identification of physical systems (applications to condition monitoring, fault diagnosis, soft...

11Fault Diagnosis ofPhysical Systems

11.1 OverviewModel-based fault diagnosis of physical systems is presented in this chapter. A physical system is aninterconnection of subsystems including the actuators, sensors, and plants. The fault diagnosis schemeconsists of the following tasks:

Selection of diagnostic parameters: The diagnostic parameters are selected so that they are capable ofmonitoring the health of the subsystems, and may be varied either directly or indirectly (using a faultemulator) during the offline identification phase [1, 2].

Identification of the system model: A reliable identification scheme is used to obtain a model of thesystem that captures completely the static and dynamic behavior under various potential fault scenariosincluding sensor, actuator, and subsystem faults [1–7].

Design and implementation of the Kalman filter: The Kalman filter is designed using the identifiednominal fault-free model. The Kalman filter residual is employed for both fault detection and isolationsince (i) the Kalman filter residual is zero in the statistical sense if and only if there is no fault, and (ii)its performance is robust to plant and measurement noise affecting the system output. The Kalman gainK0 was tuned on line (as the covariances of the disturbance and measurement noise were unknown)so that Kalman filter residual is a zero-mean white noise process with a minimal variance [8].

Estimation of influence vectors: The influence vectors, which are partial derivatives of the residual withrespect to each diagnostic parameter, are estimated offline by performing a number of experiments.Each experiment consists of perturbing the diagnostic parameters one at a time, and the influencevector is estimated from the best least-squares fit between the residual and the diagnostic parametervariations. When the diagnostic parameters are not accessible, as in the case of plant model parameters,for example, the fault emulator parameters are perturbed instead [1, 2].

Fault diagnosis: The decision to select between the hypothesis that the system has a fault and thealternative hypothesis that it does not is extremely difficult in practice as the statistics of both the noisecorrupting the data and the model that generated these data are not known precisely. To effectivelydiscriminate between these two important decisions (or hypotheses), the Bayes decision strategy isemployed here as it allows for the inclusion of the information about the cost associated with the

Identification of Physical Systems: Applications to Condition Monitoring, Fault Diagnosis, Soft Sensor andController Design, First Edition. Rajamani Doraiswami, Chris Diduch and Maryhelen Stevenson.© 2014 John Wiley & Sons, Ltd. Published 2014 by John Wiley & Sons, Ltd.

448 Identification of Physical Systems

decision taken, and the a priori probability of the occurrence of a fault. The fault detection andisolation problem is posed as a multiple hypothesis testing problem. The Bayes decision strategy isdeveloped exploiting the fact that the residual is a zero-mean white noise process if and only if thesystem and the Kalman filter models are identical, that is if there is no model mismatch and thereforeno fault. A fault in the subsystem is isolated and asserted if the correlation between the measuredresidual and one of a number of hypothesized residual estimates is maximum [1–4].

The mathematical background on fault diagnosis is given in earlier chapters on the Kalman filter and faultdiagnosis. Case studies in fault diagnosis of two-tank process control system and the position control arepresented.

11.2 Two-Tank Physical Process Control System [9]11.2.1 ObjectiveThe objective is to detect and isolate leakages, actuator faults, and liquid-level sensor faults. In order tosimulate faults in the physical system, static fault emulators are connected in cascade with the heightsensor, the flow rate sensor, the actuator, and the leakage drain pipe as shown in Figure 11.1, which isthe block diagram of the process control system. The emulator parameters 𝛾1, 𝛾2, and 𝛾3 are connected incascade with the leakage drain pipe, the actuator, and the height sensor during the offline identificationstage.

Figure 11.1 is a block diagram of the two-tank process control system where G0 is the controller,G1 is the actuator, G2 tank; r, e, u, f , w, and y are respectively the reference input, error, flow rate, themeasurement noise and the disturbance, and measured height y.

11.2.2 Identification of the Physical SystemThe physical two-tank fluid system is nonlinear with a dead-band nonlinearity and fast dynamics.

The process data was collected at sampling time Ts = 0.5 sec. The objective of the controller is toreach a reference height of 20 cm in the second tank. During this process, several faults were introduced,such as leakage faults, sensor faults, and actuator faults. Leakage faults were introduced through thepipe clogs of the system, knobs between the first and the second tank, and so on. Sensor faults weresimulated by introducing a gain in the circuit as if there was a fault in the level sensor of the tank.Actuator faults were simulated by introducing a gain in the setup for the actuator that comprises themotor and pump. A Proportional Integral (PI) was employed in order to track the desired referenceheight. Due to the inclusion of faults, the controller was unable to track the desired level. The power of

0G1G 3γ

w

yr11

Tank

Controller Actuator

Height sensor

Leakage

γ−

1γ

2Ge u

sk2γf

Figure 11.1 Two-tank process control system with emulators

Fault Diagnosis of Physical Systems 449

the motor was increased from 5 volts to 18 volts in order to provide it with the maximum throttle to reachthe desired level under various operating scenarios including faults. Due to the closed loop PI controlaction, the tracking performance under emulator parameter variations was good, but it compromised theperformance of the fault diagnosis scheme.

The physical two-tank fluid system is nonlinear with a dead-band nonlinearity and fast dynamics. Theidentified model order is different from that of the model derived from the physical laws. The identifiedmodel was essentially a second-order system with a delay, even though the theoretical model is of afourth order. Such a discrepancy is due to the inability of the identification scheme to capture the system’sfast dynamics, especially in low-SNR scenarios. Using the fault-free model together with the covarianceof the measurement noise R, and the plant noise covariance Q, the Kalman filter model was designed.To obtain an estimate, a number of experiments were performed under different operating scenarios totune the Kalman gain K to obtain an optimal performance which ensures the generation of a white noiseresidual with a minimal variance.

Various types of faults, including the leakage, actuator fault, liquid-level sensor fault, and the flowsensor fault, were all introduced. These faults were emulated by varying 𝛾𝓁 for the opening of thedrainage valve that causes a leakage fault, varying the gain block 𝛾a connected to the actuator input, andvarying 𝛾s connected to the liquid-level sensor output. The National Instruments LABVIEW packagewas employed to collect these data.

The reference input was chosen to be a step input. An offline perturbed-parameter experiment wasperformed to estimate the influence vector [1, 2].

The nominal fault-free model relating the reference input r(k) and the output y(k) was identified fromthe step response of the system. A high-order least-squares method with nh = 6 was employed. A secondreduced-order model was derived from the high-order identified model.

Figure 11.2 shows the plots of the step responses when the diagnostic parameters 𝛾𝓁 , 𝛾a, and 𝛾s

were varied one at a time to induce leakage, actuator, and sensor faults. Subfigures (a), (b), and (c) inFigure 11.2 show the leakage faults, actuator faults, and sensor faults induced by varying 𝛾1, 𝛾2, and 𝛾3,respectively. These three plots show the normal and faulty cases. Small, medium, and large faults weresimulated by varying the diagnostic parameters by 25%, 50%, and 75% respectively and are shown. Thenormal case is also shown.

11.2.3 Fault DetectionAt time instant k, the N = 189 residuals formed of the present and past (N − 1) residuals e(k − i) : i =0, 1, 2,… , N − 1, are collected. Let H0 and H1 be the two hypotheses indicating the absence and presenceof a fault, respectively. The fault detection strategy for the case when the reference input is constant wasemployed here. The fault detection strategy is:

ts(e)

{≤ 𝜂 no fault

> 𝜂 fault(11.1)

where e(k) = [ e(k) e(k − 1) e(k − 2) . e(k − N + 1) ]T , and ts(e) =||||| 1

N

k∑i=k−N+1

e(i)||||| is the statistics.

Figure 11.3 shows the Kalman filter residual and its auto-correlation for the following cases: (i)nominal (or fault-free), (ii) leakage fault, (iii) actuator fault, and (iv) sensor fault. The test statistic valueis the lowest and the auto-correlation is that of a zero-mean white noise for the nominal (fault-free) case.The subfigures (a), (b), (c), and (d) of Figure 11.3 show the residuals and their test statistics shown asstraight lines, whereas the subfigures (e), (f), (g), and (h) show the corresponding auto-correlations.


50 100 1500

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Heig

ht

A: Leakage fault

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Heig

ht

B: Actuator fault

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Heig

ht

C: Sensor fault

Figure 11.2 Liquid level for normal and fault scenarios

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A: Fault-free

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-100 0 100

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G: Actuator

-100 0 100

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Figure 11.3 The residuals and their auto-correlations for the fault-free and faulty scenarios


11.2.4 Fault IsolationThe influence vectors {Ω i} for the leakage, actuator, and sensor faults were estimated by perturbing thediagnostic parameters one at a time during the offline identification phase. During the operating phase,the faults were isolated using the Bayes decision strategy:

𝓁 = arg(max

j{cos2 𝜑j(k)}

)(11.2)

where cos2 𝜑i(k) is the cosine of the angle between the residual e(k) and its estimate 𝜓 T (k)Ωi given by:

cos2 𝜑J(k) =

[ ⟨e(k),𝝍T (k)Ωj

⟩‖e(k)‖ ‖𝝍T (k)Ωj‖

]. (11.3)

As was done before, the leakage, actuator, and sensor faults were all introduced by varying thediagnostic parameters 𝛾1, 𝛾2, and 𝛾3. Three values for 𝛾i : i = 1, 2, 3 were chosen, namely 0.25, 0.50, and0.75, with the nominal value 𝛾 0

i = 1, to simulate “small,” “medium,” and “large” faults, respectively. TheBayes decision strategy was employed to assert the fault type, that is, leakage or actuator or sensor fault,by computing their three quantities cos2 𝜑i(k), i = 1, 2, 3 associated with the leakage fault, actuator fault,and sensor fault, respectively. The decision on which fault is asserted to have occurred is made based onwhich cos2 𝜑i(k) is maximum, that is, the leakage fault is asserted if i = 1, actuator fault if i = 2, andsensor fault if i = 3. Figure 11.4 shows the plot of cos2 𝜑i(k) vs. 𝛾i for leakage, actuator, and sensor faultswhen 𝛾i : i = 1, 2, 3 were varied as described above. The maximum values of cos2 𝜑i(k) were normalizedto unity. The subfigures (a), (b), and (c) in Figure 11.4 show the plots of cos2 𝜑i(k) with respect 𝛾i for the

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

0.2

0.4

0.6

A: Leakage fault

Leakage parameter

Cosin

e-s

quare

d

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

0.20.40.60.8

B: Actuator fault

Actuator parameter

Cosin

e-s

quare

d

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

0.20.40.60.8

1C: Sensor fault

Sensor parameter

Cosin

e-s

quare

d

Leak Actuator Sensor

Figure 11.4 Plots of cos2 𝜑i(k) vs. 𝛾i for leakage, actuator, and sensor faults


leakage, actuator, and sensor faults, respectively. The maximum values of the “cos” quantities are usedto accurately isolate the faults.

Comment The leakage fault may easily be isolated from the rest of the faults, while this is not sofor both the actuator and sensor faults unless the fault size is large compared to noise floor, as theircos2 𝜑i(k) values are very close to each other.

11.3 Position Control System11.3.1 The ObjectiveThe objective is to detect and isolate a position sensor, a velocity sensor, and an actuator fault.

In order to simulate faults in the physical system, static fault emulators are connected in the cascadeposition sensor, the velocity sensor, and the actuator as shown in Figure 11.5 which is a block diagramof the position control system. The emulator parameters 𝛾1, 𝛾2, and 𝛾3 are connected in cascade with theposition sensor Ks, velocity sensor Kv, and the actuator Ka.

11.3.2 Identification of the Physical System [2, 10–12]In general, a physical system is highly complex and nonlinear and, as such, defies mathematical modeling,and this physical system is no exception. The PWM amplifier exhibits saturation-type nonlinearity andfurther, the tachometer-based velocity sensor is very noisy. Since the dynamics of the physical systemcontain uncertainty in the form of unmodeled dynamics, including nonlinear effects such as friction,backlash, and saturation, the order of the identified model was selected by analyzing the model fit fora number of different orders that range from 3 to 25. The probing input was a 0.5 Hz square wave, thesample frequency was 100 Hz, and each data record contained 1000 samples.

The model order is selected such that the identified model captures the static and the dynamic behaviorof the system not only at the nominal operating condition but also under potential fault scenarios. Varioustypes of faults, including the position sensor, velocity sensor, and the actuator were all introduced. Thesefaults were emulated by varying respectively 𝛾1, 𝛾2, and 𝛾3.

Figure 11.6 shows the estimated and the measured outputs (outputs during half the period of the squarewave) when the emulator parameter 𝛾1 of the position sensor was varied, and a 10th order model wasselected. Figure 11.7 shows the results when the model order was 5.

It can be deduced from Figure 11.6 that a 10th order model is able to capture the static and the dynamicbehavior of the system over all variations in the emulator parameters in the range 0.7 to 2.8, whereas afifth order model is unable to capture this, as can be seen from Figure 11.7.

1

11i

p

k zk

z

−

−+−

1

1

11

k z

zα

−

−−2

11

k

z−−Ak

kω2γdk

kθ 1γr e u

Actuator

PID controller

Plant

Emulator

Emulator

Sensor

Sensoryω

3γ

Figure 11.5 Block diagram of the position control system


0.2 0.25 0.3 0.35 0.4-1.5

-1

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2γ

1 = 0.7

Time

Measured output

Estimated output

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Estimated output

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21

γ = 2.8

Time

Measured outputEstimated output

Figure 11.6 Output and its estimated when a 10th order model was selected

The performance of a 10th order model, identified using the scheme outlined herein, is evaluatedconsidering the following three cases:

Case 1: The model is identified by performing a single experiment when the diagnostic parametersare at their nominal values and is validated when one of the diagnostic parameter varies and all othersare fixed. This is the conventional scheme.

Case 2: The model is identified using many experiments, perturbing one of the diagnostic parameterswhile keeping the others fixed at their nominal values.

Case 3: The model is identified using many experiments, perturbing all of the diagnostic parameters.This is the proposed scheme. The mean-squared error is minimum over a wide range of the variationsof 𝛾1.

Figure 11.8 below shows the mean-squared error for the Case 1, 2, and 3 when one of the diagnosedparameters changes. It was found that the conventional scheme of Case 1 is particularly prone to modelingerrors and the proposed scheme of Case 3 is the best.

Kalman filter designThe nominal fault-free model relating the reference input r(k) and the output y(k) was identified fromthe step response of the system. Using the fault-free identified model together with the covariance of themeasurement noise R, and the plant noise covariance Q, the Kalman filter model was designed.


0.2 0.25 0.3 0.35 0.4-1

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2γ

1 = 0.7

Time

Measured output

Estimated output

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2γ1

= 1.6

Time


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1.5

2γ

1 = 2.8

Time


Figure 11.7 Output and its estimated when a fifth order model was selected

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A: Mean squared error vs. parameter variations

Parameter

Err

or

Case 1 Case 2 Case 3

Figure 11.8 Model error: (i) Case 1, (ii) Case 2, (iii) Case 3


Table 11.1 Detectability and fault size estimation error

Emulator parameter 𝛾1 𝛾2 𝛾3

Threshold 𝜂th 0.45 0.55 0.45Minimum variation Δ𝛾i 5% 6.3% 10%Maximum fault size estimation error 0.04 0.04 0.06

11.3.3 Detection of FaultThe Bayes detection strategy is given by:

ts(e)

{≤ 𝜂th no fault

> 𝜂th fault(11.4)

Where the statistics (for a square wave input) ts(e) = 1N

N−1∑i=0

e2(k − i). A fault is detected whenever the

moving average of the residual energy exceeds a threshold value.Faults were injected by changing the emulator parameters {𝛾i} stepwise in small increments. Table

11.1 gives the performance of the Bayes detection strategy. The threshold value, 𝜂th, the minimumvariation Δ𝛾iin the emulator parameter 𝛾ithat can be detected accurately, and the maximum error inestimating the fault size. We can see that the worst-case minimal variation Δ𝛾i that can be detected is10% for the actuator fault. The maximum fault size estimation error is 0.06 for the actuator fault.

11.3.4 Fault IsolationThis says that the hypothesis of a fault in 𝛾i is correct when the angle between the residual vector e(k)and the vector, 𝚽(k)𝛀i is minimum. Alternatively we may state that hypothesis Hi is asserted (that isonly one diagnostic parameter 𝛾i has varied), if the residual and its estimate are maximally aligned in thesense defined by Eq. (11.2). The function cos2 𝝋j(k), which is derived by maximizing the log-likelihoodfunction l(𝜸 | e) for case when there is single fault, is the discriminant function for fault isolation.

The actual and the estimated faults are shown in Figure 11.9. Each of three faults, in sequence, increasein a stepwise fashion to 1.5 times the nominal value and then decrease suddenly back to the nominalvalue of unity. The implementation indicates that the methodology is able to capture incipient faults andsudden faults. The subfigure on the top shows the output y(k) when the reference input r(k) is a squarewave. During time interval [1 70], the emulator parameter 𝛾2 was varied (the value the actuator gain isdenoted by kv). During the next time interval [80 150], the emulator parameter 𝛾1 was varied (the valuethe sensor gain is denoted by ks). During time interval [170 240], the emulator parameter 𝛾3 was varied(the value of the actuator gain is denoted by ka). The actual emulator and its estimate are indicated inbold and dashed lines.

11.3.5 Fault IsolabilityIsolability is a measure of the ability to distinguish and isolate a fault in 𝛾j, from a fault in 𝛾i , i ≠ j.Isolability is improved for a larger difference between 𝜑i, and {𝜑j , j ≠ i}. The fault isolability measureis given by:

cos2 𝜑ij(k) =( |⟨𝚽 (k)𝛀i,𝚽 (k)𝛀j⟩|‖𝚽 (k)𝛀 i‖‖𝚽 (k)𝛀 j‖

)2

(11.5)


0 20 40 60 80 100 120 140 160 180 200

-101

Reference input and the output

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0.5

Actual and estimate fault size: velocity sensor

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0.5

Actual and estimate fault size: position sensor

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0.5

Actual and estimate fault size: actuator

Time

r y Parameter Estimate

Figure 11.9 Fault isolation showing the faults, (solid) and the generated fault estimates (dashed)

The isolability measures are listed in Table 11.2. It can be seen that all faults can be isolated, however,𝛾2 versus 𝛾3 has poorer isolability.

Comments The results of the evaluation on the physical systems show acceptable performanceeven though the physical system is nonlinear, complex, and is subject to uncertainty in the form ofunmodeled dynamics, including nonlinear effects such as friction, backlash, and saturation. The proposedidentification scheme plays a key role in ensuring detection and isolation capability. The identified modelis accurate over a wide range of variation in the diagnostic parameters. We have only considered asingle input and a single output system where fault diagnosis capability is constrained by the availabilityof only one output measurement for a given input.

A reliable offline identification of the physical system based on performing a number of experiments byperturbing the emulator parameters to imitate the fault operating regimes was crucial to the performanceof the fault diagnosis.

It is interesting to note that although the physical system is nonlinear and complex the fault diagnosisscheme based on linear model appears to perform very well. This is, in part, due to the fact that incipient(“small” variations in the diagnostic parameters) were faults that were introduced and highly reliableidentification based on performing a number of parameter perturbed experiments.

Table 11.2 Worst-case isolability measures

𝛾i 𝛾i max(|cos 𝜑ij|)𝛾2 𝛾1 0.6255𝛾2 𝛾3 0.8005𝛾1 𝛾3 0.7817


11.4 SummaryA fault is modeled as a variation in the diagnostic parameters. Emulator parameters play the role ofdiagnostic parameters to mimic normal and fault scenarios. A unified approach to fault detection andisolation was used based on Kalman filter residuals. Thanks to the identification scheme based onthe emulator parameter-perturbed experiments, the identified model captures not only the input–outputbehavior of the system but also the diagnostic parameter input to the system output, and as a result theperformance of the fault detection and isolation was acceptable.

References

[1] Doraiswami, R. and Cheded, L. (2013) A unified approach to detection and isolation of parametric faults usinga Kalman filter residuals. Journal of Franklin Institute, 350(5), 938–965.

[2] Doraiswami, R., Diduch, C. and Tang, T. (2010) A new diagnostic model for identifying parametric faults. IEEETransactions on Control System Technology, 18(3), 533–544.

[3] Doraiswami, R. and Cheded, L. (2012) Kalman filter for fault detection: an internal model approach. IET ControlTheory and Applications, 6(5), 1–11.

[4] Doraiswami, R. and Cheded, L. (2013) Fault diagnosis of a sensor network: a distributed filtering approach.Journal of Dynamic Systems, Measurement and Control, 135(5).

[5] Ljung, L. (1999) System Identification: Theory for the User. Prentice Hall, New Jersey.[6] Forssell, U. and Ljung, L. (1999) Closed loop identification revisited. Automatica, 35, 1215–1241.[7] Qin, J.S. (2006) Overview of subspace identification. Computer and Chemical Engineering, 30, 1502–1513.[8] Brown, R.G. and Hwang, P.Y. (1997) Introduction to Random Signals and Applied Kalman Filtering. John

Wiley and Sons.[9] Doraiswami, R.L., Cheded, L., and Khalid, M.H. (2010) Sequential Integration Approach to Fault Diagnosis

with Applications: Model-free and Model- Based Approaches.VDM Verlag Dr. Muller Aktiengesellschaft &Co. KG.

[10] Mallory, G. and Doraiswami, R. (1997) A frequency domain identification scheme for control and fault diagnosis.Transactions of the ASME: Journal of Dynamic Systems, Measurement and Control, 119(49), 48–56.

[11] Doraiswami, R., Diduch, C., and Kuehner, J. (2001) Failure Detection and Isolation: A New Paradigm. Pro-ceedings of the American Control Conference, Arlington, USA.

[12] Liu, Y., Diduch, C., and Doraiswmi, R. (2003) Modeling and Identification for Fault Diagnosis. 5th IFACSymposium on Fault Detection, Supervision and Safety of RTechnical Process, Safeprocess 2003, WashingtonD.C.

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