F. Kratz, W. NUNINGER, J. Ragot, Finite memory observer based method for failure detection in dynamic systems, SYSID'97, 11th Symp. on System Identification, Vol. 3, pp. 1189-1194, Japan, july 8-11, 1997.

Finite Memory Observer based method for Failure Detection in Dynamic Systems. F. Kratz, W. Nuninger, J. Ragot Centre de Recherche en Automatique de Nancy - CNRS UA 821 Institut National Polytechnique de Lorraine 2, Avenue de la Forêt de Haye - F 54 516 Vandoeuvre-lès-Nancy Tel. (33) 83 59 59 59 Fax (33) 83 59 56 44 E-mail : {fkratz, wnuninge, jragot}@ensem.u-nancy.fr

Abstract: In the past decade, interest has been focused on the use of analytic redundancy for the Fault Detection and Isolation (FDI) (Frank, 1990; Gertler, 1991). Many applications in the FDI area use the state estimation technique to generate the accentuated fault functions, so-called residuals. Subsequent analysis of residuals may provide an information about a faulty component localisation. It is well known that infinite observer memory leads to the so-called divergence phenomenon caused by the model errors that observer tends to accumulate during its performance [Toda et al., 1978]. Moreover, the infinite memory inflicts observer insensitivity to recent measurements. This shortcoming brings us to the conclusion that an observer structure with finite memory would be more appropriate for the FDI. This paper presents extensions and improvements on the finite memory observer. The main contribution is to incorporate the parity relation design and observer into a diagnosis scheme and how to choose optimal length of the memory. The application of the finite memory observer to the sensor fault detection problem is illustrated by a numerical example. Simulation results show that the method is very effective for robust fault detection. Key words: Fault detection and isolation, finite memory observer, sliding observation window, residual generation, robustness.

Introduction It is well know that diagnostic procedures have to be robust against process uncertainties as measurement noise and parameter variations. In general, diagnostic procedures are composed of two steps [Atherton et al., 1992]: the residual generation procedure (the residual is a signal that contains information about the fault) and the function decision generation procedure that allows the detection and the further isolation of the faults. As a consequence, to achieve good performances of the detection system, one has to ensure a good degree of robustness of each procedure with respect to system disturbances and especially against parameter variations which are the most critical ones. In the literature, many solutions have been proposed and developed with respect the robustness constraint. It is well know that robustness is the result of a compromise between distinct performances criteria as the minimal rate of no detection and the minimal rate of false alarm. It is obvious that we can not improve both criteria at the same time

and this leads to optimal solutions within some practical considerations. One can refer to the surveys by Willsky (1976), Frank (1990, 1993) and Patton (1994). Among these methods, we quote the methods based on parameter identification, the parity space approach and the observers and filter based approaches. The last two methods are known to generate structural equivalent residuals and this had been proved for the unknown input observer too [Nuninger et al., 1996]. Anyway, Luenberger observer [Toda et al., 1978] or Kalman filter are known to diverge [Heffes, 1966]. In the case of Luenberger observer, as the estimation is based on all the data available at one time (i.e. all the process history, which is referred as infinite memory), the model errors might accumulate in such a way that the estimation error will diverge. This is a real problem as fault detection systems usually use the estimation error as the residual. Indeed, this is a natural way to construct a residual based on state estimation. Besides, because of the infinite memory less influence is given to the last measured data that might contains the information about incipient faults. As a solution, some authors developed new tools. Among them, we can quote the fading filters by Sorenson et al. (1985) which is a weighted Kalman filter and the finite memory state estimators by Medvedev et al. (1991) in the continuous case. We stress on the fact that Medvedev uses a particular residual defined as the difference between two estimations at the current time based on two overlapping observation windows. This residual improve the degree of robustness [Kratz et al., 1994]. The relevance of this paper is also that we proposed solutions in order to choose the optimal size of the observation window. Indeed, if we do not take enough data, we get a bad state estimate. On the contrary, if we take into account too many data, some additive information are useless and could make the error estimation degrades. So this paper is divided into three parts. First, we give the problem formulation and define our finite memory state estimator. The solution to choose the minimal and optimal size of the horizon is given. Second, the fault detection problem is tackled and the study of the influence of data (within definition of sensitivity) is done. Finally, we give the statistical proprieties of the particular residual we choose before applying our method on a simulated fourth order system in the last part.

Part A 1. Problem formulation In this paper, we consider a linear time-invariant system described by the following discrete-time state and observation equations: x(k+1) = Fx(k) + Gu(k) (1.a) y(k) = Cx(k) + v(k) (1.b) where u(k) ∈ R mx1 is the input vector, y(k) ∈ R px1 is the output vector, x(k) ∈ R nx1 is the state vector and v(k) ∈ R px1 the measurement noise vector is a white noise process with zero mean and known variance matrix Ω. {F, G, C} are model matrices with appropriate dimensions and we assume F is an invertible matrix. Here we consider the problem of estimating the vector x(k) from the measurement vector y(i) at time instant i = k - s, k - s + 1, ..., k. A recursive solution based on the finite memory observer method developed for the discrete time case ( Kratz et al., 1994) can be proposed. 2. Finite memory observer Collecting a set of sensor outputs over a time interval (data window) from time k-s to time k, we get the following scheme:

K K 0   u(k)   C   y(k)  0  v(k)  -1 -1  CF   y(k -1)  0 CF G M  v(k -1) 0   u(k - 1) x(k) = + −  M   M  M  M  M O M   M            CF-s   y(k - s)  0 CF-s G L CF-1G  u(k - s)  v(k - s) Then the state vector x(k) could be found through solution of the linear equation: O1x(k) = Yk,k−s + O 2U k,k−s − Vk,k −s It is possible to solve equation (2.b) in the least-squares sense so that

(2.a)

(2.b)

[O1ˆx(k) − (Yk,k −s + O2U k,k−s )]T R −1[O1xˆ (k) − (Yk,k −s + O2 Uk,k −s )]

is minimised. The R is the variance matrix of Vk,k−s . The least-squares solution xˆ (k) is given by:

( ) OT R −1 (Yk,k −s + O2 Uk,k −s ) −1 (OT R−1O1) OT R −1 (Yk,k −s + O2 Uk,k −s )

T −1 xˆ (k) = O1 R O1

xˆ (k) =

−1

(3)

1

1

(3)

1

Block matrix multiplication yields to: s j   −1 −jT T −1 −(j− i+1) xˆ (k) = W ∑ F C Ω  y(k − j) + ∑ CF Gu(k − i)   j=0 i=1 where: W =

T −1 O1 R O1

s

=

∑ F− j

T T −1

C Ω CF

−j

(4)

.

(5)

j=0

The condition under which the state xˆ (k) is observable is given by the existence condition for the inverse matrix of W. From Equation (5), it is possible to rewritten matrix W in the form  s s  T −jT T −1 −j −s T  jT T −1 j −s W = F C Ω CF = F F C Ω CF = F −s ΓsF−s F   j=0  j= 0 The observability conditions of the state are given by the proprieties of the observability gramian: the minimal s is the smaller integer such that O1 is a full rank matrix (rank n) [Murthy, 1980]

∑

∑

3. Sequential observer form To clarify the influence of the number of measurements s on the observer performance, we propose in this section a sequential form for the observer with respect to s. The sequential observer form is given by an equation similar to the conventional Kalman-filter [Kratz et al., 1994]: s+1   xˆ s+1(k) = xˆ s (k) + Ts +1qs+1 y(k − s − 1) + ∑ CF-s-2+ iGu(k − i) − CF −s−1xˆ s (k) (6)   i=1

(

T

-1

where: Ts +1 = O1, s R O1,s

)

-1

-s-1 T  CF qs+1 I



+ q s+1CF

-s-1

(O

T 1, s

-1

R O1,s

)

−1

−s −1T T  CF qs+1



−1

 q Tq 1 1  0 −1 with: q s+1 defined by R s+1 =  M   0

0 qT 2 q2 L L

  O 0   0 qT s+1qs+1 L M

0 M

An appropriate value for s can be found by the designer by a systematic increase of s. On the other hand, an optimal value of s can be found through the convergence of the variance of the estimated state xˆ (k) . From equations (3) and (6), the variance of the estimation is given by: ˆ (k) = W-1 cov x As W depends on s, we can find a recursive expression of its inverse if we denote W with a subindice: −( s+1) T T T − (s +1) Ws+1 = Ws + F C q qCF Then, using the inversion bloc matrix lemma, we get: −1 -1 -1 −1 -(s+1) T T T  -(s+1) -1 -(s+1)T T T  -(s+1) -1 C γ  I + γCF Ws F C γ  γCF Ws (7) Ws+1 = Ws − Ws F with: Ω −1 = γ T γ . If sequence Ws-1 converges [Caines, 1988] when s increases, then there is no significant change in the new estimate. This implies that the observer memory is limited, and that the estimate can be calculated only on a fixed number of measurements.

Part B 1. Application to fault detection Our goal is to work out a robust diagnosis method, i.e. a diagnostic procedure which is both sensitive to failure and insensitive to model disturbances. We mean that our degree of robustness is a compromise between too criteria that have to be minimal : the non detection rate and the false alarm rate. In this part, we are going to show how finite memory based techniques can be useful in order to improve the degree of robustness of the diagnostic procedure. We use the previous finite memory observer and first study the influence of the measurement data on the estimation. Then, we will introduce a special residual defined as the difference between two estimations of the state at the same instant based on two different observation windows. We will show how this residual can improve robustness against parameter variations by comparing its statistical proprieties to the ones of the residual we usually used, i.e. the estimation error. The basic idea is that the more recent measurement might contains information about a possible fault. So, to improve the rate of detection, we have to keep a good sensitivity of the estimation to these data while the sensitivity with respect to past data decreases. This is done by finite observation window. Besides, while parameter variations occur, thanks to the finite memory, these variations are less accumulated in the estimation error. As a consequence, this error (the residual) will not diverge. Then, using the difference of two estimations based on two overlapping horizons we will give more influence to the fault. 2. Sensitivity of the estimation to data

Before we study our special residual, it is important to compute both sensitivities of the estimation to the output measurement and the input measurement. Indeed, this will be a tool to choose the optimal size of the observation window (in spite of the minimal variance of the estimator). We first define such sensitivity. In fact, we need to define the sensitivity of each component of the state (n) with respect to each component of the measurement vector, i.e. the output (p components) and the input vector (m components). So we define the following sensitivity matrix of the state estimation with respect to the output:  ∂ˆx1 (k) ∂ˆx1 (k)  K  ∂y 2 (k − q) ∂y p (k − q)  ∂xˆ (k)  (8) = M M M  ∂y(k − q)  ∂xˆ n (k) ∂xˆ n (k)  K  ∂y p (k − q)   ∂y1 (k − q) where k stands for the current instant and q from 0 to s defines the delayed instant (k-q). Note that (s+1) is the size of the observation window. We stress on the fact that a similar definition with respect to u(k-q) can be written with q from 1 to s. We want to quantify the influence of the last data taken to compute the state estimation, i.e. instant k, with respect to the influence of the previous data used, i.e. k-q. We are especially interested in comparing this influence to the one of the first data of the observation horizon, i.e. instant k-s. With our definition of the sensitivity (8) and the developed expression of the state estimation (4), we deduce: ∀k,∀q ∈ {0,K,s} T (9) ∂xˆ (k) = W −1F −q C TΩ −1 ∂y(k − q) This expression defines a matrix of dimension nxp. Each component (i,j) informs us about the influence of the corresponding output component (j) on the component of the estimation (i) at the considered instants. This equation leads to the sensitivity we are interested in, i.e. for q equals 0 and s: ∂xˆ (k) = W −1CT Ω −1 (10) ∂y(k) ∂xˆ (k) −1 −sT T −1 (11) =W F C Ω ∂y(k − s) Theses equations allow us to plot the evolution of the sensitivity for a given component of the state and of the output when (s+1), the size of the horizon, increases. We will not study this expression in the general case but we will use it on a simulated example to give some direction to think about. If we premultiply (10) by W, then we easily find from (11): ∂xˆ (k) ∂xˆ (k) −1 −s T (11) =W F W ∂y(k − s) ∂y(k)

We stress on the fact that this expression is in fact independent of the current instant k and only rely on the size of the horizon (s+1). This is true thanks to the expression of the estimation as one can verify it. It is obvious that the study of (11) is not simple in the general case. To have an easier criteria, we search some kind of ratio that could take into account simultaneously all the sensitivities defined. If we consider a norm, as for instance : Au A = sup (12) u≠ 0 u we can express a ratio like: ∂ˆx(k) ∂ˆx(k) (13) αs = ∂y(k) ∂y(k − s) for which a bound can be found using appropriate theorems on norm of matrix products. In any case, the evolution when s varies of As = Ws−1F −s Ws should be studied using the appropriate mathematical tools and a recursive expression of the matrix. Within the following definition: T −1 (14.a) B s+1 = Ws+1 F −s Ws+1 it is obvious that As+1 satisfies: −1 −1 T As +1 = Bs +1Ws+1 F Ws+1 (14.b) Thanks to the recursive expression of W (7.a) and of its inverse (7.b), we get: −1 T T   Bs+1 = In − Ws−1F− (s +1) C Tq T In + qCF − (s+1)Ws−1F − (s+1) CT q T  qCF − (s +1) As       (15) T

−1 F −s F −( s+1) C T qT qCF − ( s+1) +Ws+1 T

T

We now consider the influence of the input on the estimation. Using the same kind of definition as (1) and from the state estimation (4), we deduce: s T ∂xˆ (k) −1 (16) ∀k,∀q ∈ {1,K,s} = W ∑ F − j C T r −1CF −(j−q+1)G ∂u(k − q) j=1 From this equation, we get: s ∂xˆ (k) −1 − jT T −1 −j (17) = W ∑ F C r CF G ∂u(k − 1) j =1 ∂xˆ (k) −1 −s T T −1 −1 = W F C r CF G (18) ∂u(k − s) We note that (17) can be simplified as (19) if we add and subtract the missing term (j=0) in order to make appear the expression (5) of W. ∂xˆ (k) −1 T −1 (19) = In − W C r CG ∂u(k − 1) As for the influence of outputs, we will only give simulated results in the case of our example (see Part C). From the simulated results, we could say that there might exist an integer L which is optimal in the sense that for any integer s superior to L, the outputs y(k) and y(k-s) for any k (and respectively the inputs u(k-1) and u(k-s)) have the same influence on the estimation. In fact, we

should choose s such that the influence of y(k), that might contains information about the failure, should be more important than the influence of y(k-s). The same considerations could be expressed for the inputs. 3. A new residual for fault detection From the statistical proprieties of the estimator, we can easily deduce the proprieties of the classical residual, i.e. the estimation error. We know that the error is corrupted by parameter variations. This variations appears as a bias on the mean of the residual. This is an important drawback for fault detection as this is a reason for a performance degradation. Here, we propose a new residual that could limit this degradation. We define the following residual: (20) ε s,r (k) = xˆ s (k) − xˆ r (k) which is the difference between two estimation based on two overlapping horizons. xˆ k / s is the estimation of the state at the instant k based on the data from k-s to k whereas xˆ k / r is the estimation of the state at the same instant k based on the data from k-r to k. This concept is illustrated on figure 1. x^s(k) instant

k-s

k x^r(k) instant

k-r

k ε s,r(k) instant

k-s

k-(r+1)

k

Figure 1: ε s,r (k) = xˆ s (k) − xˆ r (k) These estimations generate two signals that might contains information about the possible changes appearing on the system. By changes we mean disturbances as the measurement noises, the faults we want to detect and the parameter variations against which the residual has to be robust. Making the difference between both signals we may eliminate some of these disturbances so that we only keep the influence of the fault that might appear between the instants k-s and k(r+1). This domain is call the detection window of size (s-r). Of course these considerations have to be proved based on the study of the statistical proprieties of the residual (20). To express this residual we have to use a proper formalism and get the expression of the two estimations by analogy from equations (4) or (6). The expression of the residual can easily being deduced from the structure of the bloc matrices appearing in the equations if we express the matrices depending on s in function of the matrices depending on r (the smallest window). Because of the number of pages required for this paper, we only give our

demonstrated results (21). We modelise the faults using an extended fault vector:

(

)T

D k,k-s = (d k K d k −s ) = Dk,k-r D k −(r +1), k-s) . As parameter variations, we consider additive matrix variations, i.e. F+∆F and so on. Nevertheless, for our demonstration we only assume variations of the matrix C. Indeed, in that case this variations lead to additive matrix variations on O1 and O2. This is the simplest case but the results are still true for additive variations on the matrix G with ∆O1 equals to the null matrix and the appropriate variations ∆O2. On the contrary these results are not true any more for variations on the matrix F because of the structure of O1 and O2 where appear F-j. In our case, we get the following expression of the residual: ε s,r (k) = Ws-1 − Wr−1 H r ∆O1,r x(k) + ∆O2,r U k−1, k-r + Vk,k-r + Dk,k -r (21.a) + Ws-1H1 ∆θ1,s,r x(k) + ∆θ2,s,r Uk,k-s + Vk−(r +1), k-s + Dk −(r +1),k -s T

(

[

) [

] ]

where: T −1 (21.b) H r = O1,r R r and the appropriate bloc matrices required when expressing the matrices (and matrix additive variations) depending on s with respect to the one depending on r, i.e.:  O1,r   O2,r 0 , O2,s = (21.c) O1,s = θ1,s,r   θ2,s, r  We can then get the statistical properties: Esp{εs,r (k)}= Ws-1 − Wr−1 Hr ∆O1,r x(k) + ∆O2,r Uk, k-r + D k,k-r

(

[

) [

+Ws-1H1 ∆θ1,s,r x(k) + ∆θ 2,s,r U k,k-s + Dk −(r +1),k-s as all the vectors and matrices are independent. Var {ε s,r (k)}= Wr-1 − Ws−1

On the contrary, for the estimation error: e s (k) = xˆ s (k) − x(k) we get: Esp{es,r (k)}= Ws-1H r ∆O1,r x(k) + ∆O2,r Uk, k -r + Dk,k -r

]

[ ] +Ws-1H1[∆θ1,s,r x(k) + ∆θ 2,s,r Uk,k-s + Dk−(r +1),k-s ]`

Var {es,r (k)}= Ws−1 )

]

(22)

(23) (24)

(25) (26)

So, if we choose s as the optimal size, we can consider that the estimation error is of minimal variance (26). Nevertheless, from (23), if we chose r as optimal but not equals to s, we can consider that Wr−1 is almost equal to Ws−1 . This result leads to a null variance for the residual (20). If we now consider (22) and (25), we can deduce: Esp(es (k)) = − Wr-1 Hr ∆O1,r x(k) + ∆O2,r U k, k-r + Dk,k -r + Esp(ε s,r (k)) (27)

[

]

This prove that the bias on the estimation error (24) is more important than the bias on (24). As a consequence, we can conclude that for a proper choice of s and then r, we can eliminate both the bias from the faults and the parameter variations occurring between the instants k-r and k. So we defined a detection window between the instants k-s and k-(r+1). Therefore the use of finite memory observer and of the residual (20) improve the robustness of the diagnostic procedure.

Of course, the same kind of residual can be defined for the extended outputs. Because of the different dimensions of Yk,k-s and Yk,k-r, one has to take the first (r+1) components of the estimation of Yk,k-s which are the estimation of Yk,k-r. The output estimation is given by: (28) Yk,k −s = -O1xˆ s (k) + O 2 U k,k −s Then, it is easy to define the residual. We proved that its statistical proprieties can be easily deduced from the ones of (20).

Part C 1. Simulated example We consider a fourth order system defined by the following matrices with respect to the state discrete-time representation (1):  0.702 0.344 0.304 −0.202 0.386 0 0   0 F=  0.304 −0.337 0.449 0.384   0 0 0 0.135   0.252 −0.405 0  1 0 0 0  0.632 G= C =  0 0 1 0  −0.271 0.717   0 0.865  The state is of dimension n equals 4, the output and the input are of dimension p and m equal to 2. We put an additive white noise on both output. This noise is of zero mean and known variance equals to 1. The system is excited by a series of step inputs. On this system we first search the minimal and then optimal size of the observation window (s+1). The minimal value is given from the study of the gramian of observability. The convergence of the algorithm is shown on figure 2 using the largest singular value as the matrix norm.

Largest singular value

15

10

5

0

2

3

4

5

6 s

7

8

9

10

Figure 2: Evolution of the largest singular value of the observability gramian.

For the optimal value of s, we consider the evolution of the variance of each component of the state estimation with respect to s. On figure 3, we only plot the variance for the 1st component as it is the most significant variance (greater than the others). From figure 3, we can chose s equal to 8. For this choice, we only plot the estimation of the first output on figure 4. But it is obvious that despite of the noise, the outputs are perfectly estimated. The estimation can be used on-line thanks to the recursive formulation of the observer and the finite memory. Then on-line fault detection can be performed. In the following, we only plot two sensitivity in order to illustrate what we can get in practice (see figures 5 and 6). The fact is that a general study of these sensitivity is not simple. Anyway, for our example, the sensitivity with respect to the output shows us that for s superior to 4, the last output data as a greater influence on the estimation than the first output data of the observation horizon. 0.8

0.7

0.6

variance

0.5

0.4 1

0.3

2 3

0.2

0.1 4 0

3

4

5

6

s

7

8

9

10

Figure 3: Variance on each component (i) of state estimation when s varies. 20

15

output(1)

10

5

0

-5

0

10

20

30

40 instant

50

60

70

80

Figure 4: First output estimation (dash) and measurement (continuous) Figure 6 illustrates what can occurs with respect to the inputs. In our case, we can conclude that s should be inferior to 8 as the influence of the 1st component of the last input data decreases in comparison of the influence of the first input data. So in this particular case, we can say that the optimal choice of s with respect to sensitivity criterion is that s belongs to the following interval [ 4 , 8 ]. Anyway, we stress on the fact that we can not conclude in the general case. All we get are quantitative results. Nevertheless, we note that this result agree the results given from the minimal and optimal choice of s presented at the beginning.

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1

1

2

3

4

s

5

6

7

8

Figure 5: Sensitivity of x3 estimation with respect to yk(1) (continuous) and to yk-s(1) (dash)

Figure 6: Sensitivity of x1 estimation with respect to uk-1(1) (continuous) and to uk-s(1) (dash) Then, in order to compare the usefulness of the finite memory for fault detection robustness we compare the estimation error (24) for the previous choice of s = 8 and the new residual (20) we defined. We choose r equal to 4 so that both estimations have the same variance. We note that this criteria agree the minimal bound of the interval deduced from the study of sensitivity. Our detection window is of (s-r) equal 4 width. When there is no failure, we plot both residuals for the 2nd component of the output on figure 7. Both are statistically close to zero but residual (20) is closer. Indeed it has a standard deviation of 0.01 for a mean of -0.005 instead of 1 and -0.01 respectively for the 2nd component of the estimation error. The same results are also true for the first component.

Figure 7: Residuals (24) (continuous) and (20) (dash) We put a bias on both outputs between the instant 40 of value equal to 60. The residuals are plotted on figure 8. They both allow detection but the estimation error is more sensitive to the bias than the residual (24) as it differs more than zero. But is this result still true when there are parameter variations ? We consider this case, with a 10% variation on each parameter of matrix F. The residuals are plotted on figure 9. It is obvious that the residual (20) allows a better detection than state estimation error (24). This is more obvious for parameter variations on matrix G. The results are plotted on figure 10. A Grubbs test should confirm this results a smaller window should be used for residual (24) than for (20).

Figure 8: Residuals (24) (continuous) and (20) (dash) when a bias occurred

Figure 9: Residuals (24) (continuous) and (20) (dash) while parameter variations on matrix F (10%)

Figure 10: Residuals (24) (continuous) and (20) (dash) while parameter variations on matrix G (10%)

Conclusion In this paper we show how the finite memory observer can be useful in order to improve the robustness of the diagnosis procedure with respect to system uncertainties. Indeed, the use of a particular residual defined by the difference between two estimations of the state at the same instant but based on overlapping observation horizons, improve the sensitivity to the fault when parameter variations occur. The main result of this paper is that we give some criteria in order to choose the size of the observation window. Indeed, the study of the observability gramian gives the minimal value of the width of the observation window while the state estimation variance give the optimal choice for it. The influence of the data on the estimation is given but more specific mathematical tools should be used in order to conclude in the general case. The statistical proprieties of the new residual defined has been studied and will be developed in a future communication. References Atherton D.P, Borne P., Concise encyclopedia of modelling and simulation, Pergamon Press, New York, 1992. Caines P.E., Linear Stochastic Systems, Wiley, New York, 1988. Frank P.M., Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy, a survey and some new results, Automatica, 26 (3), 1990, p. 459-474. Frank P.M., Advances in observer-based fault diagnosis, Tooldiag'93, Inter. Conf. on Fault Diagnosis, Toulouse, France, April 5-7 ; 1993 p 817-836. Heffes H., The effect of erroneous models on the Kalman filter response, IEEE Trans on Automatic Control, 11, 1966, p. 541-543. Kratz F., Bousghiri S., Mourot G. A finite memory observer approach to the design of fault detection algorithms, American Control Conference, Baltimore, USA, June 29-July 1, 1994, p. 3574-3576. Medvedev A.V. Three approaches towards a fault-tolerant controller structure. Report 91-3, Process Control laboratory, Abo Akademi, Abo, Finland, 1991. Medvedev A.V.,Toivonen H.T. Investigation of a finite memory observer structure. Report 91-6, Process Control Laboratory, Abo Akademi, Abo, Finland, 1991.

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