On-Line Fault Detection and Localization in Electrical DC-Drives Based on Process Parameter Estimation and Statistical Decision Methods

Copyright © IFAC Control in Power Electronics and Electrical Drives , Lausanne , Switzerland , 1983

ON-LINE FAULT DETECTION AND LOCALIZATION IN ELECTRICAL DC-DRIVES BASED ON PROCESS PARAMETER ESTIMATION AND STATISTICAL DECISION METHODS G. Geiger Institut fur Regelungstechnik, Technische Hochschule Darmstadt, 6100 Darmstadt, Federal Republic of Germany

Abstract. The monitoring of technical processes has become an increased interest due to the higher demands on reliability and safety. This paper presents a method for fault detection and -localization in elecrical d.c.-drives based on a continuoustime parameter estimation procedure, implemented on a digital micro-process-computer. Using a recursive estimation procedure, the model parameters of the differential equations describing the dynamic behaviour of the d.c.-drive are estimated. Then the physical process coefficients P of the drive (armature resistance and -inductance, magnetic flux-linkage, friction coefficient, etc.) are calculated. A subsquent modified statistical Bayestest is used to decide between (M+1) hypothesis {H. ,i=O, ... ,M } : H describes the normal operating d.c.-drive, and E. involves a f~ult-state of type i. In order to demonstrate the ~pplicability of the method, the paper will be completed by simulation studies based on a real d.c.-motor- pump system with a power consumption of 4 kW at 3000 rev / min. Keywords. Process monitoring; fault diagnosis; hypothesis test; continuous-time; parameter estimation; d.c.-motor; pump. INTRODUCTION Because of the increasing demands on reliability and safety of technical plants and their elements methods for improving the supervision and monitoring as part of the overall control of processes has been subject of an increased development. An overview of design methods for fault detection in dynamic and static systems is given by Isermann (1982). The problem can be stated by detecting and localization of faults using measurable input and output variables ~(t) and y(t) of the process which is described by

Many literature references are available for class (1). Much fewer is known about class (2). Willsky (1976) gave a survey for this class. Only very limited number of references c~n be given for class (3) and (4), for example Baikiotis, Raymond and Rault (1979), Geiger (1982a), Filbert and Metzger (1982). This paper is an extension of the publication given by the author (1982a) and presents a ~ault detection and localization method of class (3) basing on a continuous-time parameter estimation algorithm and a subsquent statistical test.

( 1)

FAULT DETECTION METHOD where n represents non-measurable noise signals, e non-measurable model parameters, x-internal state variables, depending on time t. The methods for fault detection can be divided as being based on the following variables: (1) Measurable signals ~,y, (2) Nonmeasurable state variables x, (3) Nonmeasurable model parameters- e, (41 Nonmeasurable characteristic quantities .!l .

475

It is typical for more sophisticated monitoring methods to use nonmeasurable quantities which can be obtained by process models and estimation methods. Process model parameters appear in the mathematical description of the real process. Continuoustime Multi-Input Multi-Output (MIMO) models can be formulated as differential equations in the form

G. Geiger

476

PROCESS

~B .• U{i){t),

. -11=0

r----------J-- ::l , n,

(2)

~(s) :

~O+~l·s+···+~n-ls

n-1

+(~n+!.)

F=~~'~F=~~

:,

where ~(t) and y{t) are input and output vector of dimension (pxl) and (rxl), respectively, and A. ,B. represent parameter matrices OIl dimension (rxr) and (rxp). ~(i) (t) ~nd y{i) It) denote time derivatives d 1 u{t)/dt 1 and diy{t)/dt i . ~nother system representation can be found by introducing constant matrix polynominals in the Laplace-Operator s (I:=identity matrix):

I , _ _ _ _ _ _ _ _ _ _ _ _ .JJ L...

STAGE ,

STAGE 2

n s ,

STAGE 3

( 3a) ~(s) :

~+~1·s+···+~m_1s

m-l

+~ms

m

( 3b)

with all diagonal elements of zero, which leads to ~(s)

·y{s) yts)

~(s) ·~(s),

~

-1

~n

being (4a)

Fig. 1. Fault detection based on theoretical modeling, parameter estimation and statistical test.

e:

(s)~{s)~{s)

G ( 8,s)u{s), -p -

P: physical process coefficients

(4b)

where 6 include all non-zero elements of A and B. The process model parameters 8 . are usually more or less intricat~ related to several physical process coeffici2nts Pi' e.g. mass, speed, moment of inertia, etc., so that 8 = f{P). As process faults may cause-changes of the physical coefficients, it ist necessary to determine the inverse relationship P = f- 1 (6) or e = ftP), Isermann (1980,1981b). The-remaining task is to decide if a fault happened or not, to localize the fault by deciding which physical coefficient{s) changed significantly and to characterize the kind of fault by a fault vector F. Fig. 1 shows the 3stage fault detection scheme. PARAMETER ESTIMATION The close relationship between physical characteristics and the parameters of differential equations is the stimulus for the usage of parameter estimation methods for continuous-time systems. Modern estimation theory is mostly concerned with discrete-time systems due to their close affinity to digital computers. On the other hand it seems to be straightforward to use analog or hybrid computers to estimate continuous-time parameters, but for applications this way is not recommended. Therefore the method presented in this paper can be implemented on micro-process-computers without analog elements.

process model parameters

F: fault vector

A usual approach is to define an error function E{t) and to minimize a scalar cost function J[E{t)]. The choice of E{t), reflecting-the discrepancy between the real system and the estimated model, is essential for the estimation procedure. Eq. (4a) can be used to define the Equation Error (EE) for MIMOsystems: §. (s)

(5a)

:

or equivalently e. (t): 1

(n) Yi

(l ~ i ~r)

AnT

(t)+~i·y

(n)

(t)+ ... +

y{n) {t)+y{n)T{t)a~+ ... + 1

-1

T AO T AO + y {t)a.-u {t)b. - ... -1 -1 (5b)

where

On-line Fault Detection and Localization

[ -a~j , 1

~j] ••• ,a , -r

477

( 7d)

~(k)

!:(o) 8( 0)

P

( 7e)

e

( 7f)

-0 -0

E:

The properties of the (R)LS-methods are well known for the discrete-time Eq. (5b) can be rewritten into case, see e.g. Isermann (1981a), or Strejc (1981), and attributes like simplicity and robustness promote y ~ n) (t) - ,I> ~ (t) 1 < i < r, (5 c ) these methods in system analysis con1 ~1 -1 -cepts like adaptive control. The use with of the (R)LS-methods for continuoustime systems is limited, because the (n)T T: T (m)T , ... , -y ,~ , ... , ~ ] , derivatives of the input and output [ -y measurement signals are required. Therefore, these methods can only be 8. : used for low-order systems and systems -1 with less noise and high accuracy which leads to measurement sensors and transducers. So called State Variable Filters (SVFs) represent a well known analo g ~(t) (5d) technique to circumvent this problem. The basic idea is the use of filters o to reduce noise effects from the process signal spectrum by combining filtering and derivation, see Mathews • T e. o o 8: and Seifert (1955). Assuming \jI. -1 ...-1

e. ,

-!!1

o

.

• 'T

Q..•• •~

F(s):

!:!

Q

!+!:!1s+ ... +~S ,

n

N. (s):

with respect to the parameters 8. Unlike discret~-time parameter estimation procedures, 8 in Eq. (5d) is independent of the sampling time T . o Nevertheless, for simplicity, we choose a constant sampling time T (k:=k·T ). The solution of minirnizingOEq. (5e? according to dJ/d e = OT leads to the well known least Squares (LS) estimate = [,!T(k),!(k)]-1,!T(k)'X(n)(k)

and

(8 )

where

{5e)

where,!(k): = [:t(1), ••• ,!!!(k)]

(s) ,!:!(s): =

8

The parameters e in Eq. (5d) can be estimated by minimizing

El

-1

-1

i 0 11

0

..•

n i..

JJ

0

0

1,2i '::9

.i n r leads to the structure shown in Fig. 2. 0

zlOI

_F

0

...

z lO-1I

-F

z lll _F

1

(6)

T

~(n)(k): = [y(n)(1)""';:r:(n)(k)]T

The Recursive form of the LS-method (RLS) is given by ~(k) :

..§ (k)

:

y (n) (k) _:J:T (k) ~(k-1)

(7a)

Fig. 2. State Variable Filters (SVFs) for MIMO-systerrs. Applying F(s) to input ~ and output y in Eq. (5a) leads to the Generalized Equation Error (GEE) for MIMO-systems:

(7b)

~(s) :

~(s)I(s)y(s)-~(s)I(s)~ ( s)

(9a) ~(k) :

~(k):t(k){!+:tT(k)~(k):t(k)}-1 (7c)

~ ( s ) YF ( s)

or ~(t) :

- ~ ( s) • ~F ( s) (9 b)

y~n) (t)-:tF(t) ~

(9c)

G. Geiger

478

1

where ~F(s) and ~F(s) denote the filtered process input and output.

[(2 1T )M : ~2 i 2 J -1.

f(!:(k)):

1

T -2

A

.exp { -}(!:(k)-~(k» ~

PROCESS

(!: (k) - !:!.( k) ) } 2 : = N{!: (k) , ~ (k) , ~ (k) } ( 1 Oa)

r-------J.---, D

I I

!:l

I I

A

I __tsl~(sJ

I I

t

where AM denote the number of parameters P . , 1 < k~. ~ (k): = E { ~(k) } and 0 2 (k) : ~= ET (P (k) =~ (k)) (E (k) -~ (k) ) T } represent mean and variance of probability density function f(·),

I

IL ___________ .JI

--- --,

r---

r--

( k)

I I

I I I I

- each Pi(k) is statistical independent with Pj (k), ifj, each !:(i) is statistical independent with P(j), ifj, and ~ (k), o (k) are time-Invariant for the-non-error case, implying

I I I I L ___ J

~(k) :

= const.,

~

2

(k):

const. ,

e

N

L-----------~-T---------J PARAMETER ESTIMATION

s f(P(1), ... ,P(N) = IT f(!:(i)) s i=1

( 10b)

where N

Fig. 3. GEE for continuous-time MIMOsystems. As both, ~(t) and y(t), are filtered by the same filters, parameter estimation methods can be derived from the G1E the same way then using the EE, see e.g. Young (1970) for the Single-Input Single-Output (SISO) case. Therefore the recursive form of the resulting algorithm is given by Eq. (7), y(nJ (k) and ~ (k) replaced by y : n) (k) and ~F(k), respectively (ind~x F deno te f~ltered signals). SVFs ha d been used in scalar form for analog or hybrid computer based parameter estimation methods, see e.g. Kohr ( 196 3 ) or Young and Jakeman (1980). Howe v er, we apply quasi-analog SVFs, realized on digital computers by using numerical inte rpolation methods like spline- or Newton-interpolation schemes. FAULT DECISION AND LOCALIZATION After parameter estiroation (stage 1) and calculation of the physical process coefficients in stage 2 follows the fault decision and locali~ation by generating a fault vector ~ ( k), see Fig. 1. For further discussions we assume that - E(k ) is a Gaussian vector described by t he density function

denste number of samples, s - a fault is defined as a significant deviat~on of mean ~i and/or variance 0 , of P . (k) from a non-error -2~ val ue 1.1 i' 0 i '

-=

- only one fault may time.

be happen at a

This problem is a classical hypothesis test problem, see e.g. van Trees (1968), and can be handled by formulating (M+1) hypothesis H., O
~

=

( fault of type i (significant de v iation of mean and / or variance) (1
Each hypothesis Hi can be described by a conditional density function A A Ns A f(P(1), ... ,P(N ) i H . )= IT f(p(j) iH . ) s ~ j=1 ~ N

s IT N { P(j), ~ ] =1 A

.

(H . ), o l

-

2

(H.) } l ( 11 )

where ~ (H.) and 0 2(H . ) denote conditional-me!n and vari!nce for hypothesis Hi' Therefore, the non 2error case is described by ~ (Ho)' ~ (H ), whereas ~(H~), (H . ), 1..:i ":M, de~­ cribe a fault of type i.

s:2

On-line Fault Detection and Localization

Problems arise because ~(H.), 02(H.) normally are not kncwn pfiorT. Tfierefore it necessary to estimate mean and variance; the resulting algorithm is then a combination of estimation and detection. A usual estimation approach is the Maximum Likelihood (ML) estimation to overcome this problem.

a

GENERAUZEO UKELlHOOO-PROCESSCfl

r-'....------,

A

[i, ~. (H.), o. (H.) J 1 1 1 1

F·

T

, 1
-

r---J..-, I

ei

The resulting fault vector F can be define'd by A

FAULT-VECTOR GENERATION I

I I

parometers

479

/I

FAULT VECTOR

I

j

\

J/

( ;J

----..., (12)

where i represent the type of error, characterised by ~i(Hi) and 0i(Hi)' It can be shown that the resulting algorithm is given by Eq. (13); for details, refer to Geiger (1982b): threshold t

LOGIC

a priori: Prob {no fault}:

( 13a)

= P . o

Phase I: Learning the non-error statistics from N data {P(j),1
. (H

1

0

):

N

s P. (1)

l:

:=~

1

1=1

s

o. (H ): 1

s L (P. (1) 1=1 1 A

1

N'"'=T s

0

i'

( 13b)

N

2

Fig. 4. Fault detection and localization scheme basing on hypo thesis test.

2

_

-~ . )

-2

: = 0 ..

1

1

MATHE~ffiTICAL MODEL OF A D.C.MOTOR-PUMP-SYSTEM

A real d.c.-motor-pump-system is used as a pilot process in order to apply our methods. Fig. 5 shows the scheme of the process:

(1 3c)

Phase 11: Fault detection and localization from Ns ' data { P(j),1 < j <:~ ' } - - s

-

u,

N ' s

N' s

L

( 13d)

P . (1) , 1

1=1

NS ' L 1=1 N

A2 A

S

A

N

'

_s_ {

2

o.

1

(0' 1. ) 0. 1

-~

2

1

-

.)

1

M'P A* > H ,E:,(T)l ( _ _ 0) i < HT n 1-P , o 0

, (1 3 f )

1

21n (

O. (~ .) 1_ 1

°i

arg { max( A~ )

T

,(ne)

A

(P. (1)

1=1

A* i

2

_

'

L

O. ( ~ .) : 1 1

A

(P.(l)-~.) 1 1

},

)-1},Fig. 5. Scheme of a speed controlled d.c.-motor and a centrifugal (13g) pump. ( 13h)

A

1
p: pressure U: I: tt'!: massflow It : speed of rot. R: T: torque L:

voltage current resistance inductance

(13i)

The structure of the fault detection and localization scheme is shown in Fig. 4.

First estimation results on the real process are obtained and published from Geiger ( 1982a). The new mon itoring results published in this p a pe r are till now obtained using an analog computer which simulates the behaviour of the pilot process Fi g. 5 in real time.

G. Geiger

480

The mat h ematical model of the process is derived in Isermann ( 1982) and Geiger ( 1982a), assuming

Eq.

(1 S) can be rewritten into

- constant excitat ion of the d.c.-drive,

see Eq.

~(t): ~ y(1)(t)_!T(t)£ (Sb), where

- vanishing armature reaction, - s mall deviations_( ~ ~2, @ I2' ~~ ) a steady-state (U ,I , ,, ). 2 2

around

2..

T

II

~-

Y ( t ):

(l)T

_ -y

Y -2

Assumin g further a constant mass flow r:l leads to dI L . __ 2+R · d (t)+ ';' · 61i (t) = 6 U (t) , 2 dt 2 2 ~ 2 (14a)

0 -

T'

~u

-y

T

In order to estimate the physical process coefficients, Eq. (14) must be rewri tten into a form similar to Eq. (2). This leads to v

"'-

(1) ( t) +A

v

-1 "'-

( 1) (t) +A

v

-0"'-

(t) = b u (t) , -0

Sb)

u ( t) :

(1

y

(1Sc)

(t):

( 1Sd)

b

-0

aOT i -1 I A

-0

OTJi

( 1Se)

L-2 ' <1

T'

-y ' u -

J

:

( 16 b)

T =[2.

:b 2

,

oT ~1

:b°10

T

oT '

~2

:0

~_~

,l ·e

(1)T,

T

T

T

oT

i-:l

I

L~

T

T

T

-y

0,

,

" oT] T

(16a)

-'L ' u, 0 , Q. ,

Q,2 , x [ Q. ' ~1

in Eq.

o-i

0

(1) T

b1' 0

x

T

:}

oT , ~2 , O]T,

, oT L~l

T , -y ,

re-

b

°1

oT T ] . (16d)

~2

The resulting parameter estimation algorithm is then given by Eq. (7). The relationship between process model parameters 6 and physical process coefficients P (see Fig. 1, stage 2) is given by R2 P1 (17 a) L: 8~ , 2

6

2

8

.J

8

L:

P

L2 1

3

L: 2

P

P 2

=_P 3 P 4 4 Ps Cp2 8 = - 8-:= P S 4 8

= _

!to

8·

P

3 2

2 , 8 3

( 17b)

1 8

,

( 17c)

e4 ,

( 17d)

3

e2

4 =- E · 3 =_ E2 · Ps 8 . 3

8S 8 4

, (17e)

with~: = {8 1 · .. , 8S }T and t: =.T {P , ... ,P }T: = {R ,L 2 ,1jJ, E,C J .The F1 2

1 physical ~rocess coefficient vector P will be used in stage 3 of the online fault detection and localization method as input data for the hypothesis test algorithm Eq. (13). SIMULATION RESULTS

!21

( 1 Sf)

]T

(16c)

( 1Sa)

see Eq . (2) with n=1, m=o, p=1, and r=2. The scalars, vectors and matrices are defined by

(1)

I

IT o T' O]T T IT oT ' ° [e T i2]= 'b a [ ~1 ~1 -1 : 1 -2 ~2

e

Fig. 6 shows the block-diagram for the d.c.-motor-pump-system. In most practical aDplications the d.c.-motor operates in a closed loop system, very often with 2 PI-controllers for armature current I2(t) and speed of rotation ~ (t). Therefore the simulation considered the closed loop operation in a manner shown in Fig. 7.

" :0

T'

The product ! T(t)· i duces to armature voltage in CV ] , armature current in [ A] L1 speed of rotation in [ s ], armature resistance in [~] , armature inductance in [H ] , magn. flux linkage in [wb] , moment of inertia of the 2 motor-pump-system in [kgm ] , friction coefficient of the motor-pump-system in [Nms].

TT

Q. Q. T

0 'O-v -: L

( 14b) where

( 16a)

In order to verify the theoretical approach the dynamic behaviour of a d.c.-drive-pump-system in a closed

On-line Fault Detection and Localization

loop application has been simulated on an analog computer, see Eq. (14) and Fig. 6. The physical process coefficients of the system are 1.00 5.80 c := 1. 16 LF1: = 80.0 2 8 : 1. 16 R2 : IjJ

:

ll , Wb, Nms. mH, kgm

(18)

As the time constants of the pilot process are small, the data acquisition scheme shown in Fig. 8 was used. A micro-process-computer was connected online to the analog model to perform the following computations.

481

N ' ~ s , A.(H.): = -2- {S. (H )-2ln( S (H.}}-l J , 1.

1.

1.

0

1.

( 1 9)

see Eq. (13), usinq Ns ' = 50 data in a common window technique. Fig. 12b show the subdivision of the monitoring time history in 2 phases. In phase 1 (1
\~

2

~

t

[Jj ( . t ~

2'(tl w(t)

X(t) w(t)

Fig. 8. Real-time data acquisition scheme y(t) = T = [U (t),I (t), 1l (t}] 2 2 During phase 1, the ~rocess was excited by a testsignal w(t) = w + 6w(t) and corresponding signals U (t), I (t) 2 and ll (t) were recorded. In phase 2,2 the model parameter vector 8 (1) has been computed, see Eq. (16d), and at the end of phase 2 the physical process coefficient vector P(l) has been obtained using Eq. (17). -These values are used as input data for the hypothesis test algorithm Eq. (13). Fig. 9-11 show the measured signals while phase 1 of armature voltage U2 (t), armature current I 2 (t) and speea of rotation ll (t) for a step signal excitation in the set point w(t) with an amplitud~ gf 10 % of the normal operating point w(U =150V, I =5A, 1l=25s- 1 ). 2 2 The results of the monitoring method are demonstrated by Fig. 12-13. From sample number 1 to 150 the normal operating system was simulated. At sample number 151 a 2 %-fault in the armature resistance R2(R2F=0.98 11 ) was assumed and simulated. Fig. 12a show the plot of the A1st physical process coefficiant P 1 (~) which is the armature resistance R2(k), whereas Fig. 12b show the plot of the 3 statistic quantities

S.

1.

6 i (~i) (H

0

):

S. (H.) : 1.

1.

The parameter estimation based approach to detect and localize process faults in electrical d.c.-drive s during normal operation has been extended by a statistical test decision approach. The simulation studies demonstrate that this method is able to detect small faults also under noisy conditions. Future work will be concerned with the application of the method on the real pilot process. ACKNOWLEDGEMENT This report publishes results of the research project DFG Is 14/25, supported by the Deutsche Forschungsgemeinschaft (German Research Foundation) . REFERENCES Baikiotis, C., J. Raymond, and A. Rault (1979). Parameter identification and discriminant analysis for jet engine mechanical state diagnosis. IEEE Conference on Decision and Control, Fort Lauderdale. Filbert, D. and K. Metzger (1982). Quality test of systems by parameter estimation. 9th IMEKO World Congress Berlin. Proc. pp. 43-51. Geiger, G. (1982a). Monitoring of an electrical driven pump using continuous-time parameter estimation methods. 6th IFAC Symposium on Identification and Parameter Estimation Washington. Proc. Pergamon Press, Oxford, pp. 521-526. Geiger, G. (1982b). A statistical modified Bayestest approach to detect changes in parameters. Internal report TH Darmstadt.

482

G. Geiger

Isermann, R. (1980). Methoden zur Fehlererkennung flir die Uberwachung technischer Prozesse. VDI-Bericht Nr. 364.

siqnals 12

ARMATURE CURRENT 12

Isermann, R. (1981a). Digital Control Systems. Springer-Verlag, BerlinNew York.

~j

Isermann, R. (1982). Process fault detection based on ~odeling and estimation methods. 6th IFAC Symposium on Identification and Parameter Estimation ~ashington. Proc. Pergamon Press, Oxford. Kohr, R.H. (1963). A method for the determination of a differential equation model for simple nonlinear systems. IEEE Trans. Electron. Computer, EC-12, pp. 394-400. Mathews, M.V. and W.W. Seifert (1955). Transferfunction synthesis with computer amplifiers and passive networks. Proc. Western Joint Comp. Conference, pp. 7-12. Strejc, V. (1981). In P. Eyckhoff (Ed.), Trends and Progress in system identification, Vol. I. Pergamon Press, Oxford, pp. 103-143. Van Tress, H. (1968). Detection, estimation, and modulation theory. Wiley and Sons, New York LondonSydney. Willsky, A. (1976). A survey of design methods for failure detection in dynamic systems. Automatica, 12, 601-611.

0.4

"0.0

0.8

I

1.2

1.6

2.0

srw Gi RaT."

siqnals

•

2.4

lose I

--1

~

SPEED OF ROT. SI , SETPOINT W -1 ) ( 1 dev = 5 s

5.9

5.6

i

1

i

5.4

I

IL

5.2 ~

5·~.0

0.4

o.a

ll.e 1.6

1.2

2.0

2.4

(Sp.c 1

Fig. 9-11. U2(t) ,I2(t) and N(t) during phase 1 (0~t~2,5sec). ~

rocess coeff. lonal

.\D-l

i£S15I. R2

ARMATURE RESISTANCE R2 ( 1 dev = 0.1 SI )

10.3

10.1

Young, P. (1970). An instrumental variable method for real-time identification of a noisy process. Automatica, 6, pp. 271-287.

tl~8

~~

9.9

~~,t.

9.;>

Young, P. and A. Jakeman (1980) . Refined instrumental vari able methods of recursive time-series analysis. Int. J. Control, 31, 4, pp . 74 1- 764 .

n""ber 80

120

160

200

--, ~

lEillt "I)

statistics R2

ETIIt (HI>

240 UWI\

1- )

--

(HI)

A

L~ 1.Q.11lliE1IZ--t

siQnals

8 1 (H

8

U2

0

)

6

ARMATURE VOLTAGE U2 ( 1 dev = 30 V )

10

It

Al (H ) 1 / _.--_.'---, I \ ,

4

8

,

2 6

I

~V 4

ti.o

0.4

0.8

1.2

00

li ..

1.6

2.0

2.4

I sec I

40

80

120

A

/

160

,/

X\ \

8 1 (H 1 ) .... '------

tJ.lliter

/

200

240

I-I

Fig. 12a,b. PrQcess coefficient p] (k) = R2(k) (a) and statistics 8 1 (Ho) , 8 1 (H 1 ) (H 1) (b) .

,A,.

On-line Fault Detection and Localization

likelih. ralios

1~Iff.:..:PO:.:.:H.:=:£S:.:.IS~l---.,+---------1 Fig.

I

I

483

1 3. LLRs /\ ~l fo rH. l i=1,5.

Experiment: 4

sample

1-150: normal operation sample 151-250: 2 % fault of armature r es istance R2

3

Monitoring:

2

80

40

120

150

200

240

[ -]

r-----------------------~

Ul 21

i

r---------------~

I

t 1 5

1

I

I II

I

I I

I,

I

I

I' I

I

,

U

I I

I

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Fi g. 6. Mathematical model of a d.c.-motor-pump-system with constant excitation. c : = c + c • FM1 F1 FP1

ELECTRIC

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Fig. 7. Conventional control scheme of a d.c.-motor-system.

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