Sequential Residual Design Method for Linear Systems

Sequential Residual Design Method for Linear Systems Jan M. Kościelny, Michał Syfert Institute of Automatic Control and Robotics, Św. A. Boboli 8, 02-553, Warszawa, Poland, (e-mail: {m.syfert}{jmk}@mchtr.pw.edu.pl). Abstract: The paper presents the fault isolation method for linear systems taking into account the residual dynamics. The method is based on the analysis of the sequence of observed symptoms assuming the residual models. Additionally, the method of designing the secondary residuals that allows to achieve particular symptom sequence for particular fault is proposed. Described methods are illustrated with a simple example. Keywords: fault isolation, linear systems, dynamical behavior, fault detection.

1. INTRODUCTION One of the primary indices of diagnostic system evaluation is its ability to isolate faults. The matter of fault isolation consists in separation of fault (or subset of faults) based on the premises of its occurrence carried out by the values of calculated residuals. The diagnostics is carried out in on-line mode, only with the use of working signals. The paper presents new fault isolation algorithm for linear systems based on the analysis of the sequence of observed symptoms. Known methods of structured residuals (Gertler and Singer, 2001; Gertler, 1991, 1998), and directional ones (Chen and Patton, 1999, Gertler 1998; Korbicz, et al., 2002; Kościelny, 2001) do not take into account the residual dynamics in the case of fault occurrence. The proposed method is based on the analysis of the sequence of registered symptoms. Its utilisation was suggested in (Korbicz, et al., 2002; Kościelny and Zakroczymski, 2001; Kościelny, 2001; Kościelny, et al., 2008).The knowledge of internal form of residuals is necessary to define the symptom sequence. Different residual responses to particular faults influence the symptoms forming sequence, thus its isolation is possible. The method of designing secondary residuals that allows to achieve particular (desired) symptoms sequence for particular faults is also given in the paper. There are also other works that deal with symptoms dynamics (Daigle, et al., 2005; Meseguer, et al., 2008; Pulido, et al., 2005; Syfert and Kościelny, 2009). These approaches are focused on the similar subject, however, the presented approach differs in the way how the knowledge about symptom sequence is utilised. Also the potential fields of applications are quite different and the described method is limited for relatively small processes. 2. DIAGNOSED SYSTEM DESCRIPTION WITH FAULT INFLUENCE The knowledge of the relation between the faults and the values of diagnostic signals is necessary for fault isolation. This relation can be determined based on process modelling with fault influence. Faults f are treated as separate process

inputs. The real processes are usually nonlinear, but for the purpose of simplification of the mathematical description, their linearization is conducted. It allows to formulate approximate linear description, valid in surroundings of working point on the static characteristic (this point, most often, corresponds to nominal or average working conditions). Commonly used form of a description of linear process (next to state space equations) is a transmittance model containing the set of equations defining the relationship between process outputs, inputs and faults:
s  Gs H

(1)

where:
– inputs vector, – output vector, – faults vector,  – inputs-outputs transmittance matrix,  – faults-outputs transmittance matrix. Particular equations have the following form:
 s  G s H ,

(2)

while G s: j  1, … , J includes the transmittance of inputoutput type: G s 


 

 

: p  1, … , P,

(3)

and  !: "  1, . . . , $ the transmittance for particular inputfault pairs:  %  

&' 

() 

: *  1, … , +.

(4)

In the case of fault free state the dependence
 s , G s  H   0 is fulfilled.

The residuals are calculated based on dependence called computational form: .  
 s , G s.

(5)

Equation (6) reflects general relation between particular residual and faults. It is, so called, internal residual form (Gertler and Singer, 2001; Gertler, 1991, 1998): . s  H sfs  ∑2134 H1 sf1 s.

(6)

This relation can be represented, for all the residuals, in the form as shown in Table 1. The proposed method is complementary in respect to the method of directional residuals and utilises the knowledge about the dynamics of symptom forming sequence. This knowledge is included in transmittances  % . The set of transmittances from the column of Table 1 corresponding to particular fault defines its signature.

.4

…

%

. ..

H41

H42

H4

H1

H2

H74

H71

H72

… .6

% GI I J !+ %   !+ I .

3. THE METHOD OF DEFINING SYMPTOM SEQUENCE FOR PARTICULAR FAULTS The symptoms forming sequence can be determined based on transmittance  % . The delays of symptom arising depend on: dynamical properties of the monitored part of the process, fault developing time characteristics and applied methods and parameters of detection algorithm. In the case of linear processes, the symptom sequence of the same fault depends on the transmittance  % . But, it is independent from the time development characteristic of the fault % 8.

Assume the occurrence of single faults. In the case of *th fault appearing the formula (6) simplifies to the form: 9. :

()

  %  % , ;  0, <  1, … , +, < = *.

(7)

Residual time characteristic can be determined, assuming the knowledge about function % 8, based on the inverse Laplace transformation: . % 8  >?4 .   >?4 @ %  % A.

(9)

The faults are undistinguishable (according to relation GH ) based on symptom sequence, if the corresponding to them signatures (9) are equal:

… .

!+ %   CD , ; , E , … F.

The knowledge about the symptoms forming sequence is an important information, which is worth of use in the process of diagnosing. Different symptoms forming sequence can characterise unisolable faults based on BDM.

5

H44

By putting the symptom delays in ascending order one defines pattern fault symptom sequences. As the result of the given above procedure, the signatures of symptom forming sequence are achieved for each fault and for given set of residuals. This signature includes the series of symptoms  for particular fault % written down in the sequence of their arising:

4. DIAGNOSIS BASED ON SYMPTOMS SEQUENCE

Table 1. The internal form of residuals

4

As it was already mentioned, the symptom sequence of the same fault does not depend on the time development characteristic % 8 of that fault . Assuming the shape of the function % 8 (step function is the simplest) and threshold residual value B , it is possible to determine the time, after which the symptom of the *th fault will appear. The above mentioned calculations must be conducted for all residuals sensitive for *th fault. For different symptom time developing characteristics the symptom forming delays will be different, but their sequences will not change.

(8)

(10)

Thus, the diagnostic process is based on the comparison between the recorded and pattern symptom sequences that characterise particular faults: KL  M % : !+ %   !+N,

(11)

where !+ denotes currently recorded symptom sequence. Signatures (9), achieved based on primary residuals (6), can, in many cases, enable the isolation of faults which are undistinguished based on primary structured or directional residuals. Different sequence of any pair of symptoms of particular faults is sufficient to isolate this pair of faults: !+ %   C , E F, !+ I   CE ,  F.

(12)

If the single fault % occurs and other faults I  0, U = *, then equation (17) takes the form:

5. DESIGNING OF SECONDARY SEQUENTIAL RESIDUALS Besides designing primary residuals it is also possible to design secondary ones, analogically to structured and directional residuals (Gertler, 1991, 1998; Frank, 1990). The aim is to achieve individual (unique) symptom sequence for particular faults. The symptom sequence characterising particular fault is understood as the series of consecutive symptoms ordered according to delays of theirs occurrence together with the specified delays between them. The symptom sequence can be written down in the following form: ! %   C , O ; , ; , O;E , E , … F.

(13)

The faults with different symptom sequence are isolable. To be able to distinguish any pair of faults based on symptom sequence it is sufficient that these sequences differ by one value of the delay while there is the same symptom order: ! %   C , O E  % , E F, ! I   C , O E  I , E F, O E  I  = O E  % .

(14)

For the purpose of sequential residual structurisation the secondary residuals must be designed in such a way that the sequences of symptoms for particular faults are specific. It is sufficient, that the delay time O E between  and E of the same fault % (which sequence does not have to be unique) is specific only for that fault. The single fault scenarios are assumed. Based on any pair of primary residuals sensitive for particular fault % : .   ∑5 %34  %  %  . 5 .E   ∑%34 E%  % 

(15)

. E|%    % E%  %  . .E |%   E%  % R ?ST'  % 

The transmittances of the fault % influence onto both residuals differ only in the delay element R ?S'T  . Additionally, equal threshold values for both residuals are assumed. It is appropriate to underline that the above secondary residuals are suitable only for diagnosis of fault % while the product of the transmittances of that fault  % E%  =  I EI , U = *

must be different than corresponding products for other faults present in the equations (17). The above given method can be used to design the pairs of secondary residuals for any fault and for any pair of residuals sensitive for that fault. The same pair of primary residuals can be used for generating distinct pairs of secondary residuals for two or higher number of faults, that are detected by those primary residuals. In this case distinct times of delays between symptoms must be assumed. In particular case O E  0 can be assumed which denotes simultaneous occurrence of symptoms  and E . Analogical approach can be applied for designing secondary residuals based on the whole set of primary residuals sensitive for particular fault. As a result, it is possible to achieve, for each fault % , the subset of secondary residuals which symptoms of that fault are formulated in specified sequence and specified intervals. However, the order of transmittances in equations (16) increases together with the increase of the number of secondary residual designed for particular fault. Next, it directly implies longer fault detection time.

Based on the proposed method of secondary residual design for each fault that is detected by at least two residuals the signatures including the pair of symptoms and the time period (delay) between them are obtained: ! %   C , O E  % , E F

(16)

 @QE  , E  A % R ?ST' 

The internal form of the residuals is as follows: . E|%   ∑5 ;34  ; E%  ;  . ?ST'  .E |%   ∑5

;  ;34 E;  % R

(19)

6. DIAGNOSIS BASED ON SYMPTOM SEQUENCE

it is possible to design secondary residuals, which symptoms of % fault will arise in particular sequence and with arbitrary delay O E between them. Secondary residuals that fulfil that condition are calculated from the following dependences:

. E|%   . E%   @Q  ,   AE%  .E |%   .E  % R ?ST'  

(18)

(17)

(20)

The diagnosis consists in calculating the values of the pair of secondary residuals for particular faults and in recording the delays between detected (observed) symptoms.

If the symptoms of some fault are detected and the delay between them is consistent with the delay expressed in fault signature, then the diagnosis points out this fault: KL  % : ! %   !

(21)

ZY \4

7. EXAMPLE The standard example of the three parallel tank system is considered below. It is shown on Fig. 1. FT 01

LT 03

Fig. 1. The three tank system diagram

.X   >4  , .^   >X  , .a   >^  ,

(22)

%Y

V ,

%Y >  Z[ \4 X

(23)

%Y

V ,

%] >  Z[ \4 X

(24)

Z[ \4

Z[ \4

%_ >  ZY \4 4 %b

Z] \4

,

%`

ZY \4

>^ 

>X 

(25)

(26)

While the internal form is: .4   ,ΔV *4 ΔW *d Δ! *e ΔfE .X   %g

Z[ \4

%Y

Z[ \4

ΔV , Δ>4 

ΔS4X  ,

%[i

Z[ \4

Q4 

%]

Z[ \4

%b

Z] \4

%[Y

ZY \4

Δ!X^  ,

ZY \4

ΔLX  , Δ>^ 

Δ!^  ,

%[b

Z] \4

Q ^ 

%`

ZY \4 %[]

%[_

Δ>^ 

Q X 

Z] \4

(27)

Δ>X  , (28)

(29)

ΔSX^  , (30)

The first residual utilises the relation between control signal W and flux flow V by the control valve while three others correspond to flow balance in particular tanks. The fault vector is as follows:

The linearised residuals in the computational form can be created based on the balance equations that take into account the faults influence (Kościelny, 2001):

.X   >4  ,

ΔL4  , Δ>X 

f4  ΔV n u

X  Δ>4 m t m ^  Δ>X t m a  Δ>^ t m o  ΔW t m p  Δ! t m d  ΔfE t Z  m

e  ΔfE t m  Δ! t 4X m q t

4r  Δ!X^ m t m 44  Δ!^ t m 4X  Q4 t m 4^  Q X t l 4a  Q ^ s

u

.4   V , *4 W

%_

ZY \4

ΔS4X  ,

.a   Z] \4

It must be underlined that this method cannot be applied for faults detectable by only one residual.

LT 02

%[[

%[`

where ! denotes current recorded symptom sequence.

LT 01

.^  

(31)

Table 2. Set of three tank system faults. fk f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14

Fault description Fault in the control path U Fault of the flow sensor F Fault of the level sensor L1 Fault of the level sensor L2 Fault of the level sensor L3 Fault of the servo-motor or control-valve of the actuator Fault of the pump Lack of medium Partial clogging between tanks 1 and 2 Partial clogging between tanks 2 and 3 Partial clogging of the outlet Leakage from tank 1 Leakage from tank 2 Leakage from tank 3

The following secondary residuals must be created to achieve demanded sequence:

The matrix v has the following shape: ,1 n %Y mZ[\4 vm 0 m m l 0 *4 *d *e *e …

0

0

0

0

0

0

0

0

0

0

0

0

*4

…9

0

*d 0

*e 0

0 ,1

0

0

0

0

0

0

,1

ZY \4

%b

0

Z] \4

0

0

?%g

0

ZY \4

ZY \4 %[_

Z[ \4 %[[

0

?%[Y

Z] \4

0

0

0

Z[ \4

%_

*e

0

0

0 0

%]

0

?%g

0

ZY \4

ZY \4 %[_

Z[ \4 %[[

0

%`

ZY \4

…9

,1

0

Z] \4

[

0

0

?%[Y

0

ZY \4 %[_

> z ZY \4 ^ …

(32)

Z] \4

0

0

?%[Y

ZY \4 %[_ Z] \4

u 0 t t 0 t ?%[` t Z] \4s

Table 3. Binary diagnostic matrix for the tank system. f2

f3

1 1

1 1 1

f4

f5 1

f6 1

f7 1

f8 1

1 1

f9

f10 f11 f12 f13 f14

1 1

1 1 1

.^   >X  ,

%Y

%_

V ,

>  ZY \4 4

%] >  Z[ \4 X

,

%`

ZY \4

 , X 

>^  

%_

.^   >X  ,

(35)

1

 ZY \4 X

(33)

(34)

%_

ZY \4

R ?o X 

%_ >  ZY \4 4

(36)

%Y

Z[ \4

V ,

%_ >  ZY \4 4

%] >  Z[ \4 X

,

%`

ZY \4

,

>^  

%g

 Z[ \4 q

%[[

 ZY \4 q

(37)

(38)

The computational and internal forms of secondary residuals for fault q are as follows:

%Y

1 1

Assume, that pattern sequence of the pair of residuals after structurisation characterising fault X should have the following form: ! 4   CX , OX4  5, ^ F. It means that symptom ^ is delayed by 5 sec. comparing with symptom X . Assuming the appearance of single fault X the residuals sensitive for that fault are described by the following equations:

Z[ \4

V ,

For the fault q assumed symptom sequence ! has the form: ! ^   C^ , OX4  15, X F. It means that symptom X is delayed by 15 sec. comparing with symptom ^ . Assuming the appearance of single fault q the residuals sensitive for that fault are described by the following equations:

.X,^/q  

One can notice that, among others, faults fX and fq are indistinguishable based on BDM. Their isolability can be achieved by designing secondary or directional residuals. The alternative approach is based on utilisation of the concept of sequential residuals. Such residuals for indicated pair of faults are presented below.

.X   >4  ,

,

.X   >4  ,

The binary diagnostic matrix (BDM) corresponding to the matrix v is presented in Table 2.

S/F f1 s1 1 s2 1 s3 s4

%Y

Z[ \4

Y

%`

?%[`

Z] \4

,

.^,X/X   ,R ?o .^   ,R ?o y>X 

0

0

?%[Y

%_ % .  _ y>4  ZY \4 X ZY \4 %] % > z  , _ X  Z \4 X Z \4

.X,^/X  

Z[ \4

%[[

ZY \4 %]

V ,

> z Z[ \4 X

.^,X/q   , %`

> z ZY \4 ^



%g .   Z[ \4 ^ %g %[[

,

%[[ R ?4o y>4  , ZY \4 % %[[ , g R ?4o q  Z[ \4 ZY \4

R ?4o .X  

,

%g

Z[ \4

y>X  ,

 Z[ \4 ZY \4 q

%_ >  ZY \4 4

(39) , (40)

8. SUMMARY The proposed method of fault isolation is based on the sequence analysis of recorded symptoms. To be able to define the order and eventually the delays between particular symptoms the knowledge of residuals in internal form is necessary. Thus, the method can be applied in the case of well described and recognised linear systems with limited number of inputs and outputs for which the mathematical models can be found. It is not sufficient for the large scale processes. In this case it is not possible, in practice, to achieve models with fault influence. The presented method of secondary sequential residual generation enables the increase of fault isolability in respect to diagnosing conducted with the use of only primary residuals. It is an alternative for the methods of reasoning based on secondary structured or directional residuals. Structured residuals apply binary dependence of fault influence onto residual value for diagnosis purpose. While, the directional residuals utilise static dependence between

fault and residual (i.e. gain). Whereas, the method of sequential residuals is based on the use of the knowledge about the dynamics of symptoms forming. Thus, it is complementary in respect to the method of directional residuals and more general than the method of structured residuals because the sequential signatures include also complete information about binary relation fault-residuals. REFERENCES Chen J., Patton R.J. (1999). Robust model based fault diagnosis for dynamic systems. Kluver Akademic Publishers, Boston. Daigle M., Koutsoukos X., Biswas G. (2005). Relative Measurement Orderings in Diagnosis of Distributed Physical Systems. Proc. of the 43rd Annual Allerton Conference on Communication, Control, and Computing, pp. 1707-1716. Gertler J., Singer D. (1990). A new structural framework for parity equation based failure detection and isolation. Automatica, Vol. 26, no. 2, 381-388. Gertler J. (1991). Analitical redunduncy methods in fault detection and isolation. IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes - SAFEPROCESS'91, Baden-Baden , Vol. 1, 9-21. Gertler J. (1998). Fault Detection and Diagnosis in Engineering Systems. Marcel Dekker, Inc. New York Basel - Hong Kong. Frank P.M.(1990). Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy. Automatica 26, 459-474. Korbicz, J., J.M. Kościelny, Z. Kowalczuk and W. Cholewa (2004). Fault Diagnosis: Models, artificial intelligence methods, applications. Springer. Kościelny J.M., Zakroczymski K. (2001). Fault Isolation Algorithm that Takes Dynamics of Symptoms Appearances Into Account. Bulletin of the Polish Academy of Sciences. Technical Sciences. Vol.49, No 2, 323-336. Kościelny J.M. (2001). Diagnostics of automated industrial processes. Akademicka Oficyna Wydawnicza Exit, Warszawa. (in Polish) Kościelny J. M., Ligęza A., Dziembowski B. (2008): Fault isolation in linear systems takilng into account the knowledge about symptoms forming. In: Serowanie i Automatyzacja: Aktualne problemy i ich rozwiązania. Editors: K. Malinowski, L. Rutkowski. Akademicka Oficyna Wydawnicza EXIT, Komitet Automatyki i Robotyki PAN, Polskie Towarzystwo Sieci neuronowych, pp. 481-498. (in Polish) Syfert M., Kościelny J. (2009): Diagnostic Reasoning Based on Symptom Forming Sequence. Safeprocess2009 - 7th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, Barcelona, Spain, June 30- July 3. Meseguer J., Puig V., Escobet T. (2008). Fault Diagnosis using a Timed Descrete Event Approach based on Observers. Proc. of 17-th IFAC World Congress, Seoul, Korea, pp. 6914-6919

Patton R., Frank P., Clark R. (Eds.) (2000). Issues of fault diagnosis for dynamic systems. Springer. Pulido B., Puig V., Escobet T., Quevedo J. (2005). A new fault isolation algorithm that improves the integration between fault detection and localization in dynamic systems. Proc. of 16th International Workshop on Principles of Diagnosis (DX 05), Monterey, California, USA.