ON THE APPLICATION OF A SUBSPACE-BASED FAULT DETECTION METHOD *

ON THE APPLICATION OF A SUBSPACE-BASED FAULT DETECTION METHOD * L.Mevel, M.Basseville, A.Benveniste, M.Goursat, M.Abdelghani IRISA-INRIA, Campus de Beaulieu, 35042 Rennes cedex, France! L.Hermans LMS-International, lnterleuvenlaan 68 B-3001 Leuven, Belgium. ABSTRACT In this paper, we address the problem of detecting faults modeled as changes in the eigenstructure of the associated linear dynamical system and its application to some industrial problems, such as mechanical structures. The outlined method is based on both recent stochastic subspace-based identification methods and the statistical local approach to the design of detection algorithms 1 2 ][ 16 1. The key idea is to replace regular systematic inspection by condition-based inspection, i.e. inspection decided upon the continuous monitoring of the considered system based on sensor data. The statistical local approach deals with the early detection [ 19 ] of slight deviations in usual working conditions, based on long samples of multiple sensors outputs measurements, it detects very small changes in the modal parameters [ 4, s] After a brief review of the theory, we will focus on the demonstration of the capability and efficiency of this approach using output-only samples measured on a few industrially relevant examples, such as a steel subframe structure. For each of these examples, we present the application of the method in order to detect slight changes in their eigenstructures. NOMENCLATURE {h} {Xk} ll m,. (F].(H] (<I>] (A] ('7-lpq] (p,q) E (W1],(W2] T vee( ) diag(-) Output vector at discrete time instant k State vector at discrete time instant k Number of data samples Number of system poles Number of sensors State transition and output matrices Normalized mode shapes Discret modes eigenvalues Hankel matrix Number of blocks rows/columns Mathematical expectation Weighting matrices Transpose of matrix operator column stacking operator vector of matrix diagonal This work is being accomplished in the context of Eureka project no 1562 S!!\"OPSYS (Model based Structural monitoring using inoperation system identification) coordinated by L:v1s, Leuven, Belglum. I MB. is also with Cnrs. 1 y BR CVA Size of the damage Balance Realization Canonical Variate Analysis INTRODUCTION The problem of fault detection and isolation (FDI) is a crucial issue which has been investigated with different types of approaches, as can be seen from the survey paper1 12 l and the book [! 4 ] among other references. Moreover, an increasing interest in condition-based maintenance has appeared in a large number of industrial applications. The key idea there is to replace regular systematic inspections by condition-based inspections, i.e. inspections decided upon the continuous monitoring of the considered system (machine, structure, process or plant), based on the sensors data, in order to prevent from a possible malfunction or damage before it happens and improve the maintenance cost. Condition-based maintenance typically involves the monitoring of components, and not only of sensors and actuators. It has been recognized that a relevant approach to condition-based maintenance consists in the early detection of slight deviations with respect to a (parametric) characterization of the system in usual working conditions, without artificial excitation, speeding down, or stop. Processing long samples of multi-sensor output measurements is often necessary for this purpose. It turns out that, in many applications, the FDI problem of interest is to detect and diagnose changes in the eigenstructure of a linear dynamical system. An important example is structural vibration monitoring [r.j Vibrating structures are classically modeled as a continuous time linear system driven by an excitation, and whose output vector is the set of accelerometer measurements. Their vibrating characteristics (modes and modal shapes) coincide with the system eigenstructure. The key issue is to identify and monitor vibrating characteristics of mechanical structures subject to unmeasured and non-stationary natural excitation. Typical examples are offshore structures subject to swell, buildings subject to wind or earthquake, bridges, dams, wings subject to flutter in flight, and turbines subject to steam turbulence, friction in bearings, and imperfect balancing. A relevant approach to the vibration monitoring problem has been shown to be based on the modeling of modes and modal shapes through state space representations [l s]. the use of 35

output-only and covariance-driven identification methods (such as instrumental variables or balanced realization algorithms) [ 7 1, and the computation of specific x 2 -type tests based on the socalled instrumental statistics [s]. In practice, these tests turn out to be robust w.r.t. non-stationary excitation. Moreover, the design of this \: 2 -test is a particular case of a general approach to FDI which builds a detector from a convenient estimating function [ 2. 8 19 1. On the other hand, during the last decade there has been a growing interest in subspace-based linear system identification methods [tg, 17 ] and their relationship with instrumental variables [tbj. These methods are especially well suited for capturing the system eigenstructure. Moreover, we know from practice that this family of methods is robust w.r.t. the non-stationary of the zero part. Because of what has been argued above, the question arises to design FDI algorithms based on this class of identification methods and to investigate their theoretical properties. In [ 4 ], some damage detection theoretical results are presented. It is the purpose of this paper to recall and validate them on a few laboratory examples. Some more practical aspects of this method are discussed in ll 3 l, including interface and practical implementations. 2 SYSTEM MODELS AND PARAMETERS We consider linear multi-variable systems described by a discrete-time state space model FXk + ek HX, + l/k where state X and observed output Y have dimensions m and r respectively. The state noise process (ek)k is an unmeasured Gaussian white noise sequence with zero mean. We assume noise E:k to be stationary, that is of constant covariance matrix; the issue of robustness with respect to non-stationary excitation is addressed in [to]. The measurement noise process (vk )k is assumed to be an unmeasured MA(L) Gaussian sequence with zero mean. In the sequel, we use the notational convention that 1. = -1 for no measurement noise, and L = 0 for white (i.i.d.) measurement noise. Note that, with this MA assumption for its structure, measurement noise does not affect 1 the eigenstructure of system ( 1). Let G ~~ E (X k ykt) be the cross-correlation between state X k and observation } 1,, and let : 1,...1, _- ( ZF L' ) HFp-t and Cp = ( G FG (1) for p large enough. both observability and controllability matrices have full rank m. The problem we consider is to monitor the observed system eigenstructure. The observed system eigenstructure is the collection of m pairs(.\, r >, ), where.\ ranges over the set of eigenvalues of state transition matrix F, 6:- = fl.,:>:.. and..,:>.1 is the corresponding eigenvector. In all what follows. we assume that the system has no multiple eigenvalues, and thus that the.\'s and 'P!.'s are pairwise complex conjugate. In particular, 0 is not an eigenvalue of state transition matrix F. We stress that the collection of(.\, c/ja) provides us with a canonical parameterization of the pole part of system (1). In particular, it does not depend on the state space basis [T]. In the sequel, referring to vibration monitoring, such a pair(.\, >-) is called a mode. The set of the m poles is considered as the system parameter I} I} c!_:_f (.\ ) - Yec<I> In (3), ;\is the vector whose elements are the eigenvalues.\, <ll is the matrix whose columns are the >. 's, and w c is the column stacking operator. Parameter I} has size (r + l)m. The FDI problem is to detect and isolate changes in parameter vector 1}. By isolation, we mean deciding which mode(s) has(ve) changed. The FDI problem is then to detect and isolate changes in the eigenstructure of a multi-variable. \fl::<.ia process with unknown 1\IA part. At first glance, it seems reasonable to consider that diagnosis of failures can be performed by comparing a new model signature to a reference one. However, the comparison of two different modal signatures is an extremely difficult task for the following reasons. To decide how eigenfrequencies and corresponding modal shapes have been modified, eigenfrequencies of different signatures must be compared and assigned to each other. Moreover, how do you decide if a given change in some eigenfrequency or modal shape is really significant 77 This is really a difficult point if numerical approaches are used to estimate such signatures. In the same vein, heuristic approaches lead to false alarms, since it is difficult for the human operator to evaluate how sensor noise affects the uncertainty in a considered eigenfrequency or modal shape. For this reason, the statistical approach has been developed, to overcome both of the above drawbacks. Its principle is that given a reference modal signature and a new record, evaluate how this record agrees with the signature. In particular, it is not necessary to identify a new signature for such processing, and the proposed procedure is much simpler than signature identification since it requires much less involvement of the human operator. (2) 3 SUBSPACE-BASED FDI ALGORITHMS be the pth-order observability matrix of system (1) and controllability matrix of pair ( F, G). respectively. We assume that, 1 The same would not hold true with an AR assumption for the measurement noise structure. Throughout this section, we assume that the true system order is known. Of course, this is somewhat unrealistic, but it allows us to introduce the key tools of our design of subspace-based (3) 36

tests. In situ identification of the structure provides us with a signature of the safe structures in terms of a collection of modal characteristics. When a new record is available, it is compared to the signature using a statistical measure of the distance between this new record and the signature in order to decide whether or not both are still in agreement, this gives the alarm. From (9), (6) and (7), we get that the following property characterizes whether a nominal parameter 8 0 agrees with a given output covariance sequence (R 1 ) 1 [I 8 ] : Op+I (Bo) and H.p+I,q have the same left kernel space. (10) Property (10) can be checked as follows : 3.1 System theoretic issues The arguments presented here are well known [! 8 ], we just state them here for the sake of clarity and completeness. We are given a sequence of covariances : of output Yi. of a state space model (1). For q > p + 1, let 1-lp+l,q be the block-hankel matrix: ( R,+, R,+2 R<+q ) R,+2 R,+3 R,+q+l 1-lp+l,q = : (5) R,+p+I R,+p+q As mentioned above, integer L reflects the assumed dependency in the measurement noise sequence (vk)k. It should be considered as a des'1gn parameter for the algorithms described in the paper. Choosing the eigenvectors of F as a basis 2 for the state space of model (1) yields the following particular representation of the observability matrix introduced in (2) [s] : (4) (6) 1. From 8 0 as defined in (3), form Op+ 1 (8 0 ), and pre-multiply it by some invertible weighting matrix H' 1 ; 2. Pick an orthonormal basis of the left kernel space of matrix ~VI OP+ I ( 8 0 ), in terms of the columns of some matrix S of co-rank m such that : 81's st WI Op+I (Bo) (11) (12) Matrix S has dimensions (p+ 1 )r x s, where.s = (p+ 1 )1 m, and it is not unique; it can be obtained, for example, by the SVD-factorization of lf1 01'+ 1 ( 8 0 ). 3. The parameter 8 0 which actually corresponds to the output covariance sequence ( R 1 ) 1 of model (1) is characterized by: (13) where W2 is another invertible weighting matrix for choice. This matrix is introduced here by reference to usual subspace-based identification methods [IG]_ It has been shown in [ 4 ] that any choices of such weightings should give the same test result. So any weightings, including those corresponding to the BR- and CVA- methods are acceptables. 3.2 Subspace-based tests : a simple derivation Assume we have at hand where diagonal matrix 21. is defined as 21. = diag(a), and A and <[> are as in (3). For any other state basis, the observability matrix Op+ 1 can be written as : for a suitable m x m invertible matrix U. Because of the definition of 1-lp+I,q, Op and Cq in (5) and (2) respectively, a direct computation of the Rj's from the model equations : (7) (8) A nominal model Bo. and Newly collected data Y1,..., 1;,. Form the empirical covariance sequence : R 11-] ~ d_~~ _1_ 2: 1 1 1' ) -. k+) k n- J k=l (14) Then, perform steps 1 and 2 of subsection 3.1, and replace step 3 by: leads to the following well known factorization property : (9) 3. Define the residual vector : (15) 2 This is called the modal basis in the vibration monitoring application [ 6 1. where fi.p+i.q is the empirical block-hankel matrix obtained by substituting R 1 for R 1 in (5). 37

From (13) we already know the following. Let B denote the actual value of the parameter, for the system which generated the new data sample. Then : Ee ((,(Bo)) = 0 iff B = Bo (16) Thus it is natural to replace criterion (13) by the requirement that statistics (n(bo) should have zero mean. Based on a data sample, testing if this hypothesis is valid requires the knowledge of the statistical properties of the residual (,(Bo). This is addressed next. 3.3 Subspace-based tests : theoretical design We need to know the distribution of (,(Bo) when the actual parameter for the new data sample is B. Unfortunately, this distribution is unknown, in general. One manner to circumvent this difficulty is to use a local approach, that is to assume close hypotheses : (Safe ) Ho : B = Bo and y (Faulty)H1: B=Bo+ fo (17) where vector Y is unknown, but fixed. Note that for large n, hypothesis H 1 in (17) corresponds to small deviations in B. As formally stated below, it turns out that residual (, in (15) is asymptotically Gaussian distributed, under both Ho and H1, making easy its evaluation. More precisely, let Ee and cove be the expectation and the covariance, respectively, when the actual system parameter is I}. We define the mean deviation : M(l}o) def and the residual covariance matrix : - 1 r.:; EJB a Ee 0 (n(b) I yn 8=8o (18) (19) Note that matrix M '~ 1 M (eo) in (18) can be viewed as a Jacobian matrix. The following central limit theorem ( CLT) holds, as proven in [I.B-ID,I 9 l. Also define 2: ~~ l:(l}o) in the following. Theorem 3.1 (clt.) Provided that B(Bo) is positive definite, the residual (, defined in (15) is asymptotically Gaussian distributed under both hypotheses defined in {17}; that is : (,(l}o) --+ { N( o, E(Bo)) N ( M(Bo) Y, E(Bo)) under under Ho H1 (20) where M(flo) and E(Bo) are defined in {18} and {19}, respectively. As explained in section 4, Jacobian matrix lvl can be estimated from a data sample using a simple sample average. The estimation of covariance matrix E is more tricky [l 9 ] A closed form formula for its estimation is given in 1 4 1, but we recommend to use an empirical estimate instead, based on a simple version of the jackknife method [G. II] This is clearly explained in 1 11 13 1. Let M, E be consistent estimates of M, E. The detection problem, namely deciding that residual (, is significantly different from zero can be achieved as follows. Theorem 3.2 (x 2 -test.) Assume additionally that : Jacobian matrix M(l}o) is full column rank {f.c.r.) (21) Then the test between the hypotheses Ho and H1 defined in {17} is achieved through 2 d_<ct I"T 0-I,,~I (,~IT 0-I ~~I) - 1.,~IT 0-I I" \'n - l..,n L...l ~ 11...~,\ 11...~ \,n {22) which should be compared to a threshold. In {22}, the dependence on 1} 0 has been removed for simplicity. Test statistics x; is asymptotically distributed as a x 2 -variable, with rank( M) degrees of freedom and with non-centrality parameter under H1 : yt MT l:- 1 MY (23) The isolation problem is to decide which (subsets of) components of the parameter vector- corresponding to some columns of the Jacobian matrix- are affected by the change. In the case of the present paper, this allows us to formulate the diagnosis of changes in terms of eigenfrequencies, damping coefficients, and eigenvectors. So to perform a diagnosis on some modes, it is only needed to restrict the Jacobian to the corresponding columns as presented in 1 4 1 and to perform the test as usual. 4 ESTIMATION OF THE JACOBIAN MATRIX In this section, we investigate the estimation of the Jacobian (or sensitivity) matrix M( B) of a test statistics, where U, W1, W2 denote the design matrices. Using (18), we get : (24) where S ~~ vec(sr), and 8 ~)~ 81 is obtained by differentiating constraints (11)-(12) with respect to Bo. Writing (25) we get that a~~e) is solution of the following system of equations : (OJ+ 1 (B) wr 0 Is) 0 ~~1}) + (Im C ST(I}) IF1) o;,+ 1 (1}) =(261) A consistent estimate for M(l}o). based on a data sample, is: ~ ( ~ T T ) DS ( 11) M(BJ =- w2 'HJJ+!,q wl = u CJIJ (27) where EJS(B)j8(1 is a solution of (26). It is easily checked that A-f(B) does not depend on the particular choice of the eigenvectors r/j>. contained in the parameter vector I} defined in (3). 38

safe structure I 1st damage 2nd damage J (1-4) I 6600 16600 J 3rd damage J 4th damage 5th damage~ 94000 I 400000 2000000 TABLE 1: Test result for the 1st experiment Figure 1: 1st experiment description 5 THE WHOLE PROCEDURE AS A LABORATORY EX PERIMENT In this section, we conduct several experiments in laboratory on both safe and damaged structures. Our goal is to detect, as soon as possible, a small change in the modal parameters corresponding to a slight degradation in the physics of the structure. In each of the following subsections, we address different damages of different magnitude to evaluate the quality of the test. For each of the structures (safe/damaged), we further provide a modal analysis based on a classical identification approach based on subspace methods. safe structure 35.6 77.1:3 121..5 186.7 252.9 :339.4 486.25 584.8 698.1 1st damage :3.5.55 77.01 1:26.99 1~'<6..56 251.54 :339 48.5.6.58:3.8 698 2nd damage :3.5.5 IG.9 I :.!6.~ 186.'1 2.51..5 :331.9 484.. 1 580.1 696.8 3rd damage 3.5.2 76.9 126.2 186.3 251.47 331.46 482.96 511.66 696.6 4th damage 35.1 76.8 124.9 185.61 251.34 336.35 480.2 572.6 694.5 5th damage 34.9 76.7 123.6 184.8 251 334.1 476.0 565.7 692.1 TABLE 2: Modal analysis of the 1st experiment 5.1 First experiment We use a horizontal aluminium clamped-free beam which is fixed at one end and free at the other end. The characteristics of the beam are as follows : the beam is 700 mm long, 20 mm large, 2 mm high, and we place 8 sensors on the beam (distance between sensors is 100mm), the first one is on the same location as the vibrator (see Figure 1). The beam is excited by a vibrator producing a white noise like excitation. The signal has been digitalized at 1600 Hz sampling rate, and recorded for 26 sec using all the sensors synchronously. At the end we are computing the test using 32000 samples. Two measures are done at each step (i.e. before each damaging) to measure the influence of the excitation. Once the signal is measured on the safe structure, we do a step-by-step cut on the beam to obtain damage signal data sets. The cut occurs at 24.4 em from the vibrator, and is 1mm large. The total size of the cut increases during the experiment. The beam is damaged 5 times. The first time the size of the cut was 3 mm, the second time 7.05 mm, the third time 9.15 mm, the fourth time 12.1 mm and the fifth time 15.2 mm. In the Table 1, we display the result of the damage test at each step of the experiment we plot the ratio between the value of the test obtained on the possibly damaged set and the value of the test on the reference data set. So, this value should be around 1 without change and much higher if the damage occurs. We can compare the result of the test with the modal analysis (Table 2) on each of the structures. provided by a subspacebased identification method. Modal identifications, listed in Table 2, are not needed for the test, i.e. the damage test requires only a modal identification (as well as computation of the Jacobian and covariance matrices) on the safe structure, then the computation on the test on the possibly damaged set is straightforward and quite fast (and is automatically done in contrast with the identification procedures). Notice that we display only the frequencies of the modes. Since output-only identification methods usually give unsufficiently accurate estimates for dam pings, we do not monitor them All of these identifications were made using the classical BRsubspace method, and are subject to small numerical uncertainties. So looking at these results is not sufficient to decide about the damage, especially when the modes change is very small. Remark that the (1-4) magnitude on the reference structure reflects the robustness of the test on a validation data set, i.e. a set of data provided by the same model (with possibly different excitation) but different from the reference data set. In this case, the test will give a higher value than expected, but much lower than a test value corresponding to a change in the modes. As it appears in Table 1, when we confront our reference data set with another structure with the same modes, but excited differently, the value of the test is slightly above the expected value. Therefore, to obtain reliable results with associated confidence regions, the algorithm should be trained with a large set of safe 39

Figure 2: Different locations of sensors for 1st experiment Figure 4: Modal analysis of the second experiment Change of mass I Frequency increase Figure 3: Finite elements description for 2nd experiment Figure 5: Modal analysis of the second experiment Change of stiffness I Frequency decrease structures (damaged structures) to define upper (lower) bounds for the Ho (Hl) test using some sort of monitoring method. To investigate the importance of the choice of number and location of sensors, we display in Figure 2, the different log-values of the test (normalized by the value of the test obtained on the safe structure) during the experiment with 4 sensors (instead of 8) located at different parts of the beam. On the 4 sensorstest the best results are obtained with sensors near the damaged part (x symbol) or set uniformly across the beam(* symbol). In contrast the sensors located at the other end ( o symbol) have less reaction to the damage. This stress the importance of the choice of such location for better results. Fortunately, this question has been addressed in [t;, 1 11, where some techniques were developed to find the optimal set of sensors, and this can be done with use of the safe structure alone. 5.2 Second experiment A validation test was performed on a breadboard model of a steel frame (see Figure 3). The structure resembles to subframe of a car to be connected to the body at 4 locations and on which the engine has to be mounted. The frame is approximately 720mm long and 170mm wide and the weight is about 9.8kg. The subframe was suspended on four flexible threads, the response being measured in the vertical direction at 27 points for vertical excitation at 2 points. Using dual random shaker excitation, time records of the 2 inputs and the 27 outputs were measured, sampled at 1024Hz providing 30000 samples on each sensor. A modal analysis- using only 3 sensors- was performed on this structure and is shown in Figure 4. As it can be seen in Figure 4, a mass of 0.5kg was removed shifting up the frequencies. Another experiment was made with a similar subframe modifying the stiffness of the structure shifting down the frequencies, see Figure 5. As expected, the test reacts very sharply to the change of mass (see Table 3) and detects also the change of stiffness. As expected, the biggest change in modes gives us a bigger increase in the test value. 5.3 Third experiment The third experiment is a simulation of a 30000 samples signal corresponding to the modes listed in Table 4 for a sample frequency of 60 Hz and 6 outputs. The results corresponding for the same data, same model but different excitation, and finally data corresponding to a shift of the third frequency (17.1--+ 17.6H:;) are displayed in Table 5. Finally we show some early results on damage sens1t1v1ty. In Table 6 the highest values are displayed for the sensitivity test applied on each mode. The third mode is clearly detected as the most important giving approximately a test result four times higher than the second one in the list. Similar results on real safe structure mass change (1-12) stiffness change (1-5) TABLE 3: Test result for the 2nd experiment 40

TABLE 4: Model for the Jrd experiment safe structure validation structure damaged structure 1 3 7300 TABLE 5: Test result for the Jrd experiment data are interesting only in relation with the physical model, as presented in 1 11 ] Such results will be presented later. 6 CONCLUSION In this paper, we have shown the effectiveness of this approach. The damage detection test, whatever you choose for the weightings, is very accurate, and detects very early the damage occurring during the experiments. Some results on industrial data, as well as practical considerations, are presented in [t 3 1. (6] M. BASSEVILLE, A. BENVENISTE, B. GACH- DEVAUCHELLE,.M. GOURSAT, D. BONNECASE, P. DoREY, M. PREVOSTO and M. Ou.. GNO;-;! (1993). Damage monitoring in vibration mechanics: issues in diagnostics and predictive maintenance Mechanical Systems and Signal Processing, vol.7, no 5, pp.401-423. (7] A. BENVENISTE and.j.-.j. FUCHS (1985) Single sample modal identification of a non-stationary stochastic process. IEEE Trans. on Automatic Control, voi.ac-30, pp.66-74. (8] A. BENVENISTE, M. BASSE VILLE and G. l\ioustakides (1987). The asymptotic local approach to change detection and model validation. IEEE Trans. Automatic Control, voi.ac-32, no 7, pp.583-592. (9] A. BENVENISTE,!vi. l\ietivier and P. PHIOURET (1990). Adaptive Algorithms and Stochastic Approximations. Springer, NY. [10] B. DEL YON. A. J PDITSKY and A. rlen\"e:'-iiste (1997). On the relationship between identification and local tests. Research Report lnrsa no 1104. ftp: I lftp.irisa. fr I tech reports/ 1997 /P 1-1104. ps.gz. [11] B. DEVAUCHELLE-GACH (1991). Diagnostic Mecanique des Fatigues sur les Structures Soumises a des Vibrations en Ambiance de Travail. Thesis, Paris IX Dauphine University (in French). [12] P.M. FRANK (1990). Fault diagnosis in dynamic systems using analytical and knowledge based redundancy- A survey and new results. Automatica. vol.26. pp.459-474. REFERENCES TABLE 6: Highest sensitivity test result Third experiment [1] I.V. BASAWA (1985). Neyman-Le Cam tests based on estimating functions. In Proc. Berkeley Conf. honor of Neyman and Kiefer, vol.2, Le Cam and Olshen (Eds), pp.811-825. [2] M. BASSEVILLE (1998). On-board component fault detection and isolation using the statistical local approach. Automatica, vo\.34, no 11. to appear. [3] M. BASSEVILLE and l.v. NIKIFOROV (1993). Detection of Abrupt Changes - Theory and Applications. Prentice Hall, Englewood Cliffs, N.J. (4] M. BASSEVILLE, M. ABDELGHANI and A. BENVENISTE (1997). Subspace-based fault detection and isolation methods- Application to vibration monitoring. Research Report IRISA no 1143. ftp: I lftp. irisa.fr ltechreportsl1997 IPI-1143.ps.gz [5] M. BASSEVILLE, A. BENVENISTE, G. MousTAKIDES and A. RouGEE (1987). Detection and diagnosis of changes in the eigenstructure of non-stationary multi-variable systems. Automatica, vol.23,_ no 3, pp.479-489. [13] L. HERI\IANS. L. \It:VEI..\ND H. \'.\:'-! DFH At'II"ERAER, Health monitoring and detection of fatigue problems of a sports car. In proceedings of IMAC 99. [14] R.J. PATTON, P. FH.UK and R. CLARK, Eds. (1989). Fault Diagnosis in Dynamic Systems - Theory and Application. Prentice Hall. (15] M. PHEVOSTO, :M. 0LAGNON, A. BENVENISTE,!vi. BASSEVILLE and G. LEVEY (1991). State-space formulation, a solution to modal parameter estimation. Jal Sound and Vibration, vol.l48, pp.329-342. [16] P. VA:'-1 OvERSCHEE and B. DE Moon (1996). Subspace Identification for Linear Systems Theory- Implementation -Applications. Kluwer Academic Publishers. [17] M. VIBERG (1995). Subspace-based methods for the identification of linear time-invariant systems. Automatica, vol.31, no 12, pp.l835-1853. [18].M. VIBERG, B. \". 1 AHLBERG and B. 0TTEHSTE~ (1997). Analysis of state space system identification methods based on instrumental variables and subspace fitting. Automatica, vol.33, no 9, pp.1603-1616. [19] Q. ZHANG, M. BASSEVILLE and A. BENVENISTE (1994). Early warning of slight changes in systems and plants with applic~tion to condition based maintenance. Automatica, vol.30, no 1, pp.95-114. 41