Failure Detection by Means of Singular Systems

Copyright © IFAC System Structure and Control, Nantes, France, 1995

FAILURE DETECTION BY MEANS OF SINGULAR SYSTEMS Ph. POIGNET and lVI. GUGLIELMI Laboratoire d'Automatique de Nantes, U.R.A. C.N.R.S. 823 Ecole Centrale de Nantes /F ni versi te de N antes 1 rue de la Noe, 44072 NANTES CEDEX, FRANCE Tel:(33) 40 37 16 00/Fax:(33) 40 37 25 22 E-mail [email protected] ?vlembers of G.cl.R. 134 "Traitement du Signal et des Images"

Abstract: This paper deals with the abrupt changes detection in the linear dynamical SISO systems. Our method is based on the singular state modeling which takes into account the parameter jumps in the model. The estimation is then achieved by an extended Kalman filter specifically developed for this class of models. Finally the designed detector is applied to the engine rate change detection of a water-steam engme.

Resun1e : Le travail presente porte sur la detection de ruptures dans les systemes dynamiques lineaires monovariables. La methode proposee repose sur une modelisation d 'etat singulier qui prend en compte structurellement les sauts parametriques . L'estimation de ces sauts est alors realisee grace a un filtre de Kalman etendu developpe specifiquement. Le detecteur obtenu est ensuite applique a la detection de changement de regime de fonctionnement d 'un generateur de vapeur.

Key vVords : modeling , singular systems, identification , failure detection. Mots cles : modelisation. systemes singuliers, identification, detection de defauts.

1

Introduction

tion presents the identification algorithm for this class of model. It is then easy to develop a detection algorithm based on the identification result. Finally we exhibit simulations about real process issued from identification water-steam cycle , modelized by an ARMAX model.

A usual technical way of detecting failure in dynamic systems is based on modeling input/output relations which generate residuals very useful for the decision . This paper investigates the detection of abrupt changes in dynamic characteristics of linear systems by means of a different modeling of an AR:\IAX model and a new algorithm of identification. The new modeling , exposed in the first section , takes into account the jumps parameters and the linear model is naturally transformed into an implicit system. The problem is to identify the unknown parameters of the descriptor model composed of static and dynamic equations. Th.: main difficulty concerning these static equations is that it is impossible to define a predicteJ state when applying the Kalman filter. The second sec-

2

Modeling

In this section , we present the modeling of linear systems with stiff variation parameters which leads to a singular state system (Poignet, 1995). First, let's consider the classical system described by the ARMAX model :

+ alYk-l + ... + anYk-n = b1Uk-l + .. . .. . + bnul:-n + VI: + C1Vk-l + ... + CnUk-n

.!Ik

19

(1)

where: (al • .... an,bl . .. .. bn.Cl • ... , Cn ) are parameters and ., VI: is a white centered sequence of variance (7-. In case of abrupt changes of parameters aj, the linear system is modelized with a very similar equation (1) by introducing Si the jump on the ith parameter :

wit.h :

o

-Cn

° 1

YI:

+ s!lYk- l + ... + (an + Sn)Vk-Il = bl!ll:- l + ... + b"Uk-Il + VI: + ClVk-l + .. . +

[-'-~l"]

,5=

(al

-Cl

(l0)

and : (2)

.. . +C"Vk-n

B where Si. the jump on the ith parameter. is expressed as : Sj QjH r • (k) . Qi is the jump level, r j the changing time and Hr.(k) the Heavyside function defined by :

=

1. if k 2: =

H

rj

From equation (2), it is possible, as in the classical case, to get a state model. Actually, the system (2) is equivalent to : Yk

+ alYk- l + .. . +

blUk- l

a"Yk-n

3

+ SlYk- l + ...

XE

and X k such as:

( X rl

Identification algorithm

zI

(4. )

T Yk-,,+ l, ... , x"r=)YI; k

= (x~ =

SnYk -,,+l ... . ,

= sl yJ;)I

x~

Ok = (c" • .. . , Cl , Cn

X~

The dynamic equation of

+ ...

o

In

0

0 0 0 0 0

In

n :

[

o Yk

Noting XI;

Uk

+

C

bl

Xk+

[O .. .

-

:

Cl -

-Sn ] [

an ]

[I

() )

Yk

al

Yk

(7) (8 )

\'. o -In] In . k +

+

[

[H O]X k

;\1e" S' ]

, .. . ,

0 0 0

Zk

0 0 0

In 0 0

ydn ukIn 0

0 0 0 0

In 0 0 0

is : 0 0 0 0

In

}+,~

0

In

0 0 0 0

0 0

In -y"In

0 0

(12)

-I" 0 0 0 0

Zk

(13)

In

= [H 0 0 0 0 O]Zk + Vk

(14)

In relation to the classical estimation problem, there is no prediction equation for XI.. To solve the problem of the identification, it needs a prediction of the state X I:. Since it is impossible to get it from the state space equation (13) ,we choose, a priori, the prediction equal to the previous estimation of XI. . Then the prediction of the extended

.-l e

B Uk

In

In

Yk

= Xk ~ XI: . it follows : [

ai , bn

-

and :

-SI

Ol ] X~+vk

0 0

0 0 0 0

0 [

an , .. . , Cl

b 1, sn , ... ,stlI

o

x; -

-

(5)

The system (:3) leads to the following model :

b" ]

(11 )

Here we focus on the unknown parameters {aj, bi , Ci , sd identification of the singular model (9). This will be achieved with a new algorithm based on the Extended Kalman Filtering (EKF) and especially developed for the singular systems. As the classical method , we define an extended state = (X~ , Ok, Xk) where :

... + SnYk-n = + ... + bnUk-" + Vk + ClVk-l + ... ... + CnVk-n

Let 's define

[0 .. . 0 1]

Due to the fact of abrupt changes, the state model (9) is a singular one. The structural analysis of this class of models is widely studied in Dai (1989) . In our case. the useful important properties are summarized in Poignet (1995) .

< ri .

0, if k

=

(DJ .t}k

+ VI.: 20

z" is :

state

Zk+1/1:

0

n"

In 0

0 0

ykIn 0 In

0

0

ILkI n 0 0 [n

0 0

0

know the changing time. The new modelization only consider explicitly the jump which will be computed by the identification algorithm developed in §2. With regard to the changing time, it can be obtained by tracking the jump during the recursive loop of the identification algorithm. Formally, observing the sample (YQ, ... , Yn) up to time n, we have to decide between the two hypotheses :

= ...

!

-1"1 "I'

(15)

°° ° I,. ° ° The prediction error covariance matrix related to 0

In

the singular state Xt is choosen a priori equal to 00 (i.e. no informat.ion). The prediction error covariance matrix is :

FOT

(1

:5 k :5 1) Ho Hl

For 1 :5 k FOT

(17)

This one is associated with the output equation y". The new output vector is :

_[0000 H iJ 0 °

- .I /kI n U

YPk -

1n]_ -1-[0] 0 1 -I<,

(24)

8 = 81 , i.e. 3s; #; 0(25)

Simulations

The pre\'ious algorithm is then implemented on a real process. This process, a water-steam engine, could be modelized by several linear models whose the parameters are function of the engine rate. More particularly, the water-steam engine models identified in Guglielmi (1976) are second order An.:\IAX models where the input (Ul:) is the fuel delivery and the output (Yk) is the temperature , the whole process depending on the steam delivery (Q,). We view the general diagram of slIch a process on figure (1) :

+ R)-l'HPk + l / k (20)

with:

(21 )

This new al:,!;orithm of identification for singular systems based on the Extended T\alman Filter is able to identify all the parameters of these systems (Poignet et al. , 1993).

4

:5 1) / (23)

=

P,,+l/k 1iT *

[HO °0 °U °° [~ :~]

0

Ti

\\'here ~ E[OOT]. If !If becomes greater than a threshold>' (a priori fixed by the user), a change is decided.

5

(1iPk + 1 / k 'H. T

=8

(26) (18)

-

Ti

(J

Vi<

The estimation equations for this model from the a priori conditions (pr~cliction =k+1,'k and covariance matrix P~+l/~) are:

Pk + 1 / k

(1 :5

3

In practical way, the method is the following one: In the first part, we identify the ai parameters, all the jumps Si are equal to zeros, given that the initial conditions for Si are zeros. The user's parameters of E.K.F. are choosen such as Si keep the zero value during the first step. Once the system is well identified, we track, in the second part, the estimation of each Si, Si = E(s;/measures) , estimated with the same filter. If a change occurs in the spectral characteristics, one or more Si become different of zero. The decision function gf: (quadratic norm weigthed by the covariance matrix) is designed:

where Pk+1/ i< is the usual prediction error covariance mat.rix concerning; the prediction errors about the regular states.(Xk) and the parameters (0,,) (Poignet. 1995). Then the method consists in considering the static equations of the model, i.e. the last one of the dynamic model of =10. as 3.n output measurement:

o

:5 Ti - 1 ri :5 k :5 1

8 = 80 = (Si) = (0) (22)

Detection

Generally speaking. the changes detection problem is composed of two parts: First we have to identify the jllmp level and secolldly we W<1l1t to

21

First case :

jII

o. 00

_

1

,"

Vs

11

J ~

1 '

Ilill I,,'

J ~

.

u

+ t~

-s

k

.-\

., ;.

,

v

•k

::i -, 0

Figure 1 : ARJIA.X model Issued from the models identified in Guglielmi (1976) , we compute engine rate changes by modifying the parameters :

B C

1-1.13:- +0.48:-

I,

.00

'000

' .00

20<0

nME

3000

=0

3!00

'000

I

1 - 1.06:- 1 + 0.17:- 2 4.93:- 1 - 4.74:- 2 1

!!II I

I': .1,

Figure 2 : Input

• before engine rate change (Q;) :

A

1111 III

2

'Of I

(27)

(28) (29)

• after engine rate change (Q;)

B

1 + 0.173:- 1 - 0.60:3:- 2 (30) -1.93:- 1 - 4.74:- 2 (:31)

C

1 - 1.1:3:- 1 + 0.48:- 2

A.

. -

:----,,~ OO-..,.,~ OO.,.. O -,"'"50=0-=20""CC--'-25'""00'---',""00""0-,.-:0=0=0--""'000

m..e

(:32)

Figure 3 : Output and Vk is a zero-mean white gaussian sequence. Our problem now is only to detect on-line watersteam engine model changes due to variation of Q. that. we suppose. is unmeasured and not to know what happens (engine rate change or fault ). This problem could be study further. From this set of parameters. two numerical studies were performed :

..

: !-

: ",.

!

• The first one is computed with step input signal during the complete time of experiment . vVe achieve the identification and by the way expressed in the third section the detection.

I

1

i

-: .i: ,- - - " - 00- -'-00 ' 0 - -'- '0-0-=20-00--'-25'""00'---'30""'0""0---C,.~OO=-~.OOO n .. e

• The second one considers only step input signal during the time required for the identification . When the process is correctly identified . we put the input equal to zero. Abrupt changes on parameters simulating a fault or engine rate change occur when the input is equal to zero . It is impossible to detect the changes. If the input signal is then non eq ual to zero for external reasons (engine rate change , process control, .. . ) the jump parameters inform the user on the occurence of a change.

Figure

- -." .

4 : Ju.mp

SI

o'M,,"~"-
~: ·-; "' t .: ~ L

\

~1\t-~~1

,". f ·

The following figures show inputs. Qutputs alld parameters SI and s'.! which modelize the abrupt changes for the two cases .

.• >",-

-"",~ oO-""'~ O'j)"--"","',,= ,o -=~0<":": "'--:::25=OO'---;:3O=0"" 0 ---c3500 =--:-i.OCO TIME

Figure .5 : Jump

22

S2

6

Second case :

This approach provides a new way to study failure detection. From the classical ARMAX model, we transform naturally the state system into a singular one. Applying specific algorithm for the two phases (identification and detection), we design a tool for detecting engine rate changes. Considering the system with an external point of view (i.e. only with the observation of the input and the output) , the parameter change due to the steam delivery change could be the result of an engine rate change or process fault . The tracking of this model is not able to decide between the two hypotheses but is an element in the final diagnostic. The simulations show results with an easy task on a water-steam engine and allow us to hope further development of this method.

o. o.

1I

P

o.

,idi I[ 1

o.

.s:

"

1:j; "

I' i:.'

1::,:1 1I:.

0

I '!Hl li ,'

- 02

I"i::·!;i " ,!,,I; ,.'

-0 '

r

_0.

-, 0

I ' ,"

1 " . ' 1 j' ,,:"i

111

il'1 11l,

,

1

• .;~"C

5.;0

1,/ ,W

li I,'i" 1':1.1,1'/ ",

" i',: I;" ,t:'ii i: 1:;':1: ii ii'il I

_0.

'1111

i, '

1 1 1 , .,/ /"

•!coo

~ooo

m.""

:lOoo

,,"00

,,!il ll~

I1 ' M

3500

"' oo~

Figure 6 " Input

"'I "

I~11~'rlir il i\. ~! . I~J ~. '''"'"--~ --...

!' _'0 - 1

5

7

:'':1

0

. ::J

"!\iO

2OCO ~J ME

3000

'!';O

~JO

References

Dai . L. (1989) . Singular control systems, Lecture Notes in Control and Information Sciences, Vol. 118, Springer Verlag.

!kf

o "

Conclusions

Guglielmi , M. (1976). Contribution a. la determination et a. la mise en oeuvre d'algorithmes de realisation minimale : Application a. la modelisation d'un generateur de vapeur d 'un methanier, Ph .D. Thesis, :'-iovember.

.&.000

Figure 7 " Output 1. 1

J

:·--.:'.;'v~..;.

(

",

~

Ljung, L. and T. Soderstrom (1983). Theory and pratice of recursive identification, MIT Press.

.

-

,., 1

,

;;;

oel

f

,)

!

o'" ~

/ ~--/

I

]

Poignet , Ph. and M. Guglielmi (1993). Identification of time-invariant descriptor systems by means of extended kalman filtering , I.E.E.E.E.R.K., Portoroz .

,

I

Poignet , Ph. (1995). Detection de ruptures spectrales au moyen de I'analyse de systemes singuliers, Ph.D . Thesis , January.

, 0

! ~O

':r.c

"!CO

Figure 8

2COO

2! O,)

~ooo

:!.SOa

. COOO

'"''''

.Jump

SI

O ' ~I- - - - - - - - - - - - - - - - - - - - - - - - - - , - - - - - - - -

I

~: :~

-03~ ~ ...., -Oolii _o, ~

-oa ~ ! _0 7 ~ _J 8 ~

-0 ~ 0:' ----;,-:-;: ,:; o---:'",;.'.c '7C;:---:·7;~c:-;;o - -':;;';10:;;"0--<~,,'T " ~ --V: ~ c.c;;:C"---::!:"'-:-; ' J.-­ ~IME

Figure 9 " Jump

s~

These results show the ability of th.! method track abrupt changes.

10

23