Optimal auxiliary input design for fault diagnosis

OPTIMAL AUXILIARY INPUT DESIGN FOR FAULT DIAGNOSIS...

14th World Congress ofIFAC

H-3a-03-5

Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China

OPTIMAL AUXILIARY INPUT DESIGN FOR FAULT DIAGNOSIS Toshiharu Hatanaka and Katsuji U osaki

Depar-trnent of Info1~111.ation and Knowledge Engineering

Tottori University,

Tottori 680-8552,

Japan

Phone." +81-857-31-5226, Fax: +81-851-31-0879 E-mail: [email protected]

Abstract: An optimal input design problem in frequency domain is discussed for quick fault diagnosis of dynamical systelus ,,,ithout affecting the original system in normal mode. Since the optimal auxiliary input for detecting a certain fault Inode is not

always good for detection of other fault

modes~ i.e.~

it may reduce the the distance

between the normal rnode and the other fault modes and make harder to detect true fault

mode~

the max-min approach is proposed in this paper. The auxiliary input is

designed to maximize the minimum distance measured by the Kullback discrimination

information nleasurc between system models corresponding to the normal and each fault mode in order to detect the true fault mode. Results of a numerical simulation

result indicate the applicability of the proposed auxiliary input in fault diagnosis. Copyright © 19991FAC Keywords: Optimal experiment design, input

design~

fault diagnosis, dither,

information anaJy.sis, autoregressive models, stochastic systems.

1. INTRODUCTION

During the last t,vo decades, there has been a gTo¥lillg interest in the detection probleln of de-

tection system for abrupt changes or fault in dy-

likelihood ra.tio (C LR) test, and their Inodifiea-

tions (see surveys, Basseville, 1988~ Basseville and Benveniste, 1986; Isermann, 1984; Frank, 1990; ~akamizo et al., 1979; Nikiforov, 1991; 'Villsky~ 1976).

namical systems has motivated by a ~~idc variety of applications including the detection of sensor

into account of the tradeoff

and actuator fault s, electro-cardiogram analysis,

sponse to the systeIn changes and insensitivity to

geophysical signal analysis, inlage analysis} and

system noise to avoid false alarm. It is possible

adaptive identification of time-varying systems ...A nUlnber of statistical techniques have been dp.veloped for detection of system fault SB ch &, sequential probability ratio test (SPRT), generalized

The design of fault detection system should take betVv~ecn

quick re-

that introduction of a suitably designed auxiliary input can accelerate the fault detection. Zhang (1989), and Kerestecioglu and his colleague (Kerestecioglu, 1993; Kerestecioglll and Zarrop, 1990,

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OPTIMAL AUXILIARY INPUT DESIGN FOR FAULT DIAGNOSIS...

dete<:~

1991, 1994) discussed an optirnal input design

t.hat the proposed auxiliary input in fault

problem in the course of this direction. They

tion and fault diagnosis using the backvtard se-

derived the opthnal auxiliary input for efficient

quential

prohabilit~r ratio

test (Chiell and Adams,

fault detection. Hovlever, they do not care of the

1976; Uosaki, 1985) reduces the mean detection

after effects of the additional auxiliary input to

tiIne lATithout Illaking much effect.s on the original

t.he systern hehavior. The auxiliary input rnay

system behavior and false alarm rate.

affect the original system output so much, and this

~itl1atioIl

is not suitable in certain

Cli.-'ies.

2. OPTIl\-fAL

_~UXILIA.R),- INPUT

DESIGN

Uosaki et al. (1993) have derived an optimal aux-

AND KULLB.A.CK DISCRL'v1INAT'ION

iliary input, which accelerates the detection of the

I:\FOR~IATION

systelIl fault but does not affect the original system behavior so

much~

Their key idea is to choose

the auxiliary input to enlarge the distance

mea~

Consider a linear stochastic model with an input

{Ut}1

sured by the Kullback discriruination infornlation

(KDI) between the system models corresponding

AIo (normal mDde):

to the normal and the fault nlode, but not to devi-

_4 0 {z-1 )Yt ::::: BD (Z-l )Ut

+ E~O)

(1)

ate much from the normal rnode ,\,rithout auxiliary is a delay operator, .Ao(z-l) and

input. Since their input is obtained by 'one-step-

",here

at-a-time' Inaxinlization of the time increment of the KDI in time domain~ it does not necessarily

Bo(z-l) are given by

Z-l

p

~ (0) - i A o ( Z -1) -- 1 _ L..,. ai Z ,

provides the global maximum of the distance bet\~leeIl

those Inodels. Furt.herIIlore, it is SOllletitues

i=l q

claimed that it generally offers little insight into

Bo(z-J)

the nature of the optimal auxiliary input. Hence,

=

L biO)

Z-i

i=l

Hatanaka and Uosaki (1995) have proposed the optimal auxiliary input design problem for quick

fault detection in frequency domain .

.L~n

optimal

auxiliary input iH derived by solving a Dlathemat-

ical programming problem

\\~hich

Ina.xirnize:-; the

time-average of the KDI.

and cia) is independently normally distributed ",,~ith mean

zero and variance O'"~.

Suppose that systern fault occurs at (unknown) time instant

T

and the system model changes to

one of the following models:

In this paper ~ the idea is extended to multiple faults case (fault diagnosis problem). For this case,

A-fj

(fault mode): A j (z-r )YI. == E j (z-] ) 'Ut

(j

Zhang (1989) proposed the weighted sum of the criteria between the normal nlode and each fault

mode as the criteria of optimal input design. Here the suitable choice of the

~·eight

auxiliary input to work

well~

~ 1,2,_··~~~r)

+ £~j) (2)

where

is crucial for the

""4. j (z-l) ~ 1 -

and Zhang chose

P

L a~j) z-i, i=l

Bayes a posterior probability as the \veight. But

q

Bj

its optimality is still open. And, it may give bad

('Z-l)

= L: b~j)z-i i=l

resuJts when the fault with smaller weight has

occurred.

and E~j) is independently normally distributed

Hence, max-min approach for auxiliary input de-

vlith mean zero and variance

sign is eIIlployed here. --..\.n optimal input is de-

0-1.

It is assurned

that all the system models (1) and (2) are stable.

signed to provide the maximum of minimum of

By introducing a suitable auxiliary input {u;} in

the tirue-average of the KDIs bet"v·een the system

addition to system input {Ut}, the fault detection

models corre8ponding to the normal mode and the

can be accelerated by enlarging the distance be-

fault modes. Numerical simulation results indicate

tween these rnodels corresponding to norrnal and

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Copyright 1999 IFAC

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OPTIMAL AUXILIARY INPUT DESIGN FOR FAULT DIAGNOSIS...

14th World Congress ofIFAC

fault rHodes. As a rneasurc of the distance bet\veen models, the Kullbac.k discrimination information

(KDI) is eUlployed here. The KDI in favor of the model ]tJk over the 1110del }..;ff is defined by

"Then the auxiliary input has ergodic propertY1 the time average in (8) can be replaced by the Then~

ensemble average. where Pj (yt lu t -

1

==

tion of yt

is the probability density func-

)

(Yt,Yt-l,' .. ' Yl)'l' given 'u

f

1

-

::=

(j == k, f), respectively. It is nonncF;ative and equals to

I[ .L\-fk :=.

M

f

~ y, u J(2 )

_1-E[(~4i(z--1)(B k (Z-l) _ B£(z-l) )Ut)2] (9) AIL (z-l)

2a;

under the model l\Jj

(-Ut-I, Ut-2, .. - , Ut) T

:

Ae(z-l)

N O\V, a 61tered input ii t given by

zero if and only if the models are identical.

The KDI I t [Alk posed as

:

fOlJ()\~lS

1\-1£ ; yt,u t -

1

(10)

can be decom-

]

(Kurnaluaru et al., 1988). is considered instead of 'Uf.-

It [.1\lk

Mf ~ yt, u

:

~ I o f-Alk

+It [A1k +It[.l\fk

:

-

J

Substitution of this expression into (9) leads to

]

+ It [A1k

kEf. ; Yo]

: :

t

:

Ate; yt, u t -

1

](1)

; yf',U f,-lJ(2)

Ml

ltfe ; yt, u.

t

I[Afk

1 - ](3)

(4)

:

AIt : Yr u](2) ,

== ~ 2a;

xE[((At(Z-l)Bk(Z-l) ~ A k (z-1)B e(z-1))'Uj.)2]

(11)

Vt.rhere 2

[t[NIk : A-fD ,.

t

Y,

L

'U

-

IdMk : Aft; yt, u t -

=

1

2~ Ut.

It [M

~

L4 ,(Z L.JC i=1

k :

Aft. ; y t ,

:

-

log--4)

0-;

Let

](2)

p+q _ " " ' h(kl) - i i Z

Bk(Z-l) Bf.(z-l) 21': ) ( 4 ( -1) - A ( -1) }Ut) (iJ) .. k

'It t -- ] ] (.l)

jtft,

;

ylr,

(12)

-~ i=l

i Z

Z

f( AkAe(Z-l) (z-l)

= tu~ . ~ 2a; 21r1, Here, I t (1\.1k

-1

1

a2

2

- a f. !.2 (a k (7;

t-1](1) -

where

1)2 dz

(6)

Z

uf.-l ](1)

indicates the

difference in noise conlponents, and It[Al k

Ale ; yt, u t - 1 JCJ ) and It[ .l\1k

:

n=l

l~ff. ; yt, U f - 1 ](3)

(i==2~ ..

indicate the differences in input and output re-

lations between

t.VlO

-,q)

models, respectively. They

are all nonnegative and equal to zero if models are identical. Thus, in order to find the auxiliary input, only the input-dependent term It [Afk: : 1\1e ; yt, u t - 1 J(2), or its time-average,

n::=l

(i==q+l, ... ,p+q)

Thus, the time-average of input-dependent term of KDl can be rewritten as I[Af k

:

!vIi ~ y~'U-

J(2)

.

==

(7) \vith a filtered input Ut _

can be considered.

It can be evaluated for models

!V[k

and AIl as

Ut

2 1 2 E r'~h(kt)-2 (L...,.. i Ut-i J ( 13 ) O"e i=l

1

== AO(Z-l) Ut

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OPTIMAL AUXILIARY INPUT DESIGN FOR FAULT DIAGNOSIS...

'Then, it is found tha.t the KDI is completely deterluined by the variance

rT;

and the auto-

E[Ut'Ut-i],

== 0, 1, ... ,p +

(i

q ~ 1) (14)

as fo110\',/8:

.,. . ~fe ,

y, u. )

] (2) _

-

1 (Po '""' ~ ( hi( k t) ) 2

20'2 -f

i~l

L

h).kl) h2:~)

+ ...

\~lith

the auxiliary input) should not be so

for detecting a certain fault mode, \vhich provides the maximum distance bet'iveen the normal and

p+q-r ~

-

a.void this, the auxiliary input Hhould be chosen not to affect the model so much- T'his implies that the KDI It[J.~.{o : A1~; yt, (n t - 1 ~ u· t - 1 )] (dis-

large. Furthermore; the optimal auxiliary input

i==l

+

the output process {y(t)}, \vill change.

haviors is not suitable in some practical cases. To

and

p+q-l

+ 2Pl

iors~ c.g.~

tance betvleen the normal mode models \vithout

p+q

I-[ Alk

original model AIo : and hence the system behavThe large deviations from the original system be-

covariance functions of the filtered input Ut,

Pi ==

14th World Congress of IFAC

(ki)

2pr L..J hi

(kf.)

h i +i

+ ...

i=l

+ 2hike ) h~~~jJJl+q-l)

(15)

the fault modes, usually reduces the the distance bet\veen the norrnal and other fault rnodes. And then it makes harder to detect the true fault mode. So, a luax-min approach is employed not to dete-

riorate the detection perfoTrnance for all the faui t 3. OPTI:L\1AL AUXILIARY INPUT DESIGN FOR FAULT DIAGNOSIS

modes. An auxiliary input is found; which gives

the rnaximum of the minirllum distance bet\veen the normal and each fault mode so that it works

The models of normal and fault modes with auxiliary input {u;} in addition to the original system input {1Li.} a.re given by

v,,~ell

in the Vlorst fault nlode case.

Define

1;110 [Af~ : l\l' J\;f~ (normal

~ rn~n ItP"l~

mode) :

. 4 0 (Z-1 )Yt == B O(Z-1 )(Ut + u;)

+ E~O)

J

(16)

Ak(z-l)Yt ::::: Bk(Z-l)(Ut

+

u;)

AIj ; y, (u~ U'1<)J(2)

(18)

AI' ; y, (u~ u*)] is found.

+ c~k)

respectively. Hence, the auxiliary input {u~} has

to be chosen to make the distance between the models of normal and fault Inodcs larger, or to

Thus, the optima) auxiliary input design problems

is reduced to the follo,ving mathematical programrIling problern.

Find an auxiliary input {u*} maxinlizing

make the time-averahe of the input-dependent

(PI)

term of the KDI IdAl~ : ll1~; yt,

Ilmin[.Af~ : 11// / ; y,

B~y

=

and an auxiliary input u* maximizing Irin[M~ :

ltf~ ( fall It mode) :

larger.

; y, (u~ ,u*)](2)

(ut-I, U*t-l

)](2)

(u, u*)] under the constraint

using (15) t it can be re\Vrittell as

l[.t\f~ : AI{ ; y, (u, u*)

=

1

2a 2

(( _ Po

_*)

+ Po

1

](2)

~(

llere, the constraint in (PI) can be expressed by (01»)2

~ hi

using the auto-covariance functions of the filtered

t=l

input Ut and the auxiliary input

p+q-]

+2(;31

+ p~)

L

h~Ol) h~~l{ + ...

[[Alo : M~ ; y> (it, u*')

i=l

-*) " Ili(01) h(Ul) L.-t i+r

Pr

2aJ

+ ...

. 4 o(z-1)

i==l

+ 2h 1(Ol)h<'Ol)( . p+q Pp+q-l

+

~*))

Pp+q-l

as

J(2)

== _I_ E [( BO(Z-l) U~)2]

p+q-r

+ 2( Pr +

fi;

q

== _1_ 20-2 o

(17)

t

q-l

(jJ*0 '" + ... L....t (b~U))2 + 2j5*1 """"" b~O) b\O) t+l ~ i~l

t

i=l

t

q-r

In this case~ as seen in (16) that an auxiliary input { 'Ut} makes the system model different from the

+2p-*" b~O)b~O) r L..t t J.+r

••.

+ 2b(O)b(O)p-* ) <_ 1 q q-1

L(19)

i=l

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

OPTIMAL AUXILIARY INPUT DESIGN FOR FAULT DIAGNOSIS...

14th World Congress of IFAC

This probIenl can be resta.ted in terms of auto-

==

u(t)

sin t,

covariance functions of the filtered input ii,t and

Ut

the filtered auxiliary input

as

follov.~s:

A fault occurs at the time instant

T

==

60 and one

of the fault Illodels "Till be chosen randomly \vith

(P2)

equal probability. Fault detection is carried out by applying

l\,laximize min

p;

j

t\VO

back\vard SPRT (Chien. and Adams, 1976), (lTosaki,

i=O

1985) based on the test statistics

subject to q-l

,f3ojJ~ + 2

o<

j, (j}

L PiP; :S L

t

i=l

p~

il;+f-l- 1

= maxfO, X(jJ + log (To + ! (E5\t)

Vv~here

>0

t

t-l

Ej(t) (j

:=

O~

(J"j

2

_ t':J Ct) )J aJ

aij

1,2) is the one-step output

prediction error corresponding to model

is nonnegative definite

if

\vherc

Xij) 2

mode j (j

]II/fj ,

and

K, then decide the systenl is in fault

= 1,2).

In this case, there is the possibility of false diagnosis choosing; the incorrect fault mode in addition to false alarrn and miss alarrn. The experimental (i

f30 =--

_1_

20 2

o

a.

==

conditions sueh as Land K 8,re a.."3sumed a.." same 1, ... ~ p

+q

- 1)

~(b(lIJr2 ~

n=l

Table 1 sho\vs the nlean detection tilne and the

n

number of false alarm, miss alarm and false diag-

q-i

_1_ ~

=

2a 2

fJ'l

o

~ n=l

as before.

b(O)b(O). n n+~

(i ==

1,~.~,q-l)

nosis in 100 simulation runs. Here, miss alarm rate

irnpIies the nUlnber of simulation runs where the

pZ

P;+q-I

fault cannot be detected until t

p~

P;-l-q-2

seen that introduction of the auxiliary input accelerates the detection speed

Po Here, h~Oj) (i

==

1, ... , p

+q

=

'~lithout

100. It can be much deterio-

ration of other characteristics such as false

miss ala.rm and false diagnosis

(",~rong

alarm~

decision)

rates cornpared \\rith the result without auxiliary j

1, .. _ ~ N) are

input.

same as in (13).

5. COXCL,USIONS 4. NL j\1ERIC.LL\.L EXAl\,fPLE T

An optimal auxiliary input have been derived in A nUITlerical example illustrates the usefulness of

order to accelerate the fault diagnosis of dynam-

the proposed auxiliary input in fault detection.

ical systelns. The optirnal auxiliary input. for de-

tecting a certain fault mode is not always good for

The norrnal nlode is modeled by 1\,10: Yt == O.5Yt-l u(t)

~

sin t,

+ O.4Ut-l +

detection of other fault modes since it may reduce Et

£t ~ N~(O, 0.1)

and the fault nlodes are characterized by the

models

the the distance bet\veen the normal mode and

the other fault modes and make harder to detect true fault mode. Hence, the max-min approach is proposed here to conquer it. The auxiliary input is chosen to maXIlnize the minimum distance

IIlea-

sured by the Kullback discrimination information and

bct\,,~een

the nornlal and each fault nl0dc so that

it works well for any fault. modes. It is

ShO\:llll

by

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

OPTIMAL AUXILIARY INPUT DESIGN FOR FAULT DIAGNOSIS...

14th World Congress of IFAC

Table 1 Results of fault

I\lean detection time

input \V'ithout auxiliary input

4.25

± 3.93

"Tith auxiliary input

2.72

±

det~ction

False alarm rate

1\:liss alarm rate

False diagnosis rate.

4/100 3/100

0/100

4/100 3/100

2.73

a nurnerical exarnple that t.he proposed auxiliary

0/100

Kerestecioglu, F. and 1\1. Zarrop (1990). llaycsian

input in fault detection using the back\,,·ard se-

approach to optilual input design for failure

quential proba.bility ratio test reduces the mean

detection. IFA C Adaptive Systems in Control

detection time Vv"ithout making much effects on

and Signal Processing 1989 pp. 625-630.

orig-inal system behavior and false alarm and di-

Kerest.ecioglll, F. and 1\:1. Zarrop (1991). Optimal

agnosis rate4

input design for change detection in dynamical systems. PT·oc.l,~t European ContTYJl Conference pp. 321-326. Kercstecioglu, F~ and 1"1~ Zarrop (1994). Input

ACKNOWLEDGEI\,1ENTS

design for detection of abrupt changes in This research is partially supported by Grant-

dynamical systems. Int. J. Control 59, 1063-

in-~.\.id

1084.

for Scientific Research fronl the l\Jin-

istry of Education, Science, Sport.s and Culture

Kumanlaru~

(10650432,10045043) .

K'. T. t

S6derstrom~

~1orita (1988)~

K.

S4 Sa.gara and

On-line fault detection in

adaptive control systerns by using kuHback

discrimination index. Identification and Sys-

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Somerset~

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Copyright 1999 IFAC

ISBN: 0 08 043248 4