Current Developments in the Theory of FDI

Copyright @ IF AC Fault Detection, Supervision and Safety for Technical Processes, Budapest, Hungary, 2000

CURRENT DEVELOPMENTS IN THE THEORY OF FDI Paul M. Frank· Steven X. Ding" Birgit Koppen-Seliger·

• Gerhard-Mercator- Universitiit Duisb'Urg FB 9/ MefJ- 'Und Regel'Ungstechnik Bismarckstr. 81, BB, D-47048 D'Uisb'Urg, Germany email: {p.m.jrank.koeppen-seliger}@.Uni-duisb.Urg.de .. Dept. of Electrical Eng. , La'Usitz Univ. of Applied Sciences D-01968 Senftenberg, Germany

Abstract: In this paper current developments in the FDI theory are reviewed. Attention is focused upon the analytical approaches that make use of quantitative models, knowledge based approaches using qualitative models as well as approaches using computational intelligence techniques. Besides the description of different fault diagnosis methods, a number of prospective research topics are outlined. Copyright

@

2000 IFAC

Keywords: Fault Diagnosis, Robustness, Nonlinear systems, Knowledge-based systems, Qualitative simulation, Fuzzy logic, Neural networks

1. INTRODUCTION

Starting as a special application of observer theory in the early 70's, model-based FDI theory went through a dynamic and rapid development and is currently becoming an important field of automatic control theory. As shown in Fig.1, in the first twenty years, it was the control community that made the decisive contributions to FDI theory, while in the last decade the trends in FDI theory are marked by an increasing number of contributions from the computer science and AI community associated with the integration of different techniques. This paper briefly reviews some of these approaches with emphasis on the description of basic ideas and concepts. Due to the limited space, we shall concentrate on the presentation of current results in analytical and knowledge model-based FDI as well as on FDI employing computational intelligence techniques, where, to our knowledge, the most significant progress and results have been achieved in recent years. We will finally outline some prospective research topics in the field of FDI.

1910

1980

1990

Fig. 1. Historic development of FDI theory 2. ANALYTICAL MODEL-BASED FDI The development of the analytical model-based FDI theory is characterised by its intimate relationship to control theory. Hence, besides the established framework of FDI theory for LTI systems, the trends in control theory, • development of robust system theory, • application of adaptive schemes, • development of nonlinear system theory,

17

can also be clearly observed in the field of analytical model-based FDI theory.

PI-AEd ] rank [ -C Fd

respectively, where the state-space realization of process model (1) is assumed to be [Gd(P) G f(P)] = (A, [Ed Ef] ,C, [Fd Ff]) and Ho, Hd,Hf denote the system matrices related to x(O),d(k) and f(k) respectively, which are known from the parity space approach (Frank, 1990; Gertler, 1991).

2.1 Establishment of a framework of FDI theory for LTI systems 2.1.1. Basic concepts in the framework of FDI theory Related to the tasks of fault detection and isolation, fault detectability and isolability play an important role in FDI system design. In our opinion, these concepts characterize the FDI structure of the process under consideration and should thus be expressed only in terms of the parameters given by the process model. Taking into account this, the following definitions seem the most meaningful and practical ones among the results published in the past.

2.1.2. Unified design of residual generators The first step in FDI is residual generation. In view of the structure of the residual generators, we divide the well established residual generator design approaches into (Frank, 1990; Gertler, 1998; Chen and Patton, 1999) • • • •

Given process model

y(p)

= Gu(p)u(p) + 6.y(P) + Gf(P)f(P)

(1)

where '11., y denote the process input and output with the known transfer function matrix G u (p) , 6.y(p) is used to represent disturbances and model uncertainty that are unknown but bounded, f is a vector and each of its elements represents a fault whose influence on y is described by G f(P), then a fault /; is said to be detectable when Gfi(P) f. 0 (Gfi(p) is the i-th row of Gf(p» and all faults are isolable if rank (G f (p» = dim (I) .

Suppose 6.y(P) = Gd(P)d(p) with a known transfer function matrix Gd(p), then a fault /; is detectable independent of d when and

f is isolable and decouplable from

= rank (Gd(P»

(2)

d if

rank [Gd(P) Gf(P)]

(3)

+ rank (G f(P»

r(p) = R(p) (Mu(P)y(p) - Nu (p)u(P) )

Recently, it is demonstrated (Ding et al., 1999) that conditions (2)-(3) can also be equivalently expressed by

pI - A Ed Ef; ] k [PI - A Ed] (5) ran k [ -C Fd Ff; > ran -C Fd rank [Ho Hd H f ] =

+ rank

(6)

is used as the platform, where R(P) is the so-called parametrization matrix which can be arbitrarily selected from the set of stable systems RH00, and Mu(P), Nu(P) are transfer function matrices which are stable by a suitable choice of the so-called observer gain matrix L.

rank [Ho Hd R" ] > rank [Ho Hd] <:>(4)

rank [Ho Hd]

fault detection filter (FDF) diagnostic observer (DO) parity space approach (PSA) frequency domain approach (FA)

With their steady establishment in the past, research attention has been devoted to the interconnections among these approaches, in particular, between PSA and the other three approaches. Equivalence between them has been demonstrated from different viewpoints (Gertler, 1991; Magni and Mouyon, 1994). Recently, (Ding et al., 1999) derived a one-to-one relationships among the design parameters and reveal that the real difference between these approaches lies in the fact that the on-line implementation form of PSA is nonrecursive, while the observer-based ones are implemented recursively. Making use of these results the design of residual generators can be carried out independent of the implementation form possibly used. We can use, for instance, the parity space approach for the residual generator design, then transform the parameters achieved to the parameters needed for the construction of a diagnostic observer and finally realize the diagnostic observer. In Table 1, we summarize some important properties of the above-mentioned four types of residual generators. In order to express them in a unified way, the unified form of LTI residual generators proposed by (Frank and Ding, 1994)

Another important concept is the full decoupling from unknown inputs, which is often introduced in relationship with the residual generator used. Indeed, the unknown input decoupled fault detection and isolation are also structural properties of the process and can be therefore described as follows.

rank [Gd(p) Gfi(P)] > rank (Gd(p))

+ rank [PI-AEf] -C F f

2.1.3. Main approaches to full unknown input decoupling Generating residuals fully decoupled from the unknown inputs was one of mostly studied topics in the last two decades. As a result,

[Ho Hf]

pI - A Ed Ef] <:> rank [ -C Fd Ff =

18

there exists a number of approaches to achieving full decoupling from the unknown inputs (Frank, 1994; Chen and Patton, 1999). Since they are developed using different mathematical and control engineering tools, the existence conditions and algorithms for the solutions are expressed in terms of different system properties and parameters, which may not be equivalent and thus often cause some confusion by application. Table 2 gives a brief review of the main approaches to a full unknown input de coupling and their relations to definitions and the existence conditions given in the last sub-section.

optimization problems and provided thus a satisfactory solution to the above-defined optimization problem ten years after it was first proposed. In Table 3, we briefly review the state of art of the solutions.

2.2 FDI in nonlinear systems Thanks to the rapid progress of non linear observer theory during the last decade, significant results in designing nonlinear residual generators have been achieved in recent five years. Kevertheless, a general theory for the solution of nonlinear FDI problems is still missing. Thus the development of nonlinear FDI approaches is one of the current FDI topics that are receiving much attention (Frank and Ding, 1997; Alcorta Garcia and Frank, 1997). The first works contributed by (Seliger and Frank, 1993; Wiinnenberg, 1990) are a logic extension of LTI-FDI technique to nonlinear systems using the so-called concept of Linearization via State Feedback (LSF). Due to the strong existence conditions for LSF, there was a need to find other ways. For a class of nonlinear systems this resulted in the Theory of Nonlinear System Stabilization (NSS). We shall in this subsection describe the basic ideas of LSF and NSS, and briefly review the existing approaches.

2.1.4. Unified solution to the robust FDI problem The rapid development of robust control theory in the 80's and early 90's gave a decisive impulse for the establishment of a new theory to solve robust FDI problems. Generally speaking, the robust FDI problems can be approached in three different ways: • making use of knowledge of ~y(P) (Basseville, 1997). A typical example is the Kalman filter approach, in which it is assumed that the unknown inputs are white gaussian noise; • approximating Gd(P) by a transfer function matrix Gd*(p) which, on the one hand, satisfies the existence conditions for full unknown input decoupling and, on the other hand, provides an optimal approximation (in some sense) to Gd(P) (Chen and Patton, 1999); • designing residual generators under a certain performance index (Frank, 1990; Basseville, 1997; Nikoukhau, 1994; Wiinnenberg, 1990). This is indeed a reasonable extension of the unknown input residual generator design, in which, instead of full decoupling, a compromise between the robustness and sensitivity is made.

2.2.1. FDI scheme via LSF-theory and its extension Consider nonlinear systems of the form

Imll

0<

IDmJ

~ 11:11 with 11;;11>

(7)

+ Fd(X)d + Ff(x, u, f)

(8)

y = C(x, u)

with known vector fields A(x, u), C(x, u), Ed(X) and Fd(X) . The basic idea of designing a residual generator using LSF is to construct a residual generator in such a way that its dynamics is governed by a linear system. To this end, we first

In this subsection, we concentrate ourselves on the last scheme, due to its important role in theoretical studies and its relationship to the residual evaluation and integrated design of FDI systems. Since the goal of residual generation is to enhance the robustness of the residual to the model uncertainty without loss of its sensitivity to the faults, the minimization of performance index (Frank, 1990; Wiinnenberg, 1990) h

x = A(x, u) + Ed(X)d + Ef(X, u, f)

• transform system (7)-(8) into

i = Az + B(u, y)

+ Ef(z, u, f) + Edd (9) y = Cz + Ff(z,u, f) + Fdd (lO) • and then design the residual generator based upon (9)-(10) using one of the known approaches for LTI systems. A natural extension of this technique is to replace (9)-(10) by a class of nonlinear systems , e.g. bilinear systems up to output injection, for which a solution to the residual generation problem is available (Hammouri et al., 1998). It is evident that the key of this scheme consists in finding a transformation z = T (x) such that

Q

is widely recognized as a suitable design objective. Associated to the norm used, the type of the residual generator and the mathematical tool adopted, a number of optimization approaches have been developed. Most recently, (Ding et al., 2000 b) have derived a unified solution for a number of

8T(x) ax-A(x,u) = A(T(x),u)

19

-

+ B(u,y)

(11)

8T(x) Ed(X) 8x

= O,r(T(x),C(x,u)) =

and parameters 0 , '" are chosen such that P is positive definite and the Lipschitz constant E holds the following inequality

0 (12)

Condition (11)-(12) can be interpreted as a general form of the well known Luenberger conditions.

E

e = (N -

+ T(.A(x , u) -

A(x, u))

r=Ce

It becomes evident that r carries information about fault, which allows a fault detection. 2.2 .3. Review of the existing approaches Table 4 gives a brief review of the existing approaches, from which we can see that the current development in nonlinear FDI theory is strongly influenced by the nonlinear observer theory.

r = y - f)

The essential idea of this approach is the design of matrix L(x, u) ensuring that the equilibrium e = 0 is asymptotically stable, which may be very difficult for the reason that there exist no systematic approaches to finding out such a L(x, u) when no assumptions on A(x, u, Ic), c(x, u, Is) are made. Recently, (Garg and Hedrick, 1995; Frank et al., 1999) proposed to use the theory of nonlinear observer design via Lipschitz condition approaches to the design of nonlinear residual generators , which can be briefly described as follows. Consider the process model

2.3 Parameter identification and adaptive observer based FDI

Parameter identification (Isermann, 1993) and adaptive observer approaches are an alternative way to approach FDI problems. The significant development in this field in the past decade is the nonlinear adaptive observer based FDI scheme that integrates and extends the nonlinear observer based and parameter estimation approaches (Alcorta Garcia and Frank, 1997; Frank and Ding, 1997; Yang and Saif, 1995; Zhou and Frank, 1999). The core of such kind of FDI systems is an adaptive observer which is used both to improve the robustness against model uncertainty due to parameter changes and to detect and identify the faults. In general, for a given process

= Ax + Bu + .A(x, u) + Ed(x, u)d + E,(x, u, I)

y=Cx which satisfies the following assumptions • (C, A) is observable • the nonlinearity .A(x, u) satisfies a Lipschitz condition, i.e. there exists a positive constant E such that

± = A(x, u,O) + E,(x, u, I), y = h(x)

e,

with f representing process parameters and faults, an adaptive observer based FDI system consists of an observer

11.A(Xl,U) - .A(x2,u)11 ~ Ellxl - x211 It has been proven that the following system is stable and delivers a residual signal

+ Ly + Gu + T.A(x,u)

r=Cz+(CK-I)y,

p-1C T C)e

-TEd(x,u)d-TE,(x,u,l),

+ L(x, u)(y - y)

dx = A(x, u , leo) dt f) =c(x,u, Iso),

z= Nz

2a max (PT)

y = c(x, u, Is)

where le, Is represent component and sensor faults respectively, and proposed the following system for the purpose of residual generation

:i;

-

The dynamics of the residual generator is governed by

2.2.2. FDI scheme using stabilization theory In their early work, (Hengy and Frank, 1986) studied the residual generation problem for nonlinear processes of the form

± = A(x, u, Ic),

Amin((2 - o)CTC + ",P) < .....:.:.:.:..::.:~-......:..----~

z

V = 1lJ(V,y,u) dt with V denoting an auxiliary variable. The FDI system delivers an estimation for those faults that are presented in form of process parameters and further provides information about faults described by the vector f. - = E(r, V),

N - p-1CTC is stable, G = TB T+KC=I

matrix P, depending on the parameters the solution of the Lyapunov equation

0, "',

= y -h(z)

diJ

x=z+Ky

where matrices G, K, L, Nand T have appropriate dimensions and satisfy

NT-TA=LC,

r

with r as residual and a parameter estimator

- p-1CT(f) - y)

f)=Cx,

= .A(z, y, u,O, V),

In comparison with the observer based FDI approaches, the parameter identification and adaptive observer based approaches provide a deeper insight into the process which can advantageously

is

NTp+PN -oCTC+",P= 0

20

be used for fault analysis. This additional knowledge of the faults is however achieved at the cost of more on-line computation. From the viewpoint of control theory, parameter identification and adaptive observer-based FDI systems are closedloop structured, since information about the faults and parameter changes is fed back to the residual generator. Thus, the stability and convergence properties must be considered when designing such FDI systems. Recently, attention is also paid to the study on the relationships between the observer and parameter identification-based FDI approaches (Delmaire et al., 1994; Gertler, 1995; AIcorta Garcia and Frank, 1996) .

Q-cut identity principle of Nguyen (1978) allows to reduce fuzzy mappings into interval computations. Therefore, intervals are the most fundamental representation in qualitative models. The rough representations of variables lead to the imprecision in a qualitative model that relates the variables with each other. Qualitative simulation predicts the qualitative behavior using such a model and retains or even enlarges the imprecision. Another imprecision in the behavior comes from the not exactly known initial states. According to the available information about a plant, there are several different possibilities to qualitatively represent the information of the dynamic process, each of which is associated with an appropriate simulation method. Basically, a qualitative simulation method should be responsible for retaining the accuracy of the represented system behavior (so called soundness property following the definition of Kuipers (1986)), thus the fault detection approaches based on them could avoid false alarm. The representations that are relevant to the FDI approaches presented in this section are:

3. FDI APPROACHES BASED ON QUALITATIVE MODELS Since in most cases available, a priori knowledge about a process is hardly complete, or, even if this is the case, might be too complex to directly deal with, an approximation has to be made, so that models become inaccurate. Or measurements are subject to noise. Consequently, deviations between the reality and its representation, i.e. modeling errors, are unavoidable. This method has been extensively applied in science and engineering, e.g. when a nonlinear differential equation is linearized or a complex system is represented by a trained artificial neural network. These quantitative models are able to predict the system behavior precisely but more often inaccurately. Efforts have to be made through bringing more information (e.g. training data) to raise the accuracy of the prediction in the modeling stage, or through modeling error decoupling to reduce the influence of such errors when applying the models to fault diagnosis.

• qualitative differential equations (QDE) (Kuipers, 1986; Shen and Leitch, 1993) • envelope behaviors e.g (Kay and Kuipers, 1993; Bonarini and Bontempi, 1994) • stochastic qualitative behaviors (Lunze, 1994; Zhuang and Frank, 1999) Other relevant methods to qualitative models for fault diagnosis are e.g. signed directed graphs (Leyval et al., 1994), logical based diagnosis (Lunze, 1995) and structural analysis (Staroswiecki et al., 2000). Dynamic behaviors are not emphasized in these methods, their main concerns are the causality or correlativity among various parts of systems, which are especially useful for performing fault isolation and analysis.

Alternatively, incomplete knowledge can be treated via abstraction. Instead of the precise description by a quantitative model, a qualitative description of a process can be applied. By allowing the existence of a tolerance, the resolution of the representations is reduced, to emphasize primary distinctions and ignore unimportant or unknown details. Although this description is imprecise it is able to represent the system accurately. A set, rather than a single value, becomes a primitive representation in this case.

3.1 FDI using qualitative observers based on QDE Conceptually, a qualitative differential equation can be considered to be the extension of an ordinary differential equation ;i; =

g(x,u,B)

(13)

where x, U and B denote the vectors of state variables, known inputs and parameters with the dimension of n, r and s respectively. However, in a QDE, the variables take intervals as their values and the variant of the nonlinear function g(.) is allowed to include various imprecise representations: e.g. interval parameters, non-analytical functions empirically represented by IF-THEN rules and even, in the algorithm QSIM of Kuipers (1986), unknown monotonic functions. If the nonlinear function g(.) is rational, its corresponding

In the last decade, the study of applying qualitative process models to system monitoring and fault diagnosis received much attention. See e.g. (Dvorak and Kuipers, 1989; Leitch et al., 1994; Zhuang and Frank, 1997). Typical qualitative descriptions of variables are signs (de Kleer and Brown, 1984), intervals (Kuipers, 1986) and fuzzy sets (Shen and Leitch, 1993). As a fuzzy set can be devided into a series of intervals, the use of the

21

QDE can be readily derived from it by using the natural interval extension of the real function (Moore, 1979).

While qualitative behaviors here are interval values of system variables against time; qualitative simulation, aiming at producing all possible dynamic behaviors, means the generation of their envelope. Once the envelope is generated, the scheme for fault detection is nothing else than a direct comparison between the envelope and the measurements. In fault free cases, the measurements are always contained in the envelope; otherwise, it indicates fault occurrence. Therefore, a well designed simulation method is most important in this case for detecting small faults.

Through qualitative simulation procedures that are composed of the two main steps genemtion and test/exclusion and basically different from the numerical ones, the behavior of continuous variables is discretely represented by a branching tree of qualitati ve states.

= == :--;===="'=ll qualitative

Recently many efforts have been made to increase the precision of classical qualitative simulation, i.e. to avoid unnecessary conservativeness. More quantitative information is brought into a model representation (Berleant and Kuipers, 1992) and simulation methodologies tend to be more constructive. Kay and Kuipers (1993) and Verscovi et al. (1995) propose approaches based on standard numerical methods to obtain the bounding behavior. In (Bonarini and Bontempi, 1994; Keller et al., 1999) the interval parameters and the state variables are treated as a super-cube, whose evolution at any time is specified by its external surface. Armengo et al. (1999) present the computation of envelopes by means of modal interval analysis (Gardeiies et al., 1986).

output I residual Qualitative observer

Fig. 2. Qualitative observer The qualitative observer (QOB) based on QDE is an extension of a qualitative simulator and functions in further reducing the number of irrelevant behaviors (including spurious solutions) to the system under consideration (Zhuang and Frank, 1997), as illustrated in Fig. 2. The principle of observation filtering is that the simulated qualitative behavior of a variable must cover its counterpart of the measurements obtained from the system itself; otherwise the simulated behavioral path is inconsistent and can be eliminated. Since these procedures do not lead to the violation of the accuracy of the qualitative behavior under fault free condition, the output of QOB is the refined prediction behavior in this case.

3.3 Residual generation via stochastic qualitative behaviors Another qualitative representation of system behaviors is the stochastic distribution under partitioned state and output spaces. Beginning with the similar model assumptions as in section 1.2, the parameter vector is in 9 and the initial state is uniformly distributed within a prescribed area, say cell O. Xi(t) and Yi(t) denote the probabilities that the trajectories of state variables and output variables respectively, which start from all initial states in cell 0, fall into the ith cell at any time t. The behavior can be approximately represented by a Markov chain (Zhuang and Frank, 1999). It turns out that the new state and output variables X and Y can be described by the following discrete hidden Markov model (HMM):

However, when a fault occurs which causes a significant deviation of the system output such that no consistent predicted counterpart of the output could be generated, the output of the QOB becomes an empty set, which indicates the fault occurrence. Following this principle, fault detection and sensor fault isolation can be implemented (Zhuang and Frank, 1997). It is important to note that, in exchange with the advantage of requiring weaker process knowledge in this method, one has to put up with an increase in computational complexity and less sensitivity to small faults.

3.2 Fault detection based on envelope behaviors A key issue on improving the small fault detectability when applying qualitative methods is that the qualitative system behavior should be predicted as precisely as possible. Different from the qualitative model and the simulation method presented above, the model considered in this and the next sections is of less ambiguity, i.e. imprecision in equation (13) is caused only by interval parameters and interval initial states, the structure of g(.) is fully known.

X(k

+ 1) = A(u, 9)X(k) + V(k)

Y(k

+ 1) = C(9)X(k + 1)

where V represents the influence of spurious solutions. A fault detection scheme based on the HMM is shown in Fig. 3 (Zhuang and Frank, 1999). A qualitative observer (QOB) aiming at attenuating the effect of V and watching over the possible abnormal behavior of measurements is applied. The

22

residual r and its credibility v can be calculated, the latter reflects the degree of spurious solutions. lI(k + I)

y(k + I)

Y(k + I) , .,

function, and the DMPL with a connectionist hidden layer, which has a partially recurrent structure interconnecting only the hidden units. The mixed structure is implemented in such a way that one can select either a basic architecture or a combination of them. The training of the DMPL-MIX neural network is performed by applying Dynamic Backpropagation (Marcu et al. , 1999a), while the problem of structural optimisation is solved with the help of a genetic algorithm (Marcu et al., 1999a) .

, r(k + I)

.ll ~I~f-{=JI : l·;j~ ! ~-~~~{~}=.J i

.

Y(k + Ilk)

,P,

v(k+ I)

Y(k + Ilk + I)

Fig. 3. Residual generation using HMM

Two types of observer schemes are proposed by Marcu et al. (1999b) for actuator, component and instrument fault detection: the Neural Single Observer Scheme (NSOS) and the l\eural Dedicated Observer Scheme (NDOS). While the first one is driven by all process inputs and outputs the second one is only driven by the process inputs and the output of the component to be supervised. Therefore the first scheme consists only of a single observer which is composed of a bank of MISO neural nets each estimating one output in opposite to the second scheme, which consists of a number of observers associated to each component of the plant. These neural observers in turn consist of a number of MISO neural nets each estimating one process output. In both cases the training is based on fault free process data reflecting the normal behaviour.

4. FDI EMPLOYING COMPUTATIONAL INTELLIGENCE TECHNIQt:ES In the case of fault diagnosis in complex systems, one is faced with the problem that no, or insufficiently accurate, mathematical models are available. The use of knowledge-model-based or datamodel-based techniques, either in the framework of diagnosis expert systems or in combination with a human expert, is then the only feasible way to proceed. The concepts presented in the following employ computational intelligence techniques such as • neural networks • fuzzy logic • genetic algorithms and/or combinations of them in order to cope with the problem of nonlinear processes, lacking analytical knowledge and robustness issues. The following four approaches represent the latest development in this area.

The residual evaluation part can then be performed by a well-known static MLP neural network.

4.1 Neural Observer Based Approach

4.2 Fuzzy Observer Based Approach

In the Neural Observer Based Approach (Marcu et al., 1999b; Marcu et al., 1999c) neural networks are used as nonlinear multi-input single-output (MISO) models of ARMA type to set up different kinds of observer schemes. Thereby the neural networks replace the analytical models which are usually necessary for observer-based FDI. Instead of a multi-input mUlti-output structure a separate neural network is identified for each output. Thus, a set of smaller neural networks can be identified for each class of process behaviour. The type of neural network employed for this task is of a mixed structure called Dynamic Multi-Layer Perceptron (DMLP-MIX) integrating three generalised structures of a DMPL (Marcu et al., 1998). These three are the DMPL with synaptic generalised filters, which has each synapse represented by an ARMA filter with different orders for denominator and numerator, the DMPL with internal generalised filters (Ayoubi, 1994), which integrates an ARMA filter within the neurons before the activation

The Fuzzy Observer Concept (Chen et al., 1999) actually represents a set of analytical linear observers on whose outputs a fuzzy fusion is performed based on the idea of Tagaki-Sugeno fuzzy models (Tagaki and Sugeno, 1985). Using this approach a nonlinear dynamic system is described by a number of locally linearised models. For the fuzzy observer scheme the linear models are implemented in a bank of linear observers. The final state estimation is given by a fuzzy fusion of all local observer outputs. The difference between the measured output and the estimated output provides the residual for further diagnostic evaluation (see figure 4, (Chen et al., 1999)). Although all local observers can be designed stable, the overall fuzzy observer is not necessarily stable. In order to analyse and ensure stability of the fuzzy observer scheme, the linear matrix equality (LMI) method is employed. The technique of eigenvalue assignment is used to ensure certain diagnostic performance such as speed and robustness.

23

.8

inputu

of the fault and the fuzzy clustering to provide isolation of the fault (Dalton et al., 1999). The statistical tests are based on the analysis of the mean and the variance of the residuals, e.g. the CUSUM test (Gomez, 1995). The subsequent fault isolation by means of fuzzy clustering consists of the two following steps: In an off-line phase the characteristics of the different classes are determined. A learning set which contains residuals for all known faults is necessary for this off-line phase. In the on-line phase the membership degree of the current residuals to each of the known classes is calculated. A commonly used alg~rithm is the fuzzy C-means algorithm from Bezdek (1991) .

OU!PU!y

1_Observer) _ _ _ _ I

! ----rg.-~:-------.

~t

Fig. 4. Fuzzy Observer scheme (Chen et al., 1999) 4.3 Hierarchical Fuzzy Neural Networks

4.4.2. Fuzzy Reasoning The basic idea behind the application of fuzzy reasoning for residual analysis (Ulieru, 1994; Dalton et al., 1999) is that each residual is declared as normal, hi.gh or low to a certain degree, with respect to the nominal residual value. For fault detection, the only relevant information is whether or not the residual have deviated from the fault free value, and hence it is only necessary to differentiate between normal and non-normal behaviour. However, if isolation of faults is required, it may be necessary to include the direction and magnitude of the deviation. The rules for analysing the res iduals can be written down based on knowledge of the system itself or from inspection of simulated residuals. In an iterative algorithm starting with a small number of input membership functions for each input and a single output membership function for each fault, the rules are derived and the number of inputs and/or input membership functions are increased until a satisfactory fault isolation is achieved.

The fault diagnosis system consists of a knowledge based approach coupled with a hierarchical structure of fuzzy neural networks (FNN) (Calado and da Costa, 1999). FNNs combine the advantage of fuzzy reasoning, which is the capability of handling uncertain and imprecise information, with the advantage of neural networks, which is the capability of learning from examples. They consist of a fuzzification layer, a hidden layer and an output layer. Fault detection is performed through the knowledge based system, where the detection rules are generated from knowledge obtained from the structural decomposition of the overall process into subsystems and operational process experience. After detecting a fault the diagnostic module is triggered which consists of a hierarchical structure (usually three layers) of FNNs. The number of FNNs is determined by the number of faults considered. The lower level only contains one FNN, which processes all measured variables. The FNNs on the medium level are as well fed by all measurements but also by the outputs of the previous level. The upper level consists of an OR operation on the outputs of the medium level. This hierarchical structure can cope with multiple simultaneous faults.

5. PERSPECTIVES AND CONCLUSION In the field of fault diagnosis there exists a clear trend towards an integrated use of different FDI approaches. It is further to expect that new results in control theory and the rapid development of computer science will decisively influence the development of FDI theory. Among a great variety of FDI methods and techniques, the following prospective topics, due to their closed relationship to control theory and their practical importance, will receive much attention.

4.4 Fuzzy Residual Analysis

The two main categories of recent residual analysis methods are classification or pattern recognition and inference or reasoning. In both categories fuzzy logic based approaches play an important role, namely fuzzy clustering and fuzzy reasoning. Although both approaches utilise fuzzy logic, the first one is actually data based while the second one is knowledge based.

Improving the performance of linear FDI systems In the framework of FDI theory for LTI systems, application of the well-established robust control theory like IL-synthesis or LMItechnique promises the solution of FDI problems in systems with parameter uncertainty (Chen and Patton, 1999; Sadrnia et al., 1997; Rambeaux et al., 1999).

4.4.1. Fuzzy Clustering The approach of fuzzy clustering actually consists of a combination of statistical tests to evaluate the time of occurrence

24

Towards establishment of nonlinear FDI theory Due to its theoretical and practical importance, the demand on the establishment of a framework for nonlinear FDI systems will significantly increase. Kevertheless, due to its complexity, instead of finding a unified solution, research will concentrate on three different areas: application of nonlinear theory to a class of well-modelled nonlinear systems, adaptive nonlinear FDI for systems with model uncertainty and qualitative approaches as an alternative.

IDlep'atecl CoDllOI collllDlDd

COD~~-l

Control .igual , (process input)

Process control

• i Detection and estimation

Fault c!iagnosUo

lOffaults

•

Fig. 5. Integrated controller design • Switch over between a number of different control laws depending on which or what kind of fault occurs. The task of the FDI system is in general fault detection and isolation.

Integrated design in the context of a tradeoff between false alarm rate and missed detection rate The fact that many model based FDI problems are handled in the context of control theory is a logic result of the decisive contribution of the control engineering community to the establishment of model based FDI theory and technology. In this manner, essential FDI concepts like false alarm rate and missed detection rate are expressed in terms of robustness and sensitivity and the main research attention has been devoted to the residual generator design. A current study (Ding et al., 2000a) revealed some weak points of the existing approaches and demonstrates that an integrated design of residual generator and evaluator in the context of a trade-off between false alarm rate and missed detection rate may increase the FDI system performance.

Towards an integration of different computational intelligence techniques for FDI The future application of computational intelligence techniques in FDI is clearly marked by the combination of different techniques, i.e. neural networks and genetic algorithms either in order to provide a suitable training tool or to optimise the structure of the network (Marcu et al., 1999a) . The integration of neuro, fuzzy and genetic techniques offers the chance to combine their advantages and to compensate for certain disadvantages. Acknowledgment The authors would like to thank Dr. Schreier and Dr. Zhuang for their valuable contributions.

Towards an integrated design of control and FDI systems A significant contribution in the field of FDI in feedback control systems is the attempt to integrate the FDI function in a feedback controller (Murad et al., 1996; Niemann and Stoustrup, 1997). As shown in Fig.5, in addition to the process control, the controller also delivers information about faults. The main advantages of such kind of integrated design consist in

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• improvement of system performance by fully making use of the degree of design freedom • simplification of system design by transforming the robust FDI problem into a robust control problem, for which solutions are available. Application to fault tolerant control (FTC) During the past decade, FTC has received much attention (Patton, 1997; Blanke, 1999). A typical way to approach FTC problem is to integrate an FDI system in the control system. This requires that the design of FDI system should be carried out in the context of FTC. There exist two essential ways to achieve FTC with the aid of an FDI system • the controller parameters are on-line adaptive to the process changes due to the faults . It is realized on the basis of the estimation of the faults delivered by the FDI system;

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Bezdek, J. C. (1991). Pattern Recognition with Fuzzy Objective Functions Algorithms. Plenum Press. New York. Blanke, M (1999) . Fault tolerant control systems. In: Proc. ECC '99. Bonarini, A. and G. Bontempi (1994). A qualitative simulation approach for fuzzy dynamical models. ACM transactions on Modeling and Computer Simulation 4(4),285-313. Calado, J. M. F. and J. M. G. Sa da Costa (1999). On-line fault detection and diagnosis based on a coupled system. In: ECC'99, Kar/srohe. Chen, .1 and R. Patton (1999). Robust Mode/Based Fault Diagnosis for Dynamic Systems. Kluwer Academic Publishers. Chen, .1 ., C . .1 . Lopez-Toribio and R. J. Patton (1999). Non-linear dynamic systems fault detection and isolation using fuzzy observers. Proc. Instn. Mech. Engrs. 213, Part I . Dalton, T., N. Kremer and P. M. Frank (1999). Application of fuzzy logic based methods of residual evaluation to the three tank benchmark. In: ECC'99, Karlsrohe. Paper ID: F1055-5. de Kleer, J. and J.S. Brown (1984). Qualitative physics based on confluences. Artificial Intelligence 24, 7-83. De Persis, C. and A. Isidori (1999). On residual generation in nonlinear fault detection and some related facts. In: Proc. 5th ECC '99. Delmaire, G., J.P. Cassar and M. Staroswiecki (1994). Comparison of identification and parity space approaches for fault detection in single input single output systems. In: Proc. The 3rd IEEE Conf. on Contr. Applications. Ding, S. X., E. L. Ding and T. Jeinsch (1999). An approach to analysis and design of observer and parity relation based FDI systems. In: Proc. The 14th IFAC World Congr.. Beijing. Ding, S. X., P. M. Frank, E. L. Ding and T. Jeinsch (2000a). Fault detection system design based on a new trade-off strategy. Submitted to publication to CDC'2000. Ding, S. X., P. M. Frank, E. L. Ding and T . Jeinsch (2000b). A unified approach to the optimization of fault detection systems. Int. J. of Adaptive Contr. and Signal Processing. Dvorak, D. and B. Kuipers (1989). Model-based monitoring of dynamic systems. In: Proceedings of the 11th International Joint Conference on Artificial Intelligence. pp. 1238-1243, Detroit, MI. Edelmayer, A., J. Bocor, F. Szigeti and L. Keviczky (1997). Robust Detection Filter Design in the Presence of Time-Varying System Perturbations. Automatica 33, 471-475. Frank, P. M. (1990). Fault diagnosis in dynamic systems using analytical and knowledgebased redundancy - a survey. Automatica 26, 459-474.

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Proceedings of Workshop on Applications of Interval Analysis to Systems and Control. pp. 305-313. Girona, Spain. Kuipers, B. (1986). Qualitative simulation. Artificial Intelligence 66(29), 289-338. Leitch, R., Q. Shen, G. Conghil, M. Chantler and A. Slater (1994) . Qualitative model-based diagnosis of dynamic systems. In: Colloquium of the Institution of Measurement and Control. London. Leyval, L., J . Montmain and S. Gentil {1994}. Qualitative analysis for decision making in supervision of industrial continuous processes. M athematicas and computers in simulation 36, 149-163. Lunze, J. {1994}. Qualitative modelling of linear dynamical systems with quanti zed state measurements. Automatica 30(3), 417-43l. Lunze, J. (1995). Kunstliche Intelligenz fur Ingenieure, Band 2: Technische Anwendungen. Oldenbourg Verlag. Miinchen. Magni, J. F. and Ph. Mouyon (1994). On the residual generation by observer and parity space approaches. IEEE Trans. on Autom. Contr. 39,441-447. Marcu, T ., L. Mirea and P. M. Frank (1998). Neural observer schemes for robust detection and isolation of process faults. In: UKACC Int. Conf. CONTROL '98, Swansea, UK. Vol. 2. Marcu, T., L. Ferariu and P. M. Frank (1999a). Genetic evolving of dynamical neural networks with application to process fault diagnosis. In: ECC'99, Karlsruhe. Marcu, T., M. H. Matcovschi and P. M. Frank (1999b). Neural observer-based approach to fault detection and isolation of a three-tank system. In: ECC'99, Karlsruhe. Marcu, T., M. H. Matcovschi and P. M. Frank (1999c). Neural observer-based approach to fault-tolerant control of a three-tank system. In: ECC'99, Karlsruhe. Moore, R. (1979) . Methods and applications of Interval analysis. SIAM. Philadelphia. Murad, G.A., I. Postlethwaite and D.W. Gu (1996) . A robust design approach to integrated controls and diagnostics. In: Proc. of the 13th IFAC World Congress. Nguyen, H.T. (1978). A note on the extension principle for fuzzy sets. Journal of Mathematical Analysis and Applications 64, 369-380. Niemann, H. and J . Stoustrup (1997). Integration of control and fault detection: Nominal an robust design. In: Proc. IFAC Symp. SAFEPROCESS '97. Nikoukhau, R (1994). Innovations generation in the presence of unknown inputs: Application to robust failure detection. Automatica 30, 1851-1867.

Patton, R.J. (1997) . Fault-tolerant control: The 1997 situation (survey). In: Proc. SAFEPROCESS '97. Qui, Z. and J . Gertler (1993) . Robust FDI Systems and H oo-Optimization. Proc. The 32nd IEEE CDG. Rambeaux, F. F. Hammelin and D. Sauter (1999). Robust residual generation via LMI. In: Proc. of the 14 IFAC World Congress. Sadrnia, M.A., R.J. Patton and J. Chen (1997). Robust hoof JL observer-based residual generation for fault diagnosis. In: Proc. SAFEPROCESS '97. pp. 147-153. Seliger, R. and P.M. Frank (1993). Robust residual evaluation by threshold selection and a performance index for non linear observerbased fault diagnosis. In: Proc. of International Conference on Fault Diagnosis (Tooldiag '93). Shen, Q and R. Leitch (1993). Fuzzy qualitative simulation. IEEE 1rans. SMC 23{4} . Staroswiecki, M., J. P. Cassar and P. Declerck (2000). A structural framework for the design of FDI system in large scale industrial plants. In: Issues of Fault Diagnosis for Dynamic Systems (R.J. Patton, P.M. Frank and R.N. Clark, Eds.). pp. 453-456. Springer-Verlag. Tagaki, T. and M. Sugeno (1985). Fuzzy identification of systems and its applications to modelling and control. IEEE 1rans. Systems, Man and Cybernetics 15(1), 116-132. Ulieru, M. (1994). Fuzzy Reasoning for Fault Diagnosis. 2nd Int. Conf. on InteII. Syst. Eng .. Verscovi, M., A. Farquhar and Y. Iwasaki (1995) . Nemerical interval simulation. In: International Joint Conference on Artificial Intelligence IJCAI95. pp. 1806-1813. Wiinnenberg, J. (1990) . Observer-Based Fault Detection in Dynamic Systems. VDI Verlag. Diisseldorf. Yang, H. and M. Saif (1995). Nonlinear adaptive observer design for fault detection. In: Proc. of A CC. pp. 1136-1139. Yu, D. and D.N. Shields (1996). Bilinear fault detection observer and its application to hydraulic system . Int. J. of Contr. 64 . Zhou, D. H. and P. M. Frank (1999) . Nonlinear adaptive observer based component fault diagnosis of nonlinear systems in closed-loops. In: Proc. of the 14th IFAC World Congress. Zhuang, Z. and P.M. Frank (1997) . Qualitative observer and its application to fault detection and isolation systems. Journal of Systems and Control Engineering, Proc. of Institution of Mechanical Engineers, Pt. 1211(4}, 253-262. Zhuang, Z. and P.M Frank (1999). A fault detection scheme based on stochastic qualitati ve modeling. In: IFAC World Congress 99. Beijing, China.

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Table 1: Comparison of different FDI approaches aa,rnin :

minimum observer indices. n : Order of the process. s : Order of the FDI system FDF

DO

PSA

FA

System order

s=n

s~n

s ~ aa,min(~ n)

Model form

State-space

State-space

System matrix equation

Design Parameters

R(s) = V,L

R(s) = V,L

R(s) = V,L

Implementation form

recursive

recursive

non-recursive

s ~ aa,min(~ n) Transfer functions State-space R(s) recursi ve or non-recursi ve

Table 2: A brief overview of main full decoupling approaches

Eigenstructure assignment Unknown input observer Geometric approach PSA FA

Applied technique

existence condition

Theory of eigenstructure assignment Unknown input state observer Matrix pencil technique Solution of Luenberger equation Geometric control theory Solution of linear equation Solution of rational matrix equation

rank(CEd) = rank(Ed) more strict than (5) rank(CEd ) = rank(Ed) condition (5) condition (5) condition (5) condition (4) condition (2)

Table 3: Review of the solutions of the optimization problems J2/2 =

11~112 il

2'

J

-

00/00 -

"~II J oo/min -- "illl~;n' 11~llgo Ilarll (ar)(...J.O) lIilll:' 7f! min - amin 8! r

FDF

DO

J2 / 2

J oo / oo J oo/ min

(Edelmayer et al.,97) (Ding et al., OOb) (Hou & Patton, 96) (Ding et al., OOb)

PSA

FA

(Wiinnenberg,90) (Ding et ai, 99)

(Ding & Frank, 89) (Ding et al., OOa) (Qui & Gertler, 93) (Frank & Ding, 94)

(Ding et al., OOb)

(Ding et al., 99)

(Ding et al., OOb)

(Ding et al., 99)

(Ding et aI., 00 b)

Table 4: Overview of nonlinear FDI approaches Basic idea

Technique used

System form

(Hengy & Frank, 86) (Seliger & Frank, 93)

NSS LSF

Nonlinear observer Differential geometry

(Garg & Hedrick, 95)

NSS

Nonlinear observer

(Yu & Shields, 96)

LSF Transform. to a special form & NSS Transform. to a special form

Algebr. approach Geometr. approach Algebr. approach

general form general form a class of nonlinear system bilinear system

(Frank et al., 99)

NSS

Nonlinear observer

(de Persis & Isidori,99)

LSF Transform. to a special form & NSS

Geometr. approach Geometr. approach AIgebr. approach

(Hammouri et al., 98) (Ashton & Shields, 99)

(Hammouri et al., 99)

Algebr. approach

28

state affine a class of nonlinear system a class of nonlinear system general form general form