Faults Detection and Isolation for Non Linear Hybrid Systems

9 IFAC Fault Detection, Supervision and Safety of Technical Processes, Beijing 2006

ELSEVIER

IFAC PUBLICATIONS

FAULTS DETECTION AND ISOLATION FOR NON LINEAR HYBRID SYSTEMS Mohamed el hadi LEBBAL ~a, Houcine CHAFOUK 1, Ghaleb HOBLOS ~ and Dimitri LEFEBVRE 2

IlRSEEM: Institut de Recherche en Syst~mes Electroniques Embarqu6s. ESIGELEC Technop61e du Madrillet Avenue Galil6e BP 10024, 76801 Saint Etienne du Rouvray Cedex, France (e-mail: [email protected], [email protected], Ghaleb.hoblos @esigelec.fr), 2GREAH: Groupe de Recherche en Electrotechnique et Automatique du Havre. Universit6 du Havre, 25 rue P.Lebon BP 1123, 76063 Le Havre, France (e-mail: [email protected]).

Abstract: This paper deals with the problem of FDI (Fault Detection and Isolation) for non-linear hybrid dynamical systems, with discrete inputs and continuous outputs. The objective is to discern between actuator, sensor and system faults. A continuous approach, based on Gr6bner basis for residual generation is proposed. For this computation, multimodel representation and non-linearities changes are employed. Fault isolation is obtained from the residual structuration with two stages. The first one, as usual, concerns linear dependences of the residual with respect to the faults. The second one takes advantages of the discrete inputs proprieties to improve faults isolation. (Copyright 9 2006 IFA C) Keywords: Hybrid systems, fault detection and isolation, residual structuration

1. INTRODUCTION Modern systems become more and more complex, their reliability requires the analysis of available information given by measurements and input control systems. The system is considered as hybrid when continuous behaviours and commutations on control inputs or state variables are combined (Zaytoon, 1998). The continuous part of the state evolves with respect to a given operating mode until a discrete event occurs, moving the system to another operating mode (Frisk, 2000; Nuninger, 1998). The model based diagnosis of hybrid systems, is brought back to the study either of a discrete events model (Lunze, 2000) or that of a family of continuous models (Frisk, 2000). On the one hand, continuous representations, approximations of discrete variables or local linearisations are used in order to use continuous diagnosis technique (Klein, 2000 ; Nuninger et al, 1998). These solutions lead to complex relations that are difficult to analyze and the presence of discrete inputs or states induces high values in the computation of derivatives and makes the on line algorithms inefficient. On the other hand,

discrete events representations have been proposed in (Buisson, 2001; Lunze et al 1999; Sampath et al, 2000 and Lafortune et al, 2001), that use automaton for modelling and diagnosis. These representations lead to an important simulation time and the abstraction of continuous behaviours, often involves to critical information losses. As consequence, both representations are not always adapted for hybrid systems diagnosis. To overcome these problems, continuous behaviours and discrete events must be often studied simultaneously in order to provide necessary information for diagnosis task. In this paper, a model based diagnosis method is developed for non-linear hybrid systems, with discrete input vector and continuous output vector. The faults under consideration are actuator faults, sensor faults or system faults. In the proposed technique, discrete and continuous signals are used in order to avoid some problems, depending on the hybrid nature of the considered system. In order to generate analytical redundancy relations (ARR) (Cox, et al, 1992) thanks to polynomial approach (Buchberger, 1985), intermediate variables are

986

introduced so that Gr6bner basis can be explored for residuals generation like changing of non-linearities as x 1/n or sign(x) in polynomial forms. To isolate the faults, new residual structuration is proposed based on analytical dependence relations and knowledge of the discrete inputs and magnitude of residuals. First, a table is proposed, containing all faults signatures. Second, the isolation is improved by estimating the fault magnitude, compared with fault characteristic of discrete variable. The paper is organized as follows. In the next section, some necessary definitions about hybrid systems and Gr6bner basis are reminded. Afterwards, problem statement is formulated. In section 3, our diagnosis approach is developed. Section 4 illustrates our approach on the benchmark of AS193 <>(HDS group, MACS research group).

of the polynomials ci with coefficients Oi.E R[X, Y]. The objective is to eliminate the variables X in order to carry out polynomial relations containing only the known variables Y. Buchberger's elimination algorithm (Buchberger, 1985), allows to construct a Gr6bner basis G={gl, g2..... gt} of I with respect to the elimination order of variables from Xl to yq (i.e: Xl > x2 > ..... > xp > Yl > Y2 > ..... > yq). Then,

is a Gr6bner basis of ideal Ix such that: (3b)

Ix=I:~R[Y]

This means that all polynomials of Ix, where variables X has been eliminated, are linear combinations of basis Gx 2.3 Problem statement

2. PRELIMINARIES In this section, some definitions are provided. 2.1 Hybrid systems

A hybrid system is composed by: - NM, a finite set of k possible modes Mi , i E M = { 1 ..... k} Ix, x d} respectively continuous and discrete states of the hybrid process. - u continuous and discrete inputs vector. - y continuous outputs vector. -fcontinuous and discrete faults vector. -

The studied system is non-linear and characterized by discrete inputs v resulting from thecontrol signals of actuators, u E Nu = {0, 1 }a and m continuous outputs w that lead to analogical sensors measurements y e Ny = IR m. In general, the system is sensitive to faults f e Nf where Nf is a finite set of candidate faults. This contains actuators faults fa when u ~ v, sensors faults fy when y~ w and system or functional faults f~ when S ~ Sf. S(dr/dt, x, u, y) =0 and Sf(dx/dt, x, u, y, fa, fy, f~) = 0 are respectively models of healthy and faulty systems (Fig 1). *fa

:

gi (2(t),x(t),u(t),f(t))=O E IR n defines the behaviour

-

(3a)

Gx=GnR[Y]

[U=(U,..,Ua)

of the continuous state x(t) in mode Mi. !

- h i (y(t),x(t),u(t),f(t))=O E Ig m defines the behaviour

of the continuous outputs y(t) in mode Mi. - O(X(O , Xdi(O , Xdj(O , u(t),f(t))=O defines commutation conditions from mode Mi to mode Mj.

V

Actuators ~-~1

*~s

Sy'smte

W

~

+Model

*~Y

Sensors y=(y ,..,y~)]

~[ Models S and S f f o r diagnosis ~r

Fig. 1. Diagnosis of hybrid systems The model based diagnosis consists in detecting and isolating the faults using the following elements:

2.2 GrObner basis -

Consider a system with a set X= {Xl, X 2. . . . . Xp} of p usually unknown variables to be eliminated and a set Y = {Y~, Y2. . . . . y q } of q known variables. The variables in X and Y are related by a set of m constraints C = {Cl, c2 ..... Cm} such that: Ci =~'fljX~J"...X~]'Py~ J''. . . . y b j ' q = o J

(1)

where aj,k and bj,k are integers and d: real numbers. Let R[X, Y] be the ring of the polynomials of variables {xi, x2 ....... xp, Yl, Y2....... yq}; {Cl, c2 ..... Cm}E R[X, Y]. We consider that constraints ci constitute a basis of an ideal I, of R[X, Y], which can be written by (Starowsiecki, 2000): I

-

Output measurements y = (Yl ..... Ym), collected on line with an observation window [y(t-T) ..... y(t-1)] of size T, Control inputs u = (Ul ..... Ua) of actuators, collected on line with an observation window [u(t-T) ..... u(t-1)] of size T, Models S and Sf of healthy and faulty systems.

m

This means that I is the set of all linear combinations

Based on S and Sf, residuals are generated in order to check non consistency between the theoretical information and measurements. Residuals contain only known or measured variables and are statistically nil in the absence of failure and different of zero when a failure occurs. 3. FDI APPROACH FOR HYBRID SYSTEMS In this section, our fault detection and isolation approach is developed. It is organized into four steps, which are: operating modes identification, nonlinearities change, residual generation and faults

987

detection and isolation. Two levels of structuration are proposed to improve isolation.

generated by (3) and any combination between them, are composed in two parts R res and R beh"

3.1 Step 1" Operating modes identification

R(z,u, f)=Rres(z,u)+ Rbeh(z,u, jO=O

According to our definition of hybrid systems, the problem is to define a set of models NM (operating modes) In our case, operating modes are depending on control inputs commutations. Then, the number of models is less than or equal to 2 a. The commutation conditions tr are defined by the change in discrete input vector u. After dividing the system into several operating modes, each one is defined by state and measurement equations.

3.2 Step 2: Non-linearities change In order to change non-linearities, we define a new state space representation where the functions

gi (x(t),Jc(t),u(t),f(t)) are written in polynomial form. The problem is to find an application ~q, such that, for each function

gi(x(t),JC(t),u(t),f(t))

we

can

associate Gi (zi,~i,u,3') e R[zi,~.i,u,f], where zi is a new state.

~" gi (x(t),Jc(t),u(t), f(t))

~ > Gi(zi,~i,u,f ) (4)

Here, we cite some examples about non-linearities change, used later in application.

with Rr~(z,u) = R(z,u,O) and gbeh(z,u,J)=R(z,u,jO-Rres(z,u).

Rres(z, u) and Rbeh(z, U,]) are respectively considered as, residual (nil when no fault exists) and residual behaviour containing the faults for which the residual is sensitive.

3.4 Step 4: Residuals structuration After using threshold for fault detection, isolation is obtained according to two levels of residuals structuration. The first one is to build the table of signature (aij) such that a0=l indicates that residual i is sensitive to fault j and aij=O otherwise, nf and nr stand respectively for the numbers of faults and residuals. In this table, each row corresponds to a residual equation and each column to a candidate fault. The residuals are selected such that faults or subsets of faults can be isolated. To discern the faults of a subset, a second level of structuration is proposed. Indeed, the second level uses polynomials R(z, u, fa)=0 and discrete actuators properties. These actuators and corresponding failures satisfy table 1. Table 1" Actuator faults

-sign(x) is defined by sign(x)=-I if xO. Hence if l(x)-sign(x).r(x)=O and r(O)=O where r(x) is linear or in polynomial form then l(x)2-r(x)2=O is in polynomial form.

- Xr(X)--n~(X)=O becomes f(x).Xr(X)-n.r(x).JCr(x)=O, as a polynomial form. For the measurements function, new known variables z depending on the system outputs are also defined. These new variables are linear or non linear combinations of the system outputs. They are considered as a new measurement system. The problem is reduced to find an application r/, such that: for each function hi (y( t) ,x( t),u( t) ,f( t) ) , we can

Control input u

Actuator with fault

Fault magnitude

u=1 u=O

v = u +fa =0 V = U + fa =l

fa = -1 fa = l

Then, the problem is to estimate the magnitude of candidate fault f in an isolated subset using polynomial R(z, u, f)=O . 4. APPLICATION In this paragraph, our approach is applied on the benchmark of A S 1 9 3 , Diagnosis and Supervision of HDS >> (HDS group, MACS research group) and some results are showen.

4.1 Benchmarkdescription

associate Hi (z,zi,u, f) e R[z, zi, u,f].

I]" h i (y,x,u, f) rl-'-> H i (z,zi,u, f)

(6)

(5)

3.3 Step 3: Residuals generation As discussed above, the model is a set of polynomials included in R[zi,~i,u,f]w R[zi,z,u,f]. The residuals generation problem leads to compute analytical redundancy relations (ARR) using Gr6bner basis. Indeed, according to definition 2.2, the variables should be ordered into two subsets X and Y, such that, X only contains the variables to be eliminated. For each fault f, we can associate an elimination order to have polynomials depending or not on this fault. These polynomials R(z, u, 39

The system has two identical tanks R/ and R2 that communicate with two identical pipes C/and 6;2 (Fig 2). It is controlled with logical actuators, four valves V1, V2, V3 and V4, with two states { open = 1, closed = 0} and two pumps P/ and P2. The dynamic of actuators is assumed to be fast enough in order to neglect the transitory behaviours. The state variables h/ and h2 are continuous time variables, corresponding to the tanks levels. The sensors give measurements Yl and Y2 of the levels h/ and h2 and also measurements of the flow Q1 and Q3 through the valves V1 and I13.

988

P1

1

PipeC2.

,r

4.2 FDI for benchmark

~2

a) Determination of operating modes: The operating modes are defined according to the state of logical actuators (P1,P2,I11,V2,V3,V4). As a consequence, there a r e 26=64 models, each one of them corresponds to a non-linear systerrL

hi

m

V1

I

R1

h2

V2

ipe

b) Change of non linearities. In order to change non linearities, variables zi are used as new state variables. Then, we choose application ~3 as follows :

Fig 2 : Benchmark of AS 193

Zil ,

The non linear model of the system without fault (S) is given by (7,a) and (7,b):

QpI=D'P1 , Qp2=D'P2

PI e{O,l~ P2 ~{O,1}

Ql =Ok~.V1, Q2=~.~-~.V2

V1 ~{O,l~V2~{0,1} V3~{0'1}

zi, =~max(h1,0.5)-max(h2,0.5)I

(8)

zis=sign(hl -h2)'dhl -hE I

zi6=sign(ht -h2)-dmax(hl,0.5>-max~h~,0.5~I Using properties of sign and square root (see 3.2), the state equation (7,a) and (7,b) can be written by:

(7,a)

04=a4.sign(~ -h2).

~ max(hi,0.5~-max(h2,0.5~] "V4 V4~{0,1}

2.S.Zil "zil -D.P1 +ct.V1 zil +tr3 "V3"zi5+tZa'V4 "zi6=0 2"S.zi2 "~i2-D.P2 +ct.V2 zi2-0t3 "V3 "zi5 -tZa'V4 "zi6 =0

a=~ 3 =a 4 =A.2~.g

zi2-j2=O

y~=h~

zi2-zi2=O

Yz =h2

(9)

)'1 - z i?=0

(7,b)

s.i~, =Q~-Q~ -03 -04

y2--zi2=0

s.i~ =0,,2-02 +03 +04

01 -a. v, zi, =0

with S = 0.0154 m 2 the section of both tanks, A= 0.36 10-4 m 2 the section of two pipes; g = 9.81 m/s 2 and D = 1 0 - 4 m 3 / s the flows of pumps P1 and P2.

03 -a~ % .zi~ =0 On the other hand, the outputs yl and Y2 are used to build other variables z, considered as measurable. Application r/is defined as follows:

The following faults are considered : Additive faults on the measurements Yl, Y2, Q1, Q3. The pumps P1 o r P2 can be untimely opening, untimely closing, locked open and locked closed The valves V1, V2, V3 or V4 can be untimely opening, untimely closing, locked open and locked closed A leakage can be produced on R1 or R2 - The pipes C1 o r C 2 can be blocked or partially blocked.

Zl = ~ 1 ,

-

Z2 = ~ 2 ,

Z3 =~/I yl

-Y:I'

(10)

-

Z4

= ~/Imax(Yl, 0.5) - max(y2,0.5)[

-

If a fault3') (resp. f2) occurs on output measurement Yl (resp. Y2), the corresponding sensor gives a faulty measure Yl =hi+f1 ( r e s p . Y2 = h2 + f2). We assume that the derivative of 3') ands~ are identically nil. Thus:

-

Y~ =/~ , 3'2 =/'~ , 2-z~" Zl

Table 2 : Table of faults Component, associated variable Level sensor of R1, Yl Level sensor of R2, Y2 Flow sensor, Q3 Flow sensor, Q1 Input pump of R1, P1 Input Pump of R2, P2 Evacuation valve of R1, V1 Evacuation valve of R2, V2 Valve of transmission pipe C1, V3 Valve of transmission pipe C2, V4 Connection pipe, ct~ Connection pipe, a4 Possible leakage of R1 Possible leakage of R2

Symbol

fl f2 f9 flo f3

f4 f5

f6 f7 f8

fll 3')2

f13 f14

Fault type Sensor Sensor Sensor Sensor Actuator Actuator Actuator Actuator Actuator Actuator System System System System

989

--

~71 and 2-Z 2 "Z,2=

3"2 "

When Zl, Z2, Z3 and Z4 are faulty, they are given according to :

Zl =Zil +fll Z2=zi2 +f 22

Z3=Zi3+fl2 Z4=zi4 +f 21

(11)

w h e r e ill, f12, fll and f12 are the effects of 3~ and f2 on the variables zl, z2, z3 and z4:

- fl l = o if f l = O fl l r if f l ;,~O -/2-'0 if3~=0 a n d S = 0 , / 2 ~ 0 ifflr or~;~0 and~;9~ - f l = o iffi=0 ands~=0,/17~0 if3~r or3~r a n d s ~ -3d2=O if f2=O f12=O if f270 In presence of failures, the faulty non linear model (Sf) is given by equations (11) and (12).

p~es=(2.S. z1 "7.1-D. PI +r 1 .z1 -t-03 ) 2 +2.O~-V~2 .2za +Q~2 -a'2 .V~2 .2z 3

2.S.zi~ "zi~ -D.(P~ + f3 )+a.(V~ + f~ ).zi~

+(as +f~,).(v~ +f~).zi~+(a4+f~9.(v4+A )'zi6-f~3=o

-(2"S'z2"z2-D'P2 +~'V2 .z2-Q3 )2

2.S.zi2.zi2-O.(P2 + f4 )+a.(V2 +f6 )'zi2 - ( ~ + f 0"(V3 +f7 ).zi5 -(ot4 +f12).(V4 +A )'zi6 -f14 =0

zi~-.zi~--o

(12)

According to section 3.4, 9 groups of faults can be isolated using the following table of signatures. Table 4: Signatures table

zig-.zi~--O Q3-(tr3 +ftl)'(V3 +f7 )'Z/3-f9 =0

Q1-ot.(V1 + f~ )'zi 1- f~o--O c) Residuals generation: According to section 2.2, the set of constraints C, given by (11) and (12), is defined. These constraints are included in the ring of polynomials R[ zi~ , ziz ,zil, zi2, zi2, zi4, zis, zi6, f3, f4, f5, f6, fz, f8, f9, flo, f1,, f12, f13, f14, / ' , / 2 , / 1 , / 2 , Zl , Z2 , z~, Ze, Ze, z4, Q~, Q3, as, or4]. For each fault, we associate an elimination order. For example: Zi~ > Z~ >Zil>zi2 >zi3 >zi4 >zi5 >zi6>z>z4>OtS>t~4> /1>/2>f3>f4>f6>f7>f8>fg>fl, >f12>f~3>fH>/2> V3> V4 > V 2 > V I > P 2 > P I > zl > z2 z2>a3>fi2>f5>f~o>fl~>Vl>Z

1>Q~ leads to: Q~ -ot.v~ z~-ot.f5 zx +ot.v~ f ~ + o t . f 5 - fxo=O

(]3)

with: (14)

R~" =01 -a.v~ z~ gbeh=--~'A Zl +~'V1 f l l + a ' A

Fault isolation is not perfect because all columns are not different. In fact, it is only possible to isolate the groups of faults {f3, f13], {f4, f6, f141, {f7, f11}, {f8, f121, and not a fault in a given group. Thus, a second level of structuration is used, where discrete properties of inputs are investigated. In fact, the state of these inputs belongs to set {0, 1 } and the fault values (table 5) belongs to { 1, -1 }. We can estimate the fault magnitude with the residual sensitive to this fault. Let us consider the group V4, f6, fI4 } as an example.

(15)

-A0

The computation of other residuals is similar. Residuals are chosen in order to be non sensitive to a given fault. We assume that the derivative is approximated by :

Table 5: Interpretation of fault Last Desired Present Value value of value of value of o f f actuator actuator actuator

0 0 1 1

~(k)=.z(k)-z(k-1) , with Te is sampling period.

0 1 1 0

1 0 0 1

Interpretation of fault

1 -1 -1 1

untimely opening locked closed untimely closing locked open

Te

d) Detection and isolation operation: The residuals chosen from polynomials computed are according to: res

2

2 2 2

=D.O.5).

Ri =03 --tr'3 "V3 "Z3 Rres 2 = Q 1 - 0 ~ .V1 .z 1 gres = 0 2 _ 0 ~ 2 .V2.z 2 + Q 1 - ~ 3

We assume that fault 3~ occurs in the range of time [350s, 400s], f6 in [650s, 700s] and ~ 4 in [1450s, 1500s], with value equals 50% of pump flow V14

"Zl -O~2 . v ? . z 2 2

(2.S.z2.~2-D.P2 +a'.V2 .z2 +Q3 )2 Rres

4 =(2"S'Zl "Z'l-D'P1 +Q1-03

)2

I

,

1[.......1..

+Q1-6~'v2 "Zl

-ct.v 1 .z 1 -(2.S.z2.z2-D.P2 +a,.V2 .z 2 +Q3 )2 R~es=(2.S.Zl .~I -D.P1 +g-V1 .Zl -Q3)2 +Q1-ct.V1 .z 1 +02 -g2.V?.z2 -(2.S.z2.~2-O.P2 +a"V2 "z: +03 )2 R8es=2"S'Zl "Zl-D'P1 +2a'.V z1+2.S.z2~2-D.P2 +cr.V2 z2-Q 1

, fl.015.

~'~ I

,

.

i

0 0i5,

I--~,L/

................... ~

0il,

;

,:

e.........

t

,

,

t--T']

............. ;

,0.0t5, R~. /

; ,,

~.~L...... -[-........~-,'1/ i I-,:~,L I...... +__~, .'1 ~~ ...... -,:~ ...... t ~ ....'F ......l~ ......I-"t- ......[

l~5es=(2"S"Zl "Zl -D'P1 +Q1 -Q3 ) 2-( 2"S"z2"z2-D" P2 +t~'V2 "z2-Q3 ) 2

6res=-(2"S'Zl "Zl -D'P1 +a"V1 "Zl -03 )2-2g2"V?'z2 +Q1

.

~....._ _01 ...... .... ii

0,

+Q~2 --a,2.V~2 2.zs -E a'2.V~2 2.z4

i: S ' --

t i m e (s)

t i m e (s)

"i~

R~

t i m e (s}

Fig 3.a: Residuals with fault3~ We can observe in Fig 3.a, Fig 3.b and Fig 3.c that absolute value of residuals R(es ' "'2~res and R~es are lower than the threshold of detection t~-1.5"10 -3 in and o"'5res to o1"9res are bigger our case. Whereas ores 1"3 than the threshold. For all this fault, the same signature vector equal to [001011111] T is obtained. Then, the fault belongs to the group {3~,f6, fu}.

990

1f i.~,".,t,~, -~" 10.,...... 1 ! ~ i

0 ---

~

4

r~,u~,,, . y m b o ,

I

0

o

513o

113oo temps

(s)

1~

~

Fig 4: Faults isolation in group {fi, f6, o.,.

o.5

o.~

0

, r~

-e.

't

i 7~

-0 "

z~

i ~o

time (s)

i 7~o

r~

.

.

~o

.

1,

z~

z~o

6. CONCLUSION In this paper, fault detection and isolation for non linear hybrid systems with discrete control inputs and continuous outputs has been investigated. After changing non-linearities such x 1/" and sign(x), we suggested to combine continuous and discrete information to improve fault isolation9 In future works, adaptive thresholds will be proposed for improving detection, taking into account simultaneous faults.

.

~o

lime (s)

,

time (s)

Fig 3.b" Residuals with fault f6 x 104

0

- - - ---------,---------~ . . . . .

._

2.d~c]o

f14 }.

.

REFERENCES 0

.

.

.

.

.

.

.

.

if............. ti ............. t:t ......

~

......

time (s)

.04 . . . . . . . . . . . . . . . . . . . .

time (s)

t

time l[s)

Fig 3.c 9 Residuals with faultj")4 Thus, the signatures table is not sufficient to isolate a fault in group {f4, f6, f14}. To determine which fault occurs in this group, magnitude of faults is used. Let us consider the assumption that no simultaneous fault

Ioresl "8 > a and

occurs.

R~eh=-D'f4 +O~'z2"f6-f14 are

used to estimate magnitude of fault. Indeed, we can observe that residuals of f4, f6 and 3')4 have not the same magnitude and behaviours. Then R~e' satisfies one of equations (16.a), (16.b) or (16.c). R res =

8

D

.

f4

(I6.a)

g res 8 =-a'z2"f6

(16.b)

R res

(16.c)

8 =f14 R, reS

From (16.a), fi is estimated s.t." f4 =8_L.. With a D

tolerance of estimation error equal to 10.2 and according to table 1, if (Pe, f4) is equal to (0, 1) or (1,-1), one can conclude that pump Pe has a fault (table 5). Similarly, from (16.b) we can estimate f6 s.t."

R 8res

f 6 = ~

9 Then, if (1/2, ~ ) is equal to (0, 1) or

-~'Z 2

(1, -1), a fault on Ve is occurred (table 5). We can denote that: iff4=-~ when D=trz2, f4 and are not isolable. But, this case does not hold a long time for a fault on Pe or Ve. In fact, the values of Ze and R~es must vary in presence of these faults. Finaly, if (16.c) is satisfied with 3')4<0 (the leakage leads to a negative flow), then the fault is a leakage on Re with magnitude fi4. Isolation result is shown in Fig. 4, after the second level of structuration using equation (16), table 1 and table 5. A similar analysis can be used to discern faults in other subsets.

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