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A Hybrid Neuro-Fuzzy and De-Coupling Approach Applied to the DAMADICS Benchmark Problem
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A HYBRID NEURO-FUZZY AND DE-COUPLING APPROACH APPLIED TO THE DAMADICS BENCHMARK PROBLEM Faisel J Uppal, Ron J Patton and Marcin Witczak Control & Intelligent Systems Engineering, University of Hull, Hull, HU6 7RX, United Kingdom E-mail: [email protected] [email protected]. [email protected]
Abstract: A novel multiple-model fault detection and isolation scheme for non-linear dynamic systems, that of a Neuro-Fuzzy Decoupling Fault Detection Scheme (NFDFDS) is presented, a hybrid scheme incorporating both neuro-fuzzy and model-based methods. The FDI scheme employs local optimal observers designed according to minimum state estimation variance. An application of FDI for an electro-pneumatic valve actuator in a sugar factory is presented. Key issues of finding a suitable structure for detecting and isolating nine realistic actuator faults are described. Copyright CO 2003lFAC Keywords: Fault detection and Isolation, Multiple model observers, Neuro-Fuzzy networks, Unknown input observers for FDI, Actuators
I . INTRODUCTION
'uncertainty' and this usually means that the robust linear and non-linear estimators need to be considered. The main advantage of the optimal observer is that it can be used to detect faults in the presence of both modelling errors and noise. The FDI problem then follows as an extension to the global estimation problem. The paper is structured as follows. Section-2 gives an overview of the proposed FDI scheme. Section-3 describes the formulation of NF multiple-model identification and observer parameter generation. Section-4 presents methods of determination of the disturbance and fault distribution matrices. Section-5 overviews the global disturbance de-coupling observer and an application of this scheme for an electro-pneumatic valve under DAMADlCS is presented in Section-6.
lt is well known that for fault detection and isolation (FDI) of non-linear dynamic systems, the problem of discriminating between the uncertain model behaviour and the faults presents a significant challenge. This paper presents a novel FDI scheme using the neuro-fuzzy (NF) multiplemodel paradigm and robust optimal de-coupling observers. This new scheme is called the 'NeuroFuzzy and De-coupling Fault Diagnosis Scheme' (NFDFDS) in which the multiple operating points are taken care of through the NF modelling framework . The structure also provides residuals that are de-coupled to 'unknown inputs', making use of the earlier research on unknown input decoupling (Chen, Panon, 1996; 1999 (and the references therein» . The NF paradigm exploits the combined abilities of neural networks and fuzzy logic and is an efficient modelling tool for non-linear dynamic systems because of its approximation and reasoning capabilities. Most real application systems suffer from disturbances and noise and varying operating conditions, leading to a challenging modelling requirement. The model-reality differences have an associated
2. FDI SCHEME OVERVIEW The propose scheme comprises two main parts: a set of NF based de-coupling observers and diagnostic logic. The first part generates a set of fault symptoms ( r1 ... rt) at the k-th sampling instant, in the form of a structured residual set, whilst the second part performs an analysis of the residuals to determine the nature and location of the faults . In this scheme, each residual is
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designed to be sensitive to a subset of faults. Ideally. each residual is sensitive to all but one fault. called a 'Generalized Residual Set' (Based on the generalized observer scheme of Frank, 1987; Chen & Patton 1999). Each 'Fault Diagnosis Observer' in Fig. (I) is a non-linear system comprising a number of linear (integrated by fuzzy fusion) sub-observers each one corresponding to a different operating point of the process. The input to each observer is the process input and output. Ut and YPt respectively and the number of sub-observers depends on the number of operating points needed to achieve the required approximation. Their outputs are combined by fuzzy fusion to generate the optimum output estimates. Computing the difference between the actual and estimated outputs generates a set of residuals. The set of fuzzy observers together with the NF multiple-model and diagnostic logic forms the new scheme called the 'NFDFDS·. The NFDFDS is based on the identification of a particular type of neuro-fuzzy model (a simpler form of TS structure) that can generate the parameters for the observers together with a fuzzy measure of the validity of each of them. It also gives a linguistic description of the system.
In the proposed approach a special form of the TS fuzzy model is used to combine the NF identification strategy and model-based observer FDI synthesis in a single framework to formulate the so-called 'NeuroFuzzy and De-coupling Fault Detection Scheme' (NFDFDS). The conventional fuzzy observer design Wang et al.• (1995) involves an iterative procedure. selecting eigenvalues arbitrarily using pole-placement and solving inequalities using LMI to find a single positive definite Lyapunov matrix P common to all sub-models. This common matrix ensures stability of the global observer system. If such a matrix P does not exist another set of arbitrary eigenvalues are selected and the whole procedure is repeated until a solution is found. These computational limitations and stability constraints are not present in the proposed scheme. which simplifies and speeds up the joint identification and FDI problem. The reason for the improvements in design complexity can be attributed to the fact that the global state estimation is not necessary for FDI and hence there is no need for unnecessary global feedback. A method to generate local optimally de-coupled observers automatically from the NF model is proposed. This NF model operates in parallel to the local observers so that they can be modified or tuned on-line. In this way the reasoning and approximation capabilities of the neuro-fuzzy networks can be exploited. The global output estimate of the proposed fault detection observer is obtained by a fusion of all linear sub-observer outputs. It should be noted here that for the FDI. the residual signal that describes the fault symptoms is the main requirement. Hence. the global state estimation is not computed. However. each observer has a different state variable model from all other local observers. The diagnostic signal (the residual) is the difference between the estimated and real system outputs. It is shown that if all the local observers are stable. it can be expected that the global observer is stable. This represents a significantly relaxed stability requirement over the TS based observer case. The equivalent fuzzy logic description of the NF network consists of a set of rules in the following form :
Fault Info. Diagnostic Logic
Rule-I: IF
L
UI"
is PI.;
••.
U::" is POG .} THEN ym = yml
(3.1)
where ym =JP is the process output estimate. lIt.o is a constant (for I-lh local model). 1=1.2•... .N. N is the number of rules. u" =[ uj ' " u::"] is the antecedent
I I I _____ I
input vector.
WC
=[ u• • ... •
u~]
Pu'" P"".} are
sub-model) input vector.
Fig. I: NFDFDS overview
is the consequent (linear the input
membership functions for the I-lh rule (with 1< i < nl
3. FORMULATION OF NFDFDS
where ",...
• I < j < n... V i. j Elt).
The idea of fuzzy fusion of observers based on the local affine dynamic linear models. known as the 'fuzzy observer'. has been described by many authors but attributable originally to Wang et al.. (1995). Tanaka et al.• (1996). proposed fuzzy observers based on relaxed stability conditions and LMI-based design. Chen and Patton presented fuzzy observer scheme for fault diagnosis of nonlinear dynamic systems (Chen & Patlon. 1999).
nOG
•
are the
number of input membership functions for uj '" u::" vectors respectively. The neuro-fuzzy model output and the output of the I-lh local model are denoted by ym and yml respectively. Here a discrete-time formulation is used for computational convenience. The output of the I-lh sub-model can be expressed as: I
ItV
I
n..
ym t = al,O + Lal.jymt_j + Ebl.jul.1_j j=1
958
j=1
(3 .2)
fault distribution matrices boils down to the determination of the disturbance term dl. in the following equation for the normal and faulty system operation. Note that the disturbance term dl. for normal and faulty system operations corresponds to the possible solutions for E~ and FI~ I. respectively. Also
where bIJ is the parameter vector of the size of u', a,.o and a'J are parameter vectors of the size of ym' and nu and ny are order of delays for UC and ym'. An initial fuzzy model is created using the a combination of CART (Jang, 1994) and GK clustering (Gustafson & Kessel, 1979). This model is then transformed to a form described by rules in Eq. (3.1). The model obtained is a special form of the TS fuzzy structure that can be tuned online using a neural network representation of the model. The consequents in Eq. (3.2) can be expressed in the state space form (Eq. 4.1) with faults. Note that one of the main advantages of using the NF model is that the operation points and the linear local models are determined using this approach.
note that the noise sequences wl. and w2. are ignored here for simplicity however, these are taken into account for the design of optimal local observers. X!+I = A!x! + B!Uk +dl~ (4.2) y! = c~xi + w2~ Witczak et al., (2003) presented an approach for estimation of the unknown input for the following class of non-linear systems: Xk+1 = g(Xt) + h(Ut) + dl.
Determination of the fault and disturbance distribution matrices constitutes one of the most important steps in NFDFDS scheme. Without this step it is impossible to achieve the robust residual generation and fault isolation. This step is explained in the next section.
Yk+1 = Ck+lxk+1 where gO and hO are the non-linear functions.
(4.3)
The above-mentioned approach assumes a constant distribution direction for the system described by Eq. (4.2). In this reserach an approach is suggested as part of the NFDFDS scheme to determine an individual distribution direction for each sub-model. It is assumed that the overall disturbance can be de-coupled by only the i-th sub-model corresponding to the most dominant rule i.e. dl~ =0 Tt IE[I" · Nl , /~i. The problem of unknown input estimation can be viewed as an unconstrained optimisation task of the form:
4. DETERMINATION OF THE DISTURBANCE AND FAULT DISTRIBUTION MATRICES In order to design a robust observer it is possible to use the unknown input approach by expressing the consequent linear models (Eq. 3.2) in the following state space form with faults: xi+, = A~xi +B~Uk +E~dt +FI~ft +wl~ (4.1)
J( =arg
ym~ = c!x! + w2~
where Ut E 9\', Xt E 9{" and ymt E 9\"' are the l-th model's input, state and output vectors respectively, each entry of h E 9{' corresponds to a specific fault, At. Bt and Ct are the system matrices with appropriate dimensions, FI. is a matrix with appropriate dimensions representing the effect of faults on the system and wit and w2. are independent zero mean white noise sequences with covariance matrices Qt and Rt , assumed to be known. The disturbance and fault distribution matrices are denoted by Et and Ft .
.min cf+, c
diE'"~
.+.
(4.4)
where CI+I = YI+I - YI+I' YI+I = E:':,a,(Il%+,)Y!+, is the process output estimate and Y'+I = YP.+I is the process output. The optimisation task (Eq. 4.4) can be realised by solving the following equation: il T Od t
- . Ck+IC'+1 =0
(4.5)
that is equivalent to the set of linear equations: aj(wl+l)[cl+ l( c1+,JI~ =[Ct +l( YI+I-[cl+ l (
Ea,(wk+')Y!+' (4.6) '·II,,·N)#j -aj(wk+1 )[c1+,( [cl+I(Ajx, + Bt"k)J The Eq. (4.6) can be solved explicitly only when rank(C'+I)=n. This condition is usually difficult to attain in practice (similar to the augmented observer approach). However, an approximate solution can be employed as follows:
In order to determine the distribution matrices a number of techniques can be employed including & 'Evolutionary Algorithms' (Dasgupta Michalewicz, 005., 1997), 'Gradient Based Methods' (Hagan et aI., 1996) or random search. The 'augmented observer' described in (Chen & Patton, 1999) is often not suitable as the necessary existence condition of this observer is that the rank(C)=n, where n is the number of states in the system. This technique is thus limited, as it requires that the system has n independent measurements.
j(Wk+I)[Cl+IIT Ct+ldl~ -([Ct +l( YI+I
J( = arg
.min
d~"c;K'
r
[Cl.dT
Ea,(wt+,)yi+,
(4.7)
11E(1 ... N},ht;
a j(wt+,)lcl+ , IT rc1+,(Ajx. + ~"kl1l
This optimisation problem can be solved via a number of well-developed non-linear optimisation methods.
The underlying problem is to determine the disturbance distribution matrix El and fault
In the above the unknown input
distribution matrix FI~ (where FI represents input and component faults) for I=I .. .N. The problem of determination of the disturbance distribution and
d~
is selected in such a
way as to minimise the residual or the output error
"hi
instead of the state estimation error "1+1' which
959
simplifies (or even makes it possible) the estimation of the unknown input. This is due to the fact that the output error can be directly obtained based on the system and model output. This can be justified. as the main purpose of robust FDI is to make the residual robust to the unknown input while the same for the state estimation error it is not necessary. From the estimates of d;. k=1 •... nd. i=I •.. . .N. (nd=no. of data points. N=no. of sub-models) the
the global observer can be generated as 'k =YPk - YPk • Note that although it is enough to use only one observer. i.e. an observer for the global model (5.3), described in the next section. However. from the numerical point of view it is better to use observers for each sub-model. Moreover. each observer is associated with a linear model in the NF network, which may be used for the online training.
A note on the global stability: Let us consider the l-th sub-system with faults, noise and input (Eq. 7.20). The state. and output equations can be written in a global way as:
E;
can be obtained disturbance distribution matrix as mentioned in (Chen & Palton. 1999). For a clear isolation. while finding the distribution direction for a particular fault an improvement to the above methods will be to consider the decoupling effect on the other faults . It should be noted here that usually the unwanted damping effects cannot be avoided completely. This is due to the fact that the fault directions are rarely orthogonal to each other. However it should be the part of the FDI task to minimise these.
XI +1 = i\ xk + Bk"k + Ekdk + f\ It + wtk Ym k =~Xk+w2k
(5.3)
where the above have following meaning:
xk =~!
x;
[A:0
At
0 0
0 0
0
Ak
= ~
0
0
5. THE GLOBAL OBSERVER
-
e
k
For each sub-model an optimal disturbance decoupling sub-observer for stochastic systems (Chen and Patton. 1999 (and the references therein» can be designed based on the local submodels in the state space representation together with the disturbance and fault distribution matrices. This can be used to detect faults in the presence of both modelling errors and noise.
xt"
[Cl0 et
0 0
0 0
0
= ~
I .Ymk = ~m!
ym;
... ymt"I,
1
oo ,Bk=~k.Bk. I 2 N · ··.BkI . At"
1
o I 2 N o • El = ~1.EI .. ··.Ek
ef
r. r
f'lt = fILFI~ . ...• Flf w2 1 =~2~ . w2: .· .·. w2f
wlk =
T,
~I~.wl~ . .... wlt"
r,
Remark·l: As AI is a block diagonal matrix with diagonal elements A! , A; , ... A: . it can be shown that
The global model can be described by the fuzzy IF-THEN rules that represent the local linear models of the non-linear system. The rule in Eq. (7.10) can be written in the following form:
if all Ai ,i =1,2, ··. N are stable matrices then AI is also stable.
Rule-I: O=I.2 •...•N):IF(Uk is MI THEN ymk = ym~ (5 . 1)
The global output of the fuzzy model is obtained by de-fuzzification (weighted sum of the outputs of the sub-models) as ymk = E~,a/(alk)ymf . In the above equationa/(wk) '1 I=I •.. .,N is the firing strength of I-th rule which depends on the antecedent variable (Uk (or u' ). M/ is a fuzzy set and N is the total number of linear sub-models. The process output estimate by the model can be described by YPk = ymk . Note that the global model can also be represented by Eq. (5.3). The stability of this model is discussed later. For each sub-model an optimal disturbance de-coupling observer is designed. The global observer can be described by the following set of rules: Rule-I:(I=1.2 •.. .•N): IF (Uk is MI THEN >'PI
where
)PI
Fig. 2:
Observer output space
Remark·2: If 0 1 (see Fig. 2) is the output sub-space corresponding to the I-th linear sub-model and if each E! ,I =1,2, ... ,N is obtained in such a way that observer-I converges (behaves in a similar way to the real system) to l-th subspace then it can be shown that the difference between the real output and its estimate depends only on white naise. Remark·3: If the switching rules (fuzzy selection) are obtained in such a way that while operating point the observers/models are selected changes. appropriately then it can be expected that the global output estimate of the observer approximates the system output with required accuracy.
= >'P~ (5.2)
is the global output estimate of the
observer and YP~ is the output estimate of the I-th sub-observer. The output estimate can be expressed as a weighted sum of the sub-observers estimates i.e. JP, = E~,a/(all )YP~ ' The residual for
960
A note on observer convergence: It can be said that discrete-time observer with converging output estimation is a stable observer. It is shown that if all the local observers are stable. the global observer can be expected to be stable. Two main types of observer convergence can be distinguished. The main idea is that the output estimate YPt approximately stays in the vicinity to
the real output YPt . convergence:
variable while the cons5uents are linear models with variables [yl.l. u l• u2• U. u4]. Figs. (3 & 4) show the model performance using the mean-squared-error (MSE) as performance measure. Table I: Fault Scenarios for the FDI task Control valve f.ulU FI : Valve clogging F2: Valve plug or valve seat sedimentation F3: Valve plug or valve seat erosion F4: Internal leakage (valve tightness) F5: Medium evaporation or critical flow Positioner f.ult F6: Rod displacement sensor fault General faults I external faults F7: Unexpected pressure drop across valve F8: Fully or partly opened bypass valves F9: Row rate sensor fault
This can be described as
dyP, -
(5.4)
.YPtl
While asymptotic convergence can be viewed in the light of the following relation:
Lim~Pt
t __
(5.5)
- YPt)= 0
which means that the output estimate converges (in time) to the real output. Whilst this may be achievable for deterministic systems. in practice is a demanding asymptotic convergence requirement and it is rather vain to expect a method that will guarantee asymptotic convergence. If all local observers converge to the local models of the system then:
t.,:!.vm! - vp!l < E,
,E, < 00
)
(5.6) Limlymf -ypfl
SI..., diapIacemenI r _ l : reel .... - esIimaled YIIIue, MSE • 0 0002564 " r-~-~~----~--"""",:--.......,
. 0.4
...... : ....... ..1 ...........: ........ 1-)
ju
. t:
lOot 1
I~'
...
~~
k~_
...;;--=...:.c--=_;;;-7,;,_;;--;:;,... =---;-;,_=---;;,.,.!;;;--;;,.,. !;;;-"_!;;;-'
~·G:---;"':.c--;;:
Dota ......,... (.....,.. _'sec)
Fig. 3 NF model performance for Xsd (5.7)
Liquid now residual : real value - estimated value. MSE. 0.000720.7 G.',..._-~_-~~-~~.......:.~~-....-,
or
LimEat({J)k)!ymi- yM!
.£<00
(5 .8)
G'
.,
4----+-1=1
where
E = t~at({J)k
)!ymi - YPq
(5 .9)
Remark 1: If the neuro-fuzzy model approximates the non-linear system well (with local affine dynamic linear models) . it can be expected that the global observer output will converge to the system output.
..
=-·...=-·...::.-..'...=--;'200.:..-,'-=_;;-;'=IIII;;-;':...::;;--:_=-'
.. ·O;--:;200:.--;...
6. CASE STUDY: FAULT DIAGNOSIS OF A VALVE ACTUATOR
Dolo .....,... (..mpIe Iim.. ,sec)
Fig. 4 NF model performance for F
This section presents the results of applying the NFDFDS scheme to a non-linear electeopneumatic valve actuator in a sugar plant. This study arises from a benchmark study described by Syfert et al.• (2003). Table-I shows the list of the faults detected and isolated using this scheme. A non-linear SIMUUNK model (tested against the real data) is used to generate the fault data. A 4input 2-output NF model is constructed to generate residuals (rt •. ..• rN) as described in section 4. The inputs of the NF model are: the control value (u'). inlet pressure (u 2). outlet pressure (u 3) and temperature (u 4 ). while the outputs are: the stem displacement of electro-pneumatic servo malOr (y') and I-liquid flow through the valve (I) . In antecedents Double Gaussian membership functions are used with u' being the antecedent
In the proposed approach for fault isolation. information is collected from the residuals (r, ... ..rN) in terms of 'fault effect at different operating points' . where Tt) = yM - YPt .1 = I... ·. N. As an example. Fig. 5 shows the Observer-I residual signal (in the presence of fault-I) without de-coupling. The vertical lines in the figure distinguish between different operating points according to the most dominant rule in that region. The most dominant rule is the one with the maximum firing strength. Note that the residual is mainly affected in operating points four and five (OP-4. OP-5). The effect in OP-2 when the valve is being closed can be attributed to the un-modelled dynamics and therefore it is ignored. The information collected from this residual is a set of operating points at which fault has an effect i.e. [4.5] (first entry of column-I in Table-2). It is interesting to note that the fault has a different effect at
961
7. ACKNOWLEDGEMENTS
different operating points. as expected for a nonlinear system.
...
The authors acknowledge funding support for this
research under the EU Framework 5 Research Training Network "DAMADICS" under contract HPRTN-Cf2000-00 IW. Thanks are expressed to the management and staff of the Lublin sugar factory. Cukrownia Lublin SA. Poland for their collaboration and provision of manpower and access to their sugar plant. Faise1 Uppal acknowledges funding support through an ORS (Overseas Research Scholarship) award, together with the Hull University's Open Scholarship .
ON
U t
i··i"
" ,
lOA
02
_
.':').:.... ... ....... ~ .-
. -~- . - ..
8. REFERENCES ZIOOMOOaDlIOCIaoooa.,. . . . . . . . . . .
OoIo......,...{AntpIe_, ...., ,
The information from the residuals is collected in the form of fault symptoms Si.j E [I ... .. N). where
Cben J. Pallon R J (1996). Optimal filtering and robust fault-diagnosis of stochastic systems with unknown disturbances. lEE Proc.-D: Contr.Theory & App\. 143 (1):31-36.
i=l •... ,N denotes the observer and j denotes the fault. A set of fault symptoms for all observers. forms a fault signature SjE [I .... ,N). i=l •. ..• N.
Cben J .• Pallon R. I .• (1999). Robust Model Based Fault Diagnosis For Dynamic Systems. Kluwer Academic Publishers ISBN 0-7923-8411-3 .
When a fault occurs in the system. the signatures are compared with the previously collected signatures for isolation. Table 2 shows the fault symptoms Si.j E D.. ··,N) collected by the NF FDI
Dasgupta D. and Michalewicz Z.. eds.. (1997). Evolutionary Algorithms in Engineering Applications. Spring-VerIag. New York, NY.
Fig. 5 NF de-coupling observer residual for FDI
Frank P. M .• (1987). Fault diagnosis in dynamic system via state estimation - a survey. in Tzafestas. Singh and Schmidt (eds). System Fault Diagnostics. Reliability & Related Knowledge-based Approaches. D. Reidel Press. Dordrecht. pp. 35-98 (Vo\. I).
observer for fault strengths 0.5. For fault strengths 0.4 and 0.25 , similar symptoms were observed. The symbol '! . in the tables denotes that the residual is non-zero when the valve is open. The shaded entries show that the residual is designed to be ideally zero-valued. The symbol '-' denotes that the residual is close to zero. It can be seen from Table-2 that each fault has a unique signature and therefore it can be isolated. Table 2: Fault Symptoms F,
RO
.,'
RI
U
F;
F,
F.
F,
F.
F,
F.
J
J.o4,S
I,2.J.
J.<~
~." . ~
4.' 1.2.3. <..
•
1.2.3. <.5 1.lJ. 5 1.2.3. <,' 1.2..' . ' ,5 1.2J.
1 .2 .~
R3
4, ~
R4
4,5
1.2,3,
4.> l,l.J,
U'
. ,S
R8 R9
' ,5
..
~
u
1,2•.1.
4.5
!A
'-'
•
1.2..1.
1.2.J ,
•
J.l_'.
•1.2
•
1.1..' .
•
.~
1.2.3, 4,5
1.2
1.2
-
Jang J. S. R (1994). Structure determination in fuzzy modelling: a fuzzy CART approach. In Proceedings of IEEE International Conf. On Fuzzy Systems. Orlando. Aorida. June 1994. Syfert M .• Pallon R I. Bartys M. & Quevedo J .• (2003). Development and Application of Methods for Actuator Diagnosis in Industrial Control Systems (DAMADlCS): A Benchmark Study. Submitted to IFAC Symposium SAFEPROCESS 2003. Washington DC.
1.2
2.J ....
• •5
...
F.
5
1.2•..1. 1.2..1.
R7
,
".:\
' ,5
R5
Hagan M. T., Demuth H. & Beale M. (1996). 'Neural Network Design'. PWS Publishing Company. Boston. MA. ISBN 0-534-94332-2.
~. ·.· .Nl
D .•
R2
Rn
Si.jE
Gustafson D. and Kessel W.• (1979). Fuzzy clustering with a fuzzy covariance matrix. Proc. IEEE CDC. San Diego. CA. USA. pp. 761-766.
1.2
a....
Tanaka K. Takayuki I & Wang H (1996). Design of fuzzy control systems based on relaxed LMI stability conditions. Proc. 35th CDC. Kobe 598-603.
6. CONCLUSIONS
Wang H. 0 .• Tanaka K & Griffin M F (1995). Parallel distributed compensation of a non-linear systems by Takagi and Sugeno fuzzy models. Proc. FUZZ-
In this paper a novel NF de-coupling fault diagnosis observer approach based on a special form of TS model is proposed for fault detection and isolation of non-linear dynamic systems. The approximation and reasoning capabilities of NF models are combined with the de-coupling capabilities of optimal observers to perform reliable fault detection and isolation. The proposed FDI scheme is applied to a non-linear valve model identified from the real data from a sugar factory.
IEEElIFES·95. 53\-538. Witczak M .• Pallon R. J .• & Korbicz J .• (2003). fault detection with observers and genetic programming: application to the DAMADICS benchmark problem. (accepted). IFAC Symposium SAFEPROCESS 2003. Washington DC.
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Copyright Cl IFAC Fault Detection, Supervision and Safety of Technical Processes, Washington, D.C., USA, 2003
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Publications www.elsevier.comllocatelifac
A HYBRID NEURO-FUZZY AND DE-COUPLING APPROACH APPLIED TO THE DAMADICS BENCHMARK PROBLEM Faisel J Uppal, Ron J Patton and Marcin Witczak Control & Intelligent Systems Engineering, University of Hull, Hull, HU6 7RX, United Kingdom E-mail: [email protected] [email protected]. [email protected]
Abstract: A novel multiple-model fault detection and isolation scheme for non-linear dynamic systems, that of a Neuro-Fuzzy Decoupling Fault Detection Scheme (NFDFDS) is presented, a hybrid scheme incorporating both neuro-fuzzy and model-based methods. The FDI scheme employs local optimal observers designed according to minimum state estimation variance. An application of FDI for an electro-pneumatic valve actuator in a sugar factory is presented. Key issues of finding a suitable structure for detecting and isolating nine realistic actuator faults are described. Copyright CO 2003lFAC Keywords: Fault detection and Isolation, Multiple model observers, Neuro-Fuzzy networks, Unknown input observers for FDI, Actuators
I . INTRODUCTION
'uncertainty' and this usually means that the robust linear and non-linear estimators need to be considered. The main advantage of the optimal observer is that it can be used to detect faults in the presence of both modelling errors and noise. The FDI problem then follows as an extension to the global estimation problem. The paper is structured as follows. Section-2 gives an overview of the proposed FDI scheme. Section-3 describes the formulation of NF multiple-model identification and observer parameter generation. Section-4 presents methods of determination of the disturbance and fault distribution matrices. Section-5 overviews the global disturbance de-coupling observer and an application of this scheme for an electro-pneumatic valve under DAMADlCS is presented in Section-6.
lt is well known that for fault detection and isolation (FDI) of non-linear dynamic systems, the problem of discriminating between the uncertain model behaviour and the faults presents a significant challenge. This paper presents a novel FDI scheme using the neuro-fuzzy (NF) multiplemodel paradigm and robust optimal de-coupling observers. This new scheme is called the 'NeuroFuzzy and De-coupling Fault Diagnosis Scheme' (NFDFDS) in which the multiple operating points are taken care of through the NF modelling framework . The structure also provides residuals that are de-coupled to 'unknown inputs', making use of the earlier research on unknown input decoupling (Chen, Panon, 1996; 1999 (and the references therein» . The NF paradigm exploits the combined abilities of neural networks and fuzzy logic and is an efficient modelling tool for non-linear dynamic systems because of its approximation and reasoning capabilities. Most real application systems suffer from disturbances and noise and varying operating conditions, leading to a challenging modelling requirement. The model-reality differences have an associated
2. FDI SCHEME OVERVIEW The propose scheme comprises two main parts: a set of NF based de-coupling observers and diagnostic logic. The first part generates a set of fault symptoms ( r1 ... rt) at the k-th sampling instant, in the form of a structured residual set, whilst the second part performs an analysis of the residuals to determine the nature and location of the faults . In this scheme, each residual is
957
designed to be sensitive to a subset of faults. Ideally. each residual is sensitive to all but one fault. called a 'Generalized Residual Set' (Based on the generalized observer scheme of Frank, 1987; Chen & Patton 1999). Each 'Fault Diagnosis Observer' in Fig. (I) is a non-linear system comprising a number of linear (integrated by fuzzy fusion) sub-observers each one corresponding to a different operating point of the process. The input to each observer is the process input and output. Ut and YPt respectively and the number of sub-observers depends on the number of operating points needed to achieve the required approximation. Their outputs are combined by fuzzy fusion to generate the optimum output estimates. Computing the difference between the actual and estimated outputs generates a set of residuals. The set of fuzzy observers together with the NF multiple-model and diagnostic logic forms the new scheme called the 'NFDFDS·. The NFDFDS is based on the identification of a particular type of neuro-fuzzy model (a simpler form of TS structure) that can generate the parameters for the observers together with a fuzzy measure of the validity of each of them. It also gives a linguistic description of the system.
In the proposed approach a special form of the TS fuzzy model is used to combine the NF identification strategy and model-based observer FDI synthesis in a single framework to formulate the so-called 'NeuroFuzzy and De-coupling Fault Detection Scheme' (NFDFDS). The conventional fuzzy observer design Wang et al.• (1995) involves an iterative procedure. selecting eigenvalues arbitrarily using pole-placement and solving inequalities using LMI to find a single positive definite Lyapunov matrix P common to all sub-models. This common matrix ensures stability of the global observer system. If such a matrix P does not exist another set of arbitrary eigenvalues are selected and the whole procedure is repeated until a solution is found. These computational limitations and stability constraints are not present in the proposed scheme. which simplifies and speeds up the joint identification and FDI problem. The reason for the improvements in design complexity can be attributed to the fact that the global state estimation is not necessary for FDI and hence there is no need for unnecessary global feedback. A method to generate local optimally de-coupled observers automatically from the NF model is proposed. This NF model operates in parallel to the local observers so that they can be modified or tuned on-line. In this way the reasoning and approximation capabilities of the neuro-fuzzy networks can be exploited. The global output estimate of the proposed fault detection observer is obtained by a fusion of all linear sub-observer outputs. It should be noted here that for the FDI. the residual signal that describes the fault symptoms is the main requirement. Hence. the global state estimation is not computed. However. each observer has a different state variable model from all other local observers. The diagnostic signal (the residual) is the difference between the estimated and real system outputs. It is shown that if all the local observers are stable. it can be expected that the global observer is stable. This represents a significantly relaxed stability requirement over the TS based observer case. The equivalent fuzzy logic description of the NF network consists of a set of rules in the following form :
Fault Info. Diagnostic Logic
Rule-I: IF
L
UI"
is PI.;
••.
U::" is POG .} THEN ym = yml
(3.1)
where ym =JP is the process output estimate. lIt.o is a constant (for I-lh local model). 1=1.2•... .N. N is the number of rules. u" =[ uj ' " u::"] is the antecedent
I I I _____ I
input vector.
WC
=[ u• • ... •
u~]
Pu'" P"".} are
sub-model) input vector.
Fig. I: NFDFDS overview
is the consequent (linear the input
membership functions for the I-lh rule (with 1< i < nl
3. FORMULATION OF NFDFDS
where ",...
• I < j < n... V i. j Elt).
The idea of fuzzy fusion of observers based on the local affine dynamic linear models. known as the 'fuzzy observer'. has been described by many authors but attributable originally to Wang et al.. (1995). Tanaka et al.• (1996). proposed fuzzy observers based on relaxed stability conditions and LMI-based design. Chen and Patton presented fuzzy observer scheme for fault diagnosis of nonlinear dynamic systems (Chen & Patlon. 1999).
nOG
•
are the
number of input membership functions for uj '" u::" vectors respectively. The neuro-fuzzy model output and the output of the I-lh local model are denoted by ym and yml respectively. Here a discrete-time formulation is used for computational convenience. The output of the I-lh sub-model can be expressed as: I
ItV
I
n..
ym t = al,O + Lal.jymt_j + Ebl.jul.1_j j=1
958
j=1
(3 .2)
fault distribution matrices boils down to the determination of the disturbance term dl. in the following equation for the normal and faulty system operation. Note that the disturbance term dl. for normal and faulty system operations corresponds to the possible solutions for E~ and FI~ I. respectively. Also
where bIJ is the parameter vector of the size of u', a,.o and a'J are parameter vectors of the size of ym' and nu and ny are order of delays for UC and ym'. An initial fuzzy model is created using the a combination of CART (Jang, 1994) and GK clustering (Gustafson & Kessel, 1979). This model is then transformed to a form described by rules in Eq. (3.1). The model obtained is a special form of the TS fuzzy structure that can be tuned online using a neural network representation of the model. The consequents in Eq. (3.2) can be expressed in the state space form (Eq. 4.1) with faults. Note that one of the main advantages of using the NF model is that the operation points and the linear local models are determined using this approach.
note that the noise sequences wl. and w2. are ignored here for simplicity however, these are taken into account for the design of optimal local observers. X!+I = A!x! + B!Uk +dl~ (4.2) y! = c~xi + w2~ Witczak et al., (2003) presented an approach for estimation of the unknown input for the following class of non-linear systems: Xk+1 = g(Xt) + h(Ut) + dl.
Determination of the fault and disturbance distribution matrices constitutes one of the most important steps in NFDFDS scheme. Without this step it is impossible to achieve the robust residual generation and fault isolation. This step is explained in the next section.
Yk+1 = Ck+lxk+1 where gO and hO are the non-linear functions.
(4.3)
The above-mentioned approach assumes a constant distribution direction for the system described by Eq. (4.2). In this reserach an approach is suggested as part of the NFDFDS scheme to determine an individual distribution direction for each sub-model. It is assumed that the overall disturbance can be de-coupled by only the i-th sub-model corresponding to the most dominant rule i.e. dl~ =0 Tt IE[I" · Nl , /~i. The problem of unknown input estimation can be viewed as an unconstrained optimisation task of the form:
4. DETERMINATION OF THE DISTURBANCE AND FAULT DISTRIBUTION MATRICES In order to design a robust observer it is possible to use the unknown input approach by expressing the consequent linear models (Eq. 3.2) in the following state space form with faults: xi+, = A~xi +B~Uk +E~dt +FI~ft +wl~ (4.1)
J( =arg
ym~ = c!x! + w2~
where Ut E 9\', Xt E 9{" and ymt E 9\"' are the l-th model's input, state and output vectors respectively, each entry of h E 9{' corresponds to a specific fault, At. Bt and Ct are the system matrices with appropriate dimensions, FI. is a matrix with appropriate dimensions representing the effect of faults on the system and wit and w2. are independent zero mean white noise sequences with covariance matrices Qt and Rt , assumed to be known. The disturbance and fault distribution matrices are denoted by Et and Ft .
.min cf+, c
diE'"~
.+.
(4.4)
where CI+I = YI+I - YI+I' YI+I = E:':,a,(Il%+,)Y!+, is the process output estimate and Y'+I = YP.+I is the process output. The optimisation task (Eq. 4.4) can be realised by solving the following equation: il T Od t
- . Ck+IC'+1 =0
(4.5)
that is equivalent to the set of linear equations: aj(wl+l)[cl+ l( c1+,JI~ =[Ct +l( YI+I-[cl+ l (
Ea,(wk+')Y!+' (4.6) '·II,,·N)#j -aj(wk+1 )[c1+,( [cl+I(Ajx, + Bt"k)J The Eq. (4.6) can be solved explicitly only when rank(C'+I)=n. This condition is usually difficult to attain in practice (similar to the augmented observer approach). However, an approximate solution can be employed as follows:
In order to determine the distribution matrices a number of techniques can be employed including & 'Evolutionary Algorithms' (Dasgupta Michalewicz, 005., 1997), 'Gradient Based Methods' (Hagan et aI., 1996) or random search. The 'augmented observer' described in (Chen & Patton, 1999) is often not suitable as the necessary existence condition of this observer is that the rank(C)=n, where n is the number of states in the system. This technique is thus limited, as it requires that the system has n independent measurements.
j(Wk+I)[Cl+IIT Ct+ldl~ -([Ct +l( YI+I
J( = arg
.min
d~"c;K'
r
[Cl.dT
Ea,(wt+,)yi+,
(4.7)
11E(1 ... N},ht;
a j(wt+,)lcl+ , IT rc1+,(Ajx. + ~"kl1l
This optimisation problem can be solved via a number of well-developed non-linear optimisation methods.
The underlying problem is to determine the disturbance distribution matrix El and fault
In the above the unknown input
distribution matrix FI~ (where FI represents input and component faults) for I=I .. .N. The problem of determination of the disturbance distribution and
d~
is selected in such a
way as to minimise the residual or the output error
"hi
instead of the state estimation error "1+1' which
959
simplifies (or even makes it possible) the estimation of the unknown input. This is due to the fact that the output error can be directly obtained based on the system and model output. This can be justified. as the main purpose of robust FDI is to make the residual robust to the unknown input while the same for the state estimation error it is not necessary. From the estimates of d;. k=1 •... nd. i=I •.. . .N. (nd=no. of data points. N=no. of sub-models) the
the global observer can be generated as 'k =YPk - YPk • Note that although it is enough to use only one observer. i.e. an observer for the global model (5.3), described in the next section. However. from the numerical point of view it is better to use observers for each sub-model. Moreover. each observer is associated with a linear model in the NF network, which may be used for the online training.
A note on the global stability: Let us consider the l-th sub-system with faults, noise and input (Eq. 7.20). The state. and output equations can be written in a global way as:
E;
can be obtained disturbance distribution matrix as mentioned in (Chen & Palton. 1999). For a clear isolation. while finding the distribution direction for a particular fault an improvement to the above methods will be to consider the decoupling effect on the other faults . It should be noted here that usually the unwanted damping effects cannot be avoided completely. This is due to the fact that the fault directions are rarely orthogonal to each other. However it should be the part of the FDI task to minimise these.
XI +1 = i\ xk + Bk"k + Ekdk + f\ It + wtk Ym k =~Xk+w2k
(5.3)
where the above have following meaning:
xk =~!
x;
[A:0
At
0 0
0 0
0
Ak
= ~
0
0
5. THE GLOBAL OBSERVER
-
e
k
For each sub-model an optimal disturbance decoupling sub-observer for stochastic systems (Chen and Patton. 1999 (and the references therein» can be designed based on the local submodels in the state space representation together with the disturbance and fault distribution matrices. This can be used to detect faults in the presence of both modelling errors and noise.
xt"
[Cl0 et
0 0
0 0
0
= ~
I .Ymk = ~m!
ym;
... ymt"I,
1
oo ,Bk=~k.Bk. I 2 N · ··.BkI . At"
1
o I 2 N o • El = ~1.EI .. ··.Ek
ef
r. r
f'lt = fILFI~ . ...• Flf w2 1 =~2~ . w2: .· .·. w2f
wlk =
T,
~I~.wl~ . .... wlt"
r,
Remark·l: As AI is a block diagonal matrix with diagonal elements A! , A; , ... A: . it can be shown that
The global model can be described by the fuzzy IF-THEN rules that represent the local linear models of the non-linear system. The rule in Eq. (7.10) can be written in the following form:
if all Ai ,i =1,2, ··. N are stable matrices then AI is also stable.
Rule-I: O=I.2 •...•N):IF(Uk is MI THEN ymk = ym~ (5 . 1)
The global output of the fuzzy model is obtained by de-fuzzification (weighted sum of the outputs of the sub-models) as ymk = E~,a/(alk)ymf . In the above equationa/(wk) '1 I=I •.. .,N is the firing strength of I-th rule which depends on the antecedent variable (Uk (or u' ). M/ is a fuzzy set and N is the total number of linear sub-models. The process output estimate by the model can be described by YPk = ymk . Note that the global model can also be represented by Eq. (5.3). The stability of this model is discussed later. For each sub-model an optimal disturbance de-coupling observer is designed. The global observer can be described by the following set of rules: Rule-I:(I=1.2 •.. .•N): IF (Uk is MI THEN >'PI
where
)PI
Fig. 2:
Observer output space
Remark·2: If 0 1 (see Fig. 2) is the output sub-space corresponding to the I-th linear sub-model and if each E! ,I =1,2, ... ,N is obtained in such a way that observer-I converges (behaves in a similar way to the real system) to l-th subspace then it can be shown that the difference between the real output and its estimate depends only on white naise. Remark·3: If the switching rules (fuzzy selection) are obtained in such a way that while operating point the observers/models are selected changes. appropriately then it can be expected that the global output estimate of the observer approximates the system output with required accuracy.
= >'P~ (5.2)
is the global output estimate of the
observer and YP~ is the output estimate of the I-th sub-observer. The output estimate can be expressed as a weighted sum of the sub-observers estimates i.e. JP, = E~,a/(all )YP~ ' The residual for
960
A note on observer convergence: It can be said that discrete-time observer with converging output estimation is a stable observer. It is shown that if all the local observers are stable. the global observer can be expected to be stable. Two main types of observer convergence can be distinguished. The main idea is that the output estimate YPt approximately stays in the vicinity to
the real output YPt . convergence:
variable while the cons5uents are linear models with variables [yl.l. u l• u2• U. u4]. Figs. (3 & 4) show the model performance using the mean-squared-error (MSE) as performance measure. Table I: Fault Scenarios for the FDI task Control valve f.ulU FI : Valve clogging F2: Valve plug or valve seat sedimentation F3: Valve plug or valve seat erosion F4: Internal leakage (valve tightness) F5: Medium evaporation or critical flow Positioner f.ult F6: Rod displacement sensor fault General faults I external faults F7: Unexpected pressure drop across valve F8: Fully or partly opened bypass valves F9: Row rate sensor fault
This can be described as
dyP, -
(5.4)
.YPtl
While asymptotic convergence can be viewed in the light of the following relation:
Lim~Pt
t __
(5.5)
- YPt)= 0
which means that the output estimate converges (in time) to the real output. Whilst this may be achievable for deterministic systems. in practice is a demanding asymptotic convergence requirement and it is rather vain to expect a method that will guarantee asymptotic convergence. If all local observers converge to the local models of the system then:
t.,:!.vm! - vp!l < E,
,E, < 00
)
(5.6) Limlymf -ypfl
SI..., diapIacemenI r _ l : reel .... - esIimaled YIIIue, MSE • 0 0002564 " r-~-~~----~--"""",:--.......,
. 0.4
...... : ....... ..1 ...........: ........ 1-)
ju
. t:
lOot 1
I~'
...
~~
k~_
...;;--=...:.c--=_;;;-7,;,_;;--;:;,... =---;-;,_=---;;,.,.!;;;--;;,.,. !;;;-"_!;;;-'
~·G:---;"':.c--;;:
Dota ......,... (.....,.. _'sec)
Fig. 3 NF model performance for Xsd (5.7)
Liquid now residual : real value - estimated value. MSE. 0.000720.7 G.',..._-~_-~~-~~.......:.~~-....-,
or
LimEat({J)k)!ymi- yM!
.£<00
(5 .8)
G'
.,
4----+-1=1
where
E = t~at({J)k
)!ymi - YPq
(5 .9)
Remark 1: If the neuro-fuzzy model approximates the non-linear system well (with local affine dynamic linear models) . it can be expected that the global observer output will converge to the system output.
..
=-·...=-·...::.-..'...=--;'200.:..-,'-=_;;-;'=IIII;;-;':...::;;--:_=-'
.. ·O;--:;200:.--;...
6. CASE STUDY: FAULT DIAGNOSIS OF A VALVE ACTUATOR
Dolo .....,... (..mpIe Iim.. ,sec)
Fig. 4 NF model performance for F
This section presents the results of applying the NFDFDS scheme to a non-linear electeopneumatic valve actuator in a sugar plant. This study arises from a benchmark study described by Syfert et al.• (2003). Table-I shows the list of the faults detected and isolated using this scheme. A non-linear SIMUUNK model (tested against the real data) is used to generate the fault data. A 4input 2-output NF model is constructed to generate residuals (rt •. ..• rN) as described in section 4. The inputs of the NF model are: the control value (u'). inlet pressure (u 2). outlet pressure (u 3) and temperature (u 4 ). while the outputs are: the stem displacement of electro-pneumatic servo malOr (y') and I-liquid flow through the valve (I) . In antecedents Double Gaussian membership functions are used with u' being the antecedent
In the proposed approach for fault isolation. information is collected from the residuals (r, ... ..rN) in terms of 'fault effect at different operating points' . where Tt) = yM - YPt .1 = I... ·. N. As an example. Fig. 5 shows the Observer-I residual signal (in the presence of fault-I) without de-coupling. The vertical lines in the figure distinguish between different operating points according to the most dominant rule in that region. The most dominant rule is the one with the maximum firing strength. Note that the residual is mainly affected in operating points four and five (OP-4. OP-5). The effect in OP-2 when the valve is being closed can be attributed to the un-modelled dynamics and therefore it is ignored. The information collected from this residual is a set of operating points at which fault has an effect i.e. [4.5] (first entry of column-I in Table-2). It is interesting to note that the fault has a different effect at
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7. ACKNOWLEDGEMENTS
different operating points. as expected for a nonlinear system.
...
The authors acknowledge funding support for this
research under the EU Framework 5 Research Training Network "DAMADICS" under contract HPRTN-Cf2000-00 IW. Thanks are expressed to the management and staff of the Lublin sugar factory. Cukrownia Lublin SA. Poland for their collaboration and provision of manpower and access to their sugar plant. Faise1 Uppal acknowledges funding support through an ORS (Overseas Research Scholarship) award, together with the Hull University's Open Scholarship .
ON
U t
i··i"
" ,
lOA
02
_
.':').:.... ... ....... ~ .-
. -~- . - ..
8. REFERENCES ZIOOMOOaDlIOCIaoooa.,. . . . . . . . . . .
OoIo......,...{AntpIe_, ...., ,
The information from the residuals is collected in the form of fault symptoms Si.j E [I ... .. N). where
Cben J. Pallon R J (1996). Optimal filtering and robust fault-diagnosis of stochastic systems with unknown disturbances. lEE Proc.-D: Contr.Theory & App\. 143 (1):31-36.
i=l •... ,N denotes the observer and j denotes the fault. A set of fault symptoms for all observers. forms a fault signature SjE [I .... ,N). i=l •. ..• N.
Cben J .• Pallon R. I .• (1999). Robust Model Based Fault Diagnosis For Dynamic Systems. Kluwer Academic Publishers ISBN 0-7923-8411-3 .
When a fault occurs in the system. the signatures are compared with the previously collected signatures for isolation. Table 2 shows the fault symptoms Si.j E D.. ··,N) collected by the NF FDI
Dasgupta D. and Michalewicz Z.. eds.. (1997). Evolutionary Algorithms in Engineering Applications. Spring-VerIag. New York, NY.
Fig. 5 NF de-coupling observer residual for FDI
Frank P. M .• (1987). Fault diagnosis in dynamic system via state estimation - a survey. in Tzafestas. Singh and Schmidt (eds). System Fault Diagnostics. Reliability & Related Knowledge-based Approaches. D. Reidel Press. Dordrecht. pp. 35-98 (Vo\. I).
observer for fault strengths 0.5. For fault strengths 0.4 and 0.25 , similar symptoms were observed. The symbol '! . in the tables denotes that the residual is non-zero when the valve is open. The shaded entries show that the residual is designed to be ideally zero-valued. The symbol '-' denotes that the residual is close to zero. It can be seen from Table-2 that each fault has a unique signature and therefore it can be isolated. Table 2: Fault Symptoms F,
RO
.,'
RI
U
F;
F,
F.
F,
F.
F,
F.
J
J.o4,S
I,2.J.
J.<~
~." . ~
4.' 1.2.3. <..
•
1.2.3. <.5 1.lJ. 5 1.2.3. <,' 1.2..' . ' ,5 1.2J.
1 .2 .~
R3
4, ~
R4
4,5
1.2,3,
4.> l,l.J,
U'
. ,S
R8 R9
' ,5
..
~
u
1,2•.1.
4.5
!A
'-'
•
1.2..1.
1.2.J ,
•
J.l_'.
•1.2
•
1.1..' .
•
.~
1.2.3, 4,5
1.2
1.2
-
Jang J. S. R (1994). Structure determination in fuzzy modelling: a fuzzy CART approach. In Proceedings of IEEE International Conf. On Fuzzy Systems. Orlando. Aorida. June 1994. Syfert M .• Pallon R I. Bartys M. & Quevedo J .• (2003). Development and Application of Methods for Actuator Diagnosis in Industrial Control Systems (DAMADlCS): A Benchmark Study. Submitted to IFAC Symposium SAFEPROCESS 2003. Washington DC.
1.2
2.J ....
• •5
...
F.
5
1.2•..1. 1.2..1.
R7
,
".:\
' ,5
R5
Hagan M. T., Demuth H. & Beale M. (1996). 'Neural Network Design'. PWS Publishing Company. Boston. MA. ISBN 0-534-94332-2.
~. ·.· .Nl
D .•
R2
Rn
Si.jE
Gustafson D. and Kessel W.• (1979). Fuzzy clustering with a fuzzy covariance matrix. Proc. IEEE CDC. San Diego. CA. USA. pp. 761-766.
1.2
a....
Tanaka K. Takayuki I & Wang H (1996). Design of fuzzy control systems based on relaxed LMI stability conditions. Proc. 35th CDC. Kobe 598-603.
6. CONCLUSIONS
Wang H. 0 .• Tanaka K & Griffin M F (1995). Parallel distributed compensation of a non-linear systems by Takagi and Sugeno fuzzy models. Proc. FUZZ-
In this paper a novel NF de-coupling fault diagnosis observer approach based on a special form of TS model is proposed for fault detection and isolation of non-linear dynamic systems. The approximation and reasoning capabilities of NF models are combined with the de-coupling capabilities of optimal observers to perform reliable fault detection and isolation. The proposed FDI scheme is applied to a non-linear valve model identified from the real data from a sugar factory.
IEEElIFES·95. 53\-538. Witczak M .• Pallon R. J .• & Korbicz J .• (2003). fault detection with observers and genetic programming: application to the DAMADICS benchmark problem. (accepted). IFAC Symposium SAFEPROCESS 2003. Washington DC.
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