STRUCTURAL ANALYSIS FOR RESIDUAL GENERATION: IMPLEMENTATION ISSUESCONSIDERATIONS

Dilek Düştegör, Vincent Cocquempot, Marcel Staroswiecki 1

LAGIS Université des Sciences et Technologies de Lille Bât. P2 59655 Villeneuve d’Ascq Cedex Francedilek.dustegor@univ-lille1.fr1 École Polytechnique Universitaire de Lille 59655 Villeneuve d’Ascq Cedex France

KeywordsFDI, residual generation, implementation, matching al-gorithms, complexity

Abstract

In that paper an innovative way of dealing with the gener-ation of residuals for fault-detection and isolation basedon structural information is presented. The developedtechnique considers implementation issues therefore ithas a more realistic point of view compared to classicalstructural approaches. First practical issues that can beencountered such as computational complexity or imple-mentation considerations are introduced. Then the wayof incorporating them to the existing structural analysisframework is explained. Finally, we show how the mini-mum weight maximum cardinality matching problem canbe used in order to choose the most suited matching thatleads to residual computational sequences. The methodis then illustrated using an industrial application.

1 Introduction

To meet the increasing demand for safer and more re-liable dynamic systems, early detection of faults usingFault Detection and Isolation (FDI) procedures is manda-tory.

In model-based FDI approaches, mathematical mod-els are taken as the basis for diagnostic algorithms. Evenwhen the system to diagnose is a well-known industrialplant, model building will require a major effort. There-fore, there is a recognized need for simple but efficientmethods for overall analysis, before going to any detaileddiagnostic algorithm design.

Structural analysis [1, 2, 3] enables to evaluate mod-els with respect to the properties of detectability andisolability of faults (FDI) by means of graph-based tools.Moreover if the intended diagnosability properties are notsatisfactory, some ways to improve them by extra sensorplacement [4] or by further fault modelling [5] can beproposed.

The first step of model-based FDI is to generate sig-nals called residuals that reflect the consistency between

c© 2004 IEE, CONTROL 2004 – Bath, UK

actual data and the model. Structural analysis identifiesthe part of the system which is monitorable, that is tosay the set of constraints that have to be used to gener-ate residuals. If residual generation issues are consid-ered, structural analysis may also provide the computa-tion sequences of these residuals by means of matching.There can be many different computation sequences lead-ing to residuals of equivalent structural fault sensitivity.However, when implementation issues of residual gen-eration are considered, computation sequences may notstay equivalent due to some practical considerations.

The major contribution of that paper is the incorpo-ration of available extra knowledge in the structural anal-ysis framework in order to choose among the equivalentmatchings the most suitable to generate residuals.

2 Structural analysis for FDI

Structural analysis only deals with the structural infor-mation contained in the model, i.e. which variables ap-pear in each equation. The system’s structural modelcan be represented by a bipartite graph G = (V

⋃

C,Γ)where V = {v1,v2, ...,vk} is the set of nodes corre-sponding to the variables appearing in the constraints(unknown variables, inputs and outputs variables) andC = {c1,c2, ...,cm} is the set of nodes corresponding tothe constraints, Γ = {(ci,v j)|v j appears in ci}is the setof edges.The corresponding adjacency matrix M is aboolean matrix where rows correspond to C, columns toV and M = {mi, j|mi, j = 1 if (ci,v j) ∈ Γ,0 otherwise }

Example: consider the following system:

c1(x1,x2,y) = 0 (1)c2(x1,u) = 0 (2)

c3(x1,x2) = 0 (3)

where u and y are known variables, x1 and x2 unknownvariables. Its bipartite graph and the corresponding adja-cency matrix are below:

c1 c3 c2

y x2 x1 u

x1 x2 y,uc1 1 1 1c2 1 1c3 1 1

�

Control 2004, University of Bath, UK, September 2004 ID-112

The canonical decomposition of M is performed in or-der to obtain the monitorable part (also called over-determined) of the whole system, that is to say the set ofconstraints from which residual can be generated by un-known variable elimination [1]. Then different ways ofmatching unknown variables in the over-determined partare investigated.

However, some sets of constraints are mutually de-pendent and correspond to systems of equations that haveto be solved simultaneously. These sets are called König-Hall components. In order to compute an unknown vari-able that is matched in a König-Hall block, all the con-straints in the block are required. There can be morethan one way to match unknown variables in a König-Hall block, but they are all equivalent with respect to thestructural sensitivity of the corresponding residuals.

Example (continuing): The whole system((1),(2),(3)) is over-determined. The block {c1,c3,x1,x2}is a König-Hall block of dimension 2. The redundantconstraint c2 enables to generate a residual. Even if x1and x2 can be matched to c1 and c3 in two different ways,the set of constraints required to generate this residualis {c1,c2,c3} in both cases. As a consequence, the twocorresponding residuals have the same structural faultsensitivity.

Matching (1):

yc1

x1

x2 c3

uc2 x1 x2 u,y

c1 1© 1 1c3 1 1©c2 1 1

Matching (2):

yc1

x1

x2 c3

uc2 x1 x2 u,y

c1 1 1© 1c3 1© 1c2 1 1

�

The next section shows that, for instance the two residualsfrom the example, that are structurally equivalent with re-spect to fault sensitivity, are not equivalent if implemen-tation issues are considered.

3 Towards Implementation: practicalconsiderations

In order to choose the best suited matching with respect toimplementation issues, further information, if available,may be considered and incorporated to actual structuralanalysis.

Example (Continuing): Consider again the system((1),(2),(3)). Suppose that c1 : 7x1 + x22 = u. Matching(2) leads to computations that require to handle squareroot computation since x2 is computed from c1 whereasmatching (1) leads to simpler computations. Therefore,the two matchings are not equivalent according to imple-mentation issues. �

Residual generation problem considering implemen-tation issues can be formalized as follows:

- Decide which considerations have to be taken intoaccount for implementation purposes (section 3.1)

- Incorporate them in the structural framework (sec-tion 3.2)

- Choose the best matching that fulfills the definedrequirements (section 3.3)

3.1 Which practical considerations

Practical considerations that affect accuracy and effi-ciency of the residual generation algorithm can be vari-ous and depend on the available knowledge on the sys-tem. The followings can be mentioned for illustrationpurpose:

- Computational complexity: non-linear models arehighly candidate to this kind of consideration. Forinstance, some functions are harder to compute ifthey have to be inverted (see previous example).

- Uncertain relations due to perturbations, unknownor varying parameters: in complex systems, it canbe difficult to obtain detailed and accurate model.Some parameters can be uncertain. As a conse-quence, corresponding constraints should be usedin a given way in order not to emphasize the effectof the uncertainty on the residual value.

Based on the available knowledge, a matching that max-imizes both accuracy and efficiency is desired.

3.2 Incorporation of practical considerations in thestructural framework

Based on the practical considerations, an ordered prefer-ence list ranking each of the constraints for each variablecan be created. If there is no such information about prac-tical considerations or there is no preferred matching fora given constraint, equal ranking is allowed.

The preference lists can be formally represented by apartial order < such that:c j

Example (continuing): Suppose system ((1),(2),(3))is:

c1 : 7x1 + x22 = y

c2 : x31 = u

c3 : x21 + x22 = 0

If the way of computing x1 in order to decrease computa-tional complexity is considered, it may be considered thatfirst c1 is preferable, then c3 and c2. Therefore, c1 is moresuited than c3, and c3 is more suited than c2 to computex1.

The following suitability orders can then be obtained:

x1 : c1

ControllerE/P

P2 QT1 P1

Ps

x

Figure 1: Schematic figure of the DAMADICS valve

not included in this presentation, only the structure of themodel is described. Readers interested in details of thismodel are referred to [11]. A model with 19 equationshave been used. The structure of the model is shown inTable 1. In this table, unknown variables stand for inter-nal states, known variables stand for measurements andcontroller outputs.

1 19

23

11

11

1 1 1 1 1

111

1

1

1

1 11

1

1

1

111

1

111

1

1

11

1

1

1

1

11

1

1

1

1

1

1

11

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

11

11

11

1 1

1

1

FaultsUnknown variables Known var.

Con

stra

ints

38 47

11

11

Table 1: Structural model of the DAMADICS valve

The structural analysis, in the classical sense, of theDamadics valve (see [5]) determines the monitorable partof the system, and allows to know the structure of residu-als that can be generated. However, the important numberof different matching possibilities in König-Hall compo-nents makes the residual generation task rather hard toperform.

Consider the configuration in table (2). We can gen-erate 3 residuals by replacing in c12, c14 and c18 thecomputed values of xh, P1 and T1 respectively. How-ever, there exists 17 different matchings enabling to com-pute xh and P1 due to the König-Hall block of size 10:{c1;c2;c3;c4;c6;c7;c8;c11;c20;c21}. All the 17 match-ings are giving residuals that are equivalent according tofault signature but with different computation sequences.

In order to apply the method proposed in section (3.3)preference lists of involved constraints and variables haveto be defined first. Based on the system equations (seetable 3), the suitability matrix (see table 4) has been ob-tained.

17

16

15

5

13

2

1

4

7

6

3

8

1120

21

12

1418

11

11

1

0

1

11

1

11 1

11

111

1

11

1

1111

11

111

1

1 1

11

1

cccccccccccccccccc

KH

−Blo

ck o

f siz

e 10

Ps T1 P2 Pv Q x x xh Qv3 Qv P P1 Pa Fvc x.. .

Table 2: Over-Constrained Subsystem

PsAe =kvẋ+mẍ+Fvc +(ks + kd)x+ (c1)

(ks + kd)x0 −mg

xh =hyst(x) (c2)

Qv =100Kv(xh)

√

∆pρ

(c3)

∆p−allow =Km(xh)(P1 − rc(P1)Pv) (c4)∆p =(P1 +P2)[P1 −P2 < ∆p−allow] ∨ (c6)

∆p−allow[P1 −P2 > ∆p−allow]

Fvc =πr2(P1 −∆p

Km(xh))[P1 −P2 < ∆p−allow] (c7)

∨πr2Pv[P1 −P2 > ∆p−allow]

Qv3 =Kv3(x3)

√

P1 −P2ρ

(c8)

Q =Qv +Qv3 (c11)

ẍ =dẋdt

(c20)

ẋ =dxdt

(c21)

Table 3: System Equations

c1 c2 c3 c4 c6 c7 c8 c11 c20 c21x 1 2

Fvc 1 1∆P−allow 2 1 3

P1 2 1 1 3∆p 2 1Qv 1 1Qv3 1 1xh 1 3 2 4ẋ 2 1ẍ 2 1

Table 4: Suitability matrix

Control 2004, University of Bath, UK, September 2004 ID-112

c1 c7 c6 c4 c3 c11 c8 c2 c20 c21

x Fvc ∆p−aP1 ∆p Qv Qv3 xh ẋ ẍ

Figure 2: Minimum weight complete matching

The main ideas to define the suitability partial orderare:

- to avoid square root operations (c3 and c8),

- to avoid to inverse hyst() in c1

- to avoid computing ∆p−allow, P1 and P2 in con-straints (c6) and (c7) because they appear in thelogical expressions. It is a difficult task since itis like we have to know them in order to computethem.

The matching given at figure (2) has been found, withan overall weight of 12. Among the 17 possible match-ings, this is the minimal weight possible.

5 Conclusion and Perspectives

The aim of this paper is to show how available informa-tion can be incorporated in structural analysis in order todetermine the most suitable matching to generate resid-uals. It is explained how to incorporate the availableknowledge by defining costs based on partial orders onvariables and constraints. Then, existing algorithms aresuccinctly reminded. Finally, the proposed method is ap-plied to a real life benchmark model. The algorithm en-ables to find the most suited matching among 17 differentmatchings.

Note however that the proposed method need to beimproved in order to handle dynamic residuals generationproblem.

References

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[3] V. Cocquempot, R. Izadi–Zamanabadi,M. Staroswiecki and M. Blanke, ”Residual

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[5] E. Frisk, D. Düştegör, M. Krysander and V. Coc-quempot, ”Improving fault isolability propertiesby structural analysis of faulty behavior models:application to the DAMADICS benchmark prob-lem”, Proc. of IFAC/Safeprocess’03, (2003).

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[7] H. W. Kuhn, The hungarian method for the assign-ment problem, American Mathematical Monthly,69(1), pp. 9–15, (1962).

[8] J. Munkres, Algorithms for assignment and trans-portation problems, J. Soc. Ind. Appl. Math., 5, pp.32–38, (1957).

[9] L.R. Ford and D. R. Fulkerson, Maximal FlowThrough a Network, Canadian J. Math., 8, pp.399–404, (1956).

[10] Z. Galil, Efficient algorithms for finding maximummatching in graphs, ACM Comp. Surv., 18, pp.23–38, (1986).

[11] M. Syfert, R. Patton, M. Bartys and J. Quevedo,”Development and application of methods for ac-tuator diagnosis in industrial control systems: abenchmark study”, Proc. of IFAC/Safeprocess’03,(2003).

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