On-Line Residual Compensation in Robust Fault Diagnosis of Dynamic Systems

Copyright © IFAC Artificial Intelligence in Real-Time Control. Delft. The Netherlands. 1992

ON-LINE RESIDUAL COMPENSATION IN ROBUST FAULT DIAGNOSIS OF DYNAMIC SYSTEMS R.J. Patton and J. Chen Department a/Electronics. University a/York. York. YOJ 5DD. UK

Abstract: This paper deals with the robust fault diagnosis problem for systems with unstructured uncertainties, for which disturbance de-coupling methods are not applicable. A technique for compensating the residual is proposed. By compensation, the effects of modelling errors on the residual will not adversely affect the detection and isolation of faults. The idea is to estimate approximately the bias term in the residuals due to modelling errors, then compensate it on-line. These estimates are used to form a compensated residual to decrease the effect of modelling errors on the residuals. The compensated residuals are then used to make the fault diagnosis decision. Simulation results illustrate the effect of incipient faults in sensors of a complex jet engine can be reliably detected with the method developed.

1 Introduction

system dynamics is assumed to be know a priori (Frank, 1991; Patton & Chen, 1991a, 1992; Gertler, 1991). This is often referred to as structured uncertainty. For unstructured uncertainty, no mature robust FD methods are available. Perfect solutions for this problem are almost impossible, although an approximation solution can be achieved. For example, one way (Patton & Chen, 1991b, 1991c) is to estimate an approximation structure of uncertainties and then to deal with this using disturbance de-coupling methods. Another way is to make use of adaptive thresholds via socalled "threshold adaptors" or "threshold selectors" (Emami-Naeini et ai, 1988; Ding & Frank, 1991).

Modelling uncertainty is the main problem impeding the progress of applying model-based fault diagnosis (FD) techniques to real systems. All model-based FD methods employ system models to generate fault indicating signals (residuals) to detect and isolate the presence of faults. Here, we consider that fault diagnosis includes the detection, isolation and estimation of each fault in the components of a dynamic system. Residuals are normally based on a comparison between the measured and anticipated responses. The anticipated response is obtained using a model of the monitored system. Fault diagnosis would be straightforward if an exact system model were available and if the system could be considered noise-free. Exact modelling of real systems however, is impossible. Hence, a practical and applicable fault diagnosis scheme must be made robust with respect to modelling errors and uncertainties. As a definition, the robustness of an FD scheme is the degree to which its diagnosis is unaffected by (or remains insensitive to) modelling uncertainties (Frank, 1991; Patton & Chen, 1991a, 1992; Gertler, 1991).

In practical situations, the residual is never zero even when no faults occur. A threshold must then be used in the residual evaluation stage. Normally, the threshold is set slightly larger than the largest magnitude of the residual evaluation function for the fault-free case. The smallest detectable fault is a fault which drives the residual evaluation function to just exceed the threshold. Any fault which produces a residual response smaller than this magnitude is not detectable. From our point of view, the purpose of the robust design is to decrease the magnitude of the fault-free residual and maintain (even increase) the magnitude of faulty residuals.

In recent years, a great deal of research effort has been paid towards the improvement of robustness in FD problems. Up to now, the most successful ways to achieve robustness in FD is the use of "disturbance de-coupling" ideas (Frank, 1991; Patton & Chen, 1991a, 1992; Gertler, 1991). In these approaches, all uncertainties are considered

This paper deals with unstructured uncertainties. An on-line compensation method for residuals is presented. The idea is to estimate approximately the bias term in the residuals due to modelling

as disturbance terms acting on a nominal model of

errors, then compensate it on-line. These estimates

the monitored system. Although the magnitude of the uncertainty is unknown, its distribution into the

are used to form a compensated residual to decrease the effect of modelling errors on 221

residuals. The compensated residuals are then used to make the FD decision. The proposed method has been used to detect faulty sensors in a simulated jet engine system. The results demonstrate the effectiveness of this approach.

U(z) System

2 Basic concepts of residual generation

r(z) ,,

The model-based FD process consists of two stages: (1) Residual generation: In which, outputs and inputs of the system are processed by an appropriate algorithm (a processor) to generate residual signals. (2) Decision making: The residuals are examined for the likelihood of faults, and a decision rule is then applied to determine if any faults have occurred. The residuals are quantities that represent the inconsistency between the actual plant measurements and the mathematical model outputs. If multiple faults may occur and have to be isolated, a set of stmctured residuals is required, so that different faults which are reflected in the residuals in different ways can be isolated.

!, Residuals Residual Generator Figure 1: The transfer function structure of residual generator A general form of the residual generator can then

be expressed as: u(z) ] !(z) = [Hu(z)

=

Hy
~(z)

Hu(z)!!(z) + Hy
(2)

Here, Hu(z) and Hy
=

0

and

(3)

In order to satisfy this requirement, the transfer function matrices H (z) and H (z) must satisfy the . u y equatIon:

The discrete-time input-output description of the monitored system is: Gu(z)!!(z) + Gt
'

1_- __________________________ :

Now, we consider the mathematical description of the diagnosis problem. Though the plants are usually continuous, the diagnostic computations are normally performed on sampled data. Hence, we consider discrete (discretized) plant models. Note that most of the diagnostic methods can be applied to both discrete and continuous models. Plant linearity will be assumed throughout; in the case of a non-linear plant, this implies model linearization around an operating point.

~(z) =

,

,

(1)

(4)

where ~ is the mx1 output vector and!! is the rx1 input vector, whilst f is a qx1 fault vector. The transfer function matrix G u (z) represents the nominal system model which is assumed known. Each element f.(k) (i = 1, 2, .... , q) corresponds to a special fault ~ode. From a practical point of view it is unreasonable to make further assumptions about the characteristics but consider these as unknown time junctions. The transfer function matrix G~z) ~epresent the effect of faults on the system which IS normally known a priori.

or

(5)

Equation (2) is a unified and generalized representation of all residual generators. The design of the residual generator results simply in the choice of the transfer function matrices H (z) and Hy
Residual generation plays an important role in FD. In order to be useful indicators of 'faults, the residuals should be small in the absence of faults and one or more of them should become large i~ the presence of a fault. The residual generator is shown in Fig.1 which involves the processing of the input and output data of the system.

When faults occur in the monitored plant, the response of the residual vector is: (6)

222

Indeed, it can be difficult to distinguish the effects of faults from the effects of modelling errors. The effects of modelling error obscure the performance of fault diagnosis and act as a source of false and missed alarms. Therefore, in order to minimize the false and missed alarm rates, one should design re si duals which are insensitive to uncertainties.

In order to detect the i-th fault in the residual r(z), the i-th column [H (z)G.{z»).l of the transfer matrix [HV
If the above conditions are satisfied, the i-th fault can be said to be detectable using the residual. This is the detect ability problem. The fault detection problem can be stated in terms of some decision function, J(r), and threshold, J th . ~

J th

for

f(k) = 0

J(r) > J th

for

f(k) :t: 0

J(r)

For the case where the modelling error can be considered as structured uncertainty (Ding & Frank, 1991; Patton & Chen, 1991a, 1992), i.e. .1G(z)!!(z) = G iz)g(z)

(10)

where g(z) is a unknown disturbance vector, the transfer function Giz) is known. In this situation, the effect of the uncertainty on the residual can be nulled by disturbance de-coupling design, i.e. to make:

Evidently, the ideal case would be J th = 0 which is however, normally impossible because of the presence of admissible errors.

(11) In practice, no nominal models can be describe a physical plant perfectly, the model uncertainty is inevitable and should be taken into account. Model uncertainty refers to the mismatch between the nominal model and the actual system. An uncertain system can be expressed by: y(z)

=

G(z)!!(z) + Gt
This can be done either in the time domain, e.g. the unknown input observer approach (Frank, 1990; Wiinnenberg, 1990), the eigenstructure assignment approach (Patton & Chen, 1991a, 1991b, 1991c, 1992), or the frequency domain approach (Ding & Frank, 1991).

(7) Unfortunately, most uncertainties are unstructured. In this situation, one way to achieve reliable fault diagnosis is the threshold selection or adaption method «Emami-Naeini et aI, 1988; Ding & Frank, 1991». In this method, assuming that:

where G(z) is the actual transfer function matrix which can not modelled exactly. For multivariable systems, there are three commonly used forms of representing uncertainty: G(z)

Gu(z) + .1G a (z)

(8a)

G(z)

Gu(z)(I + .1G i(z»

(8b)

G(z)

(I + .1G o (z»G u (z)

(8c)

II.1G(z) 11 ~ 0 then, an adaptive threshold is assigned as:

where .1 Giz), .1 Gi(z) and .1 Go(z) represent additive, mput multiplicative and output multiplicative perturbations, respectively.

We have classified this approach as a passive approach for robust fault diagnosis (Patton & Chen, 1992) as no effort is made to achieve robustness in the residual design. Because the threshold is relatively large in this method, the smallest detectable fault is also large. This cannot meet the requirement of incipient fault diagnosis which is the special concern in the current fault diagnosis literature. So-called incipient faults may not be very serious when they occur but might lead to serious situations. To be specific, this class of faults is quite small in magnitude and the developing speed is slow. In order to deal with the unstructured uncertainty, Patton & Chen (1991b, 1991c) proposed the approximate de-coupling approach. In their approach, an (estimated) approximate structure is used to describe the unstructured uncertainty. Hence, approximate decoupling is achievable.

When we apply the residual generator (2) to the uncertain system (7), the actual residuals are: r(z)

= H/z).1G a(z)!!(z) + Hy(z)GrCz)f(z)

r(z) = H/z)G u (z).1G j (z)!!(z) + H/z)Gr(z)f(z) r(z) = H/z).1G o (z)G/z)!!(z) + H/z)Gr(z)f(z) These three situations can be summarized as:

From equation (9), it can be seen that the residual is not zero, even if no faults occur in the system. 223

3 On-line residual compensation

From Section 2, it can be seen that the modelling uncertainty gives rise to the following term in the residual:

(12) which is not zero and called the bias tenll. This term is related with the input. Assume that this bias term can described as:

which will be used for robust fault diagnosis. As the residual compensation can be done on-line, this approach for robust fault diagnosis is called the on-line residual compensation approach. This idea is illustrated in Fig.2. As the effect of the modelling uncertainty on the residual has been compensated, the compensated residual C(k) is only affected by the faults and can be used for robust incipient fault diagnosis. The problem is that the parameter vector e is unknown. Based on equation (14), the leastsquares method (Goodwin & Payne, 1977) can be used to identify this parameter vector. When the system is in the normal condition, the bias term !1(k) is equal to the residual !(k). Where !(k) is a primary residual which is generated using any normal residual generation method (e.g. observerbased methods, parity equation methods). The fault-free residual r(k) {k=l, 2, ... , N) will be used to identified the parameter vector e. This stage is called as the training stage. Then, the estimated parameter e* is used to compensate the residual. The model order n is not known a priori which is determined in the identification procedure. In practice, a reasonable model order is less than 5 (Goodwin & Payne, 1977).

(13) where G ru (z) is an unknown transfer function matrix which represents the combination of all the effect of all uncertain factors on the residual, ego parameter perturbations, dynamic errors (e.g. model reduction error) etc. This relationship can also be described in the time domain as: !1(k) + a 1!1(k-l) + ... + a n!1(k-n) = bo!!(k) + b 1y(k-l) + ... + bny(k-n)

(14)

or

(15)

where the parameter vector

e is:

The parameter identification can be also carried out adaptively based on the recursive least-squares equation as follows:

<1>(k) = [-!l(k-l) ... -!1(k-n); y(k) y(k-l) ... y(k_n)]T

If e is known, i.e., the transfer function Grd(z) is known, the bias term !l(k) in the residual can be estimated on-linely as:

(16) where: " A

A

<1>(k) = [-!l(k-l) ... -!l(k-n) ; y(k) y(k-l) ... y(k-n)]

-.1..

- ... u

--.

Primary Residual Generation

Bias Estimation

r

(18a)

p(kr1 = p(k-lr1 + <1>(k)<1>T(k)ja(k)

(18b)

K(k) = P(k)<1>(k)ja(k)

(18c)

where a(k) is a forgetting factor. The tracking speed can be improved by changing the forgetting factor.

T

The question ansmg here is that the effects of modelling uncertainties and faults are mixed up in !(k), i.e. !1 (k) is a part of !(k). Hence, the residual compensation method is divided into two stages:

,

On-line Residual Compensation

(1) Adaptive training stage: In this stage, the system is assumed normal. The residual is only due to the modelling uncertainty and !l(k) = !(k). This residual is used for identifying the parameter vector e based on equations (18). After the estimation 8(k) converge to a constant value e*, the procedure will move to the next stage. The convergence can be tested using:

t r1

8(k) = 9(k-l) + K(k)[!1(k) - 9(k-l)<1>(k)]

r*

Figure 2: The on-line residual compensation

Hence, a compensated residual is defined as: C(k) = !(k) - I1 (k)

or

(17)

224

11

8(k+ 1)

8(k) 11

$

JJ.

where € and J.1. are small positive constants which are chosen a priori.

incipient faults may be mis-interpreted as a model variation. For the hard faults (the magnitudes are relatively large and occur abruptly), the detection is also possible even if the adaptive parameter identification IS not switched off as the compensated residual and/or the parameter estimation may jump rapidly.

(2) On-line residual compensation stage: In this stage, equations (16), (17) and the estimated parameter 9* will be used to compute the compensated residual r*(k). The idea of the adaptive parameter identification and the on-line residual compensation method is illustrated in Figure 3.

4 An example: Robust detection of inCipient faults in jet engine sensors

A complex thermodynamic simulation model of a jet engine is utilised as an example to illustrate the method developed in this paper. This model has 17 state variables; these include pressures, air and gas mass flow rates, shaft speeds, absolute temperatures and static pressure. This is a highly non-linear dynamic system which has grossly different steady-state operation over the entire range of spool speeds, flow rates and nozzle areas. The jet engine has the measurement variables N , L NH , T 7' P 6' T 29· N denotes a compressor shaft speed, P denotes a pressure, whilst T represents a measured temperature. The main engine fuel flow rate is considered as a control input. The linearized 17th order model at one operating point has been used to simulate the system. For practical reasons and convenience of design, we choose to employ a 5th order model to approximate the 17th order model. The model reduction errors and other errors are inevitable and present a real challenge to fault diagnosis. The simulation data are generated using 17th order linear model with 10% perturbation in the model matrices. The input is a step signal combined with a muti-frequency sinusoid signal. In order to give meaningful magnitudes in the final results a per-unit scaling of the engine dynamics has been used.

y u

Primary Residual Generation

Adaptive On-line Parameter I -.........-~ Residual Compensation Identification r1 r*

Figure 3: Adaptive parameter identification and on-line residual compensation

When the system operating conditions change or the system characteristics vary, the parameter identification will be started again. Frank, Ding & Wochnik (1991) used the adaptive observer approach to generate robust residuals for unstructured uncertain systems. A parameter identification procedure is also involved in their method. This is a re-identification of the system model. The disadvantage of their method is that it is only useful for the parameter variation situation as they considered that the modelling uncertainty is only due to parameter perturbation. In this paper, we consider the combined effects of all uncertainties on the residual which includes: parameter perturbations and model reduction errors etc. For a range of operating conditions, we can use a fixed system model together with a residual model, the latter may vary according to the system operating conditions.

An observer based on the 5th order model is used to generate the output estimation 2(k). The primary residual is defined as: r(k)

= W~(k) = W(X(k)

X(k»

where the weighting matrix is W = [1 1 1 1 1]. In this case, the primary residual is a scalar variable. The Fig.4 shows the primary residual for the faultfree case. It can be seen that the primary residual is too large and a small threshold cannot be used to detect faults. This primary residual is now used as training data to identify the parameter vector 9 . In this paper, a 2nd order model has been chosen, i.e. n = 2 (in equation (14). Using the MATLAB Identification Toolbox, we obtain the identification results are:

For all adaptive methods, a problem is that the fault effects may be compensated as well as modelling error effects. This makes the detection

impossible. Hence, the adaptive parameter identification procedure must be switched off before the possible faults occur otherwise the 225

a1 = -1.430808

a2 = 0.432895

bo = -0.002161

b = 0.002153 1

b2 = 0

compensated residual

primary residual -30 r-------,- - - - - - - - , - -- -

4.-------~------~------_,

-35

-2 -4 ~------~------~--------

-45L---~---~---~

o

5

10

o

15

5

10

15

time (seconds)

time (seconds) Figure 4: The primary residual for fault free case

(b) compensated residual

These parameters have been used for residual online compensation. In turn, the compensated residual will be used to detect faults. A particular emphasis is the power of the method to detect soft or mClplent faults which are otherwise unnoticeable in the measurement signals. These attributes are well illustrated in the following graphical time response results. As the fault detection scheme has been made robust against modelling errors, the scheme is able to detect incipient faults under conditions of modelling uncertainty.

Figure 6: The primary and compensated residuals when a fault occurs on the temperature sensorT7

Fig.6 shows the faulty residual signals. It can be seen that the compensated residual has a very significant increase when a fault has occurred in the system, a small threshold then can easily be placed on the residual signal to declare the occurrence of faults. But, one cannot be sure whether a fault has even occurred in the system when using the information from the primary residual.

Fig. 5 shows the faulty output of sensor TT The fault signal is very small compared with the output signal, and consequently, the fault cannot be detected directly in the output. ~T7

~P6 (KPa)

80r-------~------~----_,

(OK)

150 r-------,.---'-----,---

-----,

20

100

5

50

10

15

time (seconds) 5

10

compensated residual

15

2r-----~~~--~----~

time (seconds) Figure 5: The faulty output of the temperature sensor (T7)

o

primary residual -30 r - - - - - - - , - - - - - , - - - - - ,

-2

-35

-4~------~------~------

o

5

10

15

time (seconds) Figure 7: The faulty output the compensated residual when a fault occurs in the pressure sensor P6

____~ 10 15

-45L-----~L-----~--

o

5

time (seconds)

A fault signal with the same time profile is now added to the pressure sensor signal for P6 in order

Fig.6 (a) primary residual

226

Frank P M (1990), Fault Diagnosis in dynamic system using analytical and knowledge -based redundancy - A survey and some new results, Automatica, 26 (3), 459-474

to assess the fault detection performance further. The result is shown in Fig. 7. All simulation results demonstrate the efficiency of the compensated residual in the role of robust fault detection.

Frank P M (1991), Enhancement of robustness in observer-based fault detection, Proc.

IFAC/lMACS Symposium SAFEPROCESS'91, Baden-Baden, Sept 10-13, 1991, VoU, 275-2B7

5 Conclusion

Frank P M, Ding X & Wochnik J (1991), Model based fault detection in diesel-hydraulically driven industrial trucks, Proc. of 1991 Amer. Control COil!, 152B-1433, Boston, USA

This paper has provided a study of the on-line residual compensation method for robust modelbased fault diagnosis. Using an identification approach, the combined effects of all uncertainties on the residual are modelled as a transfer function between the residual and the input. Then, the effect of uncertainties can be compensated on-line. The compensated residual provides a basis for robust fault diagnosis.

Gertler J (1991), Analytical redundancy methods in Failure detection and isolation, Proc. of

IFAC/IMACS Symposium SAFEPROCESS'91, Baden-Baden, Sept., VoU, 9-21 Goodwin G C & Payne R L (1973), Dynamic

system identificatioll: Experiment design and data allalysis, Academic Press, 1977

The paper has also provided an interesting simulation of the application of the residual compensation approach to detect sensor faults. The approach used considers robustness in terms of the modelling uncertainty present when a low order model is used to approximate a complex non-linear high order system. The results have shown that an incipient fault which has a barely detectable effect on anyone measurement, can be detected very easily using a compensated residual. This illustrates the potential of using a modelbased method which does not actually require a detailed and accurate model of the dynamic system considered. Further studies are being carried out to evaluate the effectiveness of the approach applied to real engine data. Especially, the stochastic case will be considered. This may involve noise modelling etc.

6

Patton R J (1991), Fault detection and diagnosis in aerospace systems using analytical redundancy,

lEE Computing & Control Engineering Joumal, ~

(3),127-136

Patton R J & Chen J (1991a), A review of parity space approaches to fault diagnosis, Proc.

IFAC/lMACS Symposium SAFEPROCESS'91, Baden-Baden, Sept.lO-13, 1991, VoU, 239-255 Patton R J & Chen J (1991b), Robust fault detection of jet engine sensor systems by using eigenstructure assignment. Proc. of AIAA Guidance, Navigation & Control COil!, New Orleans, August, AlAA-91-2797 Patton R J & Chen J (1991c), Optimal Selection of unknown input distribution matrix in the design of robust observers for fault diagnosis, Proc.

Acknowledgements

IFAC/lMACS Symposium SAFEPROCESS'91, Baden-Baden, Sept.lO-13, 1991, VoU, 221-226, also to be published in Automatica

The authors acknowledge the funding support for this research from the UK Science & Engineering Research Council through grant (GR/G2586.3). Thanks are extended to Professor H . Y. Zhang for helpful comments during his sabbatical leave at the University of York.

Patton R J & Chen J (1992), Robustness in modelbased fault diagnosis, In Concise Encyclopedia of Simulation and Modelling, (Eds: D. Atherton, P. Borne), Pergamon Press, January 1992,379-392

7 Reference

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