Industrial Actuator Benchmark Test - A Systematic Approach To An Optimal FDI

Copyright @IFACFaultDetection. Supervision and Safety for Technical Processes. Espoo. Finland. 1994

INDUSTRIAL ACTUATOR BENCHMARK TEST - A SYSTEMATIC ApPROACH To AN OPTIMAL FDI Th. H()fiing, Th. Pfeufer, R. Deibert, R. Isermann Institute of Automatic Control. Laboratory for Control Engineering and Process Automation, Technical University Darmstadt, Landgraf-Georg-Str. 4, D-64283 Darmstadt, Germany, Phone: +496151163114, Fax: +496151293445, e-mail: [email protected]

In order to apply the best fault detection and diagnosi s scheme, it is a must to investigate the process model profoundly and the kinds of faults to be detected. Especially the process excitation and the effect of a considered fault play an important role. This is the starting point for the choice of one of the various model based fault detection methods. According to this strategy two different approaches, an observer based and a signal based approach, are selected for the two given faults of the benchmark task. Moreover it is shown that the application of adaptive thresholds can improve the performance of the fault detection scheme significantly with respect to false alarm rate and delay of detection. ABSTRACT.

KEYWORDS.

1.

Fault detection; model based; observer based; adaptive thresholds

Processes with lumped parameters which can be linearized around one operating point are usually described by a differential equation in continuous time domain

INTRODUcnON

The process under investigation in this benchmark test is a brushless synchronous d.c.motor with a control configuration which is used in many actuators (cascade of control loops for current control, speed control, position control). The faults to be detected are modelled as additive faults. No deviations in the physical parameters of the actuator have to be treated as fault sources, which means that no multiplicative faults have to be considered. Parameter variations are to be regarded as model uncertainty only and the fault detection should be robust with respect to them . The main problem, however, for the application of a fault detection method will be the non-measurable load which acts on the system at the same addition point as one of the fault signals.

yet) +a.y(1)(t) + ••• + G,.Y
= boU(t) +b.u(1)(t) +•.. +b..u(")(t) yet) = Yet) - Yoo

u(t)

= U(t) - Uoo

(1)

(2)

where Uoo, Yoo are steady-state (or d.c.) values and y(n)(t) = d"y(t)/dtn • Process model representations in discrete time domain can be used for sampled data. This avoids the numerical calculation of time derivatives of the measured data.

As for fault detection any kind of redundancy is required, here this is provided analytically by a mathematical model of the given technical process (including actuators and sensors). Faults shall be detected by evaluating the relationship between the available input and output variables U(t) and Y(t) using the underlying model. In general a distinction can be made between static and dynamic, linear and nonlinear process models (Isermann, 1984, 1991). Here only linear dynamic models are considered to describe briefly some basic methods.

Several publications have shown how to use these basic model equations in order to establish a fault detection scheme, Patton et al. (1989), Frank (1990), Gertler (1991). As each of these methods possesses certain advantages and disadvantages, the user has the choice to select and to tailor the optimal strategy for his specific process with its specific faults . The blockdiagrams, models, estimators and generated residuals of model based fault detection methods are summarized in Isermann (1993).

495

2. STRAIGIITFORWARD ApPROACH

ds 1 and ds2, Fig. 1 (In the sequel ds 1 is used as abbreviation of data set I, likewise ds2, ds3 and ds4 is used). But no unique threshold for small signal (ds3) and large signal (ds4) excitation can be determined (note that all plots have the same scaling). Therefore some improvements are required. One possibility is to use a low pass filter in order to get rid of noise. Applying filters, however, slows down the detection of the fault and converts primary parity equations as developed for r,! into residuals which can be obtained by observer-based approaches.

Due to the complexity of the detailed actuator model and the character of the given faults it is proposed to solve the fault detection problem by the design of different methods which work in parallel, each of them sensitive to one of the faults . In order to keep the methods used transparent, for the first results the apparently easiest approach is applied. Instead of developing complicated equations the goal of this benchmark solution is to modify the more simple approaches slightly and to emphasize the residual evaluation by using additional knowledge, e.g. the signature of the faults.

At this point further analysis of the given model structure is performed leading to the result that the measured speed nm and the measured position So can be regarded as input and output of a linear singleinput-single-output-system. It consists only of an integrator, the gear ratio N as well as measurement scaling factors . Note that the speed reference input 1\.:( is not required and this part of the model is a priori also independent of the load input.

2.1 Detection of Fault Ilso

First the fault ~so is shortly analyzed according to the main benchmark task of fast detection and isolation of the faults considering also robustness issues. It appears as additive fault on the sensor output So, so that parity space or observer based methods should be preferred to parameter estimation, because a parameter estimator copies the underlying fault into a change of the gear ratio N and acts slower in most cases. The fault signal &o(t) is not known, but in the considered closed loop a seized sensor signal So results in a ramp signal of ~so, because the tracking error is integrated.

so'(s)

1.945610-'n,Jk-l)

A very easy approach is to compute the difference rS2 between the measured position and the integrated speed. t

r,2 =

slit 10-41

~2

.C).Ol

,

Oh,, ,

,, ,,

0.01

l'

0

-0.01

-0.02 0

°hl,, ,,

,, ,,

(5)

No

Dam-s.t 1 0.01

+ so'(k-l) - so'(k)

SIC 10-41

So , - -ex'In",(t) d't

But this residual is drifting due to noise and the relative large sampling interval, Fig 2.

0

o.ta.s.t 1

(4)

n (s) Ns '"

The big advantage is that this relation is valid in both models, the simple and the complex one. Therefore good results for all the four test cases are expected.

As a linear fault detection scheme shall be designed systematically, the next step is the computation of a parity space (e.g. according to Chow and WiIlsky, 1984). The simple linear model is used and the residual should be made insensitive to changes at the load input. This leads for example to the following residual equation: r$l(k) = 4.59 10-5 n",(k-l) + 8.309 10-' n",(k) (3) +

ex.

= -

0100-5012

l'

-0.02 0

o..s.t ..

Oeta-5M3

0.01

,, ,,

0 .01

,,

·s0

oS 0

o.c.-U3

slit 10~

0-.501 •

-0.01

,, ,,

Fig. 2: Direct integration and comparison Using this technique the problem of detection appears to be solved if the drift in the large signal case (ds4) can be controlled. But the application of a threshold reveals the fact that the detection IS delayed because of the slow integration.

Fig. 1: Residuals developed with parity space methods The results of this approach are only sufficient for

496

Both problems can be tackled using a stabilizing observer feedback.

This trade-off is illustrated for three different chosen factors h of 5, 12 and 50, Figs 4, 5 and 6. These residuals provide fast and reliable detection of the fault ~so' In case of ds4 the detection is delayed due to the fact that the faulty sensor shows the right value as long as there is no change in the process signals.

So

o..·s.t2

O.t.. ~l

0.005

o'O 'FB' o , I

,

~ .005

Fig. 3: Observer for position fault

~so

-0.01

:

I

o

,

,

-0.005

,,

o

O 'O '[tB'

0.005

1

2

3

-0.01

:

,,

o

1

2

3

o.ta-Set 3

The residual r. 3, according to Fig. 3, can be calculated as follows with both the measured speed nm and the measured position so' as inputs to the observer and residual generator:

0.01,,------,..------

..Q.OOS

(6)

Moreover so' in terms of the observables yields

ha. ,la. --::....-s + n s + ha. 0 N s + ha • ..

Fig. 4: Observer-based residuals (h=5)

(7) 0 .005

Then one obtains (

0

a. n.. ha.)s'_ s + ha. 0 s + ha. N

1 -

"'.005

(8)

s

1 +

ha.

-0.01

a) N"

q( , ,

s

+

a h

•

0

o.t.. s.t 4

-0.005

. ,.O"OL

_..L...l.._ _ __

Fig. 5: Observer-based residuals (h=12) Data-Set 1

0 . 0 ',,,- - - - - - - ,

,, I

I

oWI'r---~ , "'.005

. ,.O,!-O

, , ,

DII.-s.t 2

O.O,,.--~-~-,,

,,

0.005

I

I

OWl', ,

------I

.a,COS

I

_.L...L.._~_-!

0.0 ' , . - - - - - - - ,

Ils

0

0.005

In the fault free case the term in parenthesis in eq.(8) remains close to zero and is only a random noise process. In case of a fault ~o its effect on the residual rS3 is: s

-0.005

0 .01

0.005

=

0.005

-0.01

0

0.01

A closer look at the term in parenthesis shows the reconstruction of the integral relationship between speed and position. And this is filtered by a first order low pass filter which contains one freely assignable design parameter h. In order to find a good choice the effect of a fault on the residual is examined.

r&J

0.01

Data-Set 3

( s 's - ....!. n 0

Dc.-Set 2

Datli·Sttt

O.Ot

(9)

0.005

,, ,,

0~1'~ ~~ ,

This is a differentiating filter with respect to the fault signal ~so. The parameter h has to fulfil two tasks. First, h determines the cut-off frequency of the filter. The lower the cut-off frequency the better is the suppression of high frequent noise but the slower is the fault effect gets visible. If this frequency is chosen too low the system becomes stiff, meaning it behaves similar to the direct integration as used for the computation of r.2 (see Fig. 2). Second, h fixes the steady state gain of the time derivative of the fault signal &0, which means the smaller parameter h the bigger the effect of the fault ~so on the residual.

"'.005

, , ,

-0.0'0'---'-'------'

,

0.0

0.005

,, ,,

O~ "'.005 -0.0

,

..

, , ,

0

Fig. 6: Observer-based residuals (h=50) At last a threshold I must be chosen which does not lead to false alarms, but detects an occurred fault as soon as possible. The same fixed threshold must be applicable to all the four testcases ds 1 to ds4 (Emami-Naeini et al., 1988). As an observer based approach with a choice of h=12 leads to the highest fault to noise ratio (see Fig. 5), for this case the

497

threshold shall be detennined. The maximum value of the fault free residual rs3 is 0.0033 (in ds4 at time 2.Ss). Therefore the threshold I can be chosen conservatively to I.S times that value which yields lfix=O'ooS . This permits no false alarms (at least in the considered testcases). But the detection of the fault especially in the small signal cases (dsl to ds3) is delayed according to the high limit value lfi ., see Fig. 7. ,,

0.5

I

,

.,

:

,

·1

,0

O . Ol~~---~

0.D1~

0.005

o

-{).oos 1

2

3

Fig. 8: Residual with adaptive thresholds oma-Set4

,, Data-Set 1

I

Data-Se12

0.5

o~

:U

.0.5

I I

-,

"

-1 .5 0

-0.0

3

Data-Set 3

-0.0'0

0eta-Sel3

'U

0:

2

., .5 0

,,

-0.5

1

-0.01 0

:

I

0.5

0.5

.0.5

.0.5

-1 .5

-1. S 0

.,

.,

-1.S 0

0

Fig. 7: Fault signatures with a fixed threshold

o.ta-s.t 3

o.sa-&.14 I I

A more promising approach, however, can be the application of an adaptive threshold lad taking noise and model uncertainty into account (Clark, 1989; Frank, 1991). One may think there is no model uncertainty, but the low sampling rate coupled with noise introduces some uncertainty. It is assumed that the model structure (direct integration) is perfect, but an unknown gain-factor ':\K is admitted. This leads to another description of residual rs3:

~t:.s s

+

",h

__ 1_

= ~ Dd

t:.Kn

Is

.,

.0.5

-1.5 0

-1.5 0

oH ., ,, W I

Fig. 9: Fault signatures with adaptive thresholds

- ("I N

+ t:.K)n)

•

(noise effects) (uncertainty)

..

aK +

a.sh

n

..

~

I

+

2 •

4rms.

3

4

5

11

7

B

11

10

11

~ fIIoId IhreshaId

dptIvw IhreshaId

Fig. 10: Detection time of the position fault • 'O ~

Now a limit value lad should be set up which takes noise and uncertainty into consideration. To handle the noise effects a fixed value is chosen which is four times higher than the rms (root-mean-square) of the high frequent noise in the fault-free part of residual rs3' Moreover a second term is added taking the model uncertainty into account which is driven by the input signal nm • ':\K is selected as 4% of ajN which is O.OOO4S . The obtained threshold runs as follows: I

.0.5

o

+

a.,h

0.5

6'-'(10)

V'

__ 1_ (" s s s + ",h , .

+

0.5

,

fFault si-nil

0

+

s

/"

I1

.0.00 5

~1.005

, , ,,

.,

"

-1 .5 0

op

I

.0.5

I

I

0.005

o

o~,

o~,

-0.5

0. 01~

,,, ,

,,

o.t.-s.t 2 0 .0 1

0.005

oma-s.t 2

Dtlta-Se!l

0.5

om-Set 1

o..-s.t F3

o....s.F3

:a 0

x 10"

,

2

3

IIOISe

0

The application of lad leads to a much improved performance of the fault detection. There is a smaller risk of false alarms and it is faster (see Figs. 8 and 9).

.0.5 ·1

· 1.5

0

Da..s.tF3

DatI-set F3

.~

(11)

0.5

,

0 .5

-0.5

.,

-1.5 0

Fig. 11: Left: Residuals with thresholds. Right: Fault signatures

498

Fig. 10 shows the significant improvement provided by the application of an adaptive threshold compared to a fixed one.

error ril is now checked for significant non-zero deviations . Because of the three possibilities mentioned above. appropriate thresholds have to be used in order to avoid false alarms. These thresholds take into consideration the changes of the position reference. the noise of the measured data. and model uncertainties. The position reference is first filtered by a DTJ-fiiter

Furthennore the designed fault detector is applied to dsF3 in order to verify the obtained results. Fig 11. Neither the fixed threshold nor the adaptive threshold lead to false alarms

S rqF

2.2 Detection of the current fault

1 +T1s S rq 1 11!E.!....

(13)

to detect abrupt changes and to eliminate the effect of their transient response on the residual. After the transient response the threshold is switched to a fixed value k (see Fig. 12).

To cope the problem of the detection of fault aim. the overall structure of the system under investigation has to be regarded more detailed. It is a triple control loop cascade (position. velocity. current). Related to classical control techniques. the inner current control loop is very fast whereas the outer position control loop is of comparatively rather low dynamic. Both the current fault aim and the load enter the actuator at the same point and can be considered as disturbances of the velocity control loop. It is obvious that no common model-based method like parameter estimation or unknown input observers will be able to distinguish these disturbances. Therefore. deeper investigation of the available signals of the velocity control loop is proposed.

I

where

0"

=

SrqFo(srqF-k) + ko(k-s re/)

(14)

denotes the step function o(x)

= {I for x ~

o

0 } for x < 0

(15)

"r--~-~--~-~-----,

'0 • .._ _ _ __" _ -_ _ _ .,_ _ ".-----.J '.

There are three reasons for a non-zero deviation of the error signal n,..r - Om: 1) The position reference has changed . It is assumed that only abrupt changes of the position reference occur. This is a typical operation for such actuators which is shown in the data sequences. Such changes force a change of nrer immediately (the position controller is a simple proportional controller). 2) There is a change of the load. This can be regarded as a disturbance of the velocity control loop which wiII be managed by the velocity controller and the underlying current control loop and wiII have only slight effects on the position signal. 3) The current fault appears. Due to the rectifying characteristic of the current fault the velocity controller is not able to handle this fault unless the position reference changes and the required current reference has the other sign.

·'0 .1S0~-_;t.O.•, - - - - : - - - - ; " , ' ;.•----:--2.;;;.--'--!

Fig. 12: Example for the residual and the special threshold The threshold parameter k is designed to match noise. load changes and model uncertainty and model errors while regarding the nonlinear models. The parameters of the DTJ-filter are chosen to eliminate the magnitude of the position reference steps adequately. So. for the small signal models. the parameters T D is nearly the same. The value of the constant threshold is lowest for the linear model. Applying the thresholds to the nonlinear model. noise and model errors (the residual generator is linear) have to be taken into account. for the large signal model the different position controller gain is additionally involved. The residual parameters for the different data sequences are summarized in Table 1. The differences are caused by different measurement noise (ds 1) and different controller gains (ds4).

In contrast to the benchmark description the position reference signal is assumed to be known because it is an internal computer signal. Based on the simple linear model. the following procedure is applied:

Fig. 13 and Fig. 14 show the results of the presented strategy in logical fonn . If the residual violates the threshold. the alarmsignal is set 1. otherwise it is zero (solid line). For comparison. the fault pattern (dotted line) and the load pattern (dash-dotted line) are also shown. One can see that the algorithm is

The signal rjJ(z.) = GI-z.)(nrJz.)-n... (z.»

=

(12)

is computed using a 4 1110rder digital lowpass filter in order to smooth single peaks. The filtered velocity

499

methods can be applied to cope with the benchmark restrictions.

Table 1: Threshold Parameters Data Set

To

TI

k

I

10

0.09

2.6

2

15

0.09

5

3

15

0.09

5

4

125

0.09

20

F3

15

0.09

5

However, this benchmark problem does not cover the whole field of fault detection and diagnosis because only additive faults are considered and all model parameters are assumed to be known exactly. Most of the real faults of d.c.motors appear either as parameter changes or as changes from vibration.

4. REFERENCES

able to detect the current fault while being robust against load changes for the ds 1 to ds4. Regarding dsF3 there seem to be some alarms and no robustness, respectively. But there is no possibility to distinguish between current fault and load changes if only the load change forces the current fault to appear.

Chow, E.Y.; Willsky, A.S. (1984). Analytical redundancy and the design of robust failure detection systems, IEEE-Trans. on Aut. Control, AC-29, pp. 603-614. Clark, R.N. (1989). State estimation schemes for instrument fault detection, in Patton et al.(eds), (1989). Deibert, R. (1994). Model-Based Fault Detection of Valves in Flow Control Loops. IFACSymposium SAFEPROCESS '94, Espoo. Emami-Naeini, A.; Akhter, M.M.; Rock, S.M. (1988). Effect of model uncertainty on failure detection - The threshold selector, IEEE-Trans. on Aut. Control, AC-33, 12, pp. 1106-1115. Frank, P.M. (1990) Fault detection in dynamic systems using analytical and knowledge-based redundancy - A survey and some new results, Automatica, Vol. 26, No. 3, pp. 450-472. Frank, P.M. (1991). Enhancement of robustness in observer-based fault detection. IFAC-Symposium SAFEPROCESS '91, Baden-Baden. Proc. Pergamon Press, Oxford. Gertler, J. (1991). Analytical redundancy methods in fault detection and isolation. IFAC-Symposium SAFEPROCESS, Baden-Baden. Proc. Pergamon Press, Oxford. Gertler, J.; Monajemy, R. (1993) . Generating directional residuals with dynamic parity equations. 12th IFAC World Congress, Sydney. Hofling, Th.; Pfeufer, Th. (1994). Detection of additive and multiplicative faults - Parity space vs. parameter estimation. IFAC-Symposium SAFEPROCESS '94, Espoo. Iserrnann, R. (1984). Process fault detection based on modelling and estimation methods - a survey. Automatica, Vol. 20. Iserrnann, R (1991). Fault diagnosis of machines via parameter estimation and knowledge processing. IFAC-Symposium SAFEPROCESS '91, BadenBaden. Proc. Pergamon Press, Oxford. Iserrnann, R. (1993). On the applicability of model based fault detection for technical processes. 12th IFAC World Congress, Sydney. Patton, R; Frank. P.M.; Clark, RN. (1989). Fault Diagnosis in Dynamic Systems - Theory and applications. Prentice Hall, New York London.

Fig. 13: Detection of current fault dsl..ds4 , , , , , ,

0 .• 0 .• 0.7 0 .• 0.5 0.'

i;

;,

0.3 0.2

~ ...

0.1 0 0

,., ,

,

, , , , , , , ,, ,, ,, ,, ,, ,, ,., , , "i ' , ,.

r

,:,: j:i I: ,

,. '. . ' .'

I

•

0 .5

1.5

2.5

Fig. 14: Detection of current fault, dsF3

3. CONCLUSIONS Two different methods were presented to detect the position fault and the current fault. It was shown that the choice of an optimal fault detection method depends very much on the structure of the considered plant and on the character of the faults which should be detected. The main result here dealing with this specific plant is first to look very carefully on the kind of the faults and where they enter the plant. So, rather simple

500