# Unbiased Minimum-Variance Filter for State and Fault Estimation of Linear Time-Varying Systems with Unknown Disturbances

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 recursive filter to joint fault and state estimation of a linear time-varying discrete systems in the presence of unknown disturbances. The method is based on the assumption that no prior knowledge about the dynamical evolution of the fault and the disturbance is available. As the fault affects both the state and the output, but the disturbance affects only the state system. Initially, we study the particular case when the direct feedthrough matrix of the fault has full rank. In the second case, we propose an extension of the previous case by considering the direct feedthrough matrix of the fault with an arbitrary rank. The resulting filter is optimal in the sense of the unbiased minimum-variance (UMV) criteria. A numerical example is given in order to illustrate the proposed method.


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## Body

## 1. Introduction

This paper is concerned with the problem of joint fault and state estimation of linear time-varying discrete-time stochastic systems in the presence of unknown disturbances. In spite of the presence of the unknown inputs, the robust estimate of the state and the fault enables us to implement a Fault Tolerant Control (FTC). A simple idea consists of using an architecture FTC resting on the compensation of the effect of the fault; see, for example, [1].Initially, we refer to the unknown input filtering problem largely treated in the literature by two different approaches. The first approach was based on the augmentation of the state vector with an unknown input vector. However, this approach assumes that the model for the dynamical evolution of the unknown inputs is available. When the statistical properties of the unknown input are perfectly known, the augmented state Kalman filter (ASKF) is an optimal solution. To reduce computation costs of the ASKF, Friedland [2] developed the two-stage Kalman filter (TSKF). This latter is optimal only for a constant bias. Many authors have extended Friedland's idea to treat the stochastic bias, for example, [3–5]. Recently, Kim et al. [6, 7] have developed an adaptive two-stage Kalman filter (ATSKF). The second approach treats the case when we do not have a prior knowledge about the dynamical evolution of the unknown input. Kitanidis [8] was the first to solve this problem using the linear unbiased minimum-variance (UMV). Darouach et al. [9] extended Kitanidis's filter using a parameterizing technique to have an optimal estimator filter (OEF). Hsieh [10] has developed an equivalent to Kitanidis's filter noted robust two-stage Kalman filter (RTSKF). Later, Hsieh [11] developed an optimal minmum variance filter (OMVF) to solve the performance degradation problem encountered in OEF. Gillijns and Moor [12] 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 [13] 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, Cheng et al. [14] proposed a recursive optimal filter with global optimality in the sense of unbiased minimum-variance over all linear unbiased estimators, but this filter is limited to estimate the state (i.e., no estimate of the unknown input). In [15], the author has extended an RTSF-noted ERTSF, where he solved a general case when the direct feedthrough matrix has an arbitrary rank.In this paper, we develop a new recursive filter to joint fault and state estimation for linear stochastic, discrete-time, and time-varying systems in the presence of unknown disturbances. We assume that the unknown disturbances affect only the state equation. While, the fault affects both the state and the output equations, as well, we consider that the direct feedthrough matrix has an arbitrary rank [15].This paper is organized as follows. Section2 states the problem of interest. Section 3 is dedicated to the design of the proposed filter. In Section 4, the obtained filter is summarized. An illustrative example is presented in Section 5. Finally, in Section 6 we conclude our obtained results.


## 2. Statement of the Problem

Assume the following linear stochastic discrete-time system:(2.1)xk+1=Akxk+Bkuk+Fkxfk+Ekxdk+wk,yk=Hkxk+Fkyfk+vk,
where xk∈ℜn is the state vector, yk∈ℜm is the observation vector, uk∈ℜr is the known control input, fk∈ℜp is the additive fault vector, and dk∈ℜq is the unknown disturbances. wk and vk are uncorrelated white noise sequences of zero-mean and covariance matrices are Qk≥0 and Rk>0, respectively. The disturbance dk is assumed to have no stochastic description and must be decoupled. The initial state is uncorrelated with the white noises processes wk and vk and x0 is a Gaussian random variable with ℰ[x0]=x̂0 and ℰ[(x0-x̂0)(x0-x̂0)T]=P0x where ℰ[·] denotes the expectation operator. The matrices Ak, Bk, Fkx, Ekx, Hk, and Fky are known and have appropriate dimensions. We consider the following assumptions:(i)
A1: (Hk,Ak) is observable,(ii)
A2: n>m≥p+q,(iii)
A3: 0<rank(Fky)≤p,(iv)
A4: rank(HkEk-1x)=rank(Ek-1x)=q.The objective of this paper is to design an unbiased minimum-variance linear estimator of the statexk and the fault fk without any information concerning the fault fk and the unknown disturbances dk. We can consider that the filter has the following form:(2.2)x̂k/k-1=Ak-1x̂k-1+Bk-1uk-1+Fk-1xf̂k-1,(2.3)f̂k=Kkf(yk-Hkx̂k/k-1),(2.4)x̂k=x̂k/k-1+Kkx(yk-Hkx̂k/k-1),
where the gain matrices Kkf∈ℜp×m and Kkx∈ℜn×m are determined to satisfy the following criteria.Unbiasedness
The estimator must satisfy(2.5)ℰ[f̃k]=ℰ[fk-f̂k]=0,(2.6)ℰ[x̃k]=ℰ[xk-x̂k]=0.Minimum-Variance
The estimator is determined such that(i)
the mean square errorsℰ[f̃kf̃kT] is minimized under the constraint (2.5);(ii)
thetrace{Pkx=ℰ[x̃kx̃kT]} is minimized under the constraints (2.5) and (2.6).


## 3. Filter Design

In this section, the fault and the state estimation are considered in the presence of the unknown disturbance in two cases with respect to assumptionA3. Section 3.1 is dedicated to deriving a UMV state and fault estimation filter if matrix Fky has full rank (i.e., rank(Fky)=p). A general case will be solved by an extension of the UMV state and fault estimation filter in Section 3.2.


### 3.1. UMV Fault and State Estimation

In this subsection, we will study a particular case when therank(Fky)=p. The gain matrices Kkf and Kkx will be determined as that (2.3) and (2.4) can give an unbiased estimation of fk and xk. In the next, the UMV fault and state estimation are solved.


#### 3.1.1. Unbiased Estimation

The innovation error has the following form(3.1)ỹk:=yk-Hkx̂k/k-1=Fkyfk+HkEk-1xdk-1+ek,
where(3.2)ek=Hkx¯̃k/k-1+vk,(3.3)x¯̃k/k-1=Ak-1x̃k-1+Fk-1xf̃k-1+wk-1.
The fault estimation error and the state estimation error are, respectively, given by(3.4)f̃k:=fk-f̂k=(I-KkfFky)fk-KkfHkEk-1xdk-1-Kkfek,(3.5)x̃k:=xk-x̂k=(I-KkxHk)x¯̃k/k-1-KkxFkyfk-(KkxHkEk-1x-Ek-1x)dk-1-Kkxvk.
The estimators x̂k and f̂k are unbiased if Kkf and Kkx satisfy the following constraints:(3.6)KkfGk=ℱk,(3.7)KkxGk=Γk,
where Gk=[FkyHkEk-1x], ℱk=[Ip0] and Γk=[0Ek-1x].Lemma 3.1.
Letrank(Fky)=p; under the assumptions A2 and A4, the necessary and sufficient condition so that the estimators (2.3) and (2.4) are unbiased as matrix Gk is full column rank, that is,
(3.8)rank(Gk)=rank(FkyHkEk-1x)=p+q.Proof.
Equations (3.6) and (3.7) can be written as
(3.9)[KkfKkx]Gk=[ℱkΓk].
The necessary and sufficient condition for the existence of the solution to (3.9) is
(3.10)rank[ℱkΓkGk]=rank(Gk).
We clarify   (3.10), and we obtain
(3.11)rank[Ip00Ek-1xFkyHkEk-1x]=rank(FkyHkEk-1x).
However, the matrix on the left of the equality has a rank equal to p+q. According to assumptions A2, A4 and rank(Fky)=p, this can be easily justified by considering that the faults and the unknown disturbances have an independent influences. The condition to satisfy is thus given by (3.8).


#### 3.1.2. UMV Estimation

In this subsection, we propose to determine the gain matricesKkf and Kkx by satisfying the unbiasedness constraints (2.5) and (2.6).(a) Fault Estimation
Equation (3.1) will be written as
(3.12)ỹk=Gk[fkdk-1]+ek.
Since, ek does not have unit variance and ỹk does not satisfy the assumptions of the Gauss-Markov theorem [16], the least-square (LS) solutions do not have a minimum-variance. Nevertheless, the covariance matrix of ek has the following form:
(3.13)Ck=ℰ[ekekT]=HkP¯k/k-1xHkT+Rk,
where P¯k/k-1x=ℰ[x¯̃k/k-1x¯̃k/k-1T].
For that,fk can be obtained by a weighted least-square (WLS) estimation with a weighting matrix Ck-1.Theorem 3.2.
Letx¯̃k/k-1 be unbiased; the matrix Ck is positive definite and the matrix Gk is full column rank; then to have a UMV fault estimation, the matrix gain Kkf is given by
(3.14)Kkf*=ℱkGk*,
where Gk*=(GkTCk-1Gk)-1GkTCk-1.Proof.
Under thatCk is positive definite and an invertible matrix Sk∈ℜm×m verifes SkSkT=Ck, so we can rewrite (3.12) as follow. 											(3.15)Sk-1ỹk=Sk-1Gk[fkdk-1]+Sk-1ek.
If the matrix Gk is full column rank, that is, rank(Gk)=p+q, then the matrix GkTCk-1Gk is invertible. Solving (3.15) by an LS estimation is equivalent to solve (3.12) by WLS solution:
(3.16)f̂k*=ℱk(GkTCk-1Gk)-1GkTCk-1ỹk.
In this way, we can consider that Sk-1ek has a unit variance and  (3.15) can satisfy the assumptions of the Gauss-Markov theorem. Hence, (3.16) is the UMV estimate of fk.In this case, the fault estimation error is rewritten as follows:(3.17)f̃k*=-Kkf*ek.
Using  (3.17), the covariance matrix Pkf is given by(3.18)Pkf*=ℰ[f̃k*f̃k*T]=Kkf*CkKkf*T=ℱk(GkTCk-1Gk)-1ℱkT.(b) State Estimation
In this part, we propose to obtain an unbiased minimum variance state estimator to calculate the gain matrixKkx which will minimize the trace of covariance matrix Pkx under the unbiasedness constraint (3.7).Theorem 3.3.
LetGkTCk-1Gk be nonsingular; then the state gain matrix Kkx is given by
(3.19)Kkx*=P¯k/k-1xHkTCk-1(I-GkGk*)+ΓkGk*.Proof.
Considering (3.7) and (3.5), we determine Pkx as follows:
(3.20)Pkx=(I-KkxHk)P¯k/k-1x(I-KkxHk)T+KkxRkKkxT=KkxCkKkxT-2P¯k/k-1xHkTKkxT+P¯k/k-1x.
So, the optimization problem can be solved using Lagrange multipliers:
(3.21)trace{KkxCkKkxT-2P¯k/k-1xHkTKkxT+P¯k/k-1x}-2trace{(KkxGk-Γk)ΛkT},
where Λk is the matrix of Lagrange multipliers.
To derive (3.21) with respect to Kkx, we obtain
(3.22)CkKkx*T-HkP¯k/k-1x-GkΛkT=0.
Equations (3.7) and (3.22) form the linear system of equations:
(3.23)[Ck-GkGkT0][Kkx*TΛkT]=[HkP¯k/k-1xΓkT].
If GkTCk-1Gk is nonsingular,   (3.23) will have a unique solution.


#### 3.1.3. The Filter Time Update

From   (3.3), the prior covariance matrix P¯k/k-1x=ℰ[x¯̃k/k-1x¯̃k/k-1T] has the following form:(3.24)P¯k/k-1x=[Ak-1Fk-1x][Pk-1x*Pk-1xf*Pk-1fx*Pk-1f*][Ak-1TFk-1xT]+Qk-1,
where Pkxf*:=ℰ[x̃k*f̃k*T] is calculated by using  (2.3) and (2.4):(3.25)Pkxf*=-(I-Kkx*Hk)P¯k/k-1xHkTKkf*T+Kkx*RkKkf*T.


### 3.2. Extended UMV Fault and State Estimation

In this section, we consider that0<rank(Fky)≤p. To solve this interesting problem we will use the proposed approach by Hsieh in (2009) [15]. If we introduce (3.2) and (3.3) into (3.4), then we will be able to write the fault error estimation as follows:(3.26)f̃k:=(I-KkfFky)fk-KkfHkEk-1xdk-1-Kkf(Hkx¯̃k/k-1+vk)=-KkfHkFk-1xf̃k-1-KkfHkAk-1x̃k-1+(I-KkfFky)fk-KkfHkEk-1xdk-1-KkfHkwk-1-Kkfvk.
Assuming that ℰ[x̃k-1]=0 we define the following notations:(3.27)Φk=KkfFky=Ip-Σk,Gkf=KkfHkFk-1x,Gkd=KkfHkEk-1x,
where Σk=I-(Fky)+Fky.Using the same technique presented in [15], the expectation value of the f̃k is given by(3.28)ℰ[f̃k]=Σkfk-GkfΣk-1fk-1+Gkf(Gk-1fΣk-2)fk-2+⋯+(-1)kGkf×⋯×G2f(G1fΣ0)f0-Gkddk-1+GkfGk-1ddk-2+⋯+(-1)kGkf×⋯×G1fG1dd0.
When we assume that GifΣi-1=0 and Gid=0 for i=1,…,k, then we obtain(3.29)ℰ[f̃k]=Σkfk.
To obtain an unbiased estimation of the fault, the gain matrix Kkf should respect the following constraints:(3.30)KkfFky=Φk,KkfHkFk-1xΣk-1=0,KkfHkEk-1x=0.
Equation (3.30)  can be written as(3.31)KkfG¯k=ℱ¯k,
where(3.32)G¯k=[FkyHkFk-1xΣk-1HkEk-1x],ℱ¯k=[Φk00].
Using (3.31), we can determine the gain matrix Kkf as follows:(3.33)Kkf*=ℱ¯kG¯k*,
where G¯k*=(G¯kTCk-1G¯k)+G¯kTCk-1 and X+ denotes the Moore-Penrose pseudoinverse of X.The state estimation error is given by(3.34)x̃k:=(I-KkxHk)x¯̃k/k-1-KkxFkyfk-(KkxHkEk-1x-Ek-1x)dk-1-Kkxvk=(I-KkxHk)Ak-1x̃k-1+(I-KkxHk)Fk-1xf̃k-(KkxHkEk-1x-Ek-1x)dk-1-KkxFkyfk+(I-KkxHk)wk-1-Kkxvk.
To have an unbiased estimation of the state, the gain matrix Kkx should satisfy the following constraints:(3.35)KkxFky=0,KkxHkFk-1xΣk-1=Fk-1xΣk-1,KkxHkEk-1x=Ek-1x.
From (3.35), we obtain(3.36)KkxG¯k=Γ¯k,
where(3.37)Γ¯k=[0Fk-1xΣk-1Ek-1x].
Refering to (3.34), we calculate the error state covariance matrix:(3.38)Pkx=(I-KkxHk)P¯k/k-1x(I-KkxHk)T+KkxRkKkxT=KkxCkKkxT-2P¯k/k-1xHkTKkxT+P¯k/k-1x.
The gain matrix Kkx is determin by minimizing the trace of the covariance matrix Pkx such as (3.36). Using the Kitanidis method [8], we obtain(3.39)[Ck-G¯kG¯kT0][Kkx*TΛkT]=[HkP¯k/k-1xΓ¯kT],
where Λk is the matrix of Lagrange multipliers.IfG¯kTCk-1G¯k is nonsingular,   (3.39) will have a unique solution. So, the gain matrix Kkx is given by(3.40)Kkx*=P¯k/k-1xHkTCk-1(I-G¯kG¯k*)+Γ¯kG¯k*.
The filter time update is the same as that given by (3.24) and (3.25). The obtained filters will be tested by an illustrative example in Section 5.


## 3.1. UMV Fault and State Estimation

In this subsection, we will study a particular case when therank(Fky)=p. The gain matrices Kkf and Kkx will be determined as that (2.3) and (2.4) can give an unbiased estimation of fk and xk. In the next, the UMV fault and state estimation are solved.


### 3.1.1. Unbiased Estimation

The innovation error has the following form(3.1)ỹk:=yk-Hkx̂k/k-1=Fkyfk+HkEk-1xdk-1+ek,
where(3.2)ek=Hkx¯̃k/k-1+vk,(3.3)x¯̃k/k-1=Ak-1x̃k-1+Fk-1xf̃k-1+wk-1.
The fault estimation error and the state estimation error are, respectively, given by(3.4)f̃k:=fk-f̂k=(I-KkfFky)fk-KkfHkEk-1xdk-1-Kkfek,(3.5)x̃k:=xk-x̂k=(I-KkxHk)x¯̃k/k-1-KkxFkyfk-(KkxHkEk-1x-Ek-1x)dk-1-Kkxvk.
The estimators x̂k and f̂k are unbiased if Kkf and Kkx satisfy the following constraints:(3.6)KkfGk=ℱk,(3.7)KkxGk=Γk,
where Gk=[FkyHkEk-1x], ℱk=[Ip0] and Γk=[0Ek-1x].Lemma 3.1.
Letrank(Fky)=p; under the assumptions A2 and A4, the necessary and sufficient condition so that the estimators (2.3) and (2.4) are unbiased as matrix Gk is full column rank, that is,
(3.8)rank(Gk)=rank(FkyHkEk-1x)=p+q.Proof.
Equations (3.6) and (3.7) can be written as
(3.9)[KkfKkx]Gk=[ℱkΓk].
The necessary and sufficient condition for the existence of the solution to (3.9) is
(3.10)rank[ℱkΓkGk]=rank(Gk).
We clarify   (3.10), and we obtain
(3.11)rank[Ip00Ek-1xFkyHkEk-1x]=rank(FkyHkEk-1x).
However, the matrix on the left of the equality has a rank equal to p+q. According to assumptions A2, A4 and rank(Fky)=p, this can be easily justified by considering that the faults and the unknown disturbances have an independent influences. The condition to satisfy is thus given by (3.8).


### 3.1.2. UMV Estimation

In this subsection, we propose to determine the gain matricesKkf and Kkx by satisfying the unbiasedness constraints (2.5) and (2.6).(a) Fault Estimation
Equation (3.1) will be written as
(3.12)ỹk=Gk[fkdk-1]+ek.
Since, ek does not have unit variance and ỹk does not satisfy the assumptions of the Gauss-Markov theorem [16], the least-square (LS) solutions do not have a minimum-variance. Nevertheless, the covariance matrix of ek has the following form:
(3.13)Ck=ℰ[ekekT]=HkP¯k/k-1xHkT+Rk,
where P¯k/k-1x=ℰ[x¯̃k/k-1x¯̃k/k-1T].
For that,fk can be obtained by a weighted least-square (WLS) estimation with a weighting matrix Ck-1.Theorem 3.2.
Letx¯̃k/k-1 be unbiased; the matrix Ck is positive definite and the matrix Gk is full column rank; then to have a UMV fault estimation, the matrix gain Kkf is given by
(3.14)Kkf*=ℱkGk*,
where Gk*=(GkTCk-1Gk)-1GkTCk-1.Proof.
Under thatCk is positive definite and an invertible matrix Sk∈ℜm×m verifes SkSkT=Ck, so we can rewrite (3.12) as follow. 											(3.15)Sk-1ỹk=Sk-1Gk[fkdk-1]+Sk-1ek.
If the matrix Gk is full column rank, that is, rank(Gk)=p+q, then the matrix GkTCk-1Gk is invertible. Solving (3.15) by an LS estimation is equivalent to solve (3.12) by WLS solution:
(3.16)f̂k*=ℱk(GkTCk-1Gk)-1GkTCk-1ỹk.
In this way, we can consider that Sk-1ek has a unit variance and  (3.15) can satisfy the assumptions of the Gauss-Markov theorem. Hence, (3.16) is the UMV estimate of fk.In this case, the fault estimation error is rewritten as follows:(3.17)f̃k*=-Kkf*ek.
Using  (3.17), the covariance matrix Pkf is given by(3.18)Pkf*=ℰ[f̃k*f̃k*T]=Kkf*CkKkf*T=ℱk(GkTCk-1Gk)-1ℱkT.(b) State Estimation
In this part, we propose to obtain an unbiased minimum variance state estimator to calculate the gain matrixKkx which will minimize the trace of covariance matrix Pkx under the unbiasedness constraint (3.7).Theorem 3.3.
LetGkTCk-1Gk be nonsingular; then the state gain matrix Kkx is given by
(3.19)Kkx*=P¯k/k-1xHkTCk-1(I-GkGk*)+ΓkGk*.Proof.
Considering (3.7) and (3.5), we determine Pkx as follows:
(3.20)Pkx=(I-KkxHk)P¯k/k-1x(I-KkxHk)T+KkxRkKkxT=KkxCkKkxT-2P¯k/k-1xHkTKkxT+P¯k/k-1x.
So, the optimization problem can be solved using Lagrange multipliers:
(3.21)trace{KkxCkKkxT-2P¯k/k-1xHkTKkxT+P¯k/k-1x}-2trace{(KkxGk-Γk)ΛkT},
where Λk is the matrix of Lagrange multipliers.
To derive (3.21) with respect to Kkx, we obtain
(3.22)CkKkx*T-HkP¯k/k-1x-GkΛkT=0.
Equations (3.7) and (3.22) form the linear system of equations:
(3.23)[Ck-GkGkT0][Kkx*TΛkT]=[HkP¯k/k-1xΓkT].
If GkTCk-1Gk is nonsingular,   (3.23) will have a unique solution.


### 3.1.3. The Filter Time Update

From   (3.3), the prior covariance matrix P¯k/k-1x=ℰ[x¯̃k/k-1x¯̃k/k-1T] has the following form:(3.24)P¯k/k-1x=[Ak-1Fk-1x][Pk-1x*Pk-1xf*Pk-1fx*Pk-1f*][Ak-1TFk-1xT]+Qk-1,
where Pkxf*:=ℰ[x̃k*f̃k*T] is calculated by using  (2.3) and (2.4):(3.25)Pkxf*=-(I-Kkx*Hk)P¯k/k-1xHkTKkf*T+Kkx*RkKkf*T.


## 3.1.1. Unbiased Estimation

The innovation error has the following form(3.1)ỹk:=yk-Hkx̂k/k-1=Fkyfk+HkEk-1xdk-1+ek,
where(3.2)ek=Hkx¯̃k/k-1+vk,(3.3)x¯̃k/k-1=Ak-1x̃k-1+Fk-1xf̃k-1+wk-1.
The fault estimation error and the state estimation error are, respectively, given by(3.4)f̃k:=fk-f̂k=(I-KkfFky)fk-KkfHkEk-1xdk-1-Kkfek,(3.5)x̃k:=xk-x̂k=(I-KkxHk)x¯̃k/k-1-KkxFkyfk-(KkxHkEk-1x-Ek-1x)dk-1-Kkxvk.
The estimators x̂k and f̂k are unbiased if Kkf and Kkx satisfy the following constraints:(3.6)KkfGk=ℱk,(3.7)KkxGk=Γk,
where Gk=[FkyHkEk-1x], ℱk=[Ip0] and Γk=[0Ek-1x].Lemma 3.1.
Letrank(Fky)=p; under the assumptions A2 and A4, the necessary and sufficient condition so that the estimators (2.3) and (2.4) are unbiased as matrix Gk is full column rank, that is,
(3.8)rank(Gk)=rank(FkyHkEk-1x)=p+q.Proof.
Equations (3.6) and (3.7) can be written as
(3.9)[KkfKkx]Gk=[ℱkΓk].
The necessary and sufficient condition for the existence of the solution to (3.9) is
(3.10)rank[ℱkΓkGk]=rank(Gk).
We clarify   (3.10), and we obtain
(3.11)rank[Ip00Ek-1xFkyHkEk-1x]=rank(FkyHkEk-1x).
However, the matrix on the left of the equality has a rank equal to p+q. According to assumptions A2, A4 and rank(Fky)=p, this can be easily justified by considering that the faults and the unknown disturbances have an independent influences. The condition to satisfy is thus given by (3.8).


## 3.1.2. UMV Estimation

In this subsection, we propose to determine the gain matricesKkf and Kkx by satisfying the unbiasedness constraints (2.5) and (2.6).(a) Fault Estimation
Equation (3.1) will be written as
(3.12)ỹk=Gk[fkdk-1]+ek.
Since, ek does not have unit variance and ỹk does not satisfy the assumptions of the Gauss-Markov theorem [16], the least-square (LS) solutions do not have a minimum-variance. Nevertheless, the covariance matrix of ek has the following form:
(3.13)Ck=ℰ[ekekT]=HkP¯k/k-1xHkT+Rk,
where P¯k/k-1x=ℰ[x¯̃k/k-1x¯̃k/k-1T].
For that,fk can be obtained by a weighted least-square (WLS) estimation with a weighting matrix Ck-1.Theorem 3.2.
Letx¯̃k/k-1 be unbiased; the matrix Ck is positive definite and the matrix Gk is full column rank; then to have a UMV fault estimation, the matrix gain Kkf is given by
(3.14)Kkf*=ℱkGk*,
where Gk*=(GkTCk-1Gk)-1GkTCk-1.Proof.
Under thatCk is positive definite and an invertible matrix Sk∈ℜm×m verifes SkSkT=Ck, so we can rewrite (3.12) as follow. 											(3.15)Sk-1ỹk=Sk-1Gk[fkdk-1]+Sk-1ek.
If the matrix Gk is full column rank, that is, rank(Gk)=p+q, then the matrix GkTCk-1Gk is invertible. Solving (3.15) by an LS estimation is equivalent to solve (3.12) by WLS solution:
(3.16)f̂k*=ℱk(GkTCk-1Gk)-1GkTCk-1ỹk.
In this way, we can consider that Sk-1ek has a unit variance and  (3.15) can satisfy the assumptions of the Gauss-Markov theorem. Hence, (3.16) is the UMV estimate of fk.In this case, the fault estimation error is rewritten as follows:(3.17)f̃k*=-Kkf*ek.
Using  (3.17), the covariance matrix Pkf is given by(3.18)Pkf*=ℰ[f̃k*f̃k*T]=Kkf*CkKkf*T=ℱk(GkTCk-1Gk)-1ℱkT.(b) State Estimation
In this part, we propose to obtain an unbiased minimum variance state estimator to calculate the gain matrixKkx which will minimize the trace of covariance matrix Pkx under the unbiasedness constraint (3.7).Theorem 3.3.
LetGkTCk-1Gk be nonsingular; then the state gain matrix Kkx is given by
(3.19)Kkx*=P¯k/k-1xHkTCk-1(I-GkGk*)+ΓkGk*.Proof.
Considering (3.7) and (3.5), we determine Pkx as follows:
(3.20)Pkx=(I-KkxHk)P¯k/k-1x(I-KkxHk)T+KkxRkKkxT=KkxCkKkxT-2P¯k/k-1xHkTKkxT+P¯k/k-1x.
So, the optimization problem can be solved using Lagrange multipliers:
(3.21)trace{KkxCkKkxT-2P¯k/k-1xHkTKkxT+P¯k/k-1x}-2trace{(KkxGk-Γk)ΛkT},
where Λk is the matrix of Lagrange multipliers.
To derive (3.21) with respect to Kkx, we obtain
(3.22)CkKkx*T-HkP¯k/k-1x-GkΛkT=0.
Equations (3.7) and (3.22) form the linear system of equations:
(3.23)[Ck-GkGkT0][Kkx*TΛkT]=[HkP¯k/k-1xΓkT].
If GkTCk-1Gk is nonsingular,   (3.23) will have a unique solution.


## 3.1.3. The Filter Time Update

From   (3.3), the prior covariance matrix P¯k/k-1x=ℰ[x¯̃k/k-1x¯̃k/k-1T] has the following form:(3.24)P¯k/k-1x=[Ak-1Fk-1x][Pk-1x*Pk-1xf*Pk-1fx*Pk-1f*][Ak-1TFk-1xT]+Qk-1,
where Pkxf*:=ℰ[x̃k*f̃k*T] is calculated by using  (2.3) and (2.4):(3.25)Pkxf*=-(I-Kkx*Hk)P¯k/k-1xHkTKkf*T+Kkx*RkKkf*T.


## 3.2. Extended UMV Fault and State Estimation

In this section, we consider that0<rank(Fky)≤p. To solve this interesting problem we will use the proposed approach by Hsieh in (2009) [15]. If we introduce (3.2) and (3.3) into (3.4), then we will be able to write the fault error estimation as follows:(3.26)f̃k:=(I-KkfFky)fk-KkfHkEk-1xdk-1-Kkf(Hkx¯̃k/k-1+vk)=-KkfHkFk-1xf̃k-1-KkfHkAk-1x̃k-1+(I-KkfFky)fk-KkfHkEk-1xdk-1-KkfHkwk-1-Kkfvk.
Assuming that ℰ[x̃k-1]=0 we define the following notations:(3.27)Φk=KkfFky=Ip-Σk,Gkf=KkfHkFk-1x,Gkd=KkfHkEk-1x,
where Σk=I-(Fky)+Fky.Using the same technique presented in [15], the expectation value of the f̃k is given by(3.28)ℰ[f̃k]=Σkfk-GkfΣk-1fk-1+Gkf(Gk-1fΣk-2)fk-2+⋯+(-1)kGkf×⋯×G2f(G1fΣ0)f0-Gkddk-1+GkfGk-1ddk-2+⋯+(-1)kGkf×⋯×G1fG1dd0.
When we assume that GifΣi-1=0 and Gid=0 for i=1,…,k, then we obtain(3.29)ℰ[f̃k]=Σkfk.
To obtain an unbiased estimation of the fault, the gain matrix Kkf should respect the following constraints:(3.30)KkfFky=Φk,KkfHkFk-1xΣk-1=0,KkfHkEk-1x=0.
Equation (3.30)  can be written as(3.31)KkfG¯k=ℱ¯k,
where(3.32)G¯k=[FkyHkFk-1xΣk-1HkEk-1x],ℱ¯k=[Φk00].
Using (3.31), we can determine the gain matrix Kkf as follows:(3.33)Kkf*=ℱ¯kG¯k*,
where G¯k*=(G¯kTCk-1G¯k)+G¯kTCk-1 and X+ denotes the Moore-Penrose pseudoinverse of X.The state estimation error is given by(3.34)x̃k:=(I-KkxHk)x¯̃k/k-1-KkxFkyfk-(KkxHkEk-1x-Ek-1x)dk-1-Kkxvk=(I-KkxHk)Ak-1x̃k-1+(I-KkxHk)Fk-1xf̃k-(KkxHkEk-1x-Ek-1x)dk-1-KkxFkyfk+(I-KkxHk)wk-1-Kkxvk.
To have an unbiased estimation of the state, the gain matrix Kkx should satisfy the following constraints:(3.35)KkxFky=0,KkxHkFk-1xΣk-1=Fk-1xΣk-1,KkxHkEk-1x=Ek-1x.
From (3.35), we obtain(3.36)KkxG¯k=Γ¯k,
where(3.37)Γ¯k=[0Fk-1xΣk-1Ek-1x].
Refering to (3.34), we calculate the error state covariance matrix:(3.38)Pkx=(I-KkxHk)P¯k/k-1x(I-KkxHk)T+KkxRkKkxT=KkxCkKkxT-2P¯k/k-1xHkTKkxT+P¯k/k-1x.
The gain matrix Kkx is determin by minimizing the trace of the covariance matrix Pkx such as (3.36). Using the Kitanidis method [8], we obtain(3.39)[Ck-G¯kG¯kT0][Kkx*TΛkT]=[HkP¯k/k-1xΓ¯kT],
where Λk is the matrix of Lagrange multipliers.IfG¯kTCk-1G¯k is nonsingular,   (3.39) will have a unique solution. So, the gain matrix Kkx is given by(3.40)Kkx*=P¯k/k-1xHkTCk-1(I-G¯kG¯k*)+Γ¯kG¯k*.
The filter time update is the same as that given by (3.24) and (3.25). The obtained filters will be tested by an illustrative example in Section 5.


## 4. Summary of Filter Equations

We   suppose to know the following:(i)
the known inputuk,(ii)
matricesAk, Bk, Hk, Fkx, Fky and Ekx,(iii)
covariance matricesQkx and Rk,(iv)
initial valuesx̂0 and P0x.We assume that the estimate of the initial state is unbiased and we take the initial covariance matrixP¯0/-1x=P0x.Step 1.
Estimation of fault is(4.1)Ck=HkP¯k/k-1xHkT+Rk,G¯k=[FkyHkFk-1xΣk-1HkEk-1x],ℱ¯k=[Φk00],G¯k*=(G¯kTCk-1G¯k)+G¯kTCk-1,Kkf=ℱ¯kG¯k*,f̂k=Kkf(yk-Hkx̂k/k-1),Pkf=KkfCkKkfT.Step 2.
Measurement update is(4.2)Γ¯k=[0Fk-1xΣk-1Ek-1x],Kkx=P¯k/k-1xHkTCk-1(I-G¯kG¯k*)+Γ¯kG¯k*,x̂k=x̂k/k-1+Kkx(yk-Hkx̂k/k-1),Pkx=(I-KkxHk)P¯k/k-1x(I-KkxHk)T+KkxRkKkxT,Pkxf=-(I-KkxHk)P¯k/k-1xHkTKkfT+KkxRkKkfT.Step 3.
Time update is(4.3)x̂k+1/k=Akx̂k+Bkuk+Fkxf̂k,P¯k+1/kx=[AkFkx][PkxPkxfPkfxPkf][AkTFkxT]+Qk.Remark 4.1.
Ifrank(Fky)=p, then we have Σk=0 for all k≥0 and it is easier to use the filter obtained in  Section 3.1. In this case, the gain matrices Kkf and Kkx are given by (3.14) and (3.19), respectively.Remark 4.2.
These remarks give the relationships with the existing literature results.(i)
IfEkx=0 and 0<rank(Fky)≤p, the obtained filter is equivalent to ERTSF developed by [15].(ii)
IfEkx=0 and rank(Fky)=p, then we have Σk=0 for all k≥0 and the obtained filter is equivalent to RTSF proposed by [13].(iii)
In the case whereFkx=0 and Fky=0, the filter of [8] is obtained.(iv)
In the case whereFkx=0, Fky=0 and Ekx=0, we obtain the standard Kalman filter.


## 5. An Illustrative Example

To apply our proposed filters we will treat different cases to respect assumptionA3. The parameters of the system (2.1)  are given by(5)xk=[x1,kx2,kx3,k],Ak=[ak0.10.20.10.60.30.50.10.25],ak=0.4+0.3sin(0.2k),Bk=[2-1.50.5],Fkx=[0.50.71.51.10.80.9],Ekx=[021],Hk=[1-100100-1-1],Qk=0.1I3×3,Rk=0.01I3×3,x0=[1-21],x̂0=[000],P0x=I3×3.
In this simulation, four cases of Fky will be considered as follows:(5.2)(Fky)1=[21.40.60.30.21.6],(Fky)2=[210.60.30.20.1],(Fky)3=[200.600.20],(Fky)4=[01.400.301.6].
We assume that the fault and the disturbance are given by(5.3)[f1,kf2,k]=[5us(k-10)-5us(k-70)4us(k-30)-4us(k-65)],dk=4us(k-15)-4us(k-55),
where us(k) is the unit-step function.Figure1 presents the input sequence of the system (2.1). The simulation time is 100 time steps.Figure 1
Known inputuk.In Figure2, we have plotted the actual and the estimated value of the first element of the state vector xk=[xk1xk2xk3]T. Figures 3 and 4 present the actual and the estimated value of the first element and the second element of the fault vector fk=[fk1fk2]T, respectively. The convergence of the trace of covariance matrices Pkx and Pkf is shown, respectively, in Figures 5 and 6.Actual statexk1 and estimated state x̂k1.
(a)(b)(c)(d)Actual faultfk1 and estimated fault f̂k1.
(a)(b)(c)(d)Actual faultfk2 and estimated fault f̂k2.
(a)(b)(c)(d)Trace of the covariance matrixPkx.
(a)(b)(c)(d)Trace of the covariance matrixPkf.
(a)(b)(c)(d)The simulation results in Table1 show the average root mean square errors (RMSEs) in the estimated states and faults.Table 1
RMSE values.
RMSEx1,kx2,kx3,kf1,kf2,k(Fky)10.44960.15310.51140.38750.4139(Fky)20.52480.13460.18630.49410.9612(Fky)30.63010.15420.21290.25652.3664(Fky)40.33780.09380.36473.35410.4139According to Figures2–6 and Table 1, we can conclude that if the matrix Fky has full rank, then we obtain a best estimate of the state and the fault (Figures 2(a), 3(a) and 4(a)). On the other hand, when the matrix Fky has not full rank, it is not possible to obtain a best estimate of the various components of the fault (Figures 3(b), 3(c), 3(d), 4(b), 4(c) and 4(d)), but the state estimation remains acceptable (Figures 2(b), 2(c), and 2(d)).


## 6. Conclusion

In this paper, the problem of the state and the fault estimation are solved in the case of stochastic linear discrete-time and varying-time systems. A recursive unbiased minimum-variance (UMV) filter is proposed when the direct feedthrough matrix of the fault has an arbitrary rank. The advantages of this filter are especially important in the case when we do not have any priory information about the unknown disturbances and the fault. An application of the proposed filter has been shown by an illustrative example. This recursive filter is able to obtain a robust and unbiased minimum-variance of the state and the fault estimation in spite of the presence of the unknown disturbances.



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*Source: 343586-2010-04-06.xml*