Optimal fault detection in linear stochastic systems with nuisance parameters

Optimal fault detection in linear stochastic systems with nuisance parameters

IFAC Copyright 0 IFAC Fault Detection, Supervision and Safety of Technical Processes, Washington, D.C., USA, 2003 l:Ql> Publications www.elsevier.co...

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Copyright 0 IFAC Fault Detection, Supervision and Safety of Technical Processes, Washington, D.C., USA, 2003

l:Ql> Publications www.elsevier.comllocatelifac

OYflMAL FAULT DETECI'ION IN LINEAR STOCHASTIC SYSTEMS WITH NUISANCE PARAMETERS Mitra Fouladirad • and Igor Nikiforov •

• LM2SITJIT; 12 rue Marie Curie ]0010 Troyes, France E-mail: {mitra.fouladirad.igor.nikiforov}@utt.fr

Abstract: The goal of this paper is to propose an optimal statistical tool to detect a fault in a linear stochastic (dynamical) system with uncertainties (nuisance parameters). It is supposed that the nuisance parameters are unknown but non random; practicaUy, this means that the nuisance can be intentionally chosen to maximize its negative impact on the monitored system (for instance, to mask a fault). An example of GPS integrity monitoring illustrates the proposed method. Copyright © 2003 IFAC Keywords : fault detection,linear systems, stochastic systems, decision theory, tests, nuisance parameter, parity space.

(dynamical) system with nuisance parameters which are supposed to be unknown and non random. The paper is organized as follows. We start with the problem statement in section I. A simple linear model (without nuisance) is treated in section 2. Next. the nuisance parameters are introduced and the problem of optimal fault detection with nuisance is discussed in section 3. Section 4 is devoted to a dynamical model with a deterministic state equation governed by an unknown input. An example of ground station based GPS integrity monitoring illustrating the relevance of the proposed tools is described in section 5.

INTRODUCTION Fault detection is necessary for security, conditionbased maintenance, supervision. Usually, two types of faults can be distinguished: instruments (sensors) and process faults. Instruments faults can often be modeled by an additive change in a regression or state-space model, whereas process faults are more often non additive changes in such models. The aim of this paper is the detection of additive faults. A crucial issue is to state the significance of the observed changes with respect to noises, uncertainties (nuisance parameters), and changes in the environment of the monitored process (Basseville and Nikiforov, 2002; Frank. 1990; Staroswiecki, 2(01). The problem of nuisance parameters rejection is traditionally treated in the framework of the analytical redundancy (AR) approach. This approach is based on some natural geometric properties of systems. The theory of AR is deterministic. The design of the "optimal parity space" remains the subject of extensive research. On the other hand, there exist mathematical statistics theories and tools to handle noises and uncertainties, to reject nuisance parameters, to decide between two hypotheses ?io (no faults) and 'HI (there exists a faul!). The goal of this paper is to propose an optimal statistical tool to detect a fault in a linear stochastic

I . PROBLEM STATEMENT

The two following situations are distinguished in fault detection. The first one is hypotheses testing. i.e. a fixed size sample of observations YI , . .. , Yk is available and supposed to be generated by one of two alternative hypotheses . The null hypothesis 71.0 corresponds to the fault-free case. The alternative hypothesis ?i1 corresponds to a fault mode. The second one is change detection, here. the mode (fault-free or fault) of monitored system can change within the data sample at an unknown instant v (1 ~ v ~ k) and thus is a function of time. In this paper. we concentrate on the first situation. Any detection procedure should perform a tradeoff between two incorrect decisions : false

215

alann (false rejection of the null hypothesis) and non detection (missed acceptance, or equivalently false rejection of the alternative hypothesis). Two types of hypotheses have to be distinguished : simple hypothesis 'Hi, i = 0,1, is defined by a unique value of the parameter vector: 'Hi : 0 = Oi. A composite hypothesis refers to a set of parameters 'Hi : 0 E i with 9 i C IRr. We assume that eo e] = 0. Composite hypotheses are more relevant than simple ones in practice, because of limited available amount of information, especially for the alternative (fault mode). The quality of a binary statistical test 5 is defined with the probability of false alarm: a = Pro (5", 'Ho), where Pr; stands for observations Y] , . .. , Yk being generated by distribution Pi, and the power function: /30(0) = Pr(;l (5 = 'H]) . In case of a vector parameter 0, the crucial issue is to find an optimal solution over a set of alternatives which is rich enough. Unfortunately, uniformly most powerful (UMP) tests scarcely exist (Borovkov, 1998; Lehmann, 1986). As we have mentioned in me introduction, anomer important issue is dealing with the nuisance parameters. To solve the composite hypomeses testing problem with nuisance parameters we use me theory developed by Wald in his paper (Wald, 1943) and we adapt it to the case where the parameter of interest (fault) 0 and me nuisance X belong to different subspaces of the observation space. Therefore, the goal of mis paper is twofold. First, we develop an optimal statistical tool to solve me problem of fault 0 detection in the following linear gaussian model :

n

by one of two gaussian distributions: N(O,~) and

N(O '" 0, ~), where 0 is the mean vector and ~ is a positive definite covariance matrix. The hypotheses testing problem consists in deciding between 'Ho : {O = O} and 'HI : {O =f:. a}.

e

(2)

The astuteness of the above problem is me absence of uniformly most powerful (UMP) tests in case of a vector parameter O. To overcome this difficulty, Wald propose to impose an additional constraint on the class of considered tests, namely, a constant power function over a family of surfaces S defined on the parameter space 9, in order to avoid the existence of UMP tests over a subspace of which are very inefficient over

e e

9\9.

Definition 1. A test o· E K.cx. = {o: Pro( 0 =f:.'Ho)::; a}, where a is the prescribed probability of false alann, is said to have uniformly best constant power (UBCP) on the family of surfaces S, if the following conditions are fulfilled (Wald, 1943) : Cl) for any pair of points OI and 82 which lies on the same surface Sc E S, /30· (OI) = /30.(0 2 ), where f3o(O) = Pro(5 = 'Hd is me power function of the test O. (2) for another test 0 E K.cx., which satisfies the previous condition, we have /30" (0) 2:: /30(0). To solve me hypotheses testing problem (2), Wald defines me following family of surfaces (ellipsoids) : (3)

(1) Y = HX + MO +~, where Y E IRn is the measured output, 0 E IRr is me parameter (fault) of model (10), M is a full column rank matrix of size (n x r) with r < n, X E lRm is an unknown and non random state vector (nuisance parameter), H is a full column rank matrix of size (n x m) with m < n, and ~ is a zero mean gaussian noise ~ rv N(O , 0'2 In), 0'2 > O. Second, the developed tool is used to examine several more practical questions, for instance, how to assess the impact of nuisance dynamics, injected in the model via a state equation, or how to solve the problem of "optimal residual generation".

The UBCP test defined on the family of surfaces (3) is given by o*(Y) = {'Ho if A(Y) = yT~-ly < h(a) (4) 'HI if A(Y) = yT~-]y 2:: h(a) ' where the threshold h = h( a) is chosen to satisfy the definition of the class K.cx. . Let us assume the following gaussian linear model :

(5) lR r is the

where Y E IRn is the observation vector, 0 E parameter of model (5), i.e. the vector of faults, M is a full column rank matrix of size (n x r) with r < n and { is a zero mean gaussian noise { rv N(O, 0'2 I r ), ~ > O. As in the previous case, the problem consists in deciding between the hypotheses 'Ho : {O = O} and 'Hl : {O =f:. a}. Let us apply the general Wald's idea to design the UBCP test :

2. LINEAR GAUSSIAN MODEL The goal of this section is to apply me Wald's theory of binary hypotheses testing to a linear gaussian model, to adapt the optimality criterion to this case and to design me optimal algorithm that realizes mis optimality criterion. We mainly use the technique developed by (Wald, 1943) in this section.

5* (Y)

=

{'HO if A(Y) = (fI':Ft < h(a)

'HI if A(Y) = (fI':FeB 2:: h(a) ,(6)

2. I Uniformly most powerful test with constant power function

where '0 = yT M(MT M)-I MTy is the least square (LS) estimator of 0 and :Fo = ;}r MT M, is the Fisher matrix. It is necessary to prove that the test 0·, given

Originally, Wald has proposed a solution to the following problem : the observation Y E !Rn is generated

216

by equation (6), is UBCP over the following family of ellipsoids:

extensively used now to adapt the optimality criterion to this case and to design a decision rule that realizes the optimality criterion.

S = {Se: (FFiB = :2BTMTMB = c2 } .(7) The main result of this section is established in the following Theorem 2. Theorem 2. Let us consider the regression model (5) . The test (Y) E Ko, given by equation (6), is UBCP for deciding between the hypotheses 'Ho : {B = D} and 'HI : {B i- D} over the family of ellipsoids (7).



2.2 Discussion o/the UBCP test



og

sUPofo(Y) yTM(MTM)-IMTy A(Y) fo (Y) 0'2 '

where A(Y) is given by equation (6). Because the power is constant on the same surface Se : ~BT MT MB = c2 , it is reasonable to present the power as a function of c2 : 2 f-+ f30 (2). The statistic A(Y) = ~yT M(MT M)-l MTy is distributed according to the X2 law with m degrees of freedom. The law X2 is central under 'Ho and noncentral under 'HI . It is easy to see that c2 is the parameter of non centrality of the X2 law under {{I . Hence, the power function is given by 0

J 00

f3oo(c 2) = Pr cl (0· A

wheref>,,(y)=fo(y)e-"iG

= 'HI) =

fcl(y)dy, (8)

(r2'4AY) ' fo(Y) h

e

yi- I -!

2'r~)'

is the density of the non central X~,.\ with r degrees of freedom, A is the noncentrality parameter, r(x) = Jooo u",-Ie-udu is the gamma function, and xl xP G( ) 1 '" K, X

h = tion:

=

+it+ K(K+l)2! + ... + K(K+J)" '(K+p-I)p! +

is the hypergeometric function. The threshold h(a) is defined by solving the following equa-

J 00

a = ProW i- 'Ho) =

Let us recall the regression model (1) with the nuisance parameter X :

Y=HX + MB+€ .

fo(y)dy,

(9)

h

'HI: {Y ",N(HX

3. LINEAR GAUSSIAN MODEL WITH

NUISANCE PARAMETERS The goal of this section is to apply the Wald's theory of binary hypotheses testing to a linear gaussian model with a nuisance parameter. As in the previous section, the Wald's idea to define a family of surfaces around the point B = 0 on the parameter space will be

217

]Rm}

and

(11)

+ MB,0'2In), B i- 0, X E Rm},

where B is the informative parameter and X is the nuisance parameter, while considering X as an unknown vector. Because the nuisance parameter X is non random and its current values are not bounded (X E ]Rm), the only solution is to eliminate the impact of X on the decision function A. First of all, let us note that the family of distributions Y '" N(HX + MB,0'2In) remains invariant (see (Lehmann, 1986) for details and definitions) under the group of translations G = {g : g(Y) = Y + H C}, C E Rm, which in the parameter space induces the group G = {g : g(Y) = Y + HB}, BERm, that preserves both = {lE(Y) = H X, X E Rm} and nI = {lE(Y) = HX + MB, X E ]Rm, B i- O}, i.e. gno = and gn 1 = I . Hence, the hypotheses testing problem (11) remains invariant under G (Lehmann, 1986). As it is mentioned in (Lehmann, 1986), the optimal invariant tests are based on the maximal invariants (principle of invariance). To apply the principle of invariance, let us define the column space R( H) of the matrix H. The standard solution is the projection ofY on the orthogonal complement R(H).l of the column space R(H). The space R(H).l is well-known under the name "parity space" in the analytical redundancy literature (Frank, 1990). The parity vector Z = WY is the transfonnation of the measured output Y into a set of n - m linearly independent variables by projection onto the left null space of the matrix H. The matrix WT = (Wl, . . . , w n - m ) of size n x (n - m) is composed of the eigenvectors WI,' .. , W n - m of the projection matrix PH = In - H(eT H)-I eT corresponding to eigenvalue 1. The matrix W satisfies the following conditions: W H = 0, = PH, WWT = In-m . It follows from the first of the above conditions that the transfonnation by W completely removes the interference of the nuisance parameter X. It can be shown that the function M(Y) = Z = WY is maximal invariant to the group of translations G = {g : g(Y) = Y + H X}. For this reason all invariant tests should depend on Y only via the vector Z = WY . The measurement model (10) can be rewritten by the following manner:

no

no

wrw

where a is the prescribed probability of false alarm.

(10)

The new hypotheses testing problem consists in deciding between

'Ho : {Y '" N(H X, 0'2 In), X E

Let us discuss now some issues in deciding between 'Ho : {B = D} and 'HI : {B =I- D}. First of all, the test (Y) coincides with the generalized likelihood ratio (GLR) test. After a simple algebra we obtain the following expression for the GLR

A(Y)=21

3.1 Nuisance parameters rejection

n

Z

= WY = WMO+ we = WMO+(,

(12)

where ( '" N(D, (72 In-m), (72 > D. Let US additionally assume that the matrix W M is full column rank: of size « n - m) x r) with r < n - m. Hence, the results of section 2 can be directly applied to the model given by equation (12) for deciding between 11.0 : {O = D} and 11.1 : {O =I D}.

01

Let us apply Theorem 2 to design the UBCP invariant test. In the case of model (12), the LS estimator of 0 is given by : B= (WM)T(WM»)-1 (WM)TZ, and the Fisher matrix is.r9 = ~ (W M) TW M. It directly follows from equation (6) that the UBCP test is given by if A(Z) < h(o:) 11.1 if A(Z) ~ h(o:) ,

= { 11.0

(13)

=

where A(Z) ~ZTWM(WM)TWM)-lMrwTZ The test given by equation (13) is UBCP over the following family of surfaces S = {Se: ~8TMTWTWMO = c2 } . the noncentrality parameter >. is equal now to MTWTW MO. c 2 = ~OT er

3.3 Discussion o/the UBCP invariant test Let us discuss some issues in deciding between 11.0 : {O = D} and 11.1 : {8 =I D} (ll) while considering X as an unknown nuisance parameter. As' in the previous case, the invariant test 0- (Y) coincides with the GLR test. It can be shown that the following equality is satisfied A(Y) = 2 log

°

~

3.2 Design o/the UBCP invariant test

o-(Z)

such a test based on such a "subset" parity vector. To obtain this subspace, let us define another nuisance rejection matrix W 1 of size (k x n), where k is so chosen that r < k < n - m, from the matrix W by deleting n - m - k rows of W. It is easy to see that W1H = and W1Wr = Ik . Finally, we obtain Zl = WIY = W1MO + WIe = W1 MO + (1, where (1 ,..., N(O, (72Ik), (72 > D. For this model the LS estimator of 8 is given by :

aUPix
=

~yT(PH - PM,H)Y = A(Y), where PM,H = Infi(fiT fi)-1 fiT, fi = (H M) and A(Y) is given by equation (10), the GLR is an optimal invariant test. The statistical properties of the GLR test have been examined in (Scharf and Friedlander, 1994). It has been mentioned in (Scharf and Friedlander, 1994) that the GLR test is (UMP) optimal because the X2 distribution of A(Y) = A(Y) is monotone in the noncentrality parameter >.. Therefore, the authors of (Scharf and Friedlander, 1994) have reduced a vector parameter detection problem to a scalar one (without any guarantee that this reduction does not change the sense of optimality). We would like to stress that in contrast with (Scharf and Friedlander, 1994) we prove the optimality of the test directly in the parameter space. Another interesting question arises in connection with the above mentioned problem of "optimal residual generation" (see, for instance, the survey (Staroswiecki, 200l» . The recommendation of the theory of invariant tests is to use the maximal invariant statistics Z = WY to design an optimal test. Let us consider now a subspace of the parity space (defined by W) and estimate the power function of

218

= (WIM) T (WIM) )-1 (WIM) T W1Y

and the Fisher matrix is

Fil =

1

(72

T

T

M W1 W1M,

and, finally,

A(ZI) =

~ZrWIM (W1M)TW1 M)-1 MTWTzI' (7

It follows from (6) that the new test is given by :

O-(Z ) - {11.0 if A(Zl) < h(o:) 1 1 11.1 if A(ZI) ~ h(o:) .

(14)

is UBCP over the family of ellipsoids S {Se: ~ 11 WIM811~= c2 }.

=

Lemma 3. Let us assume that Pro(oi =I 11.0) = Pro(o- =I 11.0) = 0: (0 < 0: < 1). Then the following inequality is satisfied for any 0 =I 0: {J6i (8) :5 {J6· (0) . The above equation shows that a "subset" parity vector cannot improve the power of the test. On the other hand the advantage of the full-set parity vector over a subset one can be somewhat limited for some directions in the parameter space. Therefore, it follows from the above results that the full-set parity space is better (at least not worse) than a subset one. But the choice of the full-set rejection matrix W of size (n-mxn) is not unique. As it is mentioned in the survey (Statoswiecki, 200]), the product AW, where A is a matrix of size (n - m x n - m ).!.uch that det A =I 0, leads to another rejection matrix W = A W . 1t is asked in (Staroswiecki, 2001) whether such an operation can impr~e the residual generation step or not. It is clear that W H = O. The following lemma shows that using the rejection matrix W = A W does not change the power function of the test.

Lemma 4. Let us assume that the UBCP invariant test (13) is designed by using a rejection matrix W = AW, where det A =I O. Then the power function of the test (13) remains independent of A . 4. STATE SPACE MODEL WITH UNKNOWN

INPUTS The goal of this section is to apply the statistical tools developed in sections 2 and 3 to a state space model where the nuisance is modeled by a deterministic state equation.

and (16). The nuisance Xi is replaced by F X i -

Consider the following state space model:

Xk = F Xk-l + BUk , Yk = H Xk + MBk + {k,

(15)

(16)

where Yk E Rn is the measured output, Xk E am it is the state vector, Uk E RP is the unknown input vector (nuisance parameter), Bk E Rr is the vector of faults at time k, and {k is a zero mean gaussian white noise, {k '" N(O, (72 In), (72 > O. It is assumed that all the matrix of (15) - (16) are known and the matrix H is full column rank of size (n x m) with m < n . The characteristic feature of the dynamical model (15) - (16) is that the nuisance X is modeled by a deterministic state equation (15) which is governed by the input vector U. For instance, the nuisance X k , say, the altitude of aircraft, certainly depends on its previous value Xk-l> via the dynamics of aircraft modeled by the matrix F, and some unknown factors like positions of control surfaces. A fixed k-size sample of measured outputs YI , ... , Yk is available and supposed to be generated by one of two alternative hypotheses. There are two methods to deal with the nuisance parameters Xl, ... , X k : • to ignore the presence of the state equation (15) and to uniquely use the measurement equation Yk = H Xk + MBk + ek (16); • to use both equations (15) and (16). It seems that in function of the relation between the ranks of matrices H and B the power of the UBCP invariant test based uniquely on the measurement. equation (16) can be improved by injection of the a priori knowledge of the nuisance parameter dynamics.

4.1 Ignoring the state equation Here, the UBCP invariant test is designed by using uniquely the measurement equation (16). Equation (15) is ignored. By putting together k measurement equations we get :

+

By putting together k measurement equations we get :

YI,k =UXo + VUI ,k + MBI,k

= (U

V)

+ 6 ,k

(;0 ) + MBI,k + 6,k' l ,k

(18)

where YI,k, BI ,k and {l,k are defined as in (17), and the vector UI ,k = (UI, " " Uk) . The matrix V and U are defined by the matrix H, F, B, and M has the matrix M on the diagonal. The nuisance is defined by the following vector X = (X6, Uf,k)T. Let us apply the UBCP invariant test (13) to the model given by equation (18), by replacing the vector Y, matrices H, M and PH by YI,k, (U V), M and p(U vl = Ikn -

(U

V )(U V)T(U V»-l( U

vf

respectively. This test o(U V) is UBCP over = the following family of ellipsoids S(U V) {Se: ~ 11 WIMBI,k ,,~= c2}, where matrix WI satisfies the following conditions W I (U V) = 0, WI Wr = Ikn-rank(U vl' and WrWI = p(U V) ·

4.3 Comparison o/the above mentioned methods Let us consider the above mentioned tests tS1i and

O(U V) · The result of their comparison is established by the following Lemma.

Lemma 5. Let us assume that Pro(o1i ::f. 11.0) = Pro(tS(U V) ::f. 1io) = 0 (0 < 0 < 1). Then the following mequality is satisfied for any BI ,k ::f. 0 : f36~ (BI,k) ::; f36 e", Y) (BI,k)' This lemma shows the importance of the a priori information on the nuisance parameter dynamics expressed by the state equation (15). The UBCP invariant test based on the state-space model (15) - (16) performs better (at least not worse) then the test based uniquely on the measurement equation (16). The importance of state equation modeled the nuisance has been heuristically discussed in the analytical redundancy literature. Lemma 5 proves this result.

= (Yl l . . . Yk), XI,k = (Xl, ' " Xk), BI,k=(BI, . . . Bk) and {U=({I , · ·· {k) . The matrix M and 1i have respectively M , and H where the vector YI,k

on the diagonal. Let us apply now the theoretical tools developed in section 3. The UBCP invariant test (13) is applicable now by replacing the vector Y , matrices H , M and PH by YI,k, 11., M and P'H = hn - 1i(11.T1i)-I?{T respectively. This test 01i is UBCP over the following family of ellipsoids: Src = {Se : ~ 11 WMBI,k ,,~= r?}, where the matrix W satisfies the following conditions W1i = 0, WTW = P'H and WWT = Ikn-km'

4.2 Using equations

=

l

F2Xi _ 2 + FBUi _ 1 + BUi = FkXo + E~:~ Fj BU,_j for i ~ 1. This implies that ~ = HF'Xo + HP-IBUI + ... +HBUi forO::; i::; k. BU,

5. APPUCATION: GROUND STATION BASED G~INTEGRITYMONITO~G

Integrity monitoring requires that a navigation system detects faulty measurement sources, and if possible isolates and removes them from the navigation solution before they sufficiently contaminate the output. For some safety critical navigation modes, landing, for instance, the GPS channels integrity monitoring is realised by using the measurements of monitoring station at a known position Xr = (xr, Yr, Zr)T close to the airport When a fault is detected, the corresponding information is transmitted via the integrity channel.

both (state-space and measurement)

In contrast with the previous case, the UBCP invariant test is designed now by using both equations (15)

219

it follows from section 3, the UBCP invariant test is based on the following statistics

The GPS solution is based upon accurate measuring the distance (range) from several satellites with known locations to a user (station or vehicle). Let us assume that there are n satellites located in three-space at the known positions Xl = (x:, yt, ztV, i = 1, .. . , n . The pseudo range r. from the i-th satellite to the user can be written as Yi = ri - lit, = e + ~i ' for 1 :::; i :::; n (19) where :

2_ZT( X --;;:2 I n -

is the true distance from the i-th satellite to the user,

e is an unknown (non random) station clock: bias e E R I, C ~ 2.9979 . 1Q8 m/ s is the speed of light and ~i is an additive pseudorange error, ~ = (6, ... , en) T is the vector of additive pseudorange errors at the ground station position. A fault is modeled by the vector () of additional pseudorange biases. Hence, we get the following measurement model with a fault : (+(),

dlx=xo

H6()

(20)

where C = (c, .. . , c) T . The problem consists in deciding between the null hypothesis Ho (no contaminated pseudoranges) and the alternative hypothesis HI (there are contaminated pseudoranges) Ho : {Y ",N(Ce,(12In ), e E R} against HI : {Y",N(Ce+(),(12In),()fO, eER} As It follows from (Lehmann, 1986), the following statistics Z = WY where Z = (ZI, Z2, ... , Zn_l)T and Y = (YI, Y2,' .. , Yn)T and W is a matrix defined by Zi = Yi - Yn (see details in (Nilciforov, 2002». translation G = {Y f-+ g(Y) = Y + Ce}, and the family P = {N(Ce + (),(12In ),B ERr} is {Y 1-+ g(Y) Y + invariant to the group G CU} and the group G is given by g(B) B+ Cx (x ER). Therefore, the hypotheses testing problemHo={Y ",N«(),(12In ), WB=O} against HI = {Y ",N(B,(12In ), W()fO} is invariant. To design an optimal test for deciding between the above hypotheses, the statistics Z = Wy '" N(WB , (12WWT) is used. If the inverse matrices given below exist and their dimensions are compatible then the following equation is available (A + BCD)-I = A-1_A-l B(C-1+DA-1B)-IDA- 1 • By using the above equation, we get : (WWT)-l = In-I - ~ln-II~_I' where In-l = (1, .. . ,IV. The hypotheses can be re-written in the following manner

Ho

= {Z

=

6. REFERENCES Basseville, M. and I. V. Nilciforov (2002). Fault isolation for diagnosis: Nuisance rejection and multiple hypothesis testing. In: Survey Paper, 15th IFAC World Congress, Barcelona, Spain 21 th 26 th July, 2002. pp. 179-190. Borovk:ov, A. A. (1998). Mathematical Statistics. Gordon and Breach Sciences Publishers, Amsterdam. Franlc, P.M. (1990). Fault diagnosis in dynamic systerns using analytical and knowledge based redundancy - a survey and some new results. Automatica 26, 459-474. Lehmann, E.L. (1986). Testing Statistical Hypotheses. Chapman and Hall. Nikiforov, I. (2002). Elements de theorie de la decision statistique IT : complements, Chap. 2. In: Decision et reconnaissance des formes en signal. pp. 43-84. Hermes Science Publications (Traite IC2), Edited by R. Lengelle. Scharf, L.L. and B. Friedlander (1994). Matched subspace detectors. IEEE Trans. Sign. Proc. 42(8),2146-2157. Staroswiecki, M. (2001). Redondance anaIytique. In: Auromatique et statistiques pour le diagnostic. pp. 43-(;8. Hermes Science Europe. Wald, A. (1943). Tests of statistical hypotheses concerning several parameters when the number of observations is large. Tram. Amer. Math. Soc. 54, 426-482.

= =

'" N(if, (12WWT), 'if = O} against HI =

{Z '" N('if,(12WWT ), 'if f O} where 'if

{;r

Y = R - ~ ~ Ho(X - Xo) + ~ (+8), (22) where ~ = (rIo, .. . , rno)T, r~ = IIXi - Xuoll2 + c~uo,e = . (6, .:. ,en)T,Ho = is the Iacoblan matrlx of sIze n x 4. As it follows from equations (22), a fault B affecting the GPS channels implies an in additional e.:r0r b = IE(X - X) = (Hl HO)-I the vector X. It is obvious that the set of undetectable biases B is expressed by the equation W() = O. All these vectors () can be expressed as : () = x In with x E R. Fortunately, the impact b x(H T H)-l HTIn of such an undetectable bias xln on the first three components :tu, flu, Zu is equal to zero, i.e. b", = bll = bz = O. Therefore, undetectable (by a ground monitoring station) pseudorange biases are not dangerous for the navigation.

e = (~t. ... ,en)T '" N(O, (12 In)

=

1 T) ~(Yi _y)2 nIn-lln-l Z= (12 (21)

where fi = ~ L:~=I Yi · Let us finish this example with a comment on the fault detectability problem. We consider a user (aircraft) at the positions Xu = (xu, Yu, zu)T. By Iinearizing the pseudorange equation with respect to the state vector X = (xu,yu,z,.,Uu)T = (XJ',Uu)T around the worlcing point X o, we get the linearized measurement equation with a fault

di = IIXi-XrIl2=V(x: -xr )2+(Yt-Yr)2+(zt- z r)2

Y=Ce+~

l -

= W() . As

1 It i~ assumed that the ground station is equipped by a lower accuracy clock. To explain why we assume this model of nuisance (~ E R) let us recall that a clock error of 10- 3 s lead~ to an additional pseudonlDge bias of 300 km ! In fact, nominal pseudorange standard deviation is approximately equal 10 12.5 m now and will be limited by 6.6 m or even by 3.8 m in the near future. Roughly speaking, for I} the space R is "reduced" to the interval [_ 10- 3 ; +10- 3 J s.

220