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Abstract—In terms of fault diagnosis, there are two general approaches: model-based and model-free. This paper presents the fault diagnosis techniques for a nonlinear dynamic system with multiple faults using the model-free approach. A new concept for fault detection by means of a real-time tracker was employed to predict the system outputs from which the residuals could be quickly generated. To classify faults and determine the degree of each fault, soft computing techniques: fuzzy logic and neural network were used. This study consists of three parts: diagnosis of single faults before the system reaches its steady state, diagnosis of simultaneous multiple faults and diagnosis of sequential multiple faults. A three-tank nonlinear dynamic system was chosen to demonstrate the presented techniques. The result showed promise in using the model-free approach for the diagnosis of multiple faults.

Index Terms—Model-free fault diagnosis, real-time tracker, soft computing, intelligent mechatronics,

I. FAULT DIAGNOSIS: MODEL-FREE VERSUS MODEL-BASED eliability, availability and safety are the key requirements in today’s world of automation. At the same time, because of increasing complexities in modern control systems,

fault diagnosis has become an issue of importance. A fault is an abnormal state of a machine or a system, including dysfunction or malfunction of a part, an assembly, or the whole system. The faults can cause serious performance degradation and eventual system breakdown, if they are not properly detected [1]. Therefore, a monitoring system is required to detect, isolate and identify the fault type early enough so that failure of the overall system can be avoided. Such a system is called fault diagnosis system. Fault detection and isolation, the most essential tasks of a diagnosis system, are often used in fault diagnosis [2]. Fault detection simply detect if system is faulty, while fault isolation determine the location or source of the fault. Sometimes it is necessary to identify the degree or size of fault in order to facilitate the system reconfiguration.

Fault diagnosis methods can be classified into two approaches: model-based [1] and model-free [2]. Different methods have been developed mostly using the model-based

P. P. Lin is the corresponding author, who is currently Interim Associate Dean of Fenn College of Engineering, and Professor of Mechanical Engineering, Cleveland State University, Ohio 44115-2214, USA. Email Address: p.lin@csuohio.edu

fault approach, some with analytical or heuristic reasoning. Being based on a mathematical model, model-based systems are sensitive to modeling errors, parameter variation, noise and disturbances etc. and are mostly applicable only to linear systems.

One way to handle the nonlinearity is to linearize the model at the operating point, and let the system closely operates around this operating point. This approach works well only when the linearization does not cause a large mismatch between linear and nonlinear model. However, this approach will fail to give satisfactory results for a system with high nonlinearity and wide range of operating points. This problem can be tackled by using a large number of linearized models for each range of operating point, but it will result in large number of fault diagnosis systems corresponding to all operating ranges, which is impractical for a real-time application [2].

The success of a model-based method depends on the accuracy and quality of model. However, for a complex system, an accurate mathematical model is difficult to build. Hence, there is a great need to develop fault diagnosis techniques that do not require a mathematical model. In contrast to model-based fault diagnosis techniques, model-free techniques do not depend on any specific model structure, but relies only on signal-based information obtained from closed loop experiments.

II. REVIEW OF SOFT COMPUTING USED IN FAULT DIAGNOSIS Fault diagnosis has been actively investigated in the past two

decades or so. But, the research in this area has been mostly using the model-based approach with traditional hard computing techniques. About a decade ago, some researchers started to apply soft computing techniques to fault diagnosis.

Soft computing is considered as an emerging approach to intelligent computing, which is parallel to the ability of human mind to reason and learn in circumstances with uncertainty and impression. The computing techniques are effective in acquiring economical and competitive solutions to real world problems [1]. The soft computing techniques generally involve fuzzy logic, neural network (NN), genetic algorithm (GA) or hybrid of fuzzy logic and neural network or genetic algorithm. The fuzzy logic can deal with imprecise measurement and uncertainty, whereas the neural network can handle the nonlinearity between the system’s input and output, and the GA

Intelligent Model-Free Diagnosis for Multiple Faults in a Nonlinear Dynamic System

Paul P. Lin and Hardeep Singh

Interim Associate Dean & Professor, College of Engineering, Cleveland State University, Ohio, USA Graduate Assistant, Dept. of Mechanical Engineering, Cleveland State University, Ohio, USA

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can be used to search for the optimal solution. Patton and Chen [2] described the application of neural

network for fault diagnosis of nonlinear dynamic systems. Neural network was used for residual generation and fault classification even in the presence of noise. Advantages of neural network strategy over model-based approaches were discussed. However, the strategy was limited to considerable faults (i.e. cannot be used to detect small faults).

Frank and Koppen-Seliger [3] conducted a survey on application of artificial intelligence to model-based fault diagnosis for dynamic processes. They proposed a novel observer called “knowledge observer” for residual generation. But, without an analytical model, the observer-based fault diagnosis cannot be achieved [1].

Gao and Ovaska [4] used neural network, fuzzy logic, neuro-fuzzy and genetic algorithms for fault diagnosis, and also compared their advantages and disadvantages. They concluded that, although soft computing provided improved solutions over classical strategies, more accurate and robust diagnosis approaches based on the fusion of soft computing and hard computing techniques should be developed.

Leanardt and Ayobi [5] gave the summary of methods that can be applied to automatic fault diagnosis. In this study classification methods, inference methods and an adaptive neuro-fuzzy system were investigated.

Drake and Difu [6] discussed multiple fault diagnosis far a machine tool’s flood coolant system using a neural network, where the neural network was used to perform pattern recognition with features extracted from the transient response of the coolant pressure on shutdown.

Skoundiranos and Tzafestas [7] used a modeling technique based on a local neural model to perform local linearization for a satisfying estimation of plant’s output within part of operating regime. The comparison was made between plant and model behavior using a three-tank system. The comparison led to the residual generation, which in turn triggered a decision mechanism to conclude the presence, degree and cause of possible faults.

Le et. al. [8] used multi-layer perception type of neural networks to detect leakages in an electro-hydraulic system. With the same type of fault (i.e. leakage or blockage alone), their study showed some promise in diagnosing single and multiple faults.

Recently, Lin and Li [9] developed model-free fault diagnosis, prognosis and self-reconfiguration techniques via soft computing. They used fuzzy logic to classify faults, and neural networks to determine the degree of each fault, Taylor series expansion and finite differences to estimate the remaining life, and instant optimization with fuzzy inference to self-reconfigure the faulty system. The techniques appear to work well, but limited to one fault at a time.

It is fair to say that the greatest difficulty in using the model-free approach is to detect faults before the system reaches its steady states. This is simply because the diagnosis system does not know what to expect, unless laborious experimental data or neural network pre-trained data were available, which is practically impossible. To overcome this major difficulty, this paper proposed a new concept for fault detection without prior knowledge about the system’s model.

The presented fault diagnosis system is designed to perform the following four tasks:

1. Fault detection: A fault is detected when the residual exceeds the threshold value.

2. Fault isolation/classification: The detected fault is classified and the location of the fault is determined.

3. Multiple-fault identification: Simultaneous faults and sequential faults are identified.

4. Degree-of-fault determination: The degree of each fault between 0 and 1 is determined.

This paper first describes the new facult detection technique by means of a real-time tracker, followed by fault isolation using fuzzy logic and neural networks. Afterward, the simulation results for the following cases are discussed:

Case I: Diagnosis of single fault before the system reaches its steady state

Case II: Diagnosis of simultaneous multiple faults after the system reached its steady state

Case III: Diagnosis of sequential multiple faults after the system reached its steady state

III. NEW FAULT-DETECTION TECHNIQUE

A. Nonlinear Dynamic System Fig. 1 shows the three-tank nonlinear system that was chosen for this study. The system was mentioned in [1] as an example for fault diagnosis of nonlinear system. It has two controlled inputs (pump’s flow rate), three measurable outputs (water levels), and six possible faults (three pipe blockages and three tank leakages).

Fig. 1 The three-tank system

Although in this study the dynamic system is assumed unknown, the dynamic equations for this system, indeed, can be derived using the Torricellies Law (see Appendix). In other words, the water level with respect to time for each tank can be expressed in terms of the three tanks’ physical dimensions as well as six possible faults: three blockages and three leakages. The blockage is in terms of degree of fault between 0 and 1. All the three leakages have negative values, which represent the actual outgoing flow rate. The maximum leakage for each tank is assumed -10 iter /min.

;10 13 ≤≤ s ;10 23 ≤≤ s ;10 0 ≤≤ s ;010 1 ≤≤− fQ ;010 2 ≤≤− fQ ;010 3 ≤≤− fQ

where “1” means no blockage and “0” means complete blockage.

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In this study, computer simulations were performed, in lieu of actual measurements. Thus, the calculated values of the three water levels for any given condition were used to train a neural network (NN). The NN-predicted values of water levels are considered as the measured data, which makes sense because they are close to the exact vales with some inaccuracy that can be treated as measurement noise. This approach is considered model free since only the system’s inputs, outputs and the physical parameters are needed.

B. Fault Detection The existing model-free fault diagnosis techniques assume no

faults until reaching the system’s steady states [9]. This is mainly due to the great difficulty of establishing the no-fault system response without the knowledge of the system’s model,

Let x(t) and u (t) be the state vector and input vector, respectively, x(t) = [h1 h2 h3]T; u(t) = [Q1 Q2]T (1) then the state space representation of the three-tank system in continuous and discrete forms can be described by

u(t)) (x(t),Fdt

dx(t)1= (2)

x (k) = F2 (x(k-1),u(k-1)) (3) where h1, h2 and h3 denote the water level of tanks 1, 2 and 3, respectively, and Q1 and Q2 denote the flow rare (liter/min.) of pumps 1and 2, respectively. More specifically, the u (t) is the input vector, and the x (t) is the output vector. The F1 and F2 represent nonlinear functions in continuous and discrete forms, respectively. The discrete form was used for this study. With the model-free approach, the output vector can be obtained from a trained neural network that uses experimental data by directly measuring the water levels using a sensor, such as piezo-resistive pressure sensors with resolution of 0.1mm.

A fault is considered detected when the residual signals exceed the preset threshold values. This essentially requires the diagnosis system to know the reference values (i.e. the expected output values) at any time. However, when using the model-free approach, generation of the expected output values before a dynamic system reaches its steady states is a very difficult task. A real-time tracker is employed to detect single or multiple faults even before the system reaches its steady states.

C. Using Alpha-Beta-Gamma Tracker for Fault Detection Tenne and Singh [10] developed the optimal design of α-β-γ

filters. Various metrics, such as noise-ratio, steady state maneuver error and transient response metrics were defined to quantify the performance of these filters. This work considered two classes of target trajectories, circular maneuvers with target moving at constant speed and straight-line maneuvers, with target moving with constant acceleration. This type of filters is particularly useful in estimating the next state variable values in real-time, without having to know the system’s model.

The α-β-γ tracker is a numerical technique that assumes constant acceleration in a dynamic system. It requires the information of three previous time steps in order to estimate the current states (i.e. position and velocity). Once the tracker has

been initiated, it will try to track the dynamic system as soon as possible.

The target is observed each period and the observed position is recorded. Then, the smoothed parameters are computed for position, velocity and acceleration. Using these values, the predictions for position and velocity are made. The prediction equations are given as:

)(21)()()1( 2 katktvkxkx sssp ∆+∆+=+ (4)

)()()1( ktakvkv ssp ∆+=+ (5) The smoothing equations are given as:

))()(()()( kxkxkxkx pops −+= α (6)

))()(()()( kxkxt

kvkv pops −∆

+= β (7)

))()((2

)1()( 2 kxkxt

kaka poss −∆

+−= γ (8)

where Xo, Xs and Xp are observed, smoothed and predicted positions, respectively; Vo, Vs and Vp are observed, smoothed and predicted velocities, respectively; as is smoothed acceleration; α, β and γ are coefficients, and ∆t is time interval.

Initial values are required to initialize the tracker. Three initial positions are given as:

)1()1( 0xx p = ; )2()2( 0xx p = ; )3()3( 0xx p = then

txx

vs ∆−

=)2()3(

)3( 00 ; 2

000 )1()2(2)3()3(

txxx

as ∆+−

=

Although the tracker uses the acceleration data, it only tracks positions and velocities. Thus,

2)3(21)3()3()4( tatvxx sspp ∆+∆+=

tavv spp ∆+= )3()3()4( Applying the z-Transform to the above equations and

solving for the ratio Xp /Xo leads to a transfer function in z- domain, which is given as:

1)3412()3

41(

)41()

412(

)(23

2

−+++−−+−+++

++++−−+==

αγβαγβα

γβαγβαα

zzz

zz

xx

zGo

p (9)

The roots of the characteristic polynomial, the denominator of the transfer function, are required to lie within the unit circle of stability, Jury’s stability test [11] yields the constraints on the

βα , andγ as: 0 < α < 2, 0 <β < 4-2α, 0 < γ < 4αβ/(2-α) (10)

As long as the values for βα , andγ are selected in the aforementioned optimal ranges, the tracking will be stable [10]. In other words, once the tracker tracks the system, it will not be off track by itself unless something is wrong in the system. The tracker can be used as the reference system at all time. Thus, if the measured signals are all of sudden off the track, at least a fault must have just occurred.

The tracking errors greatly depend upon the size of the time step. The smaller the ∆t is, the smaller the error will be. In comparison with Kalman or the Extended Kalman filter, the

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alpha-beta-gamma tracker does not require large computations and matrix inversion. It is simpler to implement the tracker for a real-time application.

D. Example of Fault Detection by Means of the Tracker In this example, it is assumed that a blockage fault (S13)

occurred at t = 5 sec. For a better view, the time unit used in Figs. 2 and 3 is 0.1 sec. Thus, the two figures indicate that the fault occurred at 50 (x 0.1) seconds, where h1, h2 and h3 denote the actual water levels of the three tanks, and h1p, h2p and h3p denote the water levels predicted by the tracker.

Fig. 2 System response to fault S13 = 0.3

47 48 49 50 51 52 53 54 55 56 57

14

16

18

20

22

24

time (0.1 sec)

wat

er le

vel (

mm

)

h1h2

h3h1p

h2ph3p

h1p: (21.25)

h1: (21.99)

h3p:(17.72)

h3: (16.98)

h2:(15.36) h2p(15.36)

dh1

dh3

Fig. 3 Close View of Fig. 2

Fig. 3 clearly indicates that the water levels, h1 and h3, was

off track as soon as the fault occurred, where dh1 and dh3 denotes the residuals for tanks 1 and 3. Since the residuals exceeded the preset threshold value of 0.5 mm, the dynamic system was diagnosed as faulty. The next task is to isolate the detected fault.

IV. FAULT ISOLATION With the model-free approach, fault isolation is much more

complex than fault detection. The following three cases were examined.

Case I: Single Fault before the System Reaches its Steady State With the tracker, single fault can be classified and isolated

even before the system reaches its steady state. The fault isolation is accomplished by using a fuzzy inference system (FIS). The FIS has six input variables (S13, S23, S0, Qf1, Qf2 and Q f3), and three output variables (∆h1, ∆h2 and ∆h3). Each input or output variable has three membership functions (N, Z and P) where N, Z and P stand for negative, zero and positive, respectively. One of the nine fuzzy logic rules is listed below:

If ∆h1 is P, ∆h2 is Z, ∆h3 is N, then S13 is Faulty and others are Normal.

Fig. 2 indicates that the actual h1 is larger than the predicted one (i.e. ∆h1 is P), h2 is virtually unchanged (i.e. ∆h2 is Z), and the actual h3 is smaller than the predicted one (i.e. ∆h3 is N). The FIS will conclude that a blockage S13 (between tanks 1 and 3) has just occurred. To determine the degree of fault, two neural networks (NN) were trained. The first NN mapped three input variables (t, Q1 and Q2) to three output variables ((∆h1, ∆h2 and ∆h3). The second NN used the normalized values of the ∆h1, ∆h2 and ∆h3 to predict the degree of fault between 0 and 1. Case II: Simultaneous multiple faults after the system reached its steady state

In this case, simultaneous multiple faults contain two simultaneous blockage faults and three simultaneous blockage faults. Simultaneous leakages, and mixtures of blockages and were not investigated because of high degree of complexity. Four FIS were built to classify the fault type and also isolate the faults. Each FIS handles different situations of ∆hi. FIS I: for ∆h1 ≥ 0, ∆h2<0 and ∆h3 ≥ 0 FIS II: for ∆h1 ≥ 0, ∆h2<0 and ∆h3 <0 FIS III: for ∆h1 ≥ 0, ∆h2 ≥ 0 and ∆h3 <0 FIS IV: for ∆h1 ≥ 0, ∆h2 ≥ 0 and ∆h3 ≥ 0 Case III: Sequential multiple faults after the system reached its steady state

In this case, it is assumed that sequential faults do not occur within 30 time steps. This is because the tracker needs time to get back on track after a fault has occurred. In addition, only sequential multiple faults of blockage were considered here. Depending upon the sequential order, six combinations of two faults, and six combinations of three faults were studied. Three different FIS (can be thought of three layers) were built to isolate the faults. The first FIS uses three fuzzy rules classify the first fault. The second FIS uses seven rules to classify the second fault, while the third FIS uses four rules to classify the third fault.

VI. RESULTS AND DISCUSSION The simulation results for each of the three cases are

summarized in this section. Result for Diagnosis of Single Fault before the System Reaches its Steady State

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Q1

Q2

Assumed Degree of

Fault

Detected Fault

/Probability of Fault

Degree of fault [0, 1] or [-10, 0]

Degree of fault Error

% 6.5 8.25

13s =0.17 S13 /96% 0.1700 0

7 9 13s =0.38 S13 /96% 0.3802 0.05

10 6.75 13s =0.63 S13 /96% 0.6306 0.10

6.5 8.25 23s =0.12 S23 /96% 0.1200 0

7 9 23s =0.44 S23 /96% 0.4403 0.07

10 6.75 23s =0.57 S23 /96% 0.5704 0.07

6.5 8.25 0s =0.23 S0 /96% 0.2301 0.04

7 9 0s =0.45 S0 /96% 0.4503 0.06

10 6.75 0s =0.70 S0 /96% 0.7009 0.13

6.5 8.25 Qf1 =-6.67 Qf1 /96% -6.67 0.11 7 9 Qf1 =-7.8 Qf1 /96% -7.803 0.04

10 6.75 Qf1 =-3.75 Qf1 /96% -3.752 0.08

6.5 8.25 Qf2 =-3.75 Qf2 /95% -3.754 0.12 7 9 Qf2 =-5.25 Qf2/ 95% -5.252 0.05

10 6.75 Qf2 =-6.5 Qf2/ 95% -6.501 0.02 6.5 8.25 Qf3 =-7.4 Qf3/ 95% -7.405 0.07 7 9 Qf3 =-2.5 Qf3/ 95% -2.5028 0.11

10 6.75 Qf3 =-5.75 Qf3 /95% -5.7515 0.03 where Q1 and Q2 are pump’s flow rates in liter per minute. The FIS generated the probability of fault based on the output membership functions that were generated by histograms. A component is considered faulty when its probability of fault exceeds 80%. For each identified fault, the trained NN was used to predict the degree of fault. Note that the degree of fault for blockage is normalized between 0 and 1, and that for leakage is between 0 and -10 liter/min which is the actual leakage flow rate. As it can be seen, the FIS corrected classified all the fault types, and the NN determined the degree of each fault with a very small error. Result for Diagnosis of Simultaneous Multiple Faults after the System Reached its Steady State

The result showing in the following table shows how the FIS assesses the simultaneous faults by probability. The assumed faults are in bold face, so is the probability of each detected fault.

Assumed Fault

“0”: complete blockage “1”: no blockage (normal)

Fault Probability Generated by FIS

Q1

Q2

13s 23s 0s 13s 23s 0s 6.45 8.5 0.12 1 1 96% 4% 4% 9.5 7 0.36 1 1 96% 4% 4% 9 6.5 0.65 1 1 96% 4% 4%

6.45 8.5 1 0.70 1 5% 95% 5% 9.5 7 1 0.23 1 4% 96% 4% 9 6.5 1 0.55 1 4% 96% 4%

6.45 8.5 1 1 0.15 4% 4% 96% 9.5 7 1 1 0.33 4% 4% 96% 9 6.5 1 1 0.67 4% 4% 96%

6.45 8.5 0.12 0.55 1 92% 92% 14% 9.5 7 0.36 0.23 1 92% 92% 8% 9 6.5 0.65 0.70 1 93% 93% 7%

6.45 8.5 0.35 1 0.45 91% 50% 91% 9.5 7 0.20 1 0.10 92% 50% 92% 9 6.5 0.65 1 0.70 95% 50% 95%

6.45 8.5 1 0.12 0.55 4% 96% 96% 9.5 7 1 0.36 0.65 4% 96% 96% 9 6.5 1 0.40 0.20 7% 93% 93%

6.45 8.5 0.34 0.43 0.25 93% 50% 93% 9.5 7 0.12 0.35 0.40 94% 94% 94% 9 6.5 0.55 0.45 0.60 94% 94% 94%

In the case of no-fault, the probability of fault is generally

very low such as 4% or 5%, except for some cases where the probability is 50%. But, that does not affect the accuracy of fault isolation. When a fault is identified, the probability of fault is generally over 90%. However, a usual case (the third last in the table) was found where the second of the three simultaneous blockage faults was not detected due to low probability of fault (50%). This might have resulted from insufficient data were used to train the NN, which led to misclassification by the FIS. Result for Diagnosis of Sequential Multiple Faults after the System Reached its Steady State The assumed faults are in bold face, so is the probability of each detected fault. The first column represents the start and end times of a set of sequential faults. For example of the first case, it is assumed that first blockage fault occurred at t=65.0 sec, followed by the second fault at t=72.5 sec.

Fault Assumed

Fault Probability

Start ↓

End time (s)

Q1

Q2

13s 23s 0s 13s 23s 0s 65.0 72.5

7.5 7.5

8 8

0.12 -

1 0.36

1 1

96% -

4% 96%

4% 4%

72.5 80.0

9 9

6 6

0.43 -

1 0.25

1 1

96% -

4% 96%

4% 4%

65.0 72.5

7.5 7.5

8 8

0.15 -

1 1

1 0.55

96% -

4% 4%

4% 96%

72.5 80.0

9 9

6 6

0.4 -

1 1

1 0.2

96% -

4% 4%

4% 96%

65.0 72.5

7.5 7.5

8 8

1 0.25

0.36 -

1 1

4% 96%

96% -

4% 4%

72.5 80.0

9 9

6 6

1 0.15

0.2 -

1 1

4% 96%

96% -

4% 4%

65.0 72.5

7.5 7.5

8 8

1 1

0.55 -

1 0.13

4% 4%

96% -

4% 96%

72.5 80.0

9 9

6 6

1 1

0.4 -

1 0.35

4% 4%

96% -

4% 96%

65.0 72.5

7.5 7.5

8 8

1 0.25

1 1

0.4 -

4% 80%

4% 20%

96% -

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72.5 80.0

9 9

6 6

1 0.3

1 1

0.2 -

4% 96%

4% 4%

96% -

65.0 72.5

7.5 7.5

8 8

1 1

1 0.2

0.3 -

4% 4%

4% 96%

96% -

72.5 80.0

9 9

6 6

1 1

1 0.35

0.6 -

4% 4%

4% 96%

96% -

65.0 72.5 85.0

9 9 9

6 6 6

0.15 - -

1 0.25

-

1 1

0.35

96% - -

4% 82%

-

4% 18% 96%

65.0 72.5 85.0

9 9 9

6 6 6

0.2 - -

1 1

0.2

1 0.35

-

0.96 - -

4% 4%

96%

4% 96%

- 65.0 72.5 85.0

9 9 9

6 6 6

1 0.4 -

0.3 - -

1 1

0.4

4% 96%

-

96% - -

4% 4%

96% 65.0 72.5 85.0

9 9 9

6 6 6

1 1

0.35

0.4 - -

1 0.3 -

4% 4%

96%

96% - -

4% 96%

- 65.0 72.0 85.0

9 9 9

6 6 6

1 0.3 -

1 1

0.4

0.25 - -

4% 96%

-

4% 4%

96%

96% - -

65.0 72.5 85.0

9 9 9

6 6 6

1 1

0.15

1 0.35

-

0.6 - -

4% 4%

96%

4% 96%

-

96% - -

In comparison with the simultanoes facults, sequential faults

are relatively easy to be diagnosed since they occur one at a time. However, approximately 30 time steps between two consecutive faults is necessary for the trackter to track the system.

Generally speaking, the simulation results for the three cases are quite good, which demonstrate the effectiveness of the presented fault diagnosis techniques in terms of fault detection and fault isolation.

VII. CONCLUSIONS The biggest challenge for diagnosis of multiple faults is fault

detection before the system reaches its steady state. The major difficulty in using a model-free approach is to generate the residuals that can be used for fault detection. Without the system’s model, generating rich data for training neural networks is very laborious. This study proposed a new fault detection technique by means of an alpha-beta-gamma tracker that estimates the next expected state values, and thus eliminates the necessity of generating rich data for NN training.

The given three-tank dynamic system is, indeed, a very

challenging case study for fault diagnosis. The system has six possible faults, but only three output values (i.e. water levels). Thus, even if the system were modeled, it would have been classified as a underdetermined problem. Furthermore, tanks are connected by pipes, which mean that the system outputs are not independent of each other. It is possible that a combination of a faulty leakage and a faulty blockage could make one of the three output variables almost unchanged, which will result in no detection of faults. For this reason, the diagnosis of multiple faults in this study was limited to blockages only. It is concluded that for the given nonlinear dynamic system, the maximum number of faults that can be diagnosed cannot exceed the number of measured outputs regardless of which approach is used, model-based or model-free.

In summary, the presented fault diagnosis techniques use an

alpha-beta-gamma tracker to detect faults, fuzzy logic to classify the detected faults, and neural networks to determine the degree of each fault. Future work includes diagnosis of multiple faults before the system reaches its steady state, and enhancement of soft computing techniques for better fault diagnosis reasoning.

APPENDIX Although the model-free approach was used in this study, the

dynamic model for the three-tank system can be derived using Torricellies law. The three dynamic equations are

1131311311 ||2)sgn( fQQhhghhsa

dtdhA ++−−−=

2220223232332 2||2)sgn( fQQghsahhghhsa

dtdhA ++−−−=

3232323331311313 ||2)sgn(||2)sgn( fQhhghhsahhghhsa

dtdhA +−−−−−=

A is the circular cross-section area of each tank (same for all) a1, a2, a3: the circular cross-section area of each pipe Q1, Q2: pump flow rates; h1, h2, h3: water level of each tank s13, s23, s0: pipe blockage; Qf1, Qf2, Qf3: tank leakage

REFERENCES [1] J. Chen, J. and R.J. Patton, Robust Model-Based Fault Diagnosis for

Dynamic Systems, Kluwer Academic Publishers, 1999, pp. 251-295. [2] R. J. Patton and J. Chen, “Neural Network in Fault Diagnosis of

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