (Case 2).


### 6.3. Case  3

In this case, the distribution matrices of the fault and the unknown disturbance are given by:(6.10)Ekx=[100]T,Eky=[100]T,Fka=[0.4252-0.00820.1813]T,Fks=[001]T.
In Figures 6 and 7, we deduce that the ARThSKF and ARTSKF have similar filtering performances. Indeed, the evaluation of the RMSE values presented in the Table 3 confirms this deduction.Table 3
Performance of the ARThSKF and ARTSKF (Case3).
RMSEARThSKFARTSKFx1,k0.45660.5315x2,k0.14240.1418x3,k0.50720.5924fka2.61484.6444fks0.51300.5946dk0.45690.5379Actual faultfk and estimated fault f̂k/k (Case 3).
(a)(b)Figure 7
Actual unknown disturbancedk and its estimated d̂k/k (Case 3).


### 6.4. Case  4

Now, we take the following values of the distrubition matrices of the fault and the unknown disturbance:(6.11)Ekx=[001]T,Eky=[100]T,Fka=[0.4252-0.00820.1813]T,Fks=[100]T.
The simulation results presented in Figures 8 and 9 and Table 4 show that the ARThSKF is a little better than the ARTSKF. In this case, we notes that the two filters ARThSKF and ARTSKF do not estimate suitably the sensor fault, since Eky=Fks.Table 4
Performance of the ARThSKF and ARTSKF (Case4).
RMSEARThSKFARTSKFx1,k2.78958.6757x2,k0.14160.1418x3,k0.15160.1516fka2.38234.6188fks3.72205.3151dk0.47060.5691Actual faultfk and estimated fault f̂k/k (Case 4).
(a)(b)Figure 9
Actual unknown disturbancedk and its estimated d̂k/k (Case 4).


## 6.1. Case  1

The matrices distribution of the fault and the unknown disturbance are taken as follow:(6.7)Ekx=[010]T,Eky=[100]T,Fka=[0.4252-0.00820.1813]T,Fks=[001]T.
In Figures 2 and 3, we focused on the presentation of the fault (fk) and the unknown disturbance (dk), respectively. We have plotted the actual value, the estimated values obtained by ARThSKF and ARTSKF.Actual faultfk and estimated fault f̂k/k (Case 1).
(a)(b)Figure 3
Actual unknown disturbancedk and its estimated d̂k/k (Case 1).The simulation results in Table1, show the average root mean square errors (RMSE) in the estimated states, fault and unknown disturbance. For example, the RMSE of the first component of state vector is calculated by (6.8)RMSE(x1,k)=1N∑k=1N(x1,k-x̂1,k/k)2.
From Figures 2, 3, and Table 1, the filtering performances of the ARThSKF appears to be much better that those of ARTSKF.Table 1
Performance of the ARThSKF and ARTSKF (Case1).
RMSEARThSKFARTSKFx1,k0.973110.0330x2,k0.14180.1418x3,k0.18270.3576fka1.17962.7494fks0.18870.3795dk0.22250.3035


## 6.2. Case  2

Here, we assume that the distribution matrices of the fault and the unknown disturbance are the followings:(6.9)Ekx=[000]T,Eky=[100]T,Fka=[0.4252-0.00820.1813]T,Fks=[001]T.
According to Table 2 and Figures 4 and 5, we note that the ARThSKF gives the best estimation of the first component of the state vector, the actuator fault and the unknown disturbance in comparison with the ARTSKF.Table 2
Performance of the ARThSKF and ARTSKF (Case2).
RMSEARThSKFARTSKFx1,k1.28935.3844x2,k0.14220.1418x3,k0.50280.5920fka2.59544.6809fks0.50870.5944dk1.28059.6895Actual faultfk and estimated fault f̂k/k (Case 2).
(a)(b)Figure 5
Actual unknown disturbancedk and its estimated d̂k/k (Case 2).


## 6.3. Case  3

In this case, the distribution matrices of the fault and the unknown disturbance are given by:(6.10)Ekx=[100]T,Eky=[100]T,Fka=[0.4252-0.00820.1813]T,Fks=[001]T.
In Figures 6 and 7, we deduce that the ARThSKF and ARTSKF have similar filtering performances. Indeed, the evaluation of the RMSE values presented in the Table 3 confirms this deduction.Table 3
Performance of the ARThSKF and ARTSKF (Case3).
RMSEARThSKFARTSKFx1,k0.45660.5315x2,k0.14240.1418x3,k0.50720.5924fka2.61484.6444fks0.51300.5946dk0.45690.5379Actual faultfk and estimated fault f̂k/k (Case 3).
(a)(b)Figure 7
Actual unknown disturbancedk and its estimated d̂k/k (Case 3).


## 6.4. Case  4

Now, we take the following values of the distrubition matrices of the fault and the unknown disturbance:(6.11)Ekx=[001]T,Eky=[100]T,Fka=[0.4252-0.00820.1813]T,Fks=[100]T.
The simulation results presented in Figures 8 and 9 and Table 4 show that the ARThSKF is a little better than the ARTSKF. In this case, we notes that the two filters ARThSKF and ARTSKF do not estimate suitably the sensor fault, since Eky=Fks.Table 4
Performance of the ARThSKF and ARTSKF (Case4).
RMSEARThSKFARTSKFx1,k2.78958.6757x2,k0.14160.1418x3,k0.15160.1516fka2.38234.6188fks3.72205.3151dk0.47060.5691Actual faultfk and estimated fault f̂k/k (Case 4).
(a)(b)Figure 9
Actual unknown disturbancedk and its estimated d̂k/k (Case 4).


## 7. Conclusion

In this paper, the problem of joint state and fault estimation for linear discrete-time stochastic systems with unknown disturbance is solved by using the robust three-stage Kalman filtering technique. We assume that the fault and the unknown disturbance affect both the system state and the output. To achieve this aim, a new robust filter named ARThSKF has been proposed by using an optimal three-stage Kalman filtering method and an augmented fault and unknown disturbance models. The unbiasedness conditions and minimum-variance property of this filter are also established. The proposed filter is applied efficiently to solve two problems: Firstly, it estimates the actuator and sensor faults simultaneously; Secondly, it establishes a comparative study with the existing literature results.



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*Source: 591639-2010-11-07.xml*