Survey of Robust Residual Generation and Evaluation Methods in Observer-based Fault Detection Systems

ELSEVIER

J. Proc. Cont. Vot. 7, No. 6, pp. 403-424, 1997 1997 Elsevier Science Ltd. All rights reserved

Printed in Great Britain 0959-1524/97 $17.00 ~- 0.00

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Papers Survey of robust residual generation and evaluation methods in observer-based fault detection systems

P. M. Frank* and X. Ding*

* Gerhard-Mercator-Universitat -GH-Duisburg, Fachgebiet Mess- und Regelungstechnik. Bismarckstrasse 81, 47048 Duisburg, Germany *FH Lausitz, FB Elektrotechnik, Grossenhainer Strasse 57, 01968 Senftenberg, Germany

Received 22 August 96; accepted 4 November 96

The paper outlines recent advances of the theory of observer-based fault diagnosis in dynamic systems towards the design of robust techniques of residual generation and residual evaluation. Emphasis will be placed upon the latest contributions using frequency domain techniques including Ho~ theory, nonlinear unknown input observer theory, adaptive observer theory, artificial intelligence including fuzzy logic, knowledge-based techniques and the natural intelligence of the human operator. Two representative examples illustrate the efficiency of the observer-based approach. © 1997 Elsevier Science Ltd

Keywords: fault detection; robustness; unknown input observer; nonlinear observer; adaptive observer; observer-based fault detection; fuzzy logic

Associated with an increasing demand for higher performance as well as for more safety and reliability of dynamic systems, fault diagnosis has received more and more attention. One area of active research is the development of model-based fault detection systems. A fault detection system processes on-line information of the process under observation, namely process input and output signals. The output of the fault detection system may be simply an alarm signal that takes two values, high for fault and low for fault-free or, more sophisticatedly, knowledge of faults such as location, spectrum or amplitude.

There exist a number of strategies to construct model- based fault detection systems. One of them, using observer techniques, has received much attention during the last years. Especially, substantial progress in control theory and computer capability has made it possible to apply observer-based fault detection techniques to complex processes including nonlinear and time-varying systems with considerable modelling uncertainty. Typi- cal for the classical observer-based approach is that one reconstructs measurements of the process with the aid of an observer using a quantitative mathematical model of the process and makes the decision on possible faults in the process on the basis of the analytical redundancy thus being created. Even though there are a number of different design procedures, the core of the resulting diagnostic systems is always observers or Kalman filters.

On the other hand, the involvement of knowledge- processing techniques leads to the concept of a knowledge-based observer that makes use of a qualitative process model (knowledge model). Its task in fault diagnosis is to reconstruct the symptoms corresponding to the observations of the process that can be used for a fault decision on the basis of the knowledge redundancy thus being created. Both analytical and knowledge redundancy may be subsumed under the term functional redundancy.

In the field of the quantitative model-based techniques, the observer-based approach is in contrast to the parameter estimation approach where the fault decision is performed by on-line parameter estimation. Both approaches have advantages and disadvantages in different respects, and there was much argument for and against each approach. Realistically one should admit that both methods are to a certain degree complemen- tary and are therefore best applied in combination 1.

The paper focuses on the observer-based approach. We also include in this framework the parity space methodology, because it was recently evidenced by several authors 2-5 that the parity space approach leads to certain types of observer structures and is therefore structurally equivalent even though the design procedures differ.

In theory, more attention has been paid to the observer-based approach than to the parameter estimation approach. This is probably due to the fact that the

403

404 Robust residual generation and evaluation methods: P. M. Frank and X. Ding

existing parameter estimation theory can readily be applied to fault diagnosis without major modifications whereas diagnostic observers are different from the well- known control observers and therefore deserve particular theoretical treatment. One of the essential differences is that diagnostic observers are primarily output observers rather than state observers as needed for control purposes. This has often been overlooked in the literature and has misled many practitioners to the erroneous opinion that for the observer-based approach the knowledge of state-space theory would be indispensable.

Another most important difference is that whilst control observers are used within a closed loop, diagnostic observers operate in an open-loop configuration. Therefore, modelling errors of the process, which can by no means be avoided in practice, are much more trou- blesome. This requires robustness with respect to model uncertainties. Actually, satisfactory robustness is the indispensable precondition for the practical application of a diagnostic observer scheme. A great deal of theoretical work is therefore devoted to the robustness problem, and the task of enhancing the robustness in the face of considerable modelling uncertainty is the subject of many publications in recent years 2'4,6-26.

Despite the deficiency of unifying mono- graphs 1~,22,27,2s, there is a solid theoretical foundation to the analytical observer-based approach as far as the time domain and linear systems are concerned. How- ever, extensions to the frequency domain design of linear observers 6--8'29-39 adaptive observers 32,36,48,58,59 and nonlinear robust observer schemes 11,36,39,43-45 have been elaborated upon only lately and research in these areas is still going on. Some relevant contributions in these areas for the design of robust residual generation and residual evaluation will be outlined in the paper.

In recent years, there is also a clear trend towards an enlarged involvement of knowledge-based and artificial intelligence methods, including qualitative modelling for residual generation and fuzzy logic for residual evaluation 9'12"40-44. This issue will briefly be addressed towards the end of the paper. Finally, we will outline the observer-based supervision of a three-tank system and an industrial robot as examples of a successful practical application of the observer-based methodology.

Background and problem formulation

Background

It is very interesting to notice that in practice, instead of residuals, output signals of the process under consideration are often directly evaluated and compared with a given threshold. Suppose that the process can be described by

y(s) = yo(s) + CAs)f( ) = 6.(s)u(s) + aAs)f(s) (1)

where u(s) is the input vector and yo(S) is the nominal observation vector, G,(s) describes the transfer behavior between the input vector u(s) and the nominal output vector yo(S). The effect of faults on the system dynamics is modelled by Gf(s)f(s) with f representing a fault vector and Gf(s) denoting a distribution transfer matrix, respectively. Under the assumption that no false alarm is allowed, the threshold should be the maximal value of the evaluated output in the nominal process operating state (fault-free) for example:

]] au u []e~-~ Jth for all possible u(s) and G,(s) (2)

with II'lle denoting some evaluation function. This yields

Jth = sup [1 y lie--- sup I1 yo lie (3) f=0

It is clear that a fault can be detected only if it causes the evaluated output y to be larger than the threshold:

[I Y lie =11 a.(s)u(s) + Gf(s)f(s) lie> Jth for all possible u(s) and G,(s)

(4)

which equivalently means (see below or Emami-Naeini et al. 8)

1t a f f [lee 2Jth = 2 sup 11 yo lie (s)

In practice, sup [[ Yo lie can be obtained by simulation, its value depending on the input signal u, and therefore could be very large so that, according to the above relationship, faults with smaller size become undetectable. A well known way to solve this problem is the utilization of knowledge of the nominal process transfer behavior. If one could model the dynamic processes accurately so that the desired process variables could be precisely estimated, the difference between the measurement y(s) and its estimation, called residual, can be used instead of the output y for the purpose of fault detection. In this case, the influence of the process input signals can be exactly eliminated so that Jth is nearly zero. This means that every fault can theoretically be detected. This ideal case is, unfortunately, rare in a real technical process. Perfect models do not exist nor are characteristics of possible model uncertainties, which are unavoidable in real technical systems, available.

Denote the residual with

r(s) = ar(s) + 6rAs)f(s) (6)

where Ar(s) represents the effect of model uncertainty on the residual r(s). Consequently, a fault is detectable only if

[I Grff ]]e~ 2Jth = 2 sup II Ar lie (7)

Thus, the core of constructing a fault detection system is to minimize the influence of disturbances and model uncertainties on the residual, and this can be achieved either

Robust residual generation and evaluation methods. P. M. Frank and X. Ding 405

• by utilizing additional information about the process such as qualitative knowledge, or

• by applying a robust technique to the fault detection system design.

Problem formulation

On the basis of the observation and discussion in the last sub-section, we formulate the tasks of constructing a fault detection system as follows:

• design a residual generator that eliminates the effects of process input signals and, if possible, also the effects of disturbances and model uncertainties on the residual generated;

• design a residual evaluator by selecting a suitable evaluation function [l[[e and determining the threshold Jth;

• if a full elimination of the effects of disturbances and model uncertainties on the residual is not possible, optimize the residual generator and evaluator to achieve the maximum set of detectable faults.

Observer-based residual generation

State of the art

In the analytical observer-based approach, the generation of residuals reflecting the faults is done by estimat- ing outputs of the process and using the estimation errors as the residuals. For the fault detection task, a single observer or Kalman filter is sufficient whereas, for the localization of the faults, properly structured sets of residuals are required. The latter can be generated by using banks of the observers, so-called dedicated and generalized observer schemes (DOS and GOS) 9. Depending on the circumstances, one may use linear or nonlinear, full or reduced-order, or fixed or adaptive observers (or Kalman filters).

The basic concept of an analytical observer-based residual generator is illustrated by the block diagram of a linear full order observer in Figure 1. By f we denote the vector of faults to be detected, represented by (unknown) time functions, and by d the vector of

PROCESS

d

MEASUREMENTS

MODEL

OBSERVER; KALMAN FILTER

Figure 1 Full order observer for residual generation

unknown inputs (disturbances, noise, modelling errors), to which the detection system should be immune. The ideal goal of a residual generator is to generate a vector r(t)such that r(t) = 0 asf(t) = 0 and

1. r(t) ~ 0 asf(t) ~ 0 for fault detection 2. ri(t) ~ 0 asf.(t) -¢ 0 for fault isolation 3. l imt_~ ~(t) - r(t)] = 0 for fault identification

where the f- represent the different faults to be isolated and ri the corresponding subsets of residuals.

A number of methods for observer-based residual generation have been proposed over the past two dec- ades. The most significant approaches are the fault detection filter, the innovation test, the dedicated and generalized observer scheme, and the unknown input observer scheme. In the face of an overwhelming literature on the subject, we simply refer to Patton et al. 22 and the papers cited therein.

In parallel, there have been similar efforts to solve the fault detection and isolation problem starting from the parity equations, The most relevant contributions to redefine the parity space approach, discover the con- nections to the observer-based approach and generate structured residuals were recently made by Gertler 4"13"~4.

In recent years, the studies concentrated more and more on the design of robust residual generators that are invariant or at least insensitive with respect to unknown inputs. These studies have converged to a well founded theoretical framework comprising a number of different approaches 3,5,9,1°J6,23.25,26.39,45. It is noticeable, how-

ever, that the established theory is almost entirely devoted to linear systems and to the design in the time domain, even though many processes in practice are nonlinear and, on the other hand, frequency domain techniques have made big progress.

In this section, we will therefore focus upon the latest attempts to approach the problem of residual generation in the frequency domain, and will then extend the theory of linear unknown input observers to the nonlinear and adaptive case. Finally, we will briefly outline the basic idea behind the concept of knowledge observer.

Basic principle of residual generator construction

Whilst the task of an observer for control purposes is to reconstruct the states of the process, there is normally no such need for diagnostic observers. Their task is to reconstruct the outputs (i.e. the subset of the state vector that is measurable) in order to create redundancy. Therefore, linear diagnostic observers can readily simply be designed as output observers.

A direct way to construct an output observer is using the input-output relation which is usually described by a transfer function in the frequency domain. This schema, also called the frequency domain approach, was introduced by Viswanadham et al. 45 and lately extended by Ding and Frank 6,29"34'35,37. The major features of this approach are, on the one hand, its potential to provide a complete solution and, on the other, its obvious physical purpose of checking the input-output transfer

406 Robust residual generation and evaluation methods." P. M. Frank and X. Ding

relation as well as the use of transfer functions instead of state-space descriptions.

To briefly outline the basic idea, consider a linear process described by

yz(s ) = G.(s)uL(s) + Ayt.(s) + Gf(s)f t(s) (8)

where Gu, Gf are known transfer matrices from the input vector u E R p, fault vector f E R q to the output vector y E R m, and the subscript L denotes the Laplace transformed time functions. Ay/.(s) is an unknown vector representing unknown disturbances and model uncertainty.

Viswanadham et al. 46 have proposed to construct the residual generator by factorization technique, which is applicable for all processes (both stable and unstable) and the residual generator can then be brought into the form:

rL (S) = (S)(y/. (S) -- G. (s)UL (S))

= t.(s)yL(s) -- g.(S)UL(S) (9)

where A?/.(s) and Ar~(s) are left coprime factors of G.(s) satisfying ~l~1(s)N~(s) = G.(s). Denoting the state space realization of the nominal transfer matrix G.(s) with

k( t ) = Ax ( t ) + Bu(t) , y(t) = Cx( t ) + Du(t) (10)

-~/'.(s) and N.(s) can be calculated as follows:

t~I(s) = I - C ( s I - A + L C ) - 1 L (11)

]Vu(S) = D + C(sI - A + LC) -1 (B - LD) (12)

with L ensuring the stability of the matrix A - LC. Note that this form for rz is identical with the

(transformed) generalized parity vector introduced by Lou et al. ~6 The relationship (9) was recently generalized by Ding and Frank 29 to

rL(S) = Q(s)[~l .(s)yL(s) - 17.(S)UL(S)] (13)

where Q ( s ) E R H ~ denotes a parametrization matrix yet free to select. The physical meaning of Q(s) is that of a postfilter as illustrated in Figure 2 which provides additional design degrees of freedom for, for example fault isolation or robust residual generation.

Recently, Ding et al. 7 have shown that

~l.(s)yI_(S) - N. (s )uz(s ) = yL(S) -- f~L(S) (14)

where )3L(S) is an output estimation delivered by an identity observer

5c(t) = AYe(t) + Bu(t) + L(y( t ) -- )3(t)),

f~(t) = eYe(t) + Du(t) (is)

Remember that Q(s) is a parametrization matrix yet free to select. Thus, it follows from the results given in Ding et al. 7 that all residual generators can be constructed by

rL (S) = Q (s) (yl. (s) - YL (s) ) (16)

with ~L(S) as an output estimation which is the output of any type of output observers in the classical sense of observer theory, e.g. Luenberger-type output observer, or identity observer or parity space estimator. This means that every residual generator is indeed an output estimation error system that is, if necessary, filtered by Q(s). This fact explains why in practice a lot of simple fault detection systems, consisting of an output estimator simply and experientially constructed, are successfully used for fault detection.

Frequency domain approach to robust residual generator

We now consider the effects of faults and model uncertainty AyL(s) on the residual rL(S). For this purpose, residual generators of form (13) are taken into account. Substituting the system Equation (8) into it gives

rL(S) = Q(s)l(4.(s)[Gf(s)fL(s) + AyL(s)] (17)

The model uncertainty Ay/.(s) can be divided into structured and unstructured uncertainty. For the purpose of fault detection, we only need to consider the following general form:

A y L ( s ) = G d ( s ) d L ( s ) (18)

with an unknown but bounded vector d

II d I1=< aa (19)

If rank Ga(s) = m and no information in the frequency domain is available, for instance Ga(s) = L the expression Ga(s)di.(s) represents unstructured uncertainty and otherwise structured uncertainty. Thus, it follows from Equation (17):

rL(s) = Q(s))f4.(s)[Gf(s)fL(s) + Gd(s)dL(s)] (20)

Y lPR°cEss 1

osT/ i r FILTER / ~ .> Q(s)

Figure 2 Residual generator in the most general form in the frequency domain

Robust residual generation and evaluation methods: P. M. Frank andX. Ding 407

This is the effectual form of the residual generator that is used to determine the parametrization matrix Q(s) for satisfying desired specifications.

Full decoupling. Perfect fault isolation and total invar- iance from the unknown inputs dL(s) require perfect decoupling not only among the faults but also between the faults and the unknown inputs.

The latter can only be achieved if the uncertainty is structured. In this case it follows from Equation (17) that the matrix Q must be chosen so that the following conditions are met:

Q(s)~l,(s)Gf(s) = diag((tl(s) ..... tq(S)) E Rnoc (21)

with ti arbitrary and

Q(s)f4.(S)Gd(S) = 0 (22)

Perfect fault isolation is achievable if

rank {Go(s) Ga(s)} = rank Gf(s) + rank Ga(s) (23)

rank Gr(s) = q (= number of faults) (24)

Clearly, no perfect decoupling is reachable when the uncertainties or parts of them are unstructured, since in this case rank Gd(S) = m.

The solution of the decoupling problem and the conditions for perfect decoupling have been derived in several other ways, e.g. by Massoumnia 17 by a geometric approach, Ge and Fang 45, Viswanadham and Srichan- der 2s, Frank and W~nnenberg 3 and Hou and Mtiller 15 in terms of unknown input observers, Patton and Kan- gethe 23 in the eigenstructure framework, Gertler and Singer 13 in the parity space and Viswanadham et al. 45 in the frequency domain (to mention only some).

It has been shown by Ding and Frank 35 and Ding 37 that for perfect decoupling the residual generator of form (9) is equivalent to that of (13). This means that by the generalized parity space approach which is equivalent t o ( 9 ) 45 , the same results can be obtained as by the generalized observer-based approach (13). In this case, there is no gain by employing a filter Q. On the other hand, it has been shown by Ding 37 that the residuals obtained by all observer-based time domain approaches with perfect decoupling (unknown input observer techniques, etc. 9"10"I7"26) can also be brought into the form (9), since the different time domain methods can just be interpreted as different factorizations of G,(s). There- fore, there is full equivalence of all such approaches as long as perfect decoupling is possible. In this sense we agree with Gertler's view 4 that the generalized parity space approach is equivalent to the observer-based.

If, however, no perfect decoupling is practicable and approximate solutions are to be found, then the generalized form (13) is indeed superior. One can then take advantage of the additional design freedom given by the matrix Q(s). From a physical point of view, the filter Q(s) allows use of the different frequency characteristics

of the effects of the faults and unknown inputs as an additional criterion for their discrimination. Therefore, from a more practical viewpoint, the generalized residual generator of form (13) can be seen as a useful extension of both the generalized parity space approach and the robust observer-based approaches in the time domain. It was one of the most remarkable achievements of advanced research in observer-based fault detection to show that there are indeed practical situations where perfect unknown input decoupling can be reached. In most situations, however, even if the uncertainty is structured this will not be the case due to the restrictive conditions (23), (24). Then only optimal approximations of perfect decoupling can be achieved.

Approximate decoupling. If the uncertainties are unstructured or if the conditions (21), (22) cannot be met, then thresholds Jth > 0 have to be established and the fault detection and isolation problem has to be redefined as follows:

(i) [I rL(s) lie< Jth(> 0) iffL = 0

[] rL(s) lie> Jth iffL # 0 and (25)

(ii) 11 riz(s) He< Jthi(> 0)if f Z, = 0

][ riL(S) lie> Jthi i f f z ¢ 0 (26)

The design goal is now to find a parametrization matrix Q(s) such that the thresholds become minimal in the face of the unknown inputs dL(s). This problem is known as robust residual generation.

Several time domain solutions to the approximation problem have been proposed. Lou et al. 16 have developed an algorithm based on a singular value decomposition in the parity space which, however, is not easy to apply. A simpler time domain algorithm was suggested by Frank and Wiinnenberg 3 (with a generalized version in W/.innenberg26). They maximize a sensitivity norm of the residual with respect to the faults, divided by an analogous norm with respect to the unknown inputs (28). The optimization problem reduces to a generalized eigenvalue - eigenvector problem which is easy to solve.

The frequency domain approach to the robust residual generation has some appealing potentialities. On one hand it offers powerful methods to tackle the robustness problem, for example, by using H~ theory. On the other hand, it allows the involvement of frequency specifications as additional criteria for better fault discrimination. Different approaches in the frequency domain have been recently proposed 18-21"29"30"34"35"3~'47-51, in which the robustness problem for structured and unstructured uncertainties has been solved on the basis of the generalized form (I 3).

To outline the basic idea of the design methodology, let us assume that we wish to maximize the performance index:

408 Robust residual generation and evaluation methods." P. M. Frank and X. Ding

J = II ar/Od II or II rL/adt I (27)

where ]].11 denotes some norm of the sensitivities with respect t o f and d, respectively. The design goal is to find a residual generator such that J becomes maximal. Using (17), J can be expressed as

J = [1 Q(s)M~(s)Gf(s) II (28) II Q(s)i~l,(S)Ga(s) II '

If we take the L2 norm for 11"11 and let Q(s) reduce to a vector denoted by q(s) then the optimization problem

JQ-~(~) max becomes, in generalized notation,

I[ q(s)G2(s)112 q(s) J = I1 q(s)Gl (s)Ilz ~ max (29)

where Gs(s), G2(s) stand for ~l.(s)Ga(s) and ~Q.(s) Gf(s), respectively. The maximization leads to

VOw)[Gz(/ co)Gr2(-j co) - .(co)G 0co)Gf(-jco)] = 0

(30)

i.e. a generalized eigenvalue-eigenvector problem. The solution to this problem in the frequency domain

finally yields 9

qopt(s) = qf(s) V(s), ,]opt = sup(,k(CO)) (31) (1)

Here, the eigenvector V(s) is the selector for the optimal residual generator and the maximal eigenvalue ~.(w0) is the value of the performance index at the optimal frequency w0. qf(s) represents a band pass that fishes out the spectral value of rL(jwo) at the frequency w0 at which J becomes maximal. Thus, the filter selects only that part of the frequency spectrum of the residual r which gives the best compromise between insensitivity to the unknown inputs and sensitivity to the faults.

As an alternative approach, consider the following optimization problem

j= [I Q(s)G2(s) ]l~ g(s) max (32)

]1 Q(s)GI(S) IIo~

which was solved in Ding et al. 34, Qui and Gertler 5° and Tyler 51 using H~theory. For the case that Gd(s) has no zeros on the imaginary axis, this leads to the result:

supj =II C;o (s)62o(S) Its, Q(s)

Qopt(S) = Qo(s)G-~lo (s)

(33)

where Glo(S) and G2o(S) are co-outer matrices of Gl(s) and G2(s) respectively, Qo(s) satisfies

Q-~ (-jCO)Qo(J'CO) < I, for all CO, Q-~ (-jCOo)Qo(jCOo) = I, (34)

COo "[I Gllo (s)G2o(S) IIo~ = max 9(G-(lo (jCO)G2o(jCO) ) tO

= ~(G{1o (jCoo)G2o(jcoo)) (35)

with 6(G(jw)) denoting the maximum singular value of transfer matrix G(jco).

This solution also has a significant physical meaning. In fact, the matrix-valued division G~lo(s)Gzo(S) com- pares the difference between the transfer matrices Gf(s) and Ga(s), and its Ho:norm defines a measure on this difference. To reach the maximal difference, which is achievable at the frequency Wo, a band pass Qo(s) is used. As a result, this strategy allows exploitation of all kinds of available frequency information no matter whether inherent in the unknown input signal d or within the structure of the path from d to y. In other words, it allows the involvement of frequency spectrum analysis, which has become a standard and powerful tool in practice, to improve the degree of robustness and fault distinguishability of the residual generator.

Recent studies 6,34 reveal that the selection of performance index J can strongly influence the performance of a fault detection system. It has been shown that the following index:

II Q(s)62(s) LI- [I Q(s)Gl(s) I1~

(36)

gives a more practical measurement of the robustness of a fault detection system, where II Q(s)Ge(s)[[_= inffe0 II Q(s)G:(s)f(s)II means the minimum influence of the faults on the residual. The corresponding optimization problem:

j = II Q(s)G2(s)II- a(~) max (37) ]] Q(s)G1 (s) IIo~

which is also discussed in the section 'Comparison of linear fault detection systems', has been partly solved by Ding et al. 6,34 and Hou and Patton 47.

An alternative performance index of the form:

Jl =11 Q(s)Gz(s) - I I 1 ~ / min,

J2 =11 Q ( s ) G l ( S ) I 1 ~ / min (38)

has been proposed and studied 18-2°,s°,51. The core of this approach is to reduce the difference between the residual r(s) and the faults f(s) as much as possible (the meaning of the performance index J1) and simultaneously to enhance the robustness of the residual against disturbance d. Thus, this approach provides us also with the possibility of identifying the faults. Using Hoe-optimization technique or #-synthesis one can solve

Robust residual generation and evaluation methods: P. M. Frank and X. Ding 409

this problem. It is worth mentioning that in Murad et al. is, Nett et al. ~9 and Tyler 51 the diagnostic problem has been solved in connection with the solution of the control problem. This leads to an integrated design of FDI and control systems.

An alternative strategy for robust residual generation in the case of unstructured uncertainties was recently suggested by Patton and Chen 24. The basic idea of this approach is to compute the distribution matrices such that an optimal approximation of disturbance decoupling is achieved. This optimization problem is solved via a singular value decomposition method for the rank- reduced approximation of a rectangular matrix, first introduced to fault diagnosis by Lou et al. 16

Nonlinear unknown input observer approach

Many nonlinear processes in practice cannot be represented by linear models, in particular when they are not operating at a fixed operation point. This is the normal case in fault detection, because at the occurrence of a fault the process runs out of its operating point. Hence, if a linear residual generator was used for a nonlinear process then, after the occurrence of a fault, the detection system would be submitted to increasing modelling errors and run out of its range of validity. As a result, it would release false alarms rather than detect and isolate the actual faults.

Even though this is a key point concerning the practical applicability of model-based fault detection, little work has been done so far to develop residual generators using nonlinear observers. Recently, Frank, Wfinnenberg, Seliger and Ding have intensively studied this problem 26"39'53-5s. They have extended the theory of linear unknown input observers for residual generation to certain classes of nonlinear systems.

For a brief discussion of this approach, consider the class of systems that can be described by

2 = Ax + B(y, u) + E1dl + KI~ (39)

y = c x + E2d,. + K2f2 (40)

The signals dl, d2 represent unknown inputs and the terms f i , f2 denote the faults• Note that the nonlinear term B(y, u) depends only upon y and u, i.e. upon signals which are directly available by measurements, 'observable nonlinearity'. It is therefore possible to compensate completely the nonlinearity by reproducing it using an observer of the form:

= F~ + JO', u) + Gy, r = LI2 + L2y (41)

The conditions which are to be met by the observer matrices in order to provide robustness to the unknown inputs and sensitivity to the faults can be stated as follows:

T A - F T = G C , Fstable, J (y ,u)= TB(y,u) (42)

TEl = 0 . GE2 = 0 , L2E2 = 0, L ~ T + L 2 C = O (43)

rank(TK~) = rank(K~),rank( ( G ) K 2 ) =rank(KQ L2

(44)

If these requirements can be fulfilled, the dynamics of the residual is governed by

b = Fe + GK2f2 - TKIflr = LIe + L2K2f2 (45)

The drawback of this elegant extension of the linear unknown input observer theory to a class of nonlinear systems is that the class of systems described by models matching (39), (40) is rather limited. Many technical or physical systems cannot be modelled this way. If this is the case, the given physical model must be transformed into the required form by a suitable nonlinear state- space transformation. The existence conditions for these transformations are very restrictive. Consequently, the class of models that are actually transformable is rather small. But even if the existence conditions can be satisfied, finding the transformation will be hampered by the necessity to solve nonlinear partial differential equations or the requirement of up to nth-order time derivatives of the input signal u, where n is the dimension of the model.

Therefore, a different approach that extends the class of transformable systems, because it requires weaker existence conditions, has been proposed by Seliger and Frank 53"55. It is based on the following more general model:

x = A(x) + B(x)u + E(x)d + K(x)~ y = C(x) (46)

where the unknown inputs are modelled as to represent parameter uncertainties. It is desirable to compute a nonlinear transformation z = T(x), separating the dis- turbed from the undisturbed portion of the model. This separation can be achieved if, and only if

OT(x) E(x) = 0 (47) Ox

This relation constitutes a system of l st-order linear partial differential equations which are to be solved simultaneously by z = T(x). The theorem of Frobenius can be applied to derive necessary and sufficient existence conditions for solutions of (47) 53̀ 55 .

Suppose solutions z = T(x) of (47) exist. On the assumption that a relation x=~o(z , y* ) exists, the model can be rewritten as

_ or (x) " Ox (A(x) + B(x)u + K(x)J) IX=,o(:,y.) (48)

where the output transformation y* = C*(y) denotes a subset of the set of available measurements y = C(x) which is subject ~o the condition


dim(y*) < dim(y) (49) k = Fe OT(x) Ox K(x)f, r = R(T(x) + e, C(x)) (57)

Suppose furthermore that a relation

R(r(x), C(x)) = 0 (50)

exists. Then a nonlinear observer can be set up in order to estimate the undisturbed portion z of the state x. The observer is of the form:

or( ) 2 = O~c (A(~c) +B(fc)u) + H ( ~ , y , u ) R ( L y ) I~=,0(.~.y.)

(51)

where the design freedom provided by the feedback matrix H(L y, u) can be exploited in order to stabilize the differential equation (53) governing the dynamics of the estimation error e = ~ - z.

This observer is called nonlinear unknown input observer or disturbance decoupled nonlinear observer 55 and the relation R(L y) can be used conveniently as a residual:

r = R(2, y) = R(z + e, y) (52)

The estimation error e is governed by

= p(e, t) - ~T~(~f) K(x) f (53)

where the system of nonlinear differential equations b = p(e, t) must be designed such that the equilibrium point e = 0 is at least locally asymptotically stable. The residual will then converge to zero in the fault-free case (i.e. f = 0).

On the other hand, all faul tsfwil l be reflected in e, if

rankCTo(~) K(x) ) = rank(K(x)) (54)

If, in addition to (47) and (54), the conditions

Or(x)

Ox at(x)

Ox

= FT(x) +

- - 8 (x) = "1 ( C(x) )

(55)

are satisfied, a residual generator with stable linear error dynamics can be designed 53. F is a stable constant matrix. ~o(C(x)) and ~l(C(x)) are suitable output transformations.

The residual generator is then given by the following equations:

= F2+ dPo(y)(+cbl(y)u, r = R(2, y) (56)

In this case, the estimation error and the residual evolve according to:

It is important to note that (41) and (56) are stable on the complete domain of the disturbance decoupling transformation. No such statement can be made regarding the observer described in Equation (51). Existence conditions for the discussed nonlinear observers and residual generators based on the Theorem of Frobenius can be found in the literature.

For the purpose of fault isolation, the set of all faults under consideration is to be divided into as many subsets of faults as are desired to be isolated. Subsequently, one must design as many residual generators as there are fault subsets. These residual generators must be robust to the unknown inputs as well as to certain fault subsets. This must be done in a way that allows a decision to be made uniquely to which subset an occurring fault belongs.

Nonlinear adaptive observer approach

One of the major advantages of the observer-based fault detection technique is that robustness with respect to model uncertainties can readily be accomplished 1°,36'56. However, this involves the risk that faults with slow time constants may not be detected ~ because the enhancement of robustness is associated with an accompanying decrease of the sensitivity of the detector to faults with slow time constants 38,57.

To overcome this difficulty, it was recently proposed to use adaptive observers 32,33,36,56,58. An adaptive observer is a dynamical system that estimates states and (slowly varying) unknown parameters of the observed system. One may expect that a residual generator based on an adaptive observer does not only maintain the important property of early detection of abrupt changes, but also delivers estimates of faults with slow time constants. Another motivation is that by applying on-line identification the process model can continuously be updated and the robustness of the residual with respect to model uncertainties can thus be enhanced.

With the construction and implementation of an adaptive observer, the realization problem becomes of primary concern. Bastin and Gevers 59 have recently introduced an alternative adaptive observer conception that is applicable not only to linear but also to a class of nonlinear systems and whose construction and imple- mentat ion is relatively simple. This type of adaptive observer also provides a novel and promising strategy to tackle the fault detection problem in nonlinear systems.

As shown by Bastin and Gevers 59, certain effects of nonlinearities may be handled as unknown parameters that can be estimated on-line. Thus, a system with dominant nonlinearity may be reduced to a nonlinear system with some unknown parameters that can be handled easily. In this way, many nonlinear observation problems become solvable.

Consider a nonlinear system governed by the following equations:

Robust residual generation and evaluation methods." P. M. Frank and X. Ding 41 1

P

Jc = a(x) + qo(X, u) + Z qi(x, u)Oi i= I

q

+ Z hi(x, u)f" + g(t) i=l

= a(x) + qo(X, u) + O(x, u)O + H(x, u) f+ g(t),

(58)

y = c(x) (59)

where x E R n is the state vector, u E R t the known input vector, y E R m the measurable output vector, and a : R"

R n, qi, hi : R n × RR l ---' R ~, g : R ---, R n and c : R ~ --* R m are assumed to be known and smooth enough. O(t) E RP is an unknown vector which represents unknown time-varying parameters, slowly varying faults or part of nonlinearities of the system. The deri- vative is bounded by

] O ] < M , O < M < < o c (60)

The vector 0 will be on-line estimated. Vector f E R q denotes abrupt changes (faults) in the system that are to be detected as early as possible.

The laws for the corresponding adaptive generator o f the residual r may be written in the following general form:

Residual generator" z = b ( L O , u,y), r = h(2,0, u,y)

(61)

Adapta t ion l a w ' 0 = d(L O, u, y) (62)

The structural diagram of the adaptive residual generator is depicted in Figure 3.

In order to apply the adaptive observer scheme to fault detection, the nonlinear system, Equations (58) and (59), has to be brought into the so-called adaptive observer canonical form (AOCF). As shown by Marino 6°, this requires the solution of the so-called

; , ° = ° °

= b(z,g.u.y)

h(z ,O,u,y)

Residual Generator

Figure 3 Adapt ive residual generator

Observer Error Linearization (OEL) problem that can briefly be formulated as finding a co-ordinate transformation that transforms a general nonlinear system into the AOCF. It is known that a one-to-one transformation exists only if some rigorous conditions are satisfied 6° which, unfortunately, do not hold in many practical cases.

In contrast, Ding et al. 3~ and Seliger and Frank 53 have shown that a reduced-order observer for which such a one-to-one transformation is not necessary can release these hard existence conditions.

It is well known 9,29 that a residual generator is the error system of an output estimator which can usually be constructed by a reduced-order system. Thus, it is reasonable to use a bank of output observers instead of a state observer• Following this. the corresponding so- called Output Observer Error Linearization (OOEL) problem can be formulated. For the sake of simplicity, we restrict ourselves to the description of the following problem.

Given a nonlinear system, Equations (58) and (59), and an initial state Xo: find (if possible) a neighborhood U of Xo, and transformations z = T(x) E R", (hi < n), F(y) E R defined on U such that

z,, = ? ( y ) = ?(c(x)), (63)

P q

- = Fz + O0(y..) + ~ ~,(y, .i0i + ~ ~,(x, u).~ i=1 i=1

:= rz + ¢'o(Y, u) + CO(y, u)O + E(x, u)f

(64)

and

rank E(x, u) = q, rank qJ()', u) = p (65)

for all

F =

z E T(U) and u, and

[i0 011 0 -.- 0

• -- 0 1

Note that the condition (65) ensures that faults are detectable 53, otherwise there exists a fault so that F,(x, u ) f= O. Condition (65) can be omitted if the parameter vector 0 only represents the model uncertainty.

I f the OOEL problem is solvable, then it is possible to construct an adaptive residual generator which may in general form be described by the following set of relations:

= F~ + . ( y , . )g + ~o(y, .) + Lr + f~(t)g,

0 = r~o-(t)r(t) (66)

(/(t) = R V t t ) + ,b(y, . ) , V(0) = 0

99(t) = K T V(t) + + ( y , u), r = ?(y) - 0 3

(67)

(68)

41 2 Robust residual generation and evaluation methods." P. lid. Frank and X. Ding

Here, F E R p×p is an arbitrary positive definite matrix, and

V(t) = [VT(t) . . . VmV(t)] T, Vi(t) E R (n'-l)xp

R = diag(R1 . . . . . Rm) E R ('-m)×(n-m)

R i

r i -~-

o

0 . . . 0 - r a

• .. 0 I - r i (m- l ) d

ril

rz2

ri(ni- 1)

EE R (ni-1)x(ni-l),

if(y, u) =

, i , ( y , u) =

u) =

u ) =

K =

KT=

[ ~ ( y , u ) . . . ~mV(y, u)] T E R ("-m)xp

~Pn (Y, u) - riqJz2(y, u) E R (m-l)xp u) ^s . . . %.(y, u)] v c R Xp,

qJa(Y, u) E R lxp

diag( K~ . . . . Kin) ~ R m~("-")

[ 0 . . . 0 1 0 . . . 0 ] E R l×("'-r)

In the above equations, L and ri, (i = 1 . . . . . m) have to be chosen such that the matrices F - L C and R are sta- bile matrices. Notice that since L and r; are arbitrary, the eigenvalues of matrices F - L C and R can be assigned arbitrarily.

The dynamics of the residual r with respect to the unknown parameter vector 0 and faults f are given by the following error system equation:

+ z(x,u)f

(69)

r = ~(y) - C2 (70)

= z - 2 , g = o - o , .2, = Q ~ - v ( t ) o

Q = diag(Ql . . . . . Qm),

Qi

1 0 . . . 0 --ril

0 1 . . . 0 - r a

0 . . . 0 1 -ri(ni_l)

0 . . . 0 0 1

E R nixni

F* = Q ( F - L C ) Q -1 , ~bT(t) = [~bT(t)...~bmv(t)],

~o 7( t ) = [O~oVi (t)]

Residual eva luat ion

Problem formula t ion

The second step of a fault detection procedure is to evaluate the residuals. This is a decision-making process which always comes down to a threshold logic of a decision function. If there are no uncompensated unknown input effects on the residuals due to a perfect decoupling, then the thresholds diminish to zero. Otherwise, thresholds different from zero have to be assigned.

In practice, there is usually such a great number of unknown inputs that, in the face of the limitations of available measurement information, a complete decoupling from all unknown inputs is hardly achievable even if the model uncertainties are structured. The situation becomes even worse if the uncertainties are unstructured, in which case a perfect decoupling in the residual generation stage is basically impossible. Hence the residuals or any decision functions built from them always deviate from zero even if no fault is present.

In this case, robust residual evaluation is the only way to keep the false alarm rate small with an acceptable sensitivity to faults. Robust residual evaluation can be accomplished in many ways, for example by statistical data processing, data reconciliation, correlation, pattern recognition, fuzzy logic or adaptive thresholds.

Here, we restrict ourselves to the adaptive threshold and fuzzy decision-making approaches.

Robus t residual evaluation using adaptive thresholds

Basic idea. The crux with fixed thresholds is that choosing the threshold too low increases the rate of false alarms, choosing it too large reduces the efficiency of fault detection• Evidently, the optimal choice of the magnitude of the threshold depends upon the nature of the system uncertainties and varies with the system input. One may therefore use thresholds that adapt to the input.

Figure 4 shows a typical situation. Consider the shape of the residual (or a decision function) with effects of model uncertainties from t = 0 on and a fault at t ~ tF. With a fixed threshold, the increase of the residual due to an input maneuver leads to a false alarm at trA and no detection of the fault. Using an adaptive threshold depending upon the system input allows the avoidance of the false alarm and the detection of the fault at tE.

The idea of adaptive thresholds was first introduced by Clark 22. He has chosen the shape of the threshold as a function of the input of the process in a more or less intuitive manner. A broader theoretical foundation of this strategy was given by Emami-Naeini et al. 8 in terms of the threshold selector.

Robust residual generation and evaluation methods." P. 114. Frank and X. Ding 413

FALSE ALARM ADAPTIVE THRESHOLD

/ \ \

Figure 4

I DECISION FCT. ]

t FA t F TIME

A d a p t i v e t h r e s h o l d t e s t o f t h e r e s i d u a l o r d e c i s i o n f u n c t i o n

Evaluation functions and thresholds. The first step of residual evaluation is to choose an evaluation function and, based on it, to determine the corresponding threshold. Among a number of residual evaluation functions, the so-called root mean square (rms) is often used in practice. The rms can be represented either in the time domain:

II r(t) t1~= J(r) = (r - t jrr(t)r(t)dt) 1/2

0

(71)

or in the frequency domain:

o) 2

II rLO'O ) = J(E) = J r*L(jo))rL(jog)d(.o) 1/2, o) I

ff = dO2 - - O} 1

(72)

where z and c denote the detection window in the time and frequency domain, respectively, and H'][e stands for an evaluation function.

The selection of residual evaluation functions often plays an important role for fault detection, especially in practice, and may strongly influence the performance of a fault detection system. Unfortunately, this problem is usually neglected by roost of the theoretical researches. In practice, the residual evaluation function may be selected according to

• how to evaluate the faults to be detected and • which kind of information we have about the pro-

cess and possible faults. Once the evaluation function has been selected, we are able to determine the threshold. A major requirement on the fault detection is to reduce or prevent false alarms. Thus, in the absence of any faults, 11 r lie should be less than the threshold value Jth, i.e.

Jth = sup ]l r lie (73) ay,f=0

Linear threshold selector

invariant model with additive unstructured modelling uncertainties can be represented by

yL(s) = (G.(s) + AG.(s))uL(s) + Gf(sff'L(s) (74)

From this we obtain the residual generator equation:

rL(s) = O(s)iif/l.(s)Gr(s)fL(s ) + ~Ids)AG.(s)uL(s)I (75)

As shown by Frank and Ding 52, one can find from (75) the following relation for the threshold

J,h -II GQMuu lie (76)

Here, II AG. tl< ~. denotes a known bound on AG. and Q again the parametrization matrix. It is seen that the threshold is no longer fixed but depends upon the input u, thus being adaptive to the system operation. A fault may be declared if II r lie> Jth.

In general, a threshold selector can be, according to (73), established as follows:

Jth = sup li Q~I.Gdd lie (77) d

Emami-Naeini et al. s have solved this problem under the assumption that the rms in the time domain is used as residual evaluation function H r lie. The similar solution based on the frequency domain rms (72) was performed by Ding and Frank 51. By a suitable choice of the frequency window ~ = a~2 - COl, one can find a threshold so that the robustness with respect to uncertainties can be increased. The expression for the threshold can be found by setting fL (s) = 0 in (20) which yields

Jth = sup I[ Q ( s ) i Q . ( S ) G d ( s ) d ( s ) lie (78) d

Notice that d(fio) can be written as

d(jo)) = [l(joo)(3d(w), II cl(jco) ]]2_< 1 (79)

This leads to

Jth = max a(3d(og)Q(jw)iQ.(jw)Gd(jw) )E -1/2 (80) ¢OE E

Assume that the input signal uL(jo)) is known before the process comes into operation. If the post-filter Q(s) is optimally chosen according to the algorithm given in the section 'Frequency domain approach to robust residual generator', then we have 6

max S(3a(w)Q(jog)J~I.(jco)Gd(jco) ) = 1 coEE

(Sl)

To outline the basic idea of the threshold selector, we consider unstructured modelling errors. A linear time

This means, somewhat surprisingly, that the threshold is a constant equal to e -~n. As a matter of fact, one may

414 Robust residual generation and evaluation methods: P. ltd. Frank and)(. Ding

expect a Jth that changes with the input signal. How- ever, if we observe 8d(W) in detail, the reason becomes evident. For instance, for a model with additive unstructured modelling uncertainties:

d(s) = [ ZXGw( ) (s) ]

so we have

I[ d(fio) [12 < ~(co) + ~(09) t1 u(j09) ]l~:= 82(c0) (82)

This shows that the information on the input signal is included in ~d(o~) which is further processed during the residual generator design. Figure 5 shows the resulting block diagram for the overall procedure for robust fault detection in the frequency domain.

Nonlinear threshold selector

Nonlinear threshold selection. In connection with the observer-based residual generation scheme for nonlinear uncertain systems, Seliger and Frank suggest a threshold selection technique allowing for a robust residual evaluation in the case that a complete disturbance decoupling of the estimation error is not possible 54. The residual will then differ from zero even if no faults occur.

In order to avoid false alarms, the residual must be evaluated by some algorithm before a decision on a fault is made. Since we do not make assumptions about the dynamic properties or the statistics of the unknown inputs, noise and faults, the only practical solution is the selection of a threshold Jth different from zero. The following simple decision logic can be employed:

, Process

:: Adaptive Generation

y(0

q Adaptive Detection Filter :,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I t r(t,~ ) Generation Transformation .. of J(v.)

~ J(s)

Generator Logic

:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I f . . . . . . . . . . . . . . . . . ? Alarm

Figure 5 Block diagram of the robust residual generation and evaluation algorithm in the frequency domain

I] r [Ioc~ Jth =:> a f au l t h a s o c c u r r e d (83)

[t r ]1~ < Jth ~ no fault has occurred (84)

Here l] r II~ denotes the Lo~ norm of the residual r which is defined by

l[ r I1~= ess sup II r II (85) t

Suppose the dynamic model of the process under consideration is given by

2 = A ( x ) + B ( x ) u + E l ( x , u ) d + K l ( x , u ) f (86)

y = C(x) + E2(x)d+ K2(x)f (87)

where d denotes the vector of unknown input signals representing, for example, modelling errors, and f denotes the vector of faults to be detected. Suppose, moreover, there exist state transformations z = T(x, u) as well as output transformations ~ = p(y) allowing for a system description according to:

= Pz + +(z, u, u) + Or(x, u_______A) el (x, u)d Ox

OT(x, u) K1 (x, u) f -+ (88)

f: = p(y) = S(x, d , f ) (89)

We assume that there are several possible state and output transformations for the above system description with, in general, different matrices F and nonlinearities ,i,(-).

For the system description (88), (89), the residual generator takes the form:

k z = F~ + * (L )5, u, u), r = *(~, )5, u) (90)

where

*( T(x, u), p(x, O, 0), u) = 0 V x, u (91)

The function q~(.) may contain a suitable feedback of the measurements or the residual in order to yield, for f = d = 0, in each case an at least locally asymptotically stable estimation error system of the form:

= Fe + l-'(e, t) - O T E * ( t ) d - O T K * ( t ) f , OX OX

r = qJ*(e, t, d , f ) (92)

where

**(0, t, 0 , 0 , ) = 0 Vt, r ' ( o , t ) = 0 v t (93)

The following abbreviations have been used in the above equations:

Robust residual generation and evaluation methods. P. M Frank and X. Ding 415

I'(e, t) = *( T(x, u) + e, S(x, d, f) , u, it)

- Yo(T(x, u), S(x, d,f), u, it) + ( f - F)z (94)

E*(t) = E, (x, u), K*(t) = KI (x, u) (95)

**(e, t, d,J) = 4,(T(x, u) + e, S(x, d, f) , u) (96)

The goal that is being pursued by introducing the transformations T(x, u), P(Y) and, if necessary, also a suitable feedback, is to obtain a partially linear estimation error system as stated in (92) which is asymptotically stable for f = d = 0.

The estimation error e can be written as

e = g F * r - - g e * \Ox ) - - g F * -~X K + eo

(97)

where

gF(t) = L-I{GF(S)} = L-I{(sI - F) -1} (98)

denotes the impulse response of the linear system portion and * is the convolution operator. The term e0 denotes the response to initial conditions. Using L~ norms one can write

O T E ' ( d lifo II e I1~ -</7~ II r(e, t) I1~ +¢~o~ II ax

aTK* t + ~ II ~ ( )fll~ + II eo I1~ (99)

where

fl~ -- i [{ gF(t) It at o

(100)

denotes the corresponding induced operator norm of gr(t). Since we assume that all functions are smooth and therefore satisfy Lipschitz conditions, we can compute the following upper bounds for the three terms on the fight-hand side of (99):

II r(e, t) I1~_< ×= Ii e t1~o, II O=-TE*(t)d I I ~ ca~ II d It~ dX

(101)

II ~ s c ( o f t l ~ < cs~ I l f l l~ (102)

which must be valid in the complete operating range of the plant under consideration. Substituting the inequalities (101), (102) into (99) one obtains

II e I1~ -< 1 - t ~ × (cd~ II d l t~ +cfo~ I I / l l~ ) 11 eo I1~ (103) +

1 - / 7 ~ y~

provided that, in accordance with the small-gain theorem, the inequality (103) holds

1 - f i ~ y ~ > 0 (104)

Suppose, moreover, that similarly an upper bound can be found for the residual, i.e.

II r U~c< O/ea c II e I1~ +~a~ IId I1~ +~c~ I l f l i~ (1o5)

Substituting Equation (103) into Equation (105) yield

Hrlioc <_ l - f l ocy~cdx+°~a~ Ildll~

+ ( 1 fl~ex_fl~g~cJ~ +offx) I l f "~

+ ~<~ tleotl~

This inequality can conveniently be used to define a detection threshold which excludes completely the possibility of false alarms due to disturbances. Notice that usually there are no disturbances or unknown inputs of arbitrarily high magnitude. It is therefore reasonable to exploit some a priori knowledge of the process under consideration in order to define the set of possible unknown inputs:

se ; {d ~ R' :]1 d II~< &~x} (106)

by means of the maximum L~ norm of d. The threshold can then be defined by the supreme of

H r [Is with respect to d in the fault-free case:

s .p I! r ll~lj=o J'* = - 7

_ fl~ Y~ + e~d~ (lO7)

This threshold guarantees that no false alarm will occur. Under certain conditions, the term

'~e~ I! eo N~ (lOS) 1 - f l~ y~

in Equation (107) can be omitted, thus reducing the threshold. I f this is done, the residual must not be evaluated before the transient due to initial estimation errors has sufficiently died out.

Nonlinear adaptive observer-based threshold selection. A residual evaluation algorithm based on the nonlinear adaptive observer approach to fault detection described in the section 'Nonlinear adaptive observer approach' was developed by Ding and Frank 32'33'36. The basic idea is to evaluate the residual vector in terms


of a norm of r. To this end, one may take the root mean square (rms) defined by

lo +'r

J(v) - ( ( l / r ) J rY(t)r(t)dt) 1/2 :=ll r I1~ Io

(109)

The condition for J(z) is that in the absence of faults it should stay less than a threshold Jrn:

J(r) < Jth (110)

F r o m Equations (69), (70) we obtain for f = 0 :

r(t) = G(t) * ((o(g)O(t) - (~(t)O(t)) (111)

t

O(t) = - ] F~oT (s)F(s)ds + O(to) + A0(t), t E [to, to + r]

to

(112)

where AO(t) = O(t) -- O(to) and G(t) denotes the inverse Laplace transformation of the transfer funct ion

G(s) := C ( s I - F*) -1

Assume that 0(t0) = 0. Re-write r(t) as

r(t) = G(t) * (~( t )O- Kr(t) + Kr(t) - V(t)O(t)) (113)

and define

e(t) = (o(t)O(t) + Kr(t) (114)

where K is some matrix whose function will be men- t ioned below. This leads to

r(t) = Gl(t) * (e(t) - fr(t)O(t)) (115)

f

e( t) = (o( t ) ( - I r~°-r (s)r(s)ds + aO( t) ) + Kr( t) to

(116)

Gj (s) = (I + 6(s)IC)-~6(s) (117)

F rom the small gain theorem we know that if the following inequality

II e II~<~ ~ II r Ib + ~ (118)

holds for some or, 3(> 0), then for the above system

II ~ I1~_<11 G~ tim ( ~ + II IS'(t) ~ I1~)/(1 - ~ II G1 I1~) (119)

Notice that to ensure that the above relation holds, one must choose the matrix K such that

1 - ~ II Gt I1~> 0

Following the inequalities

II IT"(t) ~ [[r = max 6(Vr(t) fs( t)) [] 0 lit (120) t~[to,to+r i

and [I 6 II~_< ao we have

1[ r Ib_< t~l(~ + a0Ctl)/(1 - 3 3 1 ) (121)

where 8(-) denotes the maximum singular value of a time-varying matrix and

t~l =11 al I1~,~1 = max 6( (zT ( t) ("( t) ) tE[to,to+r]

F(vT(t) V(t)) = max tE[to,to+Z]

We now show how to estimate or,/~ in Equation (118). Following Equation (116) the inequality below holds:

Ilell~ <_ l/r) ( II (o(t)r~or(s)r(s)112 ds) 2tit l0 t0

+[[~b(t)/x0lb +1[ Kr I1~

which, using Schwarz's inequality, results in

1/2

(122)

/ o + r l

11 e []r -< ( ( l / r ) I ( I 62(~°(t)F~°T(s)cp(s)FcpT(t))ds tO to

l

j [I r(s) I1~ ds)dt) 1/2+ [I K I[~fl r 113 to

+ max ~(~oT(t)~O(t))30 t~ [to, to + r]

(123)

t 0 + r t

<11 r lit ( 1[ Kl]o~ +( I J 62(~0(/)F~pT(s)~0(s) t 0 l0

F~o-r(t))dsdt) z/2) + max 6(~ov(t)co(t))3o tE[to,to+r]

(124)

Write

& =ll x" II~, tO+T l

/33 = ( I 1 6"2(~0(t)FcpT (s)~p(s)F~PT (t))dsdt)l/2 l0 t0

~2 = max t~(tpT(t)~0(t)), ot = fl0ot 2 tE[toto+r]

(125)

(126)

Robust residual generation and evaluation methods." P. 114. Frank and X. Ding 41 7

We then have an estimation for flr[l~ in fault-free case:

)~, r I1~< ~(~oa,. +c~ao) l (1 - ~ , . -/~1/33) (127)

based on which the threshold is defined by

Jth = fll (~0012 -~" ~1120) / ( ] - - ~1/~2 - - /~1~3) (]2g)

Notice that the thresholds are expressed in terms of a number of constants; some of them are known or can be off-line calculated, such as ill, f12, Oto, fin, but the others, fl3,~lcr2, are only on-line achievable, since they are dependent on y(t) and u(t) (see the definition of P(t),~o(t)). This may cause some troubles with the implementation, since the computation of singular values may be too expensive to be carried out on-line and the dependence of thresholds on the output vector y(t) may reduce the sensitivity to the faults and even prevent a successful fault detection. To overcome these difficulties, estimations for ~3~1, or2 are in some cases needed.

The block diagram of the resulting adaptive residual generation and evaluation system is depicted in Figure 6.

Fuzzy threshold logic

The problem of robust residual evaluation can be trea- ted in a different way with the aid of fuzzy logic. To outline briefly the basic idea let us again consider the case that the residual due to faults is superimposed by noise and the effects of disturbances and modelling errors due to incomplete decoupling, so that the residual will be non-zero even in the absence of faults. Typically, these effects will be time varying, that is the residual will fluctuate depending on the unknown time functions of the disturbances, noise and the input of the process. In the context of fuzzy theory, this is a typical fuzzy situation, and hence fuzzy decision logic seems to be a natural tool to handle the residual evaluation.

: . . . . . . . . . . . . . . . . . . . . . 21 ...........

O u t p u t ' , ~(t)~+ O b s e r v e r j

! : | Es~mator I- z ................................................

Adaptive Residual Generator

y(t) P

_r(t)

Generator]

_ . . _ , _ j Threshold i I II ~ ~ Alarm Selector .1

Adaptive Residual Evaluator

Figure 7 shows a characteristic shape of the residual associated with a fixed threshold. This bears the danger of false alarms due to the fact that the thresholds have to be chosen as small as possible because any increase of the threshold is associated with a loss of sensitivity to faults. As a remedy, one can replace the crisp threshold by a fuzzy threshold. This means that the line which constitutes the discrimination between zero and one in the decision logic is replaced by an interval with properly chosen upper and lower bounds and membership functions defining the variables {zero} and {one} in a fuzzy sense.

The crisp set {zero} is replaced by a fuzzy set {zero} characterized by a membership function lz(x) as, for example, shown in Figure 8. The parameter an is chosen as the level of noise to ensure that residuals within this range are definitely interpreted as zero. The parameter takes into account the effects of disturbances and modelling uncertainties beyond the noise level. In our case we have chosen for the S interval the standard triangle type of membership function.

The membership function of the fuzzy set {one} may be assigned as illustrated in Figures 9 and 10. In Figure 9, the classical way of threshold logic decision is shown

r(t)

Figure 7 threshold

t Threshold

,a . . A A v v v v V

,~, /I AA a AI ~ A . A~.

I V V V V ~ V V V V " /

! I / Threshold / v a . ~

Typical situation for residual evaluation with a fixed

fuzzy p(x) luzzy threshold threshold

x \ \ ,

a o -~ -a£ a'o ao+~ x

Figure 8 Membership function of the fuzzy set {zero) under consideration of disturbances and noise

X~ fault

disturbance /l Threshold

t

Figure 6 Concept of adaptive fault detection Figure 9 Classical threshold evaluation of a residual


fault x

........................................................................ disturbance t ~ x [ ............... Xma x

. . . . . . . p.(x) 1 0

t

Figure l0 Definition of a membership function for the fuzzy set {one}

again. Let the first maximum indicate a disturbance, and the second maximum a fault. It can be seen that the first maximum stays below the threshold but, if the residual slightly increases, it will surpass the threshold and cause a false alarm.

This type of false alarm can be avoided with a fuzzy threshold as illustrated in Figure 10. Beyond the noise level, the truth value one is characterized by the membership function/z(x), as shown on the right-hand side of Figure 10. Here again, a standard triangle type is chosen for/z(x) for ease of explanation.

It can be seen that if the residual due to the disturbance (peak 1) in Figure 10 (left side) increases slightly, this will cause just a small effect in the truth value of the alarm rate. In contrast to the conventional case where the small increase of the residual causes a false alarm, we now get a weak indication of inconsistency. Hence, we have replaced the yes-no decision by a continuous indication of a faulty situation.

The resulting membership function diagram of the fuzzy variables {zero} and {one} is shown in Figure 11. As one can see, there is an overlapping of the membership functions of the fuzzy sets zero and one, which is typical for the application of fuzzy logic. Therefore, one has to define and evaluate rules as a basis to make the final decision on the occurrence of a fault in a concrete situation. This task can be carried out either by the computer (using artificial intelligence) or by the human operator (using natural intelligence). Notice the simila- rities to the statistical decision-making concept where, instead of the membership functions, the probabilities are used and the decisions are made, for example by maximum-likelihood ratio tests. In our case it is not necessary to define the membership functions in terms of probabilities.

The fuzzy logic approach is illustrated in Figure 12. By this procedure the human operator can make the final decision involving all kinds of additional knowledge and experiences he has concerning the process.

~(x)

1

o

Figure 11

{zero} {one}

a 0 ao'l- ~ X~x

Membership function diagram

X

Since natural intelligence is still superior to artificial intelligence, this method of residual evaluation may improve the reliability of fault decision considerably. The application of the fuzzy decision logic to fault isolation was recently discussed by Sauter et al. 42

Comparison of fault detection systems

In the last two sections, we have outlined recent advances of the theory of observer-based fault diagnosis with emphasis on the latest contributions using frequency domain techniques, nonlinear unknown input observer and adaptive observer theory. In practice, we often face the problem of choosing a suitable FDI scheme from a great number of well developed approaches. This problem has been recognized and studied by Ding et al. 34'6

and Seliger and Frank 54.

Basic idea

To outline the basic idea, recall that there will always be a trade-off between the avoidance of false alarms on one hand and the detection of small faults which do not cause the residual to surpass the threshold on the other hand. That faults which exceed the given tolerance should be detected is the major practical requirement on a fault detection system. It is clear that a fault can be detected only if it causes the residual evaluation function to surpass the threshold, i.e:

II r lie > - J,h (129)

Notice, however, that a fault may, due to the model uncertainties, have different influences on the residual and, furthermore, on evaluation function. Taking this into account we say a fault is detectable if

inf ]l r Ile~ Jth (130) Ay

INPUT RESIDUAL

VISUAL

DIRECT PROCESS I FUZZYLOGIC I INFORMATION

4t

i NATURAL INTELLIGENCE I EXPERT KNCWLEGE

7 FAULT DECISION

Figure 12 Residual evaluation approach using fuzzy logic and natural intelligence

Robust residual generation and evaluation methods." P. M. Frank and X. Ding 419

i.e. a fault is detectable if it causes the residual evaluation function to surpass the threshold for all possible model uncertainties. Thus, the set of all detectable faults, denoted by Sf, can then be expressed by

Sf := { f : inf II r ]]<_> Jth} Ay

(131)

It is evident that the size of the set Sf depends on the constriction of the residual generator, i.e. Q(s), the selection of the residual evaluation function I ] lie as

well as the threshold. A reasonable evaluation basis for a certain fault detection system is certainly given by the size of set Sf because the larger it becomes, the more faults are going to be detected. In this sense, we can say that a fault detection system is optimal if the size of set Sf reaches its maximum.

Comparison of linear fault detection systems

To avoid the difficulty of determining the size of Sf, Ding et al. 6'34 suggested the use of the concept o f minimum detectable faults which was introduced by Emami- Naeini et al. 8

Minimum detectable fault, f,,i,, is defined by

IIf.,i. II< = i n f { l l f l l e : f 6 Sf} (132)

Assume that for different residual generators, QI (s) and Q2(s), we have two sets of detectable faults, S} and S 2, respectively. If

II f ~ . lie = i n f { l l f l l < : f~ S}} < l i fe / . lie = i n f { l l f l l e : fE S}}

(133)

then the set S} belongs to the set S}. Following this, it can be concluded that for a given residual evaluation function the set of detectable faults reaches its maximum, with respect to the parameterization matrix Q(s), if and only if t] fmi~ lie is minimized. The problem of comparing different fault detection systems is thus reduced to comparing minimum detectable faults. Since for systems described by (20)

inf II r li< = inf II Q~Io(GyfL + GadL) I1~ Ay

=1) Olfl~@fL I1~ - s u p 11 QM~GddL lie d

(134)

and the threshold is given by

] , h = sup II QM,,Ga& lie d

(135)

we finally have 6,8,34

II Qf4.Gffs_ I1<> 2J,h (136)

which results in

I lfmi. I1< = 2J,h( inf II Q M . G r f l l ) -~ UL=~

(137)

In case that frequency domain rms is used as an evaluation function, Ding et al. 6"34 have shown that

tl fmi. I1< = 2 maxo~, 8(,~a(a))Q(joo):~Iu(joo)Gd(jo)))

minoo~ g(Q (rio) M,, (jo))Gf(j~) ) (138)

where g(Gjeo) denotes the minimum singular value of a transfer matrix G(jo)).

On the basis of this scheme, Ding et al. 6,34 have furthermore proposed an approach to designing an optimal fault detection system. The core of it is to minimize the minimum detectable fault, which is equivalent to solving the following optimization problem:

inf II f~,;. lie = inf 2 maxo,~ 8(aa(w)R(joo)f4u(joo)Ga(jo)) ) Q(s) Q(s) min,oe~(Q(jo))~I,(joo)Gr(fio) )

(139)

Comparison of nonlinear fault detection systems

Compared to with the above discussion, the study on the comparison of nonlinear fault detection systems is much more sophisticated, and usually no analytical solution or expression like (138) can be achieved. Seliger and Frank 53 have proposed a method which allows comparison of the performance of different nonlinear residual generators.

Define the set of undetectable faults as follows:

= { f E R 1 ll r I1~ < J,h for some d E Sa} (140)

It is clear that SU and Sf describe a partition of the set R l in the sense:

(141)

Suppose now that the right-hand side of Equation (106) which is the upper bound of the L:c-norm of the residual equals the detection threshold Jth; i.e.

( .,Socae~ ) l _ fl~g ca~c + °taoc II d {{~

+ 1 7/7~--y~ cs~+°q~ I I f ( l~=J*h

(142)

Clearly, this equation describes a straight line in the (11 d [I~, I{ f [[~)-plane (Figure 13). The intersection f0 of the line with the II f l l~-axis is given by


Figure 13

II f fo

do

Upper bound of the residual norm

|

I1 d

J~oo Ctoc

fo = l-~=×= Cd~ + ade~ dmax (143) 13~tx,~ C + Olfc c 1-,8=×~ f ~

For any point (11 d i l l , ILf[l~) located below the line, the actual L~-norm of the residual stays smaller than the detection threshold. Thus, any fault f satisfying

II f H~ < f0 (144)

certainly belongs to the set ~{f; i.e.

{ f E R f : l l f l l ~ < fo} C SZ (145)

These faults are therefore undetectable. In other words, f0 defines the minimum detectable fault. As mentioned in the last subsection, in order to reduce the size of the set Sf, which is equivalent to increasing the set Sf of detectable faults, one must therefore reduce the minimum detectable fault f0 which, in this sense, can be understood as a performance index describing the qual- ity of the residual generator.

The performance index is a function of the constants which previously have been defined in order to evaluate upper bounds for the nonlinearities and the residual. These constants are in turn directly related to the transformations the residual generator is based on and are therefore strongly affected by the choice of the transformation. According to Equation (143), the performance index will be mostly affected by the constants Cd~c and c~d~ versus C f o c and ¢xfo o.

By minimizing this performance index (143), one could find an optimal residual generator. Due to the complexity of this minimization problem which is mainly caused by the nonlinearity of the models, it is virtually impossible to derive a generally applicable, Systematic optimization strategy. This restricts the performance index to compare different residual generators rather than to find systematically an optimal solution.

E x a m p l e s o f application

Finally, we want to demonstrate the applicability of the observer-based fault detection methodology with two

concrete practical examples. To this end, we have chosen a three tank system and an industrial robot MAN- UTEC r3.

Fault detection at a three tank sys tem

Figure 14 shows a draft of the experimental setup of a three tank system. This process has the typical dynamic characteristics of tanks, pipelines, etc., which are widely used in the chemical industry. Wtinnenberg z6 had successfully applied a nonlinear observer scheme to detect and isolate faults such a leaks in tanks.

Using the incoming and outgoing mass flows under consideration of Torricell's law the dynamics of the three tank system can be modelled by

AI~I = Q1 - alsl3sgn(hl -- h3) x/2g I ha - h31 + ayl

A1~2 = Q2 - a3s23sgn(h3 - h2)v/2g ] h3 - h2 ]

- a2so v/2gh2 + Qf2

Ah3 = alSl3sgn(hl - h3) x/2g i hi - h3

- a3s23sgn(h3 - h2)¢2g I h3 - h2 [ +Of3

where Q1 (t), Q2(t) are incoming mass flows, hi (t), h2(t), h3 (t) are the water levels of each tank and measured and QfI( t ) , Qf2(t) and Qf3(t) denote faults representing undesirable mass flows into the tanks caused by leaks or plugging in the various tanks or pipes. The three circular tanks have the same cross-section A and are intercon- nected via circular pipes with cross-sections s~3, s23. The outlet pipe is also circular with cross-section so. al, a2, a3 are scaling constants and g is the gravity constant.

Based on the system description given above, Wtinnenberg had developed a nonlinear observer-based scheme to achieve a successful fault detection and isolation 26. The key to this scheme is the construction of three nonlinear observers described by

1 Zl = - l l z l + A (Q1 - alsl3sgn(hl - h3) x/2g I hl -h3 ])

- l lh l , r l = ll(hl - zi)

Pumpl

Figure 14

Pump2

Ql(t) A Q 2 ( t ) ~ ~ : :

Qou:O)

Setup of three tank system

Robust residual generation and evaluation methods." P. M. Frank and X. Ding 4 21

1 zz9 = --12Z2 + -'~ (Q2 - a3s23sgn(h3 - h2)v/2g { h3 - h2 I

- a2so 2 v / ~ 2 ) - 12h2, r2 = 12(h2 - z2)

1 33 = -13z3 + -~ (alsl3sgn(hi - h3)x/2g [ h~ - h3 r

-- a3s23sgn(h3 - h2) x/2g [ h3 - h2 I) - 13h3

r3 = 13(h3 - - 23 )

where l~. 12, 13 should be positive to ensure the stability of the observers. In t roducing el(t) = hi(t) - zi( t) , i = 1,2, 3. yields

kl( t ) = - h e l ( t ) + Qf l ( t ) , r l ( t ) = lle~(t) (146)

/'2(t) = -12ez( t ) + Qf2(t), r2(t) = ~e2(t) (147)

e?(t) = --13e3(t) + Qf?(t), r3(t) = 13e3(t) (148)

It is evident tha t each residual is sensitive to a single fault only. This leads to a successful fault detection and isolation.

The fol lowing figures show results o f experiments f rom the on-l ine application o f the developed scheme to the physical system. The dynamic behavior o f the residuals with respect to different faults:

• leak in T a n k 3 from t = 8 sec to t = 30 sec; • leak in T a n k 2 from t = 70 sec to t = 92 sec; • leak in T a n k 2 from t = 120sec to t = 140sec; • plugging in the connect ion between Tank 2 and

Tank 3 for t > 3sec

is given in Figures 15 and 16. They demonst ra te a successful fault detect ion and isolation.

Fault detection at a robot

The efficiency o f observer-based fault detection method- o logy was recent ly studied by Schneider e t a / . 61"62 with

(a) 60

2O

- 2 0 = • y, -60

.... :- ~ " -:-'--':-----:'"-'---'-'- ---+ (b) ..... : "- "--.-'---.",--.'----:--' ....'-.. -' .....

. . . . . . . . . i I,!, . . . . . . : : .... : :. :.--.--.---.-:-.-.-.-: .... 60 - ,: - i'-": -;-'..:'.4-- -i -:--: ..... ....

° ; . . . . - z . . - ~ . - , ' - : ~ ; - . . 2 ' :

2-ZZZ 21 221 • -.-2-.-- + . - . " - - - ; . - -- b . - -- ; -- -- ; - . -- , ' - --4 . . . . .

.... ..... i i ! i i i i i i .....

15 Time (see) 135 15 Time (sec~ 135

Figure 15 (a) R e s i d u a l 2: leak in t ank 2 (b) Residual 3: leak in t ank 3

(a) 60~

2o

¢'~ -20

-60

..... i .+.-'--.+--.'-..--'- --i--.-!-•- -i .....

. . . . . . 2 . . . , 2 . . - ~ . . . , ~ - . . : . . . . . ~ . . . . : - . . ~ . . . . . . . . . .

. . . . " - ' . - . ; - - - • " . . . . ; . - - . ' - ' . . . , . ' . . . . ; - . - - - : ' . . - 4 . . . . .

. . . . h i t t

21 Time (scc) 189

"" -20 '----'.'--r.--'."r"'."-r":--'. --"

,~ , . - . . , : - . - : - - - - 1 : . - ~ . - . . , - . . . . : , . . . , - . - ~ . ---- ; . . . . .

_1 Time (see) 189

Figure 16 (a) R e s i d u a l 2: p lugging between tank 2 and 3 (b) Res idua l 3: p lugg ing be tween t a n k s 2 and 3

an industrial robot which was opera t ing under real-life conditions•

The mathemat ical model o f a robo t is normally given in the form:

J(q( t ) ) i j ( t ) = xd(q(t) , q(t)) + xg(q( t ) ) + f (u ( t ) ) (149)

Here, the vector q ( t ) represents generalized position co- ordinates, J is an inertia mass matr ix , X d is a vector o f Coriolis and centrifugal torque and X g is a vector o f gravitat ional torque. The driving torques are described

b y f ( u ( t ) ) . It can easily be shown 61 that this equation can be

b rough t into the following f o r m by suitable transfor- mat ions:

( o,,, x( t ) = j - z ( x d ( q , q) + xg(q) + f ( u ) ) '

-*-B(u~t),y(t)) ' " ~--'-Kt

(15o)

\0(t)) (lSl) Obviously, this state equat ion is the same as Equat ions (39) and (40) where in this special setup A, Eb E2 and K 2 a r e z e r o .

The fault f l ( t ) which is o f p r imary concern in this s tudy represents influences o f unmodel led torques on the robot where a t r ans format ion f * l ( t ) = J (q( t ) ) f l ( t ) performs a decoupling of these torques on the generalized axes.

There are considerable unmodel led torques primarily due to friction. When an observer-based residual generation is applied, the friction affects the residual evok- ing respective deviations f rom zero. Hence, in order to avoid false alarms, the threshold would have to be increased which, however, would reduce the sensitivity to faults. Practical investigations with a M A N U T E C r3 robo t have shown that in this case friction cannot be modelled in a simple form like Co lumb or viscose friction. Instead, the friction has a posit ion dependent characteristic which turns out to be exactly reprodu- cible. This fact can elegantly be exploited for fault diagnosis.

The key idea o f the fault detect ion scheme implemented here is that the residual is t reated versus posit ion ra ther than versus time. This makes it possible to adapt the threshold to the friction characteristic. One can also interpret this as compensat ing for the friction characteristic in the residual. This makes it possible to elimi- nate the effects of the friction, which means that false alarms can be avoided and the detector can be made highly sensitive to faults.

In our study, we first determine the friction characteristic with the aid o f an u n k n o w n input observer scheme. In contrast to the pa rame te r identification method or break-away experiments, this can be performed dur ing normal robo t opera t ion.

422 Robust ms/dual generation and evaluation methods. P. M. Frank and X. Ding

The resulting friction characteristics for two axes of the MANUTEC r3 robot are shown in Figure 17. Dif- ferent measurement cycles are performed and repeated and the friction force is plotted versus generalized posi- tions (positive friction branch corresponds to a positive velocity and vice versa). In Figure 17a, partial wear-out at angles of q = 40 ° and q = 100 ° can be observed. This may result from former intensive and heavily loaded robot movements at specific configurations. Figure 17b reveals a periodic oscillation which results from defects in train gears (here a ratio of n = 5) and which might be due to misplaced bearings, etc. Notice the perfect reproducibility of the friction characteristic. The threshold can therefore be selected so as to adapt to this characteristic or, in other words, the characteristic can be compensated 61"62.

The observer-based fault detection system with compensated friction characteristic is next used to gain additional process information by detecting and moni- toring external torques. If the friction characteristic is compensated in the residual, then changes are solely related to interactions with the environment. Such interactions may be contact forces during mounting or fitting operations, soft collision (no emergency shut down), loss of pay load, etc.

In the laboratory of our industrial partner, a test environment has been installed where a robot performs some standard transport and mounting operations in order to study the power of the detection system.

As a result, intentionally introduced faults could be detected. For example, a soft collision with a flexible spring brought into the regular path of the gripper as

sketched in Figure 18a could be detected reliably (Figure 18b), giving hints on critical situations and pos- sibly severe crashes.

The detection of external interaction proves to be robust enough so that it can even be used for fault identification, i.e. to measure external forces and torques. From a comparison of reconstructed (observer technique) with measured (force/torque sensor) torques that were externally applied, one can conclude that the observer-based technique is a reliable method to gain additional process information even in terms of quantitative figures. This is demonstrated in Figure 19.

Figure 19a shows the velocities of the test trajectories, and Figure 19b gives the measurement result, where the identified torques (dotted line) and measured torques (solid line) are plotted. The accuracy of the results obtained by identification lies within a :t: 2 Nm range.

In summary, this example shows that the observer- based technique can be applied successfully to robots to monitor and analyze friction and, using a compensation scheme, detect small external torques immediately. Combining both features, the observer-based technique can be used as a powerful tool for fault detection and process supervision.

Strategies of implementation

A fault detection system can be implemented using either analytical or knowledge-based techniques or combinations of both, (Figure 20). Analytical residual generators and evaluators are the core of analytical

Figure 17

(a) ~° 6~

40

zg2~

t ~ - 2 0

-°°I - 8 0

-200 -1~0 -100 -50 0 50 200

Position [deg]

(a) Friction characteristic

(b) ~

40

._g o

~ - -2Q

- 4 0

-60 -150

of axis I (b) Friction characteristic o f axis 2

- ~ oo - 50 0 50 1 oo

Position [deg]

] i bo

(a)

E b z

" o

g:

(b) 50

l o

5

2 2.5 3 3.5 4 4 5

Time Is]

Figure 18 (a) Test environment (b) Residual of soft collison

Robust residual generation and evaluation methods." P. M. Frank and X. Ding 423

(a) ~r

q _ a J , . , , . . . . .

0 1 2 3 4 5 6 7 8 9 ~0

Time [s]

Figure 19 (a)Velocities of the test trajectory (axes 1 to 3) (b) Measurement torques to axis 1

tool-boxes. The inclusion of artificial intelligence leads to the concept of diagnosis expert systems in which com- monly analytical and heuristic information and knowledge processing are combined. While expert systems have not been very successful for control, they are widely used for fault diagnosis systems 4°-42,44. An alternative approach is the implementation of combining the analytical techniques with the natural intelligence of the human operator. This leads to a computer assistant human supervisory concept as suggested by Frank and Kiupe112. This technique is the most powerful available as long as natural intelligence is superior to artificial intelligence, which is still the case.

Conclusion

The paper presents a survey of advances in the theory of observer-based fault diagnosis. Because of the current tremendous research activity in this field, it was not possible to provide a comprehensive representation of the scene. We have therefore focused on those contributions which we think close a gap in the existing theory and may gain some relevance for future research

,' MODEL-BASED • ( MODEL-BASED • ~..,..~ROA OH

• DIGITAL COMPUTER HUMAN OPERATOR EXPERT SYSTEM

i COMPUTER + COMPUTER + NATURAL HUMAN HEURISTIC ANALYTICAL RESOURCES KNOWLEDGE INFORMATION

PROCESSING PROCESSING I

i 1 NATURAL "i ( ARTIFICIAL "i (SYSTEM "1

L 'NTELLIGENCE J [ INTELLIGENCE l , ~ THEORY J

; D'EXPERTi FANA 'C I ( s Y ~ M ] j :~Di !

L joQ Box iJ t COMP~ER~AGSIS~ED i

H U M A N ; : S U P E ~ V B o R ; Jl

Figure 20 Strategies for implementation of fault diagnosis systems

] (b) 'o!

'i 6[-

E Z

"i !! -2i

g"---J;i'

/,'.,,^ ,: ,'i ,

ip I

C 2 5 4 5 6 7 8 9 ' 0

Time [s]

(solid line) and fault identification (dotted line) of externally applied

and practical applications. Though the practicability of the outlined methods is basically out of question, their practical significance is still an open question. However, the examples discussed in the paper give rise to great encouragement and may help to motivate intensive future efforts towards the practical application of these ideas.

Acknowledgements

The authors owe gratitude to Mr Kiupel, Mrs Seliger- K6ppen and Dr Seliger for their scientific contributions and to Mrs Appelt and Mr G6bel for their support in the production of the paper.

The paper is dedicated to the memory of Jtirgen Wfinnenberg, a wonderful person and most talented scientist, who was much too early taken away from us forever.

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Survey of Robust Residual Generation and Evaluation Methods in Observer-based Fault Detection Systems