Robustness in Model-Based Fault Diagnosis: The 1995 Situation

Copyright © IFAC On-line Fault Detection and Supervision in the Chemical Process Industries, Industries. Newcastle upon Tyne, Tyne. UK,1995 UK.1995

ROBUSTNESS IN MODEL-BASED FAULT DIAGNOSIS: THE 1995 SITUATION Ron J. Patton Department of Electronic Engineering The University of Hull. Hull, Hull~ HU6 7RX, 7RX~ UK 1482466006 Phone 44 1482 465878: Fax 44 148246 6006 email: [email protected]

Abstract. The robustness issues in model-based fault detection and fault isolation (fault diagnosis) have received considerable attention in recent years. due to the increasing demand for safe and reliable operation of uncertain and complex dynamic systems. The ultimate goal of robustness is to provide rapid and reliable detection and isolation of system faults when the plant under control is disturbed. and when the mathematical model upon which the diagnosis is based cannot faithfully reproduce the full dynamic operation of the plant. The aim of this paper is to give the state-of-the-art in robust fault diagnosis, based principally on residual generation. Some of the key challenges and potential for future directions in the research are drawn up. Key Words. Robust fault detection; robust fault isolation: robust fault diagnosis in dynamic systems; unlmown unknown input observers; multi-objective optimisation, 'Hoc 7-l oo optimisation: stnlctured structured uncertainty; unstructured uncertainty; analytical redundancy.

1. INTRODUCTION

(1986), Walker \Valker (1983), Isermann (1984), Basseville (1988), Gertler (1988), Frank (1990), Patton and Chen (1991a) (1991a),, Patton (1991) and the books of Hirnmelblau Himmelblau (1978), Basseville and Benveniste (1986), Patton, Frank and Clark (1989) (1989),, Brunet, Brunet~ Jaume, Labarrere M, Rault rl.., A, Verge Niki(1990), Kerestecioglu (1993), Basseville and N ikiforov (1993) and the dissertation of Chen (1995) and the references contained therein. These approaches are based upon what has come to be known as the quantitative approach to modelbased FDI. It may often be both difficult and time-consuming to develop a good mathematical model and there have been many attempts to use cruder descriptions. For example, qualitative models can be used in which the variables are quantized crudely and monotonicity variables are used. There are several approaches to qualitative modelling and qualitative reasoning for FDI as discussed by Arkin & Vachtsevanos (1990), D'ambrosio (1989), Trave-Massuyes et ai, (1990), Leitch (1993). (1993) .

The model-based approach to fault detection and isolation (FDI) in automated processes has received considerable attention. during the last two decades, both in a research context and also in the domain of application studies on real processes. processes. There are a great variety of methods in the literature, based on the use of mathematical models of the process being monitored for faults. faults . The procedures of using model information to signals, to be compared with generate additional signals, the original measured quantities are known as analytical redundancy approaches, which can be used to obviate the repetition of hardware in the alternative approach known as hardware redundancy. Analytical redundancy is a form of dissimilar redundancy, in the following sense. Given one measurement from the dynamical system under control, a mathematical model of the system is used to generate estimates of other measurements, ments , thereby developing redundancy in an analytical form.

On the other hand, quantitative modelling approaches to FDI use a dynamical model of the system being monitored. The model is used to generate estimates of measured and unmeasured system . Estimates of measured variables of the system. variables are compared \vith with the actual measurements, thereby generating error signals known as residuals.

Quantitative models used for FDI are usually linear, domains~, ear , using either the time or frequency domains however non-linear methods have also been described based upon the known non-linear difsystem. The ferential equations of a physical system. interested reader is referred to a number of survey papers by Willsky (1976), (lg76)~ Himmelblau 45

is, a faithful replica of the plant dynamics. This is~ of course, not possible in practice, as an accurate and complete mathematical description of a process is never available. Sometimes the mathematical structure of the dynamical system is not fully applications, the parameters of known. For other applications~ the system may not be fully kno\vn known or may only be kno\vn t 's operknown over a limited range of the plan plant's ation. ation . There is therefore ahvays always a "model-reality mismatch" between the plant dynamics and the FD1. ..Another model used for FDI. -\.nother idealising assumption accompanying the quantitative model-based approach to FDI is that the true characteristic (e .g. spectral shape, Yariance~ variance, mean, stationarity, (e.g. etc) of disturbance and noise signals acting upon the system are known. This can also never be true in practice.

The most common approach to quantitative model-based FDI is the use of state estimation observers or Kalman filters, which make direct use of a state-space model of the system being moniare , ho\vever however a number of other imtored. There are, portant methods, which \vhich \vhilst whilst based upon the use of a quantitative mathematical model of the system. do not use the feedback structure of a state observer. Interesting applications of quantitative methods for diagnosis are described by Duyar & Merrill Miliar et al (1993) describe an applica(1990). MilIar tion study st udy for total process surveillance using FDI. FD 1. Patton, Frank and Clark (1989) provides brief details of earlier application studies and Brunet et al (1990) describe application studies in the aerospace field.

..\s As the complexity of a dynamical system increases, the harder the task of modelling the syswe can speak of tem and its disturbances and "Ne an uncertain system, for which there is an uncertainty of knowledge of the system's structure, parameters and effect of disturbances. There are therefore robustness problems in FDI with respect to uncertainty and disturbances (Patton & Chen, 1992a). Whilst, these can be considered as ·sepasseparate problems, it is customary to lump all uncertain effects together as far as is possible and we thus speak of a general robustness problem in FDI, FDI , in which the goal is to discriminate between the effects of faults and the effects of uncertain signals and perturbations in the system. The uncertainties can of course be quite normal effects, arising for example, as a consequence of a nonlinear system.

In this quantitative framework for diagnosis, faults are detected by setting a (fixed or variable) A number of threshold on each residual signal. .A. residuals can be designed, each having special sensitivity to individual faults occurring in different locations in the system. The subsequent analysis of each residual, once a threshold is exceeded, exceeded , isolation . then leads to fault isolation.

Decision making Decision • decision function generation fault d~cision d~cision • filult logic

Faults

The robustness problem in FDI is thus defined as detect.ability and isolathe maximisation of the detect-ability bility of faults together with the minimisation of the effect of uncertainty and disturbance on the FDI procedure. The optimisation problems can be achieved using sensitivity theory, as long as due care has been paid to the robustness of the global system operation.

Faults Outputs OutPuts

Inputs

!J

y.

Reference inputs

Robustness in FDI has been recognised as a professional field for a considerable time. See for example, Frank and Keller (1980), Leininger (1981), vVatanabe & vVillsky & Watanabe & Himmelblau (1982), LOll, Lou, Willsky & Verghese (1986). Some of the earlier approaches \vere reallY were reallv based on local sensitivity requirements, rat-her rather than producing truly robust solutions. However, during the past 15 years, there has been increasing activity focused on the development of truly robust methods for FDr. FD1. wlost Most of these studies focus on the problem of creating robustness in residual generat.ion. generation. Other proposals, focus on the problem of making the actual decision-making stage of FDI robust.

Fig. 1. Quantitative model-based fault diagnosis

Fig. 1 illustrates the general concept of the structure of model-based fault diagnosis system comprising two main stages of residual generation and decision making and t.he use of a knowledge-base kno\vledge-base for improving the decision-making and assisting in residual generation. The quantitative approach to model-based mod-el-based FDI is built upon a number of idealised assumptions, one of which is that the mathematical model used is

46

decision-making stage of FDI robust.

nominal model to be used in the observer, A.t:o, .A4 ul .4 u2 ,, ~B and Bu B u represent the paAeo, u 1,, ~A, .Au2 whilst rameter variations and plant plan t uncertainty, \vhilst Go and Gu are the system (process) noise distribution matrices.

Quantitative methods for FDI are either deterministic ho\vever this paper focuses ministzc or stochastic, however on the deterministic (quantitative model-based) approaches, \vith with an emphasis on the robustness problems in fault detection and fault isolation. Whilst. \Vhilst~ it has not been possible to include all the recent \vork work on FDI. the paper attempts to bring the review of the subject up to date. The paper also gives some indication of the likely directions of future research in this field.

The equations for the dominant part of the system are: ~(t) i(t)

Section provides some descriptions of system dynamics with uncertainties in the time and frequency domains. The need for robustness in FDI is demonstrated. Section provides the state-ofFDII based on structured uncerthe-art in robust FD tain ty. Section reviews revie\vs the approaches for robust tainty. FDI for systems with unstructured uncertainty. The problems of robust decision-making in FDI are discussed in Section . Section provides some pointers as to the likely developments in the field and stresses the topics which are probably worthy of the most research activity in the future.

I

J

L\B

A~o I

Go], [

:'~i!)]

(3) (3)

f.I.(t)

From this simple. but nevertheless comprehensive, comprehensive~ model of uncertainty, \ve we can see that all uncertainty tain ty effects can be considered as additive signals when using the above time domain structure. \vhen

If we no\v now consider that the system response IS corrupted by the fault vector f(t), Ht), then Eqs. (2) & (3) can be re-written as:

i(t) A~(t) B~(t) + R1IJt) E1s!.(t) (4) A~(t) + BJf(t) Rll.(t) + ElQ(t) 1t( 1t(t) t) = = CC~(t) ~( t) + DJf(t) D~ (t) + R21.(t) R 21.(t) + E 2Q(t) s!.(t) ((5) 5) There is a conflict between the effects that the uncertainty terms EIs!. ElQ and E~d E"d and the fault terms, Rll. R11. and R21. R 21 have on the system response. Fig. 2 illustrates the system structure.

All modelling uncertainties can be described as disturbance terms acting on the dynamic model. The uncertain dynamic effects can be lumped together into one disturbance vector with a distribution matrix, both of which can be difficult to determine. The structure of an uncertain dynamic system can thus be expressed (in state space form) as:

=

JL\A

~ .4 1 (t)

2.1. Time domain model with uncertainty and faults

=

+ BJf(t) B~(t)

El

2. SYSTEM MODEL AND ROBUST FDI ISSUES

~c(t) ~At)

A~(t) A~(t)

d (t)

f(t)

d (t)

[t ] [ A+~A A +.6.A A AUl U1

A~o ko ] [

A u2 Au2

x(t) ]

~u (t) ~u(t)

Jf(t) + [ + [ B ~~B ] J£(t)

g:g~ ]

t:(t) 1:(t)

(1)

--------.4 D t---------

N where Xc E IR IRN is the state vector(N the real order of the plant dynamics), X E IR rn.nn is a partial state vector corresponding to dominant part m outof the system, u E IRrr input vector, y E IRm put vector, J.Ljj E IRq noise vector. The perturbed measuremen t equation takes the form:

Fig. 2. Dynamic system with disturbances and faults

2.2. Frequency domain model with uncertainty and faults

(2) In the frequency domain, the uncertainty picture is by no means so clear, unless the uncertainty distribution matrices El and E E22 are kno\vn known and constant. in \vhich which case the additive time tilne domain

where E Emd. m d2 is a convenient notation for the uncertain term Cu~u Cu~u and for measurement noises. .A.! B C and D matrices form the linearised A, B, 1

47

uncertainty transforms into additive transfer function perturbations in the frequency domain. On the other hand, when there is no structure to the uncertainty (when El and E E22 are not kno\vn known or vary in time), the time domain perturbations transform in a very complex way into both additive and multiplicative uncertainty transfer terms in the frequency domain.

can be a discrete-time algorithm in non recursive or recursive form. In the case of a non-linear process, model linearization around an operating point should be considered~ although non-linear models can also considered, be used for residual generation (Frank 1993) (see Section ), according to the non-linearities of the plant.

A A. simpler and (possibly) more useful s-domain representation can be developed in terms of a sinQ( s) using the additional gle disturbance input !J.( terms .6.G ~Gu(s) Gd(S)f1(S) as follows: u(s) and Gd{S)f1(S)

1{(s) ~(s)

=

with Consider here a linear dynamical system \vith no disturbances and a number of possible faults (based on Eqs. (4) & &: (5)). The input-output relation for this system is:

.6.Gu(s)J.y(s) + Gd(s)Q(s) [Gu(s) + ~Gu(s)l!f(s) Gd(S)~(S)

+G J(s)1.( s) +GJ(s)l.(s)

(8)

(6)

The general structure for all ail deterministic residual sho,vn in Fig. 3. generators, using this concept is shown

Gd{ d( S)fl.( S)Q( s) is the s-domain transfer matrix bet,veen between Q( s) the unknown un certain ty disturbance vector !l:.( and the output vector y( s). Although Q( sf:.( s) is unfu~ction matrix Gdl (s) is asknown, the transfer function sumed known (i.e. we know how the uncertainty and disturbances act upon the system) system).. .6.G ~Gu(s) u (s) is used to describe the modelling errors which are in this case, defined as the unstructured uncertainty as usually .6.G ~Gu u (s) is unknown.

Y-(s)

U(s)

:

=--;. .

[(5) ._ +6~_r_(s_)

--:...1 Hu(S)~ 1H u(s) ~

2.3. Robust residual generator

. Residuals

Methods which depend upon the deterministic generation of residuals, are very well known in the literature (Frank, 1987, 1990, 1991, 1993; Isermann, 1984, Isermann & & Freyermuth, 1991; Patton et aI, al, 1989; Patton & Chen, 1991a, 1991b; Gertler, 1991).

Fig. 3. The general structure of a residual generator

The value of each component of the residual vector

This structure is expressed mathematically as:

Residual Generator

z:(t)((E E IR.P) IR,P) generated by means of a determinis1:(t) (9)

tic model, should be insensitive to (independent from) system inputs and outputs and therefore should satisfy the condition:

1:(t) #0 z:(t) :1=

only if

l(t) =1= :1= 0

The transfer matrices H u (s) ( s) and H y (s) are realisable using stable linear systems. In order to make the residual 1:( z:( s) become zero for the faultfree case (i.e. to achieve requirements in Eq. (7)), Hu(s) and Hy(s) must satisfy the condition:

(7)

Eq. (7) is the basis for fault detection using model-based residuals. From a practical point of view it is reasonable not to make further assumptions about each fault vector except that they are unknown time functions.

(10)

Eq. (9) is a generalised representation of all linear residual generators. The design of the residual generator results simply in the choice of the transfer function matrices Hu(s) and Hy(s), \vhich which ust satisfy Eq. (10). The various \vays m must ways of generating residuals correspond to different parameterisations of H u (s) and H'J (s). One can obtain Hu(s) Hy(s). different residual generators using different. forms for H u (s) and H y (s). Using the design freedom. the desired performance of the resid ual can be residual u (s) and H y(s). achieved by suitable selection of H Hu(s) Hy(s).

We no'v now give a general definition of the residual 'vVe signal (vector) in FDJ FDI as a linear (or non-linear) function of inputs and outputs of the monitored system 1uhich which is also independent of the normal operating state of the system. In the fault-free case, the residual is zero or near to zero (very small in some sense). When a fault occurs in the system, the residual will \vill grow significantly. This processor can be worked out in continuous time or

48

FDI is achievable. achievable .

If we substitute Eq. (6) into the residual generator Eq. (9), is : (9) , the s-domain residual vector is:

residual . Both faults and uncertainty affect the residual~ and discrimination bet,veen between these effects is difficult. The main task in the design of a robust which remain FDI system is to generate residuals \vhich sensitive to the specified fault(s), \vhilst whilst remaining insensitive to uncertainties including the rest 1991 , of faults faults~, and therefore robust (Frank, 1990, 1991, 1993; Gertler, 1991, Gertler & Kunwer, 1995; Patton & Chen, 1991a, 1991c, 1992a, 1992b, 1992b , 1993, Patton, Frank & & Clark, 1989). The robustness is of course only proved if the residual of interest remains insensitive to uncertainty over the whole range of operation of the system being monitored.

Generalised approaches for obtaining the strucFDI residuals have ture of uncertainty acting upon FDr However, Patton & & Chen seldom been considered. However. (1991a, 1993), 1993) , Patton~ Patton , Chen & Zhang (1992) and Patton, Zhang & & Chen (1992) have shown (using different approaches approaches)) that approximations to the structure of uncertainty can be obtained using identification or optimisation techniques. A. A similar study has recently been carried out by Gertler & Kun\ver & Kunwer (1995) \vith with application to the generation of robust residuals in the parity space. Their results are identical to those due to Patton et al as it can be shown that given a parity space residual, ual , the same residual can be derived using observer theory (this is part of the correspondence bet,veen between the parity space and observer approaches to FDI).

If the uncertainty (disturbance) transfer matrix Gd(S) is completely known and the residual generator has been designed to satisfy:

3. ROBUST FDI FOR STRUCTURED UNCERTAINTY

r:(S) r(s)

=

Hy(s)GJ(s)l(s) Hy(s)Gj(s)f.(s) + H Hy(s)Gd(s)g(s) y(S)Gd(S)s1(s) +Hy(s)LlGu(s)jl(s) (11) +Hy(s)L::..Gu(s):y(s)

'rVhen When the distribution matrices El, £2 E2 are comde coupling designs can pletely known, disturbance decoupling be achieved using most of the well known methods of FDI, e.g. the failure detection filter, the parity space, space, the unknown input observer (Watanabe & Himmelblau, 1982; Phatak & & Viswanadham, 1988; Chen &: & Zhang, 1991; Ge & & Fang, & Wiinnenberg 1989) or alternatively, 1988; Frank & eigenstructure assignment approaches (Patton et al 1988, 1989, 1991c) and Magni & Mouyon (1991). Chen & Patton (1995) considered robust fault diagnosis in stochastic systems using the unknown input filter.. filter. . kno'wn

(12) then the residual vector r(t) r:(t) will be independent of g(t). all disturbance and uncertainty signals s1( t). The residual is then robust to the structured uncertainty. This is the principle of disturbance decoupling, when the structure of the uncertainty is known.

If perfect disturbance decoupling is impossible, one can design residuals: (a) Maximising fault detectability, (b) Maximising fault isolability, isolability, & (c) Minimising sensitivity to modelling errors & disturbances.

vVe can now review some of the mechanisms in'rVe volved when these methods are applied to the solution of the robustness problem.

do . However, we This is perhaps the best we can do. may not actually produce robustness! This means that a good design method should provide some key robustness indices of the designed FDI process. Even so, simulation work is still needed before applying the FDI process in real applications.

3.1. The unknown input observer

vVatanabe and Himmelblau (1982), Ge & Fang Watanabe (1988) and Phatak and Viswanadham (1988) developed some FDI observers based on the unknown input observer principle of Kudva et al (1980). (1980) . According A.ccording to Frank (1990, (1990 , 1993), the unkno\vn known input observer can be derived through the generalised Luenberger observer whose structure is sho\vn shown in Fig. 4 and mathematical description is given by

2.4. A determination of approximate structure for unstructured uncertainty

'rVhen vVhen the uncertainty has no exact structure, an optimal structure can sometimes be derived by approximation : considering the following approximation: (13)

jQ( t) li!(t) r(t) r(t )

\vhere Q (s)) is an unkno\vn Qu unknown vector and Gdu (s) where u (s is a known transfer function matri..x. matrix. If an approximate structure is used to design disturbance decoupling residual generators, generators , a suitably robust

=

F1Q(t) FYd.(t) + 1{~(t) K¥.(t) + J]J.(t) J:y(t) L J!.(t) + L12£(t) 4- L L3Jd(t) 1JQ(t) + L 2¥.(t) 3 y'(t)

(14)

(15 )

\vhere lQ is the estimated vector of a linear comwhere 1Q bination of the state variables given by the trans-

49

ing a transformation of the state variable system into Kronecker Canonical Canonlcal form, \vhich which leads to a block-diagonal state transition matrix.

f(t)

and 1: satisfy the If these conditions are satisfied satisfied~, ~ and! state equations:

R2 - T Rdl(t) + [J{ [I{ R:. L2R2J.(t) Ll~ + L:.R:.l.(t) F~

rank formation 3Q T~ (the subscripts (t) have been J£ = T:£ omitted, for brevity). F is a matrix with eigenvalues in the open left-half of the s-plane and K, L22 are appropriately dimensioned maJ, L 1I and L trices.

the observability of the pair (F, L Lt} 1)

F~+[FT-TA+I{C]~ F~ + [FT - TA + KCk

A residual vector r can then be generated from A the generalised observer above as: L1~ Llf.

+

+ (LIT + L2C)~ + (L (L33 + LL2D)M 2 D)]d

!..

L2R21 L2E2Q L 2 E 2sl. 2 R2 + L

J !{E KE22 -- TE TEI1 LIT + L L2C 2C L + L L33 L2D 2D L L2E2 2E 2

= = = =

I{C KC TB - !{D KD 0 0 a

=

0 a 0 a

(27)

3.2. Fault diagnosis in bilinear systems

(17)

The goal of the unknown input observer is to force indeeach of the state estimation errors to become inde.. pendent of the uncertainty. Once, the estimation error vector f.~ is de-coupled from the uncertainty, it then follows that the residual will also be decoupled from the uncertainty. To achieve this goal it is necessary to satisfy the following decoupling 1993) : conditions (Frank, 1990, 1993):

TA-FT

(26) (26)

is a further requirement, or at least in the ob(F, L 1l ) no cancellations of servable subspace of (F, different components of the fault signals may occur. This means that we need to ensure that the be sensitive to the fault vecresidualvector !1: will be. tor f, f , noting that observability and sensitivity are Vukobra&: \Tukobrasynonomous in this context (Tomovic & tovic, 1972).

+ [J - TB + I{Dly KD]M + [I{R [KR22 -- TRdl TRill + [I{ (16) [K E E22 -- T E EdQ 11!

!1:

[-aT i~] [ ~~ ] =rank [ ~~ ]

To ensure the detectability of faults,

The estimation error ~ (= J£ Y1. - T:£) T~) is given by:

=

(25) (25 )

now show that These equations no\V that~, for this case of structured uncertainty, uncertainty, the residual vector remains unaffected by the unknown inputs Q, s!:.: the state ~, and the input]d. input M. In order to avoid cancellations of different components of the fault signals, the following rank conditions must be satisfied, so will be correctly reflected in the that the faults \vill residual vector:

Fig. 4. The general structure of an observer for FDI

§. ~

(24)

Bilinear systems are a special class of non-linear systems having the form (discrete time case, Yu & Shields 1994):

:£(k+1) ~(k + 1)

Ao:£(k) Ao~(k)

BM(k) + B1J.(k)

1 I

+

I: A;1!;(k):£(k) l: Ai3!i(k)~(k) ;=1 i=l

Rll + EIs!:. Elf1 + R1f.

(18) (19) (20) (21 (21)) (22) (23)

]t(k) J..{k)

=

C~(k) C:£(k)

(28) (29)

The terms in the inputs, faults and disturbances of the measurement equation are assunled assumed to be zero. This does not affect the essential features of the problem. Yu & Shields (1994) applied the bilinear unkno,yn unknown input observer (Hac 1992) to the FDI in bilinear systems. systems. They proposed a bilinear fault. fault detection observer (BFDO)

\vhere where T is an unknown matrix to be designed. Using the discrete-time equivalent of the generalised Luenberger observer, Frank & Wiinnenberg (1989) proceed to solve for the matrix T by us-

1£(1. + 1)

50

=

FJ£(k)

+ KJl.(k) + J1!(k)

h

I{i1li(k)~(k) L Kiy.;(k)JL(k)

where H (i::yJ is a time-varying matri..x H(i.,lf) matrix determined to ensure asymptotic stability of the observer and residual solutions and i. is the correF! t) t ) is the J acosponding state estimate vector. F( bian matrix given by:

(30)

1=1 i=1

r(k)

=

LU!2.(k)

+ L 2JL(k)

(31)

To design a BFDO. it is required to solve the equawith tions (18-21) together \vith

TAi - KiG

= 0,

An explicit solution A.n (1994).

i=l.··,h IS 1S

(32) Defining the residual as rI:. f. .Q yields: f = 12 yields :

given by Yu & Shields gIven

rI:. around

~1any ~vlany systems in applications, such as hydraulic drive systems, gas-burning furnace systems and heat exchange systems, can be modelled as bilinear systems (Shields & Yu 1995). The above method offers a tool for handling FD FDrI in such nonlinear systems.

=

=Y -

~j and linearizing Y

-( 37) (37)

It should be noted that this is a very complex approach to the residual generation problem and may not be useful in other than very simple cases However, it remains an interestin terestof non-linearity. Ho\\"ever, development.. ing principle which \vhich may find further development One of the most challenging aspects is t.he the determarrLx H (i: (i, 1!) If), as mination of the time-varying matri..x there is still no systematic approach to this problem (Frank, 1993).

.A A. recent st udy (Daley et ai, 1995) sho\vs study shows that the known BFDO approach failed to give a solution for a real bilinear system, an induction motor, and a fault detection observer is indeed obtained in work. Since the BFDO conditions are known that \vork. as necessary and sufficient, sufficient , one may ask what goes wrong? We notice that the condition (32) is necessary only when the independence of the dynamics of a BFDO from the bilinear term in (28) is required. If we consider that the stability and bilinear de-coupling problems of the BFDO are independent issues, then we can design a BFDO under weaker conditions. This suggests that residual generation, even for this special non-linear case needs further investigation.

3.3. Non-linear residual generation

_.'=--=-=.. _=-==---=-=-=_.====---_._-

In keeping with the possibility of decQupling decoupling disturbance and uncertainty (when the structure of these are known), Frank and co-workers (Frank, 1991, 1991 , 1993, Seliger & Frank, 1991) have developed an extension to the unknown input observer with a non-linear residual (see Fig. 5), based on the known non-linear structure of the system being monitored. They consider the system in nonlinear form:

y

[(~):yJ + R1[(t) [(l:.,yJ RI[(t) f(~)1!) fC~,lf) + R R2l(t) 21(t)

NONUNEAR OBSERVER

Fig. 5. Non-linear observer configuration for residual generation (Frank. Innl) l!)!ll)

3.4. Unknown input observer with nonlinear decQupling decoupling

(33)

Frank (1993) described an extension to the unkno,,~n input observer by Wi.innenberg \Viinnenberg (1990) who \vho known sho\ved showed that a class of non-linear systems can be characterised by:

(34)

\vhere f(., .) and c(., where c( ., .) are non-linear functions of the state ~ and the inputs~. inputs If. They then proceed to compute the locally linearised (time-varying) residual using an identity observer and based on the system non-linearities. The estimation error is then given by:

~ Rll + B(JL,JJ.) B(~lJJJ + E1g E1Q":':R1l G3: E 2 !i. -+-:- R?l R2f + -:- D}l. Dy C~ -:~ E?si

.-1.3: .-i~

(38)

(39)

As B(J!.' B(ll,lf) .-\s ld) depends only upon measured signals, this yery \'ery special non-linearitv non-linearity can be compensated clecoupling . The resuiting resulting unfor by non-linear decoupling.

(35)

51

known input observer equations are:

_ = !!: = =

F;. + J(Jt.'~) J(Jt,.Y) F=. Ll!.. + Lll.

+ !(Jt KJt.

L 2M Jt.

The eigenstructure assignment ap3.6. proach

((40) 40) (41 )

The unknown input observer is a robust observer DI in which both the state estimation error and F FDI residuals become insensitive to uncertainty. For FDI purposes, purposes , it is not necessary to ensure that the state estimates have this insensitivity. Follo\vFollowcQ-\vorkers ing this line of reasoning Patton and co-workers (1988, 1989, 1991, 1992) have studied the use of eigenstructure assignment for linear observer design, \vith with the FDI problem as the principle goal. Eigenstructure assignment is a very powerful approach to the design of linear state-space feedback systems, as it can be shown very easily that the closed-loop system structure depends entirely upon (a) the system eigenvalues and (b) the system's left and right eigenvectors.

.A.s As it is assumed that the uncertainty is structured through El and E 2J 2 , perfect decoupling of the uncertainty can be achieved using the unknown input observer constraint Eqs. (18)-(23). The only difference is that the constraint (18) must be replaced by: (42) (42) The class of systems having the structure of Eq. restrictive, although Seliger and Frank (42) is very restrictive~ (1991) have shown that some non-linear systems can be transformed under non-linear transformation to yield the form of Eq. (42). The derivation of the required transformations is exceedingly complex in all but simple cases.

Consider an identity observer in form: form :

~(t) i(t) :Q(t) i(t)

Seliger and Frank (1993) extended the above approach to non-linear systems of the more general form: ~ .?<

= A(~) B(~)'y + El El(~)s! Rl(~)l. A(.?<) + B(.?<)~ (.?<)Q + RI (.?<)1. (43)

Jt Jt.

=

C(~) C(.?<)

+ E2Q E 2sl. + R21. R21 + Dy; D~

= Aci.(t) M(t) Aci(t) + (B - !( K D)J!;(t) D)~(t) + !( KJt.(t) = = Ci(t)+D~(t) C~(t) + D:g(t)

(48) (49)

where i:( i(t) t) E IR rn.nn is the state estimate vector and m m y(t) E IR rn. is the output estimation vector and Acc = A - !( KC. A C. In this case the state estimation error vector is ~(t) ~(t) - i(t). The error dynamics are (assuming DJ!; D~ term cancel):

=

(44)

Their derivation is based on the existence of a nonlinear transformation transformation;., T(~).. The unknown =- = T(~) input observer conditions are extended via T(.?<) T(~) (Frank, 1993). The FDI residual is then a non(Frank, R{T(~),, C(~)} C(~)} only of.?<, of~, such that linear function R{T(.?<) in the absence of faults:

=

=

1:

= R{T(~), C(~)} =Q

The residual vector 1:( t) E !RP .r:(t) IR? is defined as:

(51 ) \vhere matrix. designed weighting matrix where W is a suitable \veighting to give good isolability properties and insensitivity to uncertainty.

(45) (45)

Frank (1993) discusses ho\v how a bank of such observers can be specially designed for fault isolation purposes.

From Eqs. (50) & & (52), the complete response of the residual vector is:

3.5. FDI in general non-linear systems

Zhirabok and co-workers (see Zhirabok & Preobrazhenskaya (1994) and references therein) focues on the investigation of FDI problems for non-linear systems fl.(~(t),~(t) Q(t),D g (~( t ), .Y( t ),, sl( t ), 1)

~(t) ~ (t )

~ (t ) Jt.(t)

= (~( t ) ) = 11ll(~(t))

.r(s) .r:(s)

WR WR2f.(S) 2[(s)

+ WC(sJ WG(sI - .Ac)-l(RI 4 c )-1(R 1 -- !{R )[(s) KR 2 2 )1.(s) 4 c )-1 EIQ(S) E1sl(s) + WC(sI WC(sJ - .Ac)-l + [W - WC(sJ ~VC(sI - Ac)-llE~d(s) A c )-l]E2d(s) (53)

(46) (46) ((47) 47)

Clearly, the residual responds to both faults and uncertainty. In order to minimise the false alarm rate resulting from the uncertainty, the residuals must be de-coupled from the disturbances and uncertainty.

In contrast to the system of Eq. (4),1. (4), f is assumed to be a parameter vector which equals equaG some nominal value _0 f when \vhen no faults occur. -0

..\.lthough Although the result is far from applicable in practice due to a complex design procedure and extremely restrictive conditions, conditions , the effort to deal FDI ·problem is ap\vith the difficult non-linear FDIproblem with preciated.

The disturbance decoupling requires that in the transfer function matrix between nulled,, i.e. bances and the residual be nulled

52

entries distur-

then: H12.£ HJLi

(54) Patton (1988) noted that this problem is a special output-zeromg problem \vhich which is \vell well case of the output-zeroing kno\vn known in multivariable control theory (Karcanias &: Kouvaritakis, 1979). Alternatively, the resid& ual vector will have zero-sensitivity to uncertainty \v hen Eq. (54) is satisfied (see brief discussion at when at the en endd of Section Section..

H(sI - A,:)-l Ao)-l E == H

= WC(sI -

,.q-I El

=0

=

The observer eigenstructure assignment problem can be handled by means of a transformation of the dual control form. On assignment of the right eigenvectors to the dual control problem these eigenvectors become the left eigenvectors of the observer system. The assignability condition is f3i, the corresponding left that, for each eigenvalue 13i' row subspace eigenvector 1T must belong to the ro\v {C(f3iII -_A)-I}. spanned by: {C(j3i A)-l}. If the left eigenvector assignability condition is not satisfied, a suitable choice of right eigenvectors can be assigned as columns of the matrix El. El '

iT

Lemma 2: The resolvent matrix can be expanded in terms of the eigenstructure as:

=

columns of Theorem 2: If WC El 0 and all coluI1Uls the matrix El are right eigenvect.ors of Ac corresponding to eigenvalues of A,:, A o , Eq. (55) is satisfied.

(56)

iT

where 3Li JLi and 1T are the right and left eigenvectors \vhere of A Acc respectively, corresponding to the eigenvalue f3i.

If the desired observer eigenstructure is assignable (using the left or right eigenvectors) eigenvectors), perfect decoupling can be achieved. On the other hand, if the required eigenstructure is not perfectly assignable, the eigenvectors must be chosen to be close in a least-squares sense, sense , to the desired eigenvectors. In this situation~ situation. the residuals also have low sensitivity to uncertainties due to approximate decoupling. decoupling . I

=

Theorem 1: If WC yVCE 0 and all rows of the Ell matrix H = WC are left eigenvectors of A Acc corresponding to p eigenvalues of A CI then Eq. (55) is , c satisfied. Proof Let all ro\vs vVC are p rows of the matrix H = WC (1[, i = 1~L 2, ... ,p) left eigenvectors (1:, p) of A c , i.e. I

iTLT El = = O~ 0, for i = =

=

iT

=

p

(1)) Compute the residual weighting weIghting matrix vV W so (1 El1 = O. O. that WC vVCE obseruer: (2) Determine the eigenstructure of the observer: The observer eigenvalues are chosen according to the desired dynamic property of residuaIs. I-VC must be p left WC uals. The rows of H eigenvectors of the observer, the remaining (n - p) eigenvectors can be chosen freely. /{: .Assign (3) Compute the gain matrix !{: .\.ssign the required eigenstructure of the observer using an assignment algorithm.

A given left eigenvector 1T (correLemma 1: .A. f3i) of A Acc is always orsponding to the eigenvalue 13i) JLj correspondthogonal to the right eigenvectors Jlj f3j of A c , ing to the remaining (n-1) eigenvalues j3j where Pi f3i :/; f3j. \vhere =/; j3j.

=

i.e.~, i.e.

SS --

\\then \Vhen the residual is completely de-coupled from the disturbances (unkno\vn (unknown inputs), robust fault detection is achievable. The eigenstructure assignment approach can be summarised as:

The remaining problem is to choose the matrices !{ yV to satisfy Eq. (55), f{ and W (55) , in addition to FDII perforchoosing eigenvalues to optimise the FD mance. A.s As the observer (and hence the residual) mance. is a linear time-invariant multivariable system, it can be uniquely defined from a feedback point of view using the observer's eigenstructure. The asvie\v signment of the observer's eigenvectors and eigenvalues is a direct \vay way to solve the required FDI design problem. Two Lemmas show how the relationship between the eigenvectors and eigenvalues (the eigenstructure) of a system can be used as follows:

I

l'

sS -- 3 1

Thus: Gra(S) = VVC(sI - ,'.q-1E l = O.

(55)

IT , IT v iT v iT _n_n -_ -1-1 -n-n A c )) -1 ---r ... + ;- e -(( s II - A /31 f3n s - (31 SS - 13n

[T ] -p-~. 1El l.iJ 3p .3

V IT ~ + ... + [.Y.llf [

= H El = = O.

But ~l!C Ell lVCE L2 , .. ',p . 1.2~···~p.

When it can be assumed that the term E'2sl. E2Q is very \Vhen small (as is often the case, in practice), it is then necessary to determine a matrix W such that: Grd(S)

= 0 for i = pp++ 1... n. and L·· "·,n.

I

Hence, for a given matrix El, the disturbance decoupling problem can be solved if] ifJ a matrix matrix: H can be found which will also satisfy the follo\ving following rank condition: rank (H)

H=

:5 :s

n - rank (El)

(57)

A.s As far as the design of robust residuals is concerned, the unknown input observer and the eigen-

53

sensitivsolutions which are incompatible. The senSItIVity and robustness properties are illustrated in In a sensitivity robustness plane diagram.

structure assignment FDI observer are formally equivalent and use different mathematical tools to achieve the same goal in robustness (Gertler 1991) .

Kinnaert (1993) designed redundancy relations, based on the parity relations and structured uncertainty, using constrained stochastic optimisation. He optimises a quadratic cost function under non-convex quadratic inequality constraints. The constraints express the desired performance of the FDI system both for fault detection and isolation purposes. He shows that robustness of with respect to structured uncerthe FDI system ,vith tainties in the model parameters can be achieved with a fiby representing the process behaviour ,vith &; Zhang (1993) nite set of models. Chen, Chen~ Patton & tackled this problem based on a multi-objective optimisation procedure.

3.7. 3. i. Robustness in failure detection filters

White & &; Speyer (1987) reformulated the FDF as vVhite assignmentt problem and this an eigenstructure assignmen greatly simp lifted lified the design approach (compared with original approach due to Beard (1971) and Jones (1973)), although Patton (1988. 1989) had already shown that fault-sensitive observer residuals could be designed using eigenstructure assignment, with a special interest in robustness to uncertainty. 'When \Vhen some of the degrees of design freedom are used to provide good fault isolation properties, there are fewer degrees of freedom available for robust uncertainty de-coupling or minimisation of uncertainty. To a large extent, now supertherefore, the failure detection filter is no'v seded by the robust observer approaches used for FDI. FDL There has, however, been a fairly recent contribution to robustness design of the FDF by Olin and Rizzoni (1991), based on a frequency domain approach (see Section ). Another recent ,vork work by &; Rizzoni (1994) provides some explicit soPark & lutions to the FDF with free parameters as design freedom which can further be used to improve roFD F. bustness of the FDF.

Krishnasswami Krishnasswarni &: Rizzoni (1994) have suggested some new ideas for generating non-linear parity equations for residual generation dealing \vith with both fault detection and isolation. They consider the solution to the robustness problem for a class of non-linear systems having inverse models.

3.9. Frequency domain 11. Hco 00 approaches

When the uncertainty is structured according to Eqs. (4) & &: (5), it is perfectly feasible to attempt a maximisation of the sensitivity of the residual to faults, whilst minimising the sensitivity to the uncertainty i.e. as a mini-max problem. One suitable choice of performance index in this optimisation problem can be defined in the frequency domain as an unconstrained optimisation problem &: Frank (1989): as shown by Ding &

3.8. Parity relation methods

There have been some studies on the use of robust parity relation residuals for structured uncertainty (also valid for the unstructured case) (Lou et ai, aI, 1986; Wiinnenberg, 1990; Kinnaert, aI, 1993; Chen, Patton & &; 1993; Staroswiecki et ai, Zhang, 1993; Gertler et ai, aI, 1991, 1995). There seems to be a considerable growth in interest in al (1990, 1991, 1995) this approach. Gertler et ai used orthogonal parity relations for disturbance de-coupling design. Lou et al (1986) first studied this problem using a rank reduction method from a multiple-model approach to uncertainty. Wiinnenberg (1990) set up a performance index, based on parity relation residual generation, including all weighting matrices arising from known uncertainty distributions and faults.

=

y (jw)G d (jw)1I J = IIH IIHy(jw)Gd(jW)11 IIH IIHyy (jw)G f (jw)11

(58)

By minimising the performance index J over a specified frequency range, an approximate decoupling design can be achieved (Frank &: vViinnenberg Wiinnenberg 1989, Frank 1991). The perfect (or approximate) disturbance decoupling can also be designed using frequency domain techniques, Hco &: Frank e.g. 1i. oo optimisation (Frank 1991; Ding & 1989, 1990, 1991; Qui & &: Gertler 1993; Frank & &: Ding 1994).

Staroswiecki et al (1993) considered an unconstrained approach to multi-objective residual optimisation. They define the sensitivity of the residual to faults in terms of pulse and step sensitivities. They also define a robustness measure and aggregate these sensitivity and robustness measures together in the optimisation procedure. A particularly interesting feature of their ,vork work is the description of the Pareto P areto optimal set defining the

These approaches are based on the disturbance decoupling problem expressed in the frequency domain. In fact, most of the frequency domain approaches to robust FDI focus on the optimisation problem when the uncert.ainty uncertainty is unstructured. vVe ret.urn to this We therefore return t his topic again in Section to give an outline of the developments in frequency domain approaches. based upon sensisensItivity optimisation.

54

3.10. Time TiIne domain 1-l 71.00 00 approaches

Given t\VO two scalars 13 ;3 > Ii' > 0, observer (59-60) is called an 1-l_ /1-l 00 fault diagnosis observer if the rL /71.00 following three conditions hold:

?-loo approaches used in FDI have been deThe 71.00 veloped overwhelmingly in the frequency domain~ domain . see Section . Edelmayer et al (1994) used an 7-l 71.00 00 observer for the purpose of robust fault detection. Chen et al (1994) proposed a multi-objective approach for robust FDI (see Section ). The method can be considered a mix of the frequency and time domain approaches.

2.

IIGrd(jW)ll IIGrd(jw)lI oo oo < iI

(64)

3.

IIGrj(jw)L > ;3 IIGrj(jw)ll_ f3

(65)

=.A.4 - !{I\ C and = = = C(s 1- ]{ R'!.) R2 ) + R R2 = I - Ac)-I(RI Ac)-1(R -!(

Grd(s) = C(s 1-]{ E'2) E 2 ) + E'2 E2 Grd(S) I - .o4<)-I(E 4~)-1(EII -!(

(66)

Grj(s)

(67)

1

2

7-l_ /'H Theorem 3: An 71._ /71. 00 oo fault diagnosis observer exists if and only if there exists Y yT > 0 (L.MIs) solving the linear matrix inequalities (L?\'IIs)

=

[G(jw)]

w

CT [ Er

and

IIGII_ IIGII-

:= inf O"rnin O"min

[G(jw)]

M(t) £(t) - D 3!(t) ]!(t) J!.(t) - C i(t)

r(t)

LMIs are simultaneously solvWhen the above LMls /71.co fault diagable, the gain matrix of the 71._ H_ /HC'O nosis observer is given by !{ ]{ YeT. Y CT .

(59) (60)

=

In the above, the notation A 1. represents, assumA.L mxn is not of full ro\v ing that A E IRmxn row rank, an arbitrary matrix satisfying A.l. 1. ..A.L 4.1. T > 0 A.L A = 0, A A.l. and rankA.l. rank A.L + rank A = m.

the estimation error dynamics are obtained as: ~(t)

= +

<0

[ ATy + Y A +<7T C [ er].l. RI RTy + RIc

Using an identity observer of the form

= j!(t) + !( r(t) = A~(t) Ai(t) + B 3!(t) +]{

]l.T

and

w

where O"rnax O"max and 0" min represents the maximum and minimum singular values, respectively. It is worth noting that the use of IIGI'IIGIl_ alone does not make sense as this will be zero-valued.

~(t) i(t)

(63)

The meaning of conditions (63) and (64) is wellknown . They correspond to the requirements for known. 71.00 we 11. 1990). While, \Vhile~ as \ve 00 estimation (Shaked, kno\\y, know, condition (64) represents the \vorst-case worst-case criterion for the effect of disturbances on the residual r, condition (65) stands for the worst-case \vorst-case criterion for the sensitivity of r to faults. Clearly, these two criteria capture the most significant features of fault diagnosis observers.

71.00 71._ norms of a stable transfer function 11. 00 and 11.._ G are defined respectively by := sup O"max

~4,= is asymptotically stable .1<

\vith 4c with ...1c

A recent work of Hou & & Patton (1995) shows that criterion form (58) does not coma pure 11. 71.00 in 00 pletely coincide with the real requirements of a robust FDI observer. Besides minimising the largest disturbance effect on the residual, the criterion leads to maximise the largest fault effect on the detection signal, however the latter does not correspond to the correct sensitivity requirement for FDl. It is our hypothesis that by maximising the FDI. smallest fault effect, the true sensitivity with \vith respect to faults is achievable. This means that, 71. 00 71._ /11. /71. 00 instead of only 11. 00 00 ,, a mixed criterion 11._ should be addressed.

IIGll oo IIGlloo

1.

=

(A - ]{C)§.(t) KC)§.(t) + (RI - ]{R KR 2 ) let) l(t) (El -]{E (61) - KEz)s1(t) 2 )!l.(t)

C §.(t) + R (62) = C§.(t) R21(t) Ezs1(t) 2 Q(t) 2 1.(t) + E where §. ~ = =~ -- i.i.. The design of a robust fault de-

By using the LMI approach the resulting computational optimisation optirnisation problem of the robust FDI observer design is convex. Numerically effective algorithms for convex LMls L~IIs exist (Beck, 1991; Geromel, et a11991) and a ne\vly newly developed !vIATMATLAB toolbox is available (Gahinet~ a/ 1994). (Gahinet, et aI1994).

r(t)

tection observer, i.e. determination of the observer gain matrix L, involves three objectives: (a) stabilising the observer, observer , (b) reducing the effects of disturbances to the residual and (c) increasing the effects of faults on the residual. This can be characterised as a mixed 71._ 7-l_ /'H. /71.00 estimat.ion problem. oo estimation We call the corresponding estimator an 11._ 71._ /71.00 vVe /11. 00 fault diagnosis observer \vhich which is defined as:

55

disturbances over the practical plant operation. The optimal matrix is near all matrices "as close as possible" .

4. ROBUST FDI FOR UNSTRUCTURED UNCERTAINTY

In order to design robust FDI schemes, we need the description of uncertainties acting upon the system, system~ during typical process plant operations. Furthermore, it is necessary to find a description of these uncertainties \vhich which can be handled in a straightfor\vard \Vhilst straightforward and systematic manner. Whilst all system uncertainties can be summarised as unknown system~ kno\vn inputs (disturbances) acting on the system, their effect can be considered as bounded or unbounded and structured or unstructured. When an attempt is made to make the residuals robust \vith with respect to uncertainty, \ve we call this active ro&: Chen 1991a, 1992a). bustness in FDI (Patton & \Vhen, on the other hand, the robustness is foWhen, cused on the decision-making stage of FDI, this is termed a passive approach to robust FDI, for example using adaptive threshold techniques (see Section ).

As shown in Section , the modelling error (un.As certainty) can be lumped into a (generally timevarying) term Elsl(t). EIQ(t). The matrLx matrix El can be computed according to Eq. Eq . (3), although this is a very difficult problem to solve. In general. generaL the matrix El is a non-square matri.x matrix: with more columns than rows. The key principle behind achieving robustness (disturbance de-coupling) is to find a p x n matrix H to satisfy Eq. (55). If rank(E l ) ::5 ~ n - p, (55) has solutions and exact de-coupling is possible. If, however, rank(E rank(Er) 1 ) > n - p, (55) has no solutions and exact de-coupling is impossible. Here, we \ve consider more general problem in \vhich which approximate de-coupling is necessary. The procewill be to first compute a matrLx matri.x Ei Ej that is dure \vill ~ n - p, as close as possible to El, and rank(Ei) ::5 i.e. to determine the matrices Ej Ei and H so that:

:\1ethods FDII based on unstructured uncer~1ethods for FD tain ty can be derived in either the time or frequency domain.

HEj = 0 HEi

with

IIE l -- Eill} IIEl

minimized

(68) (69)

matri.x. The here II.II} denotes the Frobenius matri..x. optimisation optirnisation constraint condition (68) is equivalent to requiring that rank( rank(Ej) Ei) ~ n - p. If only p linearly independent rows of H are required, the constraint condition can be changed to rank( Ej) Ei) = n - p, so that Ej Ei is chosen to minEi contains imise (subject to the constraint that Ej only n - p linearly independent columns) the sum of squared distances between the columns of El and Ei.

4.1. Time domain approaches: Optimal dis-

turbance decoupling The most powerful approaches to robust FDI make direct use of disturbance decoupling, according to the most readily available estimates of the uncertainty, given an operating point of interest. In particular, approaches making use of the special design of observers are important. An observer used for fault diagnosis can be designed to be robust in the sense of disturbance decoupling; the robustness will ensure that the observer residual is insensitive to disturbances and modelling errors.

=

optimisation problem is easy to solve using This optirnisation Singular value Decomposition (SVD) of El as follows:

=

El = Z(diag{ 0"1, .•• ,, O"n}, Z(diag{O"l,···

For all disturbance decoupling methods, an important assumption is that the unkno\vn unknown input distribution matrices El and E E22 must be kno\vn, known, however their derivation is, apart from very simple examples, a very complex problem. The most significant of these terms is El!!. E lli. and P atton & Patton &: Chen (1991a, 1992, 1993) have proposed a number of methods for computing El, based upon the assumption that E E2Q 2 Q can be ignored.

O]r Ojr

(70)

\vhere where Z and fare rare orthogonal matrices, 0"1 ~ (J'2 0"2 ::5 ~ • . • ~ O"n (J'n are the Singular Values of El. As shown ... El which \vhich minin Lou et al (1986), the matrix Ej imises (69) is given by:

Ei

Z(diag{O,··· , O'O"P+l,···,O"n}, = Z(diag{O,···,O'O"P+l'··',O'n},

Qjr O]f

(71)

and an orthonormal solution of the matrLx matri.-.;: His:

In order to achieve the decoupling condition, an optimal matrix is used to approximate the original matrix. For a complex non-linear system, the operating poin pointt \vill will change according to t.he the inputs and outputs of the process. In general, ts correspond to different different operating poin points unknown input distribution matrices. Patton & Chen (19g1a, (1991a, 1992, 1993) propose the use of one matrix to represent the changing structure of the

(72) \vhere Z1,···, Cerwhere =1,·· · , zp Zp are first p columns of S. tainly, the matrix H in (72) and El Ej in (71) together satisfy the Eq. (68). An alternative statement of the optimisation problem is given by Patton & &: Chen (1991a~ (1991a, 1993).

56

=

:\n An ideal matrix H should make H ~~ ~\ 0 for .all all i= 1 ... , ni, \vhere 1,~ 2, 2,···, where ~\ ~il is the ith column of the always possible. Hence, it matrix El. This is not ahvays makes sense is to choose a matrix H that is "as orthogonal as possible" to all ~\ (i 1,2, ... , nt), nl), i.e. 1,2.... i.e . to make each of H ~i (i 1,2, ·· ·,, nl) nt) as close to :ero zero as possible. The optimisation criterion can L:7~IIIH~;II}. The optimal be defined as: as: J = 2:~lIIHf\II}. follows by minimising J, solution for H follo\vs J ~ subject show that the to H HT = I. Lou et al (1986) sho\v choice of H given in (16) also minimises J, yielding the minimum value as J = 2:f=l a}. ~1· The ne\v new statement of the optimisation problem provides some very useful insight as J* J" can be used as a robustness measure which is clearly relative to the independent row number p of the matrix H.

=

subject to: to :

Typically, some components of unknown input vector !l d:. are larger than others. Furthermore, certain components of the unknown input vecresiduaL Hence we tor have more effect on the residual. \ve must pay varied attention to the different components of the disturbance in the optimisation procedure. For example, if the ith jth component of the disturbance is significantly larger than the i - th component, the term H§.{ ~~ will \vill be more imp ortant than the term H §.\ ~\ . The criterion J must be replaced by: J = 2:7~1 L:7~1 O'iIlH§.\II}, aiI\Hf\II}, where ai O'i(i 1,2,···,nl) \vhere (i 1,2,···, nl) are positive weighting factors. The relative magnitudes of the ai O'i correspond to relative magnitudes of components of the disturbance weighting. By rewritoptimisation criterion as: J = ing the weighting optirnisation L:7~11IH(Jiii§.;II}, 2:7~1 IIH(y'(ii~~II}, the problem can be solved using the procedure described above, but with y'iii§.\ and with El replaced by ~\ replaced by y'(ii~\ r;::-: 1 J(i2 r;:;:;: e 2 ... Vanl~l r;;;-- n 1 ] . E '1 -- [kral§.l El JO'nl§.~ll· val~l Va~l

pointt of their work \vas was to consider The starting poin the frequency domain model of the nominal plant (arising from the system of Eqs. (4) & (5), with \vith the uncertainty terms EIQ E 1s1 and E2Q E'2s1 set to zero, initially), i.e.:

=

A)-I B + D Go(s) = = C(51 C(sI - A)-l

=

5 00(s) [I + C(sI - A)-l (5) = [1 A)-! !{]-l Kt!

=

=

As Marquez and Diduch point out, there are a number of serious limitations inherent in the classical state space observer structure: structure: (a) The order of the resulting observer sensitivity, 5 0 (s), is limited to the order of the plant, G Go(s). o( s). 5 0(s) (5) are identical to the poles (b) The zeros of So of the original plant, and (c) ( c) It is difficult to relate frequency domain constraints on So (s) \vith the choice of the ob(5) with matrL",{ !{. [{. server gain matrL,{

(73)

(74)

If rank(P) S ~ n - p, (74) has solutions and the exact de-coupling at all operating points is achievable. If rank(P) > n-p, approximate de-coupling must be used. This is equivalent to the solution of Eq. (55) and can be solved by defining an optimisation problem:

liP - P*ll} P"1I}

(78)

J{ is the gain matrix of the full order observer. !( From (78), it is clear that a measure of sensitivityrobustness of the FDI scheme, is the oo-norm of 1150 (jw)l!. '\vhere where 11.1100 the observer sensitivity IISo(jw)11, 11·1100 = max(u{.}) max( ~{ .}) - the maximum singular value of the sensitivity function, over all frequencies w.

pointt of the system varies according The operating poin to different plant conditions, and different operating points correspond to different unknown input · · ·, M) matrices, Ei(i E1 (i = 1,2, 1,2,···, A1).. It is attractive to be able design a single FDI scheme for a whole range (or a set) of operating points. The success of the single FD FDII design depends on its robustness properties. In order to make the disturbance de-coupling hold for all operating points, '\ve \ve must make:

min

(ii) (i7)

The sensitivity function for the ident.ity identity observer for this system is:

r ...

HEf=o, i=1,2,···,Al HEl=o, for i=1,2,···,A'1 or H[E{ H[E} Er E? ... Efl] Ef1] = HP = 0

(76)

4.2.1. The Marquez and Diduch Robust Observer. :Ylarquez ~larquez and Diduch (1992) developed a new ne\v observer structure for FDI, based on unstructured uncertainty. Their '\vork work belongs essentially to a class of frequency domain approaches, based on sensitivity optimisation. In an attempt to overcome the limitations on ~chievable achievable sensitivity imposed by the structure of classical were motistate observers, ;'vIarquez Nlarquez and Diduch \vere new generalised observer structure vated to find a ne\v which provides additional degrees of freedom in \vhich shaping the observers sensitivity to unknown inputs. They gave a parameterisation of all stabilising generalised observers, through an observer Hoo sensitivity minimisation problem, via 11 00 optimisation.

L:f=1

=

=0

4.2. Frequency domain approaches

=

=

H p. Hp·

The active approach to robustness in FDI can also be applied by means of the structured parity equations (Gertler & & Kun\ver: Kunwer, 1995).

=

=

3H :j; -::p 0 for

Limitation (a) implies small gain at high frequencies, and consequently a small lIelL Ilell, may not be achievable when reduced order models are used. Limitation (b) implies t.hat that one does not have

(75)

57

complete freedom in the frequency domain shap( s) ing of the observer sensitivity. sensitivity. Every zero of 5 0 (s) will increase the slope of the observer sensitivity by +20dB per decade at high frequencies. Consequently, if Go (s) does not have high frequency poles , then So(jw) 5 0 (jw) will inevitably be large at high poles, frequencies . frequencies.

Yi(s), "Yl(S), XI(S) , satisfying appropriate forward and Y((s), 7-{co reverse Bezout identities identities.. This is a standard 7-£00 optimisation problem. Optimisation problems of this kind can, of course. be solved using different approaches.

Marquez and Diduch gave a simple example showing that the full order identity observer \vill will actually attenuate low frequency effects. Since steadystate residuals are being used for decision-making, the observer will \viII attenuate the steady-state misbehaviour caused by a fault.

A new and 4.2.2. Multi-objective methods. .A systematic approach to the design of robust residincipient uals for detecting inci pien t faults has been given by Chen, Liu and Patton (1994), \vhich which uses mu 1tiobjective Optimisation. To reduce false and missed alarm rates in fault detection, a number of performance indices are introduced into the observer design. Some performance indices are expressed in the frequency domain to take account of the frequency distribution of faults~ faults . noise signals and modelling uncertainties uncertainties...All objectiyes are reformulated into a set of inequality constraints on the performance indices.

The partial state observer considered Minto (1988) does not imby Viswanadham \Tis\vanadham and ~1into prove matters. On considering the dual coprime [Nr(s), Dr(s)] and [lV1(S), [N/(s), Dl(S)], D/(s)], of factorisation [lVr(S), the plant (where IV1(S), N/(s), Dl(S) D/(s) and iVr(S), Nr(s ), Dr(s) are RH 00 matrices), RHco matrices) , one has: has :

Although the approach taken has been based on the identity observer, and the residual is simply the primary residual or output estimation error fly (t), the use of frequency domain optimisation ~lI becomes an even further generalisation of the optiMarquez and Diduch mised observer structure by lVIarquez (1992). The essential difference is that. instead of considering the optimisation of the estimation error, the residual is optimised over suitable frethis , the actual structure quency bands. By doing this, of the observer used is less important. The use of the residual rather than the state estimation error is, of course, more relevant to the FDI problem.

Marquez and Diduch (1992) show that the optimal partial state observer with zero estimation error in the partial state (considering the nominal or unperturbed system), is actually given by Eq. (79) and this corresponds to an open-loop observer (with gain !{ K = 0). 0) . The sensitivity for this "optimal" case is given by:

5 0 (s) = 1

(80)

As this is clearly of no value, when considering the robustness to uncertainty (via frequency-domain sensitivity), Marquez and Diduch then proceeded to develop what they call the general observer, which we will now call the "Marquez and Diduch" observer. This observer comprises the nominal plant frequency domain model Go(s) given in (77), together \vith with a stabilising generalised observer (s), yielding an observer sensitivity of: gain !{ K(s),

=

5 0 (s) = [1 [I + C(sI C(sl - A)-l A)-1 BI((s) BK(s)

Examining only the case of sensor faults fault.s and considering the identity observer FDI problem, the Laplace transform of the residual r( t) is given by:

1:(s)

Gr f ( S, ]{) I.(s) Grj(s, K)lJs) +Grd(S, K)~(s) ]()G!(s)

+ D!{(s)]-l DJ{(s)]-1 (81)

+ fl(O)] ~(O)]

(84)

I

which reduces to a problem of finding a Q( s) s) such that:

inf WE(Wl,W,] WE(Wl,W2]

IIW(jw)[Yi(jw) ~ IIW(jw)[Yi(jw ) - Nr(jw)Q(jw)]Dr(jw)11 Nr(jw)Q(jw)]D/(jw )11 S;

+ ~(O)] fl(O)]

where fl(O) ~(O) is the initial value of the state st.ate estimaerror.. Both faults and disturbances affect the tion error t.he residual, and discrimination between these two effects is difficult. To reduce false and missed alarm ' rates, rates , the effect of faults on the residual should be maximised and the effect of disturbances on the residual should be made minimised. One can maximise the effect of the faults by maximising the following follo\ving performance index, index, in the required frequency range [Wl,W2]: [Wl' W2]:

(82)

Q' 0'

[I - C(sl - Ac)-l Ae)- I I{]R K]R2l(S) 2[(s) +C(sI +C(sl - Ac)-l~(s) Ae)-I~(s )

Marquez and Diduch then proceed to show how IISo(jw)lIco of the above sensitivity sensiti\;ty functhe norm IISo(jw)lloo tion can be made appropriately small at high frequencies, to minimise the effect of modelling uncertainty. To perform this minimisation they propose the use of a weighting funct.ion function W(j:..;), ~V(jw), such that, that , for some chosen 0': 0' :

IISo(jw)lloo IISo(jw)llco ::; ~

=

S[{(I - C(jwI ~4c)-1 d[1 C(jwl - .4 K]} c )-1 !{]}

(85)

(83)

(86)

such that there exist RH yr (s),, RHco }'~ (s). ( s). 4' Xr(s) oo matrices l'~

This equation is equivalent to the mininusation minimisation of

Q' 0'

58

the folIo\ving following performance index:

=

.]1 ·h (!{) (I{) =

residual:

sup 7f{ [I - C(jwl C(jwI - 04<)-1 .4~)-1 !(]-l} O'{[I Kj-I} (87)

WE(Wl,W~] wE(wl'w~]

.Uter A.fter the transient period, the residual steady state value plays an important role in fault detection. Ideally, it should reconstruct the fault signal. The disturbance effects on residual can be follo\ving permade minimised by minimising the following formance index:

where!z'{.} 7f{.} denote the minimal and maxwhere,q:{.} and O'{.} imal singular values. One can minimise the effects of both disturbance and initial condition bv by minimising the following performance index: .

=

J 2 (J{) = h(K)

sup wE(wl wE(""

o={ C(jwI - Ac)-l} O'{C(jwl-Ac)-I}

(88)

,w~] ,,,,~]

(93)

Besides faults and disturbances, the noise in the system can also affect the residual. To reduce the noise effect on the residual, we need to minimise: 11

J 4 is minimised. matri.x J{ When 14 minimised, the matrix K is very larae large 0 and the norm 11 A~l!{ A~l f{ 11 approaches a constant value. This means that the fault effect on the residual has not been changed by reducing the disturbance effect. This is what is required for good FDI performance. ,

f. - C(jwI - A c )-l!{ 1100

This contradicts the requirement of maximising the effects of faults on the residual. Fortunately, the frequency ranges of the faults and noise are normally different. For an incipient fault signal, the fault information is contained within a low frequency band as the fault development is slow. However, the noise comprises mainly high frequencies signals. Based on these observations, the effects of noise and faults can be separated by using different frequency-dependent weighting penalties. In this case, the performance index (!{) will \vill be: 1J 1I (K)

As there are four performance indices, to achieve .As robust fault detection (in terms of minimising missed and false alarm rates), a multi-objective optimisation problem must be solved. The parameter set to be designed are the elements of the gain matrix K, which must guarantee the stability of the observer. This leads to a constrained optimisation problem which is difficult to solve. To tackle this problem, Chen et al (1994) use the eigenstructure assignment method, via the dual control problem, to yield the parameterisation of the gain matrix !{ K (Patton & & Liu, 1994). To remove the constraints, they use a simple transformation developed by Burrows and Patton (1991). The multi-objective problem has been solved using a combination of a genetic algorithm and the method of inequalities (Patton, Liu & & Chen, 1993).

(f{vV C(jwl - Ac)-l !{]-l} (89) O'{WI(jW)[I Ac)-IKtl} 1 (jw)[I - C(jwl-

sup wE("" ,w,] WE(Wl,W2]

To minimise the effect of noise on the residual, J3 (K) is introduced a new performance objective h(K) as:

(f{W C(jwI - Ac)-l 0'{W3 (jw)[I (jw)[J - C(jwJ Ac)-I !{]} KJ}

sup

(90)

WE[Wl,W'J] wE("'"w,]

offaults In order to maximise the effects of faults at low frequencies and minimise the noise effect at high frequencies, the frequency-dependent weighting factor W 1I (s) should have large magnitude in the low frequency range and small magnitude at high frequencies. The frequency effect of W 3 (s) should be oppos~te to W1(s) y~t3(S) opposite WI(s) and can be chosen as vV 3 (s) = W1I- 11 (s). (~). The disturbance (or modelling error) and input noise affect the residual in the same \vay. way. As both effects should be minimised, the performance index Jh2 does not necessarily need to be \veighted. weighted. Ho\vever, However, modelling uncertainty and input noise effects may be more serious in one or more frequency bands. The performance index should reflect this fact, and hence a frequency dependent \veighting weighting factor must also be placed on J 2 (!{), in some situations. h(K),

4.2.3. Robust faults detection filters. Interval type parametric uncertainty is very common in practice and should be considered in robust FDI. Onlin & & Rizoni (1991) studied this problem in the design of failure detection filters. They show how the time-frequency distribution of the output er§.yy (t) can be used to generate a design proror .f cedure for failure (fault) detection filters. They developed a frequency domain min-max optimisation problem, with the search being carried out at each frequency, as the worst-case effect of the unstructured uncertainty at each frequency must be found. The system to be monitored is considered as having time-varying dynamics \vith with state space matrices parameterised by:

=

=

J 2 (!{) = h(K)

sup

w

~ (t) = (8) ~ (t) i(t) = .4 o4(8)~(t)

a={~V2(jw)C(jwI - o4<)-I} 44~)-1} (91) 0'{W2(jw)C(jwI

+ B(B) ~ (t ) B(8)~(t)

(94)

\vhere where 8B E [8 1 ,8 2] define one of many frequencydependent intervals in \vhich which the parameter should lie.

WE(Wl,W-;;:] wE("",w,,]

e

Now, considering the steady state value of the

59

adaptive threshold

5. ROBUST DECISION-MAKING IN TO

FDI

!

howefPatton & Chen (1991a. 1992a) reported how efforts to enhance the robustness of FDI can be made at the decision-making stage. Due to inevitable parameter uncertainty, disturbance and noise encountered in a practical application~ application, one \viII will rarely find a situation \vhere where the conditions for a perfectly robust residual generation are met. This is especially true for unstructured uncertainties. It is therefore necessary to provide sufficient robustness not only in the residual generation stage but also in the decision-making stage. vVhen When the decision- making stage of FDI is made robust against uncertainty, we can speak of passive robustness in FDI FDJ (Patton & Chen, 1991a, 1992a), in which \vhich case it may not be necessary (or it may be difficult) to make the residual robust. Passive robustness is thus an alternative to active robustness which should be used when there is very limited plant information available.

fixed threshold

I/ fault

Fig. 6. Application of an Adaptive Threshold

an adaptive threshold for direct residual evaluation. The idea of adapting the thresholds for fault detection, to the instantaneous operation of the monitored plant has been used by several investigators (Clark, 1989; Ding & & Frank, 1991). An how should \ve we determine interesting question is ho\v the functional form of the adaptive threshold law?

The goal of robust decision-making is to minimise the false and missing alarm rates due to the effects that modelling uncertainty and unknown disturbances will have on the residuals. This can be achieved in several ways, \vays, e.g. by statistical data processing, averaging, or by finding and using the most effective threshold. .

Clark (1989) used an empirical adaptive law. Horak (1988) calculated the envelope of maximum variations in an output based on an interval description of uncertain parameters (i.e. using bounds upon the uncertainty). He chose, at every time instant and for each parameter, the value within its uncertainty range that maximises the output and also the parameter vector that would Within this range of minimise the same output. 'YVithin outputs, the variations can therefore be explained by the parameter uncertainty. In this way Horak used, at every time step, the smallest possible threshold required to prevent false alarms in the presence of the worst case possible parameter deviations. Whilst this idea is intuitively appealing, ally very complex for all but simit is computation computationally ple examples of uncertain systems.

When a fixed threshold is used, the sensitivity to faults will be intolerably reduced if the threshold is chosen too high, whereas the false alarm rate will be too large when the threshold is chosen too low. The proper choice of the threshold is a delicate problem. vValker Walker & Gai (1979) and vValker Walker (1989) proposed the determination of the optimal threshold via Markov theory. In the case of large manoeuvres these changes might be large enough so that there allows satisfactory FDI is no fixed threshold that allo\vs at a tolerable false alarm rate.

Emarni-Naeini Emami-Naeini et al (1988) in their threshold selector (or threshold adaptor) method use the same detectable strategy of characterising the set of detectab le faults as those whose smallest possible effect on the norm of the residuals (under assumptions of bounds on norms of noise and plant uncertainty) are larger than the largest possible effect of noise and uncertainty alone.

5.1. Adaptive thresholds

The methods of passive robustness in FDI \vhich which have received the most attention are based on the use of adaptive thresholds (Clark, 1989; Ding & Frank, 1991; Emami-Naeini et ai, al~ 1988; Horak, 1988; Isaksson, Isaksson~ 1993; Weiss, \Veiss, 1988), i.e. each threshold becomes a function in some way of measurable quantities quantities...A ne'v new idea makes use of fuzzy logic techniques for decision-making (Frank 1993).

Ding &: & Frank (1991) developed this concept further in connection \vith with frequency domain approaches as follows. If the disturbance decoupling condition in Eq. (12) holds true, the fault-free residual is:

In the adaptive threshold approach the residual thresholds are varied according to' the control acto the tivity of the process. This concept is illustrated which also sho'vs shows the typical shape of in Fig. 10 \vhich

(95)

Assuming that the modelling error is bounded by

60

- ------_. _. --- _. _ .. _.... --- ._. _. _. _.-

-

'i 'J. (t) :~

---- -. -.. --- ..

. _ ... I

used a test statistic in the form of a weighted sum of the open-loop residuals and compared this \vith with the time-varying threshold as a weighted sum of .fy (t) and the decision the primary residual vector fly function O(t): n(t):

~

:

Residual --. Decision ~ ~ Generator I[(t) Mechanism I: [ (t)

U(t) u. (t) :

Il ~yy (t)

Threshold

:

: L....... :-... Adaptor Adapror

O(t)

Th(t)

(Selector) (Selecror)

--

]L(t) - 2(t) 1t(t) 2(1)

(101)

=

(102) 1':-00 T=-OO

-. 0--·

Fig. 7.

with the time-varying threshThis is compared \vith old:

some limiting value 8, i.e.

Til(t)

=7~OO w(r- t) I

[

I

1J;ool(q)U(t - q)

]2

(103)

(96) \vhere where l(t) is the time domain counterpart of L(w) and w(t) provides the necessary forgetting. The method can be effective for slo\v \vhen slow changes, when a tight bound on the model is available in the low frequency range. At high frequencies it will \vill anticipate larger model mismatch and therefore will tolerate larger errors.

In this situation, the fault-free residual will \vill be bounded as:

111:(s)1I 1I1:(s)11

IIHy(s)c.G,,(s)1f(s)11 IIH y (s)LlG u (s)jf(s)1I

< 81IH 8I1Hy(s)1f(s)11 y (s)1f(s)11

(97) It is believed that a good solution to robust FDI is to combine the disturbance decoupling design with \vith adaptive threshold.

An adaptive threshold can be assigned as follows: .A.n Til(s)

= 8Hy(s)1f(s)

(98)

An FDI scheme employing the threshold adapter (or threshold selector) is shown in Fig. 7.

5.2. Robust decision-making based on fuzzy logic

Isaksson (1993) developed this a priori threshold selection into an on-line scheme. His approach differs from the studies of Horak and Frank & & Ding in the form of the assumed uncertainty description. Instead of describing the uncertainty in terms of parameter intervals, he uses an unstructured frequency domain bound:

Iima:[C.(jw)]

~

8(jw)

vVhen an FDI residual can be made robust against When uncertainty (the so-called active approach), the process of deciding as to whether or not a fault has occurred in the system is facilitated. In this case, with fixed thresholds the false and missedalarms rates are minimal. It is very difficult to achieve true robustness in residual generation and the active approach to robust FDI is an attempt to include robustness at the decision-making stage. stage. The use of adaptive thresholds is one way to achieve a low false alarm rate, as discussed in Section 6.4. The decision-making process, in one way or another, comes down to a comparison of a deciWhen the threshsion function with a threshold. vVhen old is a numerical value (either fixed or adaptive), it is said to be crisp. .An An alternative approach is to make use of fuzzy logic decision-making, as discussed by Frank and Kiupel (1993). (1993) . Fault decisions based on fuzzy logic, provide a way of avoiding the effects of uncertainty and disturbance and hence reducing the false and missed alarm rate.

(99)

(J'ma:[.] \vhere where Ii ma: [.] denotes the maximum singular value and the real (perturbed) plant transfer mat.rix is given by: trix G(s) = (I

+ .6.(s))G c.(s))Go(S) o(S)

(100)

and with Go(s) the nominal plant transfer function.

A similar approach has been described earlier by A. \·Veiss Weiss (1988) who applied a bound on the uncertainty of the magnitude response, in order to construct time variable thresholds for fault detection. He assumed that an upper bound L(w) on the Fourier transform, E (w t.ransform, E( w)),, of the difference in impulse response between the model and the real plantt can be determined, usually considered as a plan high-pass filter, since the high frequency dynamics of the plant are often ignored in the model. Weiss vVeiss

are , like In the theory of fuzzy logic, fuzzy sets are, crisp sets, defined on exact fundamen tal sets. For all values of the fundamental set, set, the membership function of a crisp set has only two t.wo elements, '0' and '1'. In contrast to the crisp set the membership function of a fuzzy set can take on all values

61

between '0' and '1'. The value '0' stands for no membership, whilst the value '1' stands for a full membership. Thus the crisp borders of the classical sets vanish and are replaced by fuzzy sets.

p(x) p(I) 1

\ve let u be the generic element of the universe If we with U = {u}, \vhere where U is the crisp of discourse U ~, \vith discrete).. set of all possible objects (continuous or discrete) A fuzzy set .4 ..\ . 4 in U is completely described by \vhich orders every its membership function J.lA, J1.A, which u E U to a membership degree of A in the interval [0,1], [0 , 1], i e : J1.A : U E [0,1] J.LA

=

---+---"'I.----+---~C> --~----~----~---~

-IO e -Ioc

XO e IOc

IX

Fig. 8. The set approximately zero as a crisp set

The following three cases can be distinguished:

=

J1..4(U) = 0: u belongs to A with membership (1) J.L.4(U) degree 0 - i.e. not (2) 0 < ,uA (u) < 1: u belongs to ..44 \vith J1.A(U) with the J1..4(U), i.e. i.e . partially, degree J.LA(U), (3) ,uA(U) 1~ i.e. J1.A (u) = 1: u belongs to A with degree 1, completely

° °-

with cases (1) The classical set theory only deals \vith and case (3), i.e.:

J1.A : U E (0,1) J.LA -IO!

With vVith this definition, every crisp set becomes a &. Kiupel, subset of the related fuzzy set (Frank & 1993). Therefore, the concept of fuzzy sets can be interpreted as a generalisation of crisps sets. In this way fuzzy interpretations can be made using typical linguistic predicates such as:

Fig. 9. The set approximately zero as a fuzzy set

Frank and Kiupel (1993) assume that~ that, if an interval is used instead of a crisp threshold (the borders of which have to be properly chosen), the residual evaluation can be derived using an upper border as the maximum of the highest value of the residual, and the lower border that of noise. By using a fuzzy set according to Fig. 10, the free parameter ao is proportional to the residual amplitude caused by noise and model uncertainties. This parameter describes the variance of the noise and uncertainties. In the case of time-varying uncertainties, tainties , ,ve we have: have:

warm, cold, small, large, little, some\varm, thing, much, etc. or more precisely as:

a little bit, much more, approximately, zero, etc. These vague linguistic predicates can be specified with mathematical exactitude as fuzzy sets &: 9 in terms of membership functions. Figs. 8 & shows the difference between a binary set (Fig. 8) and a fuzzy set (Fig. 9) {approximately zero}. zero} . The membership to the crisp set {approximately zero} can be written as:

8= disturbance, model uncer= f (noise variance, disturbance, tainty) Frank and Kiupel (1993) show sho\v further that the type of false-alarm caused by a residual slightly

x E {approximately zero} or x E {approximately zero}) zero}, using an exclusive or function.

Fuzzy supervision for lean production Fzc=y

In contrast to a crisp set, a fuzzy set allows all 0 'vs arbitrary values in the membership function from J.L(x) 1 (full J1.(x) = 0 (no membership) up to J.l(x) J1.(x) membership); that means that arbitrary small changes in the value x result in arbitrary small x). J1.(x). changes in the membership J.l(

=

p(rJ

'('J

=

/

-ao Qo - 0()

Fault evaluation in FDI can no'v in a now be given gIven In weighted form rather in yes-no terminology. This means that false alarms that make the detection system unreliable can be avoided.

II -ao Qo

I

~

~o "0

Qo

+ ()0

:: =

C> D>

Fig. 10. The fuzzy set approximately =cro zero under consideration of uncertainty

62

exceeding a crisp threshold \vill will not occur when the fuzzy threshold is used. In the fuzzy case, a slight increase in the residual produces a weak inconsistency. In the crisp threshold case, a sligh t increase in the residual can cause a false alarm. This is of considerable advantage when \vhen considering robustness to uncertainty" uncertainty. The fuzzy decision logic applied to the residuals in this way \vay falls into the class of passive methods for robust FDI (as defined by Patton and Chen (1991)).

residuals . Clearly, if non-linear or time-varying residuals. the system"s system~s dynamic structure and non-linearitv are not \veIl \ve are better off trying t~ well known then we use approximate uncertainty decoupling methods.

A.ll real life systems have uncertain characteristics All and cannot be mo delled perfectly. modelled perfectly. The paper has drawn out the essential differences between various methods for optimising robustness~ robustness. based on whether or not the uncertainty can be considered \vhether as structured or unstructured. In some cases, even systems \vith ured uncertainty can form with unstruct unstructured the basis of estimation schemes generating an approximate structure, so that multivariable decou~Iost of the robust FD FDII pling theory can be used. ~lost approaches based on unstructured uncertainty use frequency domain optimisation and several papers have discussed the importance imp ortance of various observer \vithin this framework. frame\vork. structures, within

6. CONCLUDING DISCUSSION

This paper has reviewed most of the latest advances in the field of robustness for fault diagnosis, nosis , based on model-based residual generation methods. The study has drawn out the correspondences between various approaches connected with either parity space concepts or state observers (or both together). After outlining the basic properties of the main approaches, approaches , the general problem of robustness in FDI has been described as a foundation for discussing the ways in which robustness can be dealt with either in the frequency or time domain. Very few methods domain , howhave actually made use of the time domain, ever it has been shown that the eigenstructure is a very powerful tool for bridging the time and frequency domains to provide optimal disturbance decoupling.

The paper has pointed out therefore, the importance of appropriate optirnisation optimisation and structure of residuals for robust fault detection and, to some extent, robust fault isolation (although these problems are not easily separable). Above all, it has no\v FDI problem now clear that the FOI is not an issue of \vhether whether or not one should use an observer or a generalised parity space. Future FDI \viII will continue to make use of work on robust FOI whether or not they are optimisation procedures, \vhether 11. oo based upon 1f. 00 .,

The freedom available for residual generation design and for fault isolability is determined by rules governing the design freedom of any multivariable system. An observer can, of course, be considered as a dualization of the feedback control problem, for which there are several decades of multivariable design theory available. In order to understand how to achieve the best performance of an FDI scheme in terms of robustness to uncertainty, sensitivity to faults and fault" isolability, it is essential to kno\v know something about the degrees of freedom available for these tasks, given a particular FDI approach. It is very clear from this study that the freedom issue has not been properly understood in the literature.

Of course, we \ve will never understand the true value of robust FDI theory until the methods are applied to real process plants. Developments in real application studies are eagerly awaited. The use step of bench-mark testing in the laboratory is a st.ep in the right direction and, hopefully there ,vill will be \'Vorkshops ans Symmore benchmark sessions at Workshops posia to accompany sessions on real applications of robust FOI. FDI.

ACKNOWLEDGEMENTS \vishes to ackno\vledge The author wishes acknowledge funding support for this research from the UK Science & t.hrough grants ref. Engineering Research Council through GR/J12758 & GR/J1278Q. GR/Jl278Q. Essential contribution to the research into robust FOr FDI methods have been given by Dr. ~Iing Hou (both Dr. Jie Chen and Dr. Ming of the University of Hull), are gratefully ackno\vlacknowledged.

IItt has also not been fully appreciated that the failure detection filter is no other than a full order observer with special directional sensitivity properties for actuator or sensor fault isolation. \,Vhen When trying to achieve good fault isolation \vith with one observer (or failure detection filter), it is almost impossible to achieve good robustness to uncertainty - both cannot live together perfectly, when using a linear system approach! This takes us to the point that residuals can be designed using non-linear theory, a non-linear observer, or non-linear feedback design, or just using

63

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