# Simultaneous Robust Fault and State Estimation for Linear Discrete-Time Uncertain Systems

Authors: Feten Gannouni; Fayçal Ben Hmida
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

We consider the problem of robust simultaneous fault and state estimation for linear uncertain discrete-time systems with unknown faults which affect both the state and the observation matrices. Using transformation of the original system, a new robust proportional integral filter (RPIF) having an error variance with an optimized guaranteed upper bound for any allowed uncertainty is proposed to improve robust estimation of unknown time-varying faults and to improve robustness against uncertainties. In this study, the minimization problem of the upper bound of the estimation error variance is formulated as a convex optimization problem subject to linear matrix inequalities (LMI) for all admissible uncertainties. The proportional and the integral gains are optimally chosen by solving the convex optimization problem. Simulation results are given in order to illustrate the performance of the proposed filter, in particular to solve the problem of joint fault and state estimation.


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## Body

## 1. Introduction

This paper is concerned with the problem of joint fault and state estimation of linear discrete-time uncertain systems under convex bounded parametric uncertainty. This problem is solved by using a robust filtering approach to produce a robust fault and state estimation [1–6].The proposed filter can play a significant role in several applications, for example, model based fault detection and isolation (FDI) problem [1–7] and fault tolerant control (FTC) problem [7, 8].In the past three decades, the problem of robust state estimation in the presence of uncertainties has attracted the interests of many researchers. This problem is largely treated in the literature by different approaches: the guaranteed cost design [9–12], theH∞ filtering [13–15], and the set-valued estimation [16, 17]. One limitation of the different design approach for online filter operation is that they require continuous testing of a certain existence condition. When the condition fails at any particular iteration, the proposed filters can diverge.From the point of view of minimizing the worst possible regularized residual norm over the class of admissible uncertainties, new robust filters are designed for linear uncertain systems by [18, 19]. Compared with the aforementioned robust formulations the developed filters perform data regularization rather than deregularization which represent an important property for online operation. The proposed filters in [19] are based in data regularization solution. This filter guarantees an error variance with an optimized guaranteed upper bound for any allowed uncertainty. To improve robustness against uncertainties such as disturbances and modeling errors, [20] introduced a proportional integral Kalman filter (PIKF). The proportional and the integral Kalman gains were obtained from the solution of Riccati equation leading to minimum error variance. Later, [21] developed a new robust proportional integral Kalman filter for stochastic linear uncertain systems. The filtering problem is converted into a convex optimization problem for continuous time systems with polytopic uncertainties and the filter parameters are optimally chosen by solving this problem.The problem of robust Kalman filtering and optimal filtering in the presence of unknown inputs and unknown faults has received considerable attention in the last two decades due to its significations role in many applications, for example, geophysical and environmental applications, fault detection and isolation (FDI) problems, and fault tolerant control (FTC) problems.The FDI (fault detection and isolation) problem for linear systems with unknown disturbances is largely studied in the literature by different approaches; see, for example, [2–7] and [22, 23]. By using the error innovation technique, a robust fault detection and isolation filter in continuous time is developed in [22] to generate unbiased white residuals signals. In [23] a new method is developed for linear time-invariant (LTI) stochastic discrete-time systems with unknown inputs. This method is important to detect and isolate multiple faults appearing simultaneously or sequentially in linear time-invariant (LTI) systems.The optimal filtering and robust fault diagnosis problem has been treated for stochastic systems with unknown disturbances in [6, 7]. An optimal observer is proposed for linear time-varying systems. This observer can produce disturbances decoupled state estimation with minimum-variance. The output estimation error with disturbance decoupling [6, 7] is used as a residual signal. After that, a statistical testing procedure is applied to examine the residual and to diagnose faults. Nevertheless, the simultaneous actuator and sensor faults problem is not considered in [6, 7].More recently, [3] presents a new recursive filter to joint fault and state estimation of linear time-varying discrete-time 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 unknown disturbances is available. The filter considers an arbitrary direct feedthrough matrix of the fault and it permits a multiple faults estimations. However, the obtained filter may in certain cases suffer from poor quality fault estimation.Later, in [24] the problem of joint fault and state estimation of linear systems in the presence of unknown input with uncertain noise covariances was presented. This problem was solved by using the proportional integral three-stage Kalman filter (PI-ThSKF) to estimate the state and the fault of stochastic discrete-time systems with unknown inputs. However, this approach assumes that the models for the dynamical evolution of the fault and the unknown inputs are available.Based on the assumption that no prior knowledge about the dynamical evolution of the fault is available, the same author [25] was proposed a new recursive optimal filter structure with transformation of the original system. A new recursive optimal unbiased minimum-variance filter has been developed when the direct feedthrough matrix of the fault has an arbitrary rank. However, the filtering algorithm requires the knowledge of a perfect dynamic model. Thus the developed filter may not be robust against modeling uncertainty in the state and the output matrices.One limitation of the proposed design approach [3, 25] is that they require testing of a certain existence conditions. When the condition fails at any particular iteration, the desired performance is lost and the filter can diverge. In addition the disadvantages of the existing approaches [3, 4, 24, 25] are that the filter lost its optimality in the presence of uncertainties in the state and the output matrices.In this paper, we consider the problem of robust joint fault and state estimation for linear discrete-time systems with norm bounded uncertainties in both the state and output matrices. The problem addressed is the design of robust linear filters that bound the state covariance matrix for all admissible uncertainties. It is shown that a robust proportional filter (RPF) is developed using transformation of the original system. This transformation is based on the singular value decomposition of the direct feedthrough matrix distribution of the fault which assumed to be arbitrary rank. The proposed filter guarantees that the variance of the estimation error is not more than an optimized upper bound for all admissible uncertainties. The minimization problem of the upper bound on the estimation error variance is formulated as a convex optimization problem subject to linear matrix inequalities and the filter parameters are optimally chosen by solving this problem. To improve robustness against uncertainties and to improve robust estimation of unknown time-varying fault, a new robust proportional integral filter (RPIF) is proposed. The proportional and the integral gains are optimally chosen by solving a convex optimization problem. So the resulted filter will be applied to solve a simultaneous actuator and sensor faults estimation problem.The remainder of this paper is organized as follows. In Section2 we set up the robust regularized least square problem for models with data uncertainties. Section 3 states the problem of interest. In Section 4 we design the robust proportional filter. Next in Section 5 we propose a design approach for the robust proportional integral filter (RPIF). Finally, in Section 6, the estimation performance of the proposed filters is demonstrated through an illustrative example.


## 2. Preliminaries

Consider the following optimization problem:(1)minxmaxδA,δbxΠ2+A+δAx-b+δbW2,where Ais the data matrix, b is the measurement vector which is assumed to be known,xis the unknown vector,Π=ΠT≥0andW=WT>0 are given weighting matrices, andδA,δbare uncertainties assumed to satisfy a model of the form:(2)δAδb=HΔNaNb,whereΔis an arbitrary contraction,Δ≤1, andH,Na,Nb are known quantities of appropriate dimension.Problem (1) and (2) has a unique solution x^k that is given by [17]:(3)x^=Π^+ATW^A-1ATW^b+λ^NaTNb,where the modified weighting matrices Π^,W^are defined by(4)Π^=Π+λ^NaTNa,W^=W+WHλ^I-HTWH†HTWand λ^ is a nonnegative scalar parameter obtained by the following optimization problem:(5)λ^=argminλ^≥HTWHGλ,where(6)Gλ≔xλΠ2+λNaxλ-Nb2+Axλ-bWλ2.


## 3. Problem Statement

Consider the linear stochastic uncertain discrete-time system with unknown additive fault in the form:(7)xk+1=Ak+ΔAkxk+Fkxfk+Bkuk+wk,yk=Ck+ΔCkxk+Fkyfk+vk,wherexk∈Rn is the state vector,yk∈Rp is the observation vector,fk∈Rm is the unknown additive fault vector,wkand vk are uncorrelated white noise sequences of zero-mean and with covariances matricesQk=EwkwlT≥0andRk=EvkvlT>0, respectively.The matricesAk, Bk, Ck, Fkx, andFky are known and have appropriate dimensions.The initial statex0 is a Gaussian random variable that is uncorrelated withwk,vkfor all k with Ex0=x^0 andEx0-x^0x0-x^0T=P0x, where E·denote the expectation operator.Δ A kandΔCk are unknown matrices which represent time-varying parameter uncertainties. These uncertainties are assumed to be of the following structure:(8)ΔAkΔCk=H1,kH2,kFkEk,whereH1,k, H2,k, andEkare known time-varying matrices of appropriate dimensions, whileFk is an unknown time-varying matrix satisfying arbitrary contraction,FkFkT≤I,∀k∈0N.The aim of this paper is to design a new robust proportional integral filter (RPIF) to obtain a robust fault and state estimation when0<rankFky≤min spite of the presence of parametric uncertainties.Initially, we seek to change the coordinate of system (7) by using the same technique developed in [26].Letrk=rankFky<m, and then the singular value decomposition of the matrixFkyis given by(9)Fky=U1,kU2,kΣk000V1,kTV2,kT,where Σk∈Rrk×rk,U1,k∈Rp×rk,U2,k∈Rp×(p-rk),V1,k∈Rm×rk, andV2,k∈Rm×(m-rk).U 1 , k U 2 , k and V1,kV2,k are unitary matrices.Using the notations(10)fk=V1,kf1,k+V2,kf-2,k,(11)F1,kx=FkxV1,k,F-2,kx=FkxV2,k,F1,ky=U1,kΣk,wheref1,k=V1,kTfkand f-2,k=V2,kTfk, we obtain the following equivalent system of the original system (7).(12)xk+1=Ak+ΔAkxk+F1,kxf1,k+F-2,kxf-2,k+Bkuk+wk,(13)yk=Ck+ΔCkxk+F1,kyf1,k+vk,whereF1,kyis of full-column rank due to (11).Note thatF-2,kxin (12) may not have full-column rank and the unknown faultf-2,kmay not be estimable. However it can be solved by finding a full-rank factorization ofF-2,kx, that is, F-2,kx=F2,kF˘2,k, where F2,kis n×r full-column rank matrix andF˘2,kisr(m-rk)full-row rank matrix [27].Thus by defining the notation(14)f2,k=F˘2,kxf-2,k(12) becomes(15)xk+1=Ak+ΔAkxk+F1,kxf1,k+F2,kxf2,k+Bkuk+wk.


## 4. Robust Proportional Filter Design

In this section, we propose to solve equivalent system (13), (15) for x^k/k,f^1,k/k and f^2,k-1/k, such thattraceEekekTis minimized, where ek=xk-x^k/k is the state estimation error.Here, we adopt a robust least-squares estimation approach to obtain a robust estimate for the state variable and the unknown faults by following a two-step procedure.


### 4.1. Bounded Uncertainties inCkAlone

Assume first that there are no uncertainties inAk; we will incorporate the uncertainties in Aklater. With bounded uncertainties in the output matrix Ck alone and by using the robust least-squares estimation procedure, we can transform the original system (13), (15) into the following augmented output equation (AOE):(16)yk-Ck+ΔCkx^k/k-10=Ck+ΔCkF1,ky0I0-F2,k-1xxk-x^k/k-1f1,k/kf2,k-1/k+vk-wk-1+00Ak-1,cF1,k-1xυk-1,where(17)x^k/k-1=Ak-1x^k-1/k-1+Bk-1uk-1+F1,k-1xf^1,k-1/k-1,(18)υk=-ekTf~1,kTT,where f~1,k=f1,k-f^1,k/k. Here f^1,k/k is the estimate of the fault and υk is zero-mean Gaussian and independent of wk andvk+1.The filtered estimatex^k/k, the unknown fault f^1,k/k, and the delayed unknown fault f^2,k-1/k can be obtained by solving the following robust least-squares problem:(19)minxk,f1,k,f2,k-1maxΔCkyk-Ck+ΔCkx^k/k-10-Ck+ΔCkF1,ky0I0-F2,k-1xxk-x^k/k-1f1,k/kf2,k-1/kdiagRk-1,P-k-1,where(20)P-k=Exk-x^k/k-1xk-x^k/k-1T=Ak-1F1,k-1xPk-1Pk-1xf1Pk-1f1xPk-1f1Ak-1TF1,k-1xT+Qk-1.Here Pk-1xf1andPk-1f1will be defined later.Note that problem (19) can be written more compactly in the form (1) and (2) with the identifications(21)x⟵colxk-x^k/k-1,f1,k/k,f2,k-1/k,b⟵yk-Ckx^k/k-10,A⟵CkF1,ky0I0-F2,k-1x,δA⟵ΔCk00000,δb⟵-ΔCkx^k/k-10,W⟵diagRk-1,P-k-1,H⟵H2,k000,Na⟵Ek00000,Nb⟵-Ekx^k/k-10,Δ⟵Fk. From (1)–(3) the solution to (19) is given by(22)x^k-x^k/k-1f^1,k/kf^2,k-1/k=P˘k-1CkTR-k-1F1,ky-P-k-1F2,k-1xF1,kyTR-k-1CkD˘1,k-10-F2,k-1xTP-k-10D˘2,k-1-1-1CkTR-k-1yk-Ckx^k/k-1-λ^kEkTEkx^k/k-1F1,kyTR-k-1yk-Ckx^k/k-1,where(23)P˘k-1=P-k-1+CkTR-k-1Ck,D˘1,k-1=F1,kyTR-k-1F1,ky,D˘2,k-1-1=F2,k-1xTP-k-1F2,k-1x,R-k-1=Rk-λ^k-1H2,kH2,kT-1.Moreover, λ^k is the minimizing argument in the interval λ^k>H2,kTRk-1H2,k=λl,k of the corresponding scalar-valued function Gλin (6) constructed with identification (21).Using [28, Prop. 2.8.7], we find that the inverse in (22) can be written as(24)P˘k-1CkTR-k-1F1,ky-P-k-1F2,k-1xF1,kyTR-k-1CkD˘1,k-10-F2,k-1xTP-k-10D˘2,k-1-1-1=PkPkxf1Pkxf2Pkf1xPkf1Pkf1f2Pkf2xPkf2f1Pk-1f2,(25)Pkxf1=Pkf1xT=-PkCkTR-k-1F1,kyD˘1,k-1,(26)Pkxf2=Pkf2xT=PkP-k-1F2,k-1xD˘2,k-1-1,(27)Pkf1f2=Pkf2f1T=-D˘1,kF1,kyTR-k-1CkPkP-k-1F2,k-1xD˘2,k-1-1.Note thatPk, Pkf1, andPk-1f2can be identified as the covariance matrices of x^k/k, f^1,k, andf^2,k-1; that is,(28)Pk=Exk-x^k/kxk-x^k/kT,Pkf1=Ef1,k/k-f^1,k/kf1,k/k-f^1,k/kT,Pk-1f2=Ef2,k-1/k-f^2,k-1/kf2,k-1/k-f^2,k-1/kT,where the inverse ofPk,Pkf1, andPk-1f2are given by(29)Pk-1=P˘k,1-1-CkTR-k-1F1,kyD˘1,kF1,kyTR-k-1Ck,Pkf1-1=D˘1,k-1-F1,kyTR-k-1CkP˘k,1CkTR-k-1F1,ky,Pk-1f2-1=D˘2,k-1-1-F2,k-1xTP-k-1P⌢k,1P-k-1F2,k-1x,P˘k,1-1=P˘k-1-P-k-1F2,k-1xD˘2,k-1F2,k-1xTP-k-1,P⌢k,1-1=P˘k-1-CkTR-k-1F1,kyD˘1,kF1,kyTR-k-1Ck.Finally substituting (24) in (22) yields(30)x^k/k=I-λ^kPkEkTEkx^k/k-1+PkCkTR-k-1yk-Ckx^k/k-1-F1,kyf^1,k/k,(31)f^1,k/k=Mkf1yk-C-kx^k/k-1,(32)f^2,k-1/k=Mkf2yk-C-kx^k/k-1,where(33)Mkf1=F1,kyTR~1,k-1F1,ky-1F1,kyTR~1,k-1,(34)Mkf2=F2,k-1xTR~2,k-1F2,k-1x-1F2,k-1xTR~2,k-1,(35)C-k=Ck-λ^kCkP-kEkTEk,(36)R~1,k=CkP~kCkT+R-k,(37)R~2,k-1=P-k-1-P-k-1P⌢k,1P-k-1,(38)P~k-1=P-k-1-P-k-1F2,k-1xD˘2,k-1F2,k-1xTP-k-1.Furthermore, we find thatPkf1 and Pk-1f2are given, respectively, by(39)Pkf1=F1,kyTR~k-1F1,ky-1,Pk-1f2=F2,k-1xTR~2,k-1F2,k-1x-1.Finally, it follows from (17) and (30) that(40)x^k+1/k=Ap,kx^k/k-1+F1,kxf^1,k/k+Bkuk+Kp,kyk-Ckx^k/k-1-F1,kyf^1,k/k,whereAp,kandKp,k are defined in terms of the parameterPk as (41)Ap,k=AkI-λ^kPkEkTEk,Kp,k=AkPkCkTR-k-1.Note that these expressions for Ap,kand Kp,kare defined linearly in terms ofPk and have been determined by assuming uncertainties inCk alone. We will incorporate the uncertainties in Ak in the next section.


### 4.2. Bounded Uncertainties inAkandCk

We now incorporate uncertainties intoAk. That is, we consider norm bounded uncertainties in AkandCkas in (13) and (15). We will move to select the parameter Pk by assuming uncertainties inAkalone and one that meets robustness criterion (19) when there are uncertainties in Ck.Denotingx~k=xk-x^k/k-1, we define the extended weight vectorζk=x~kx^k/k-1. Then in the absence of uncertainties in Ck, we find thatζksatisfies(42)ζk+1=A-k,c+H-k,cFkE-k,cζk+F-k,cf-k,c+G-k,cw-k,where(43)A-k,c=Ak-Kp,kCkAk-Ap,kKp,kCkAp,k,F-k,c=Fkx-Kp,kF1,ky00Kp,kF1,kyF1,kx,G-k,c=I-Kp,kF2,kx0Kp,k0,(44)H-k,c=H1,k000,E-k,c=EkEk00,f-k,c=f~kf~1,kf^1,k,w-k=wkvkf^2,k.The covariance matrix ofζksatisfies(45)Σk+1=EA-k,c+H-k,cFkE-k,cΣkA-k,c+H-k,cFkE-k,cT+EA-k,c+H-k,cFkE-k,cΓkF-k,cT+EF-k,cΓkTA-k+H-kFkE-k,cT+F-k,cΥkF-k,cT+G-k,cSkG-k,cT,where(46)Sk=Qk000Rk000Pkf2,Γk=PkxfPkxf10000000,Υk=PkfPkff10Pkf1fPkf1000Pkf1,where(47)Pkf=V1,kV2,kPkf1Pkf1f2Pkf2f1Pkf2V1,kTV2,kT,Pkxf=V1,kPkxf1+V2,kPkxf2,Pkff1=Pkf1fT=V1Pkf1+V2F˘2,k-1Pkf2f1.Now observe that the expressions forAp,k,Kp,kare parameterized linearly in terms of the parameter Pk. We shall choose Pk by minimizing an upper bound ofΣk+1 in the absence of uncertainties in Ck.Letαkbe a scalar such that αkI-E-c,kΣkE-c,kT>0 and letβkbe a scalar such that βkI-E-c,kΥkE-c,kT>0.Bearing in mind (45), we can see thatΣk+1is bounded by(48)Σk+1≥A-k,cΣkA-k,cT+A-k,cΓkF-k,cT+F-k,cΓkTA-k,cT+F-k,cΥkF-k,cT+G-k,cSkG-k,cT+αk+βkH-k,cH-k,cT.We choose Pk>0, by solving(49)minPk>0trΣk+1,(50)subjecttoΣk+1≥A-k,cΣkA-k,cT+A-k,cΓkF-k,cT+F-k,cΓkTA-k,cT+F-k,cΥkF-k,cT+G-k,cSkG-k,cT+αk+βkH-k,cH-k,cT.Or equivalently(51)-Σk+1+A-k,cΓkF-k,cT+F-k,cΓkTA-k,cT+αk+βkH-k,cH-k,cTA-k,cΣkG-k,cSk1/2F-k,cΥkΣkA-k,cT-Σk00SkG-k,cT/20-I0ΥkF-k,cT00-Υk≤0.Since inequality (51) is affine in Ak, Pkthus found will ensure minimum error covariance Σkover all possibleAkin the bounded domain. Therefore the desired robust proportional filter is given by (40)–(41) and (31)–(38). So initializing Σ0=diagεI,P0forP0>0 and a scalar ε>0, the resulting filter is listed in the following section.


### 4.3. Summary of the RPF

In this section we summarize the filter equations, we assume thatx^0is the estimate of the initial state a zero-mean and has known variance P0.Initial Condition. x ^ 0 = 0 ,   Σ 0 = d i a g ε I , P 0, whereP0>0andε>0.Step 1.
IfH2,k=0, then setλ^k=0. Otherwise, constructGλ of (6) with identification (21) and determineλ^kby minimizingGλover the interval (52)λ^k>λl,k=H2,kTRk-1H2,k.Step 2.
UsingΣk, computePk,Σk+1by solving (49) subject to inequality (51), whereA-k,c,F-k,c,G-k,c,H-k,c andSk,Γk,Υkare given, respectively, by (43) and (46).Step 3.
Robust simultaneous fault and state estimation are as follows:(53)f^1,k/k=Mkf1yk-C-kx^k/k-1,f^2,k-1/k=Mkf2yk-C-kx^k/k-1,f^2,k=F˘2,kxf-^2,k,f^k=V1,kf^1,k+V2,kf-^2,k,whereMkf1,Mkf2, andC-k are given by (33)–(37).
Updatex^k/k-1tox^k+1/kas(54)x^k+1/k=Ap,kx^k/k-1+F1,kxf^1,k/k+Bkuk+Kp,kyk-Ckx^k/k-1-Fk,1yf^1,k/k,where Ap,kandKp,kare given by (41).Remark 1.
Note that the robust filter (RPF) developed in the previous section gives a better estimation of the state and the fault; however, when the unknown fault is time-varying, the performances of the robust filter can deteriorate. So we will extend the RPF to further propose a new robust proportional integral filter (RPIF) structure, in which the integral action is believed to improve robust estimation of the unknown time-varying faults and to improve robustness against uncertainties.


## 4.1. Bounded Uncertainties inCkAlone

Assume first that there are no uncertainties inAk; we will incorporate the uncertainties in Aklater. With bounded uncertainties in the output matrix Ck alone and by using the robust least-squares estimation procedure, we can transform the original system (13), (15) into the following augmented output equation (AOE):(16)yk-Ck+ΔCkx^k/k-10=Ck+ΔCkF1,ky0I0-F2,k-1xxk-x^k/k-1f1,k/kf2,k-1/k+vk-wk-1+00Ak-1,cF1,k-1xυk-1,where(17)x^k/k-1=Ak-1x^k-1/k-1+Bk-1uk-1+F1,k-1xf^1,k-1/k-1,(18)υk=-ekTf~1,kTT,where f~1,k=f1,k-f^1,k/k. Here f^1,k/k is the estimate of the fault and υk is zero-mean Gaussian and independent of wk andvk+1.The filtered estimatex^k/k, the unknown fault f^1,k/k, and the delayed unknown fault f^2,k-1/k can be obtained by solving the following robust least-squares problem:(19)minxk,f1,k,f2,k-1maxΔCkyk-Ck+ΔCkx^k/k-10-Ck+ΔCkF1,ky0I0-F2,k-1xxk-x^k/k-1f1,k/kf2,k-1/kdiagRk-1,P-k-1,where(20)P-k=Exk-x^k/k-1xk-x^k/k-1T=Ak-1F1,k-1xPk-1Pk-1xf1Pk-1f1xPk-1f1Ak-1TF1,k-1xT+Qk-1.Here Pk-1xf1andPk-1f1will be defined later.Note that problem (19) can be written more compactly in the form (1) and (2) with the identifications(21)x⟵colxk-x^k/k-1,f1,k/k,f2,k-1/k,b⟵yk-Ckx^k/k-10,A⟵CkF1,ky0I0-F2,k-1x,δA⟵ΔCk00000,δb⟵-ΔCkx^k/k-10,W⟵diagRk-1,P-k-1,H⟵H2,k000,Na⟵Ek00000,Nb⟵-Ekx^k/k-10,Δ⟵Fk. From (1)–(3) the solution to (19) is given by(22)x^k-x^k/k-1f^1,k/kf^2,k-1/k=P˘k-1CkTR-k-1F1,ky-P-k-1F2,k-1xF1,kyTR-k-1CkD˘1,k-10-F2,k-1xTP-k-10D˘2,k-1-1-1CkTR-k-1yk-Ckx^k/k-1-λ^kEkTEkx^k/k-1F1,kyTR-k-1yk-Ckx^k/k-1,where(23)P˘k-1=P-k-1+CkTR-k-1Ck,D˘1,k-1=F1,kyTR-k-1F1,ky,D˘2,k-1-1=F2,k-1xTP-k-1F2,k-1x,R-k-1=Rk-λ^k-1H2,kH2,kT-1.Moreover, λ^k is the minimizing argument in the interval λ^k>H2,kTRk-1H2,k=λl,k of the corresponding scalar-valued function Gλin (6) constructed with identification (21).Using [28, Prop. 2.8.7], we find that the inverse in (22) can be written as(24)P˘k-1CkTR-k-1F1,ky-P-k-1F2,k-1xF1,kyTR-k-1CkD˘1,k-10-F2,k-1xTP-k-10D˘2,k-1-1-1=PkPkxf1Pkxf2Pkf1xPkf1Pkf1f2Pkf2xPkf2f1Pk-1f2,(25)Pkxf1=Pkf1xT=-PkCkTR-k-1F1,kyD˘1,k-1,(26)Pkxf2=Pkf2xT=PkP-k-1F2,k-1xD˘2,k-1-1,(27)Pkf1f2=Pkf2f1T=-D˘1,kF1,kyTR-k-1CkPkP-k-1F2,k-1xD˘2,k-1-1.Note thatPk, Pkf1, andPk-1f2can be identified as the covariance matrices of x^k/k, f^1,k, andf^2,k-1; that is,(28)Pk=Exk-x^k/kxk-x^k/kT,Pkf1=Ef1,k/k-f^1,k/kf1,k/k-f^1,k/kT,Pk-1f2=Ef2,k-1/k-f^2,k-1/kf2,k-1/k-f^2,k-1/kT,where the inverse ofPk,Pkf1, andPk-1f2are given by(29)Pk-1=P˘k,1-1-CkTR-k-1F1,kyD˘1,kF1,kyTR-k-1Ck,Pkf1-1=D˘1,k-1-F1,kyTR-k-1CkP˘k,1CkTR-k-1F1,ky,Pk-1f2-1=D˘2,k-1-1-F2,k-1xTP-k-1P⌢k,1P-k-1F2,k-1x,P˘k,1-1=P˘k-1-P-k-1F2,k-1xD˘2,k-1F2,k-1xTP-k-1,P⌢k,1-1=P˘k-1-CkTR-k-1F1,kyD˘1,kF1,kyTR-k-1Ck.Finally substituting (24) in (22) yields(30)x^k/k=I-λ^kPkEkTEkx^k/k-1+PkCkTR-k-1yk-Ckx^k/k-1-F1,kyf^1,k/k,(31)f^1,k/k=Mkf1yk-C-kx^k/k-1,(32)f^2,k-1/k=Mkf2yk-C-kx^k/k-1,where(33)Mkf1=F1,kyTR~1,k-1F1,ky-1F1,kyTR~1,k-1,(34)Mkf2=F2,k-1xTR~2,k-1F2,k-1x-1F2,k-1xTR~2,k-1,(35)C-k=Ck-λ^kCkP-kEkTEk,(36)R~1,k=CkP~kCkT+R-k,(37)R~2,k-1=P-k-1-P-k-1P⌢k,1P-k-1,(38)P~k-1=P-k-1-P-k-1F2,k-1xD˘2,k-1F2,k-1xTP-k-1.Furthermore, we find thatPkf1 and Pk-1f2are given, respectively, by(39)Pkf1=F1,kyTR~k-1F1,ky-1,Pk-1f2=F2,k-1xTR~2,k-1F2,k-1x-1.Finally, it follows from (17) and (30) that(40)x^k+1/k=Ap,kx^k/k-1+F1,kxf^1,k/k+Bkuk+Kp,kyk-Ckx^k/k-1-F1,kyf^1,k/k,whereAp,kandKp,k are defined in terms of the parameterPk as (41)Ap,k=AkI-λ^kPkEkTEk,Kp,k=AkPkCkTR-k-1.Note that these expressions for Ap,kand Kp,kare defined linearly in terms ofPk and have been determined by assuming uncertainties inCk alone. We will incorporate the uncertainties in Ak in the next section.


## 4.2. Bounded Uncertainties inAkandCk

We now incorporate uncertainties intoAk. That is, we consider norm bounded uncertainties in AkandCkas in (13) and (15). We will move to select the parameter Pk by assuming uncertainties inAkalone and one that meets robustness criterion (19) when there are uncertainties in Ck.Denotingx~k=xk-x^k/k-1, we define the extended weight vectorζk=x~kx^k/k-1. Then in the absence of uncertainties in Ck, we find thatζksatisfies(42)ζk+1=A-k,c+H-k,cFkE-k,cζk+F-k,cf-k,c+G-k,cw-k,where(43)A-k,c=Ak-Kp,kCkAk-Ap,kKp,kCkAp,k,F-k,c=Fkx-Kp,kF1,ky00Kp,kF1,kyF1,kx,G-k,c=I-Kp,kF2,kx0Kp,k0,(44)H-k,c=H1,k000,E-k,c=EkEk00,f-k,c=f~kf~1,kf^1,k,w-k=wkvkf^2,k.The covariance matrix ofζksatisfies(45)Σk+1=EA-k,c+H-k,cFkE-k,cΣkA-k,c+H-k,cFkE-k,cT+EA-k,c+H-k,cFkE-k,cΓkF-k,cT+EF-k,cΓkTA-k+H-kFkE-k,cT+F-k,cΥkF-k,cT+G-k,cSkG-k,cT,where(46)Sk=Qk000Rk000Pkf2,Γk=PkxfPkxf10000000,Υk=PkfPkff10Pkf1fPkf1000Pkf1,where(47)Pkf=V1,kV2,kPkf1Pkf1f2Pkf2f1Pkf2V1,kTV2,kT,Pkxf=V1,kPkxf1+V2,kPkxf2,Pkff1=Pkf1fT=V1Pkf1+V2F˘2,k-1Pkf2f1.Now observe that the expressions forAp,k,Kp,kare parameterized linearly in terms of the parameter Pk. We shall choose Pk by minimizing an upper bound ofΣk+1 in the absence of uncertainties in Ck.Letαkbe a scalar such that αkI-E-c,kΣkE-c,kT>0 and letβkbe a scalar such that βkI-E-c,kΥkE-c,kT>0.Bearing in mind (45), we can see thatΣk+1is bounded by(48)Σk+1≥A-k,cΣkA-k,cT+A-k,cΓkF-k,cT+F-k,cΓkTA-k,cT+F-k,cΥkF-k,cT+G-k,cSkG-k,cT+αk+βkH-k,cH-k,cT.We choose Pk>0, by solving(49)minPk>0trΣk+1,(50)subjecttoΣk+1≥A-k,cΣkA-k,cT+A-k,cΓkF-k,cT+F-k,cΓkTA-k,cT+F-k,cΥkF-k,cT+G-k,cSkG-k,cT+αk+βkH-k,cH-k,cT.Or equivalently(51)-Σk+1+A-k,cΓkF-k,cT+F-k,cΓkTA-k,cT+αk+βkH-k,cH-k,cTA-k,cΣkG-k,cSk1/2F-k,cΥkΣkA-k,cT-Σk00SkG-k,cT/20-I0ΥkF-k,cT00-Υk≤0.Since inequality (51) is affine in Ak, Pkthus found will ensure minimum error covariance Σkover all possibleAkin the bounded domain. Therefore the desired robust proportional filter is given by (40)–(41) and (31)–(38). So initializing Σ0=diagεI,P0forP0>0 and a scalar ε>0, the resulting filter is listed in the following section.


## 4.3. Summary of the RPF

In this section we summarize the filter equations, we assume thatx^0is the estimate of the initial state a zero-mean and has known variance P0.Initial Condition. x ^ 0 = 0 ,   Σ 0 = d i a g ε I , P 0, whereP0>0andε>0.Step 1.
IfH2,k=0, then setλ^k=0. Otherwise, constructGλ of (6) with identification (21) and determineλ^kby minimizingGλover the interval (52)λ^k>λl,k=H2,kTRk-1H2,k.Step 2.
UsingΣk, computePk,Σk+1by solving (49) subject to inequality (51), whereA-k,c,F-k,c,G-k,c,H-k,c andSk,Γk,Υkare given, respectively, by (43) and (46).Step 3.
Robust simultaneous fault and state estimation are as follows:(53)f^1,k/k=Mkf1yk-C-kx^k/k-1,f^2,k-1/k=Mkf2yk-C-kx^k/k-1,f^2,k=F˘2,kxf-^2,k,f^k=V1,kf^1,k+V2,kf-^2,k,whereMkf1,Mkf2, andC-k are given by (33)–(37).
Updatex^k/k-1tox^k+1/kas(54)x^k+1/k=Ap,kx^k/k-1+F1,kxf^1,k/k+Bkuk+Kp,kyk-Ckx^k/k-1-Fk,1yf^1,k/k,where Ap,kandKp,kare given by (41).Remark 1.
Note that the robust filter (RPF) developed in the previous section gives a better estimation of the state and the fault; however, when the unknown fault is time-varying, the performances of the robust filter can deteriorate. So we will extend the RPF to further propose a new robust proportional integral filter (RPIF) structure, in which the integral action is believed to improve robust estimation of the unknown time-varying faults and to improve robustness against uncertainties.


## 5. Robust Proportional Integral Filter Design

In this section, we propose to design a new robust proportional integral filter (RPIF) for stochastic linear uncertain system (13) and (15) to improve the estimation of unknown time-varying faults and to improve robustness against uncertainties such as disturbances and modeling errors.The proposed filter has the following structure:(55)x^k+1/k=Ap,kx^k/k-1+F1,kxf^1,k/k+Bkuk+Kp,kyk-Ckx^k/k-1-F1,kyf^1,k/k+Kz,kzk,zk+1=α0zk+Hkyk-Ckx^k/k-1-F1,kyf^1,k/k,f^1,k/k=Mkf1yk-C-kx^k/k-1,f^2,k-1/k=Mkf2yk-C-kx^k/k-1,f^k/k=V1,kf^1,k/k+V2,kf-^2,k/k,f^2,k=F˘2,kxf-^2,k,where the matricesKp,k andKz,k represent a proportional gain and an integral gain, respectively. The variablezkis related to the weighted integral of the output estimation error. The constant valueα0stands for a fading effect coefficient that regulates the transient response. The matrixHkis an integral effect coefficient. The two design parameters are assumed to be preselected by designers. Theses expressions for Ap,k,Kp,k,Kz,k,Mkf1, andMkf2have been determined by assuming uncertainties in Ckalone. We now move on to select the parameterPkby assuming uncertainties in Akalone. By doing so, we will arrive at a filter that minimizes a bound on the state error covariance matrix when there are uncertainties inAkalone.Proceeding in the same manner as in the previous section, we know that the expression forAp,k,Kp,k,Mkf1,Mkf2can be parameterized linearly in terms of the parameter Pk.Defining the extended weight vectorηk=x~kzk. Ignoring the uncertainties in Ck, we find that ηk satisfies(56)ηk+1=A-k+H-kFkE-1,kηk+F-k+H-kFkE-2,kf-k+G-kw-k,where(57)A-k=Ak-Kp,kCk-Kz,kHkCkαI,F-k=Ak-Ap,kFkx-Kp,kF1,ky00HkF1,ky,G-k=I-Kp,kF2,kx0Hk0,(58)H-k=H1,k000,E-1,k=Ek000,E-2,k=Ek00000,f-k=x^k/k-1f~kf~k,1,w-k=wkvkf^2,k.Suppose thatΣ-k+1satisfies(59)Σ-k+1=EA-k+H-kFkE-1,kΣ-kA-k+H-kFkE-1,kT+EA-k+H-kFkE-1,kΓ-kF-k+H-kFkE-2,kT+EF-k+H-kFkE-2,kΓ-kTA-k+H-kFkE-1,kT+EF-k+H-kFkE-2,kΥ-kF-k+H-kFkE-2,kT+G-kSkG-kT,where(60)Sk=Qk000Rk000Pkf2,(61)Γ-k=0PkxfPkxf10PkzfPkzf1000,(62)Υ-k=Pkx-Pk000PkfPkff10Pkf1fPkf1,(63)Pkzf=HkCkPkxf+F1,kyPkf1V1T+F1,kyPkf1f2F˘2,k-TV2T,(64)Pkzf1=HkCkPkxf1+F1,kyPkf1.Letα-kbe a scalar such that α-kI-E-1,kΣ-kE-1,kT>0and letβ-kbe a scalar such that β-kI-E-2,kΥ-kE-2,kT>0.Bearing in mind (59), we can see thatΣ-k+1is bounded by(65)Σ-k+1≥A-kΣ-kA-kT+A-kΓ-kF-kT+F-kΓ-kTA-kT+F-kΥ-kF-kT+G-kSkG-kT+α-k+β-kH-kH-kT.We choose Pk>0, by solving(66)minPk>0trΣ-k+1,(67)subjecttoΣ-k+1≥A-kΣ-kA-kT+A-kΓ-kF-kT+F-kΓ-kTA-kT+F-kΥ-kF-kT+G-kSkG-kT+α-k+β-kH-kH-kT.Or equivalently(68)-Σ-k+1+A-kΓ-kF-kT+F-kΓ-kTA-kT+α-k+β-kH-kH-kTA-kΣ-kG-kSk1/2F-kΥ-Σ-kA-kT-Σ-k00SkG-kT/20-I0Υ-kF-kT00-Υ-k≤0.Therefore the desired robust proportional integral filter (RPIF) is given by (55) wherePkis the positive-definite solution of (66) subject to (67) with the initial conditionΣ-0=diagP0,εI forP0>0. Note that there always exists a solution to (66)-(67). The resulting filter is listed in following section.Summary of the RPIF. In this section we summarize the filter equations; we assume thatx^0is the estimate of the initial state a zero-mean and has known variance P0.The initialization step of the filter is then given as follows.Initial Condition. x ^ 0 = 0 ,   Σ - 0 = d i a g P 0 , ε I, whereP0>0andε>0.Step 1.
IfH2,k=0, then set λ^k=0. Otherwise, constructGλ of (6) with the identification (21) and determineλ^kby minimizing Gλ over the interval(69)λ^k>λl,k=H2,kTRk-1H2,k.Step 2.
UsingΣ-k, computePk,Kz,k,Σ-k+1 by solving (66) subject to inequality (67), whereA-k,F-k,G-k,H-k andSk,Γ-k,Υ-kare given by (57) and (60)–(62).Step 3.
Robust simultaneous fault and state estimation are as follows:(70)f^1,k/k=Mkf1yk-C-kx^k/k-1,f^2,k-1/k=Mkf2yk-C-kx^k/k-1,f^2,k=F˘2,kxf-^2,k,f^k=V1,kf^1,k+V2,kf-^2,k,where Mkf1, Mkf2, and C-k are given by (33)–(35).
Updatex^k/k-1 to x^k+1/k as(71)x^k+1/k=Ap,kx^k/k-1+F1,kxf^1,k/k+Bkuk+Kp,kyk-Ckx^k/k-1-Fk,1yf^1,k/k+Kz,kzz,(72)zk+1=α0zk+Hkyk-Ckx^k/k-1-Fk,1yf^1,k/k,where Ap,kandKp,kare given by (41).


## 6. Illustrative Example

Robust estimation of simultaneous actuator and sensor faults is as follows.In this section, we propose the use of the resulting filters RPF and RPIF to solve the robust estimation of simultaneous actuator and sensor faults problem.We consider the same numerical example used in (Chen and Patton [5, 6]). The linearized model of a simplified longitudinal flight control system is as follows:(73)xk+1=Ak+ΔAkxk+Bkuk+Fkafka+wk,yk=Ck+ΔCkxk+Fksfks+vk,where the state variables are pitch angle δz, pitch rate wz, and normal velocityηy, the control inputukis the elevator control signal, andFkaandFksare the matrices distribution of the actuator faultfkaand sensor fault fks.The presented system equations (73) can be rewritten as follows:(74)xk+1=Ak+ΔAkxk+Bkuk+Fkxfk+wk,yk=Ck+ΔCkxk+Fkyfk+vk,where FkxandFky are the matrices injection of the faults vector in the state and the measurement equations.(75)Fkx=Fka0,Fky=0Fks.The system parameter matrices are(76)Ak=0.9944-0.12030.0017+δ21k0.9902+δ22k00.8187-0.4302-0.07470,Bk=0.4252-0.00820.1813,Ck=I3×3,xk=ηywzδzT,Qk=diag0.12,0.12,0.012,Rk=0.12I3×3,whereδ21kandδ22k represent parametric uncertainties in the state matrix satisfying(77)δ21k≤0.0008,δ22k≤0.05.We inject simultaneously two faults in the system:(78)fkafks=4usk-40-4usk-70-4usk-60+4usk-80,whereuskis the unit-step function. The first fault fkaoccurs in the actuator and the second faultfksoccurs in the sensor for δz.The matrices injection of the fault and the unknown disturbances is taken as follows:(79)Fka=0.4252-0.00820.1813,Fks=001.In the simulation we setuk=10,x0=0, andP0=0.012I3×3.In Figure1, we have plotted the actual and the estimated value of the first element and the second element of the fault vectorfk=fkafksT, respectively, using the RPF and the RPIF. Figure 1 presents the simulation results for the worst case (δ21k=0.0008 and δ22k=0.05). In Figure 2, we have plotted the actual and the estimated value of the state vector xk=ηywzδzTfor the two worst cases (δ21k=0.0008 and δ22k=0.05).Figure 1
Actual faultfk and estimated fault f^k.Figure 2
Actual statexkand estimated state x^k.According to the simulation results, it can be seen that both the proposed filters RPF and RPIF give a better estimation of the state and the faults. Mainly, we focus on the simultaneous estimation of the actuator and the sensor faults in spite of norm bounded uncertainties in the state and the observation matrix.Figures3 and 4 present, respectively, the actual and the estimated value of the first element of the fault vectorfka using the RPF and the RPIF. We can conclude that in the case of time-varying faults the RPIF gives a better estimation results than the RPF. So the integral action is believed to improve robustness against uncertainties and to improve the estimation of time-varying faults.Figure 3
Actual faultfk and estimated fault f^k (using RPF).Figure 4
Actual faultfk and estimated fault f^k (using RPIF).


## 7. Conclusion

In this paper, we have derived a new robust filter for linear time-varying discrete-time systems with unknown faults that affect both the system and the output. By including the unknown fault vector as a part of the augmented system state and using transformation of the original system, a new robust proportional filter (RPF) is developed to solve the robust estimation problem, further, and show how to enforce certain minimum error variance property. We design robust filters that bound the state error covariance matrix for all admissible uncertainties. The robustness criterion used is based on robust square estimation approach. We have extended the developed filter to further propose a new robust proportional integral filter (RPIF) to improve robust estimation of unknown faults and to improve robustness against uncertainties. The design procedure is through the solution of a robust weighted recursive least square problem and it enforces a minimum state error variance property. The advantages of this filter are especially important in the case when the direct feedthrough matrix of the fault has an arbitrary rank and when we do not have any prior information about the fault. An application of the proposed filter has been shown by an illustrative example. The proposed robust filter is able to obtain a robust state and fault estimation in spite of the presence of norm bounded uncertainties. In the next step, we can consider nonlinear systems and the extended Kalman filter can be used to solve the problem of simultaneous fault and state estimation.



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*Source: 9251031-2017-01-15.xml*