robust fault diagnosis and prognostics of a hoisting mechanism: a

ROBUST FAULT DIAGNOSIS AND PROGNOSTICS OF A HOISTING

MECHANISM: A SIMULATION STUDY.

S.K. GHOSHAL

Department of Mechanical Engineering & MME, ISM, Dhanbad, Jharkhand - 826004, India

S. SAMANTA

Department of Mechanical Engineering, Asansol Engineering College, Kanyapur, Asansol - 713305, West Bengal, India.

Abstract: In this article, different methodology in the area of fault isolation, robustness in fault diagnosis, parameter estimation (for root cause analysis) and prognostics are surveyed and applied to a model developed for a hoisting mechanism mounted on a vehicle with planer oscillation. The developed model is of multi energy complexity and intended to isolate the components responsible for abnormal behaviour of the system using structural analysis of some constraint relations, called Analytical Redundancy Relations (ARR), the numerical evaluation of which are residuals. Bond graph modelling, which is a unified tool for multi-energy domain system representation, is used to model the system. The fault indicators and fault signatures are derived from the model. The robustness in fault detection is addressed through passive approach to make residuals insensitive to uncertainty of system parameters. Also, multi-tier parallel simulation method is applied to isolate some structurally nonisolable faults. Then, faulty parameters are estimated to predict remaining useful life for prognostics analysis.

Keywords: Bond graphs; fault isolation; prognosis; analytical redundancy; robustness; parallel simulation.

1. Introduction

In supervision platform of safety-critical systems different approaches for condition monitoring and Fault Detection and Isolation (FDI) procedures have been developed: quantitative model-based, qualitative model-based and process history based approaches [Venkatasubramanian et al. (2003)]. The present study is focused on a particular branch, i.e the quantitative model-based approach using Analytical Redundancy Relation (ARR) for FDI, which consequently enables better fault accommodation through an appropriate decision support system. Generally, model-based methods provide superior diagnostic performance while requiring the development of mathematical model to describe the behavior of the physical system for various operating conditions.

Therefore, modeling is an important and difficult step because of the complexities of the modern industrial systems and their control equipment. Bond graph modeling [Karnopp et al. (2006), Mukherjee et al. (2006)], which is a unified multi-energy domain modeling method, is especially suitable for developing analytical models of most engineering systems. Bond graph modeling has also been used in the past for different Fault Detection and Isolation approaches [Ould Bouamama et al. (2003), Ould Bouamama et al. (2005), Samantaray et al. (2004)]. Moreover, the structural control properties (controllability, observability, etc.), which can be deduced by analyzing the causalities (cause and effect relationships) on a bond graph model [Dauphin-Tanguy et al. (1999)], have been already applied to optimize sensor placements [Samantaray et al. (2004)] and to determine hardware redundancies [Samantaray and Ghoshal (2007)].

FDI procedures are generally comprised of four stages: alarm or fault detection, isolation of fault, estimation of faulty parameter, and operational change [Venkatasubramanian et al. (2003)]. In the alarm stage, the system behavior is continuously monitored for the occurrence of process faults. Once a failure is declared, the isolation phase attempts to identify the failed system component. The estimation stage determines the extent of failure to enable the implementation of operational changes needed for fault accommodation.

Isolation of the faulty component can be done based on the structural properties of the ARR [Staroswiecki and Comtet-Varga (2001)]. It is better to write the ARRs in differential form to solve the initial condition decoupling problem, i.e., avoiding any integration [Ould Bouamama et al. (2003)]. Off course, there are pitfalls in this due to sensor noises, whose derivatives cause diagnosis problems and need specific filters. Secondly, if

S.K. Ghoshal et al. / International Journal of Engineering Science and Technology (IJEST)

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any parameter exists in multiplicative form with derivative of any state in ARR, then it would have no influence on the corresponding residual when the state is in static steady-state condition. In other words, that particular parameter cannot be estimated by using the associated residual.

In the present study, the methodology given in [Ould Bouamama et al. (2003)] is followed for deriving fault indicators (ARRs) and also for monitorability and isolability analysis of the possible faults. The fault isolation of a vehicle mounted hoisting mechanism, virtually instrumented with seven sensors, is investigated through simulation. Robustness [Samantaray and Ould Bouamama (2008)] in fault detection is achieved for structural uncertainty and for that adaptive threshold [Sié Kam and Dauphin-Tanguy (2005), Djeziri et al. (2006)] is designed. Then, Parallel simulation using multi-block model is carried out to isolate structurally nonisolable faults [Samantaray et al. (2005)]. The integral form of ARR is used to estimate rope stiffness and from that estimation, remaining useful life [Muller et al. (2008), Medjaher and Zerhouni (2009)] is predicted.

Following assumptions are made in this work: The system is state-observable [Dauphin-Tanguy et al. (1999)]. At a time, a single independent parameter of the system may be faulty i.e. single-fault-hypothesis. Sensors are considered non-faulty. For a given process input and output vector, a unique set of parameter values exists.

2. Generation of Fault Indicators

An ARR is a relationship between a set of known process variables. In a bond graph based approach, the known variables are the sources (Se and Sf), the modulated sources (MSe and MSf), the measurements from sensors (De and Df), the model parameters ( ), and the controller outputs (u). A fault indicator or residual, r, which represents the error in the constraint, is formed from each ARR and can be written as r = f (De, Df, Se, Sf, MSe, MSf, u,) = f (K) = 0, where f is the constraining function. For a system with n structurally independent residuals; ri=fi (Ki), where i=1…n and Ki is the set of known variables in the argument of function fi; the following property is satisfied : Ki Kj i j , where i, j = 1...n. Although the residuals are theoretically equal to zero, they are never so in an online application involving measurements from real sensors due to the sensor noises and the uncertainties associated with the parameters.

The evaluation of the ARR using the actual sensor data and the process parameters is used to detect the faults in the process. This leads to the formulation of a binary coherence vector C=[c1,c2,...,cn], whose elements, ci (i=1...n), are determined from a decision procedure, , which generates the alarm conditions. Robust decision procedures minimize misdetection and false alarms by treating the residual noises.

In this paper, we use a decision procedure, C = (Rd1, Rd2,.. Rdn), whereby each residual, Rdi, is tested against an adaptive threshold, )(ti , to generate the coherence vector, C. The elements of C, ci (i=1...n), are

determined from

otherwise.,1

);(by bounded is if,0 trc ii

i

(1)

The coherence vector is calculated at every sampling interval. A fault is detected, when ]0,,0,0[ C , i.e.

at least one element of the coherence vector is non-zero (alternatively, at least one residual exceeded its threshold). The isolation of the faulty component is done using the binary Fault Signature Matrix (FSM), S. The fault signature matrix describes the participation of various components (physical devices, sensors, actuators and controllers) in each residual. Thus, matrix S forms a structure that links the discrepancies in components to changes in the residuals. The elements of matrix S are determined from the following analysis:

otherwise. 0,

component; in faults tosensitive is ARR if,1 thth

ji

jiS

(2) It is important to note that the component’s faults need not be explicitly modeled in the residuals. Faults in

any physical component can be mapped to the undesirable changes in the values of the parameters of the component. A residual is sensitive to faults in a component, when the parameters or the measurements belonging to that component appear in the symbolic residual, or are causally linked in the numerical form of the residual. The elements of the fault signature matrix can as well be constructed through experimentation by introducing various faults, one at a time.


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3. Case Study: A Hoisting Mechanism

The load hoisting, the fundamental motion of mobile crane, is investigated and reported in recent literatures [Guangfu et al. (2005), Kaczmarczyka and Ostachowicz (2003a), Kaczmarczyka and Ostachowicz (2003b)]. The crane dynamic behavior during hoisting motion that is driven by a hydraulic secondary control system is simulated by Guangfu et al. (2005) using a nonlinear finite element model for the hoisting system. Kaczmarczyka and Ostachowicz (2003a,b) employed the classical moving co-ordinate frame approach, Hamilton’s principle, and later the Rayleigh–Ritz procedure to derive a nonlinear distributed-parameter mathematical model for deep mine hoisting cables, and presented numerical simulation in 2 parts. Chang et al. (2008) proposed a new method based on principal component analysis (PCA) and support vector machines (SVMs) for fault diagnosis of mine hoists, which includes faults associated with the gearbox, the hydraulic system and the wire rope. Zhang et al. (2003) performed a series of experiments on the fretting friction and wear of mine hoisting steel wires on an elastic beam oscillation test rig. The research results demonstrated that the fretting wear depth of the steel wires increased with the increasing fretting cycles and contact loads.

In our earlier paper [Samanta et al. (2009)], we made a bond graph model (linear) of a hoisting system and studied (through simulation) the dynamic response and residual behavior for fault detection and isolation. There, we considered bilateral constant threshold, which may not decouple the fault with model uncertainties and process or measurement noise (i.e. fault detection is not robust). Secondly, issues like isolation of structurally nonisolable faults, root cause analysis (parameter estimation), and estimation of remaining useful life (RUL) were not addressed therein. Now, in this work, we have surveyed different approaches available and applied to the same linear model for complete robust fault disambiguation and prognostics.

3.1. System modeling and model-based FDI

A schematic of simplified hoisting mechanism mounted on a truck having planer oscillation with pitch and bounce motion is shown in Fig. 1 and the bond graph model for the system is given in Fig. 2. Junction 1 and

y1 represent the pitch and bounce velocity, to which are attached the inertia terms, J and M, respectively. The

current-torque relation for the DC motor is idealised as: ai. .

Seven number of sensors have been installed with the system: Df( mf ), Df(fym), Df(fpm), Df(fhm), De(efm),

De(erm), and De(ehm). In [Samanta et al. (2009)], the model in Fig. 2 was simulated with the parameter values given in Table 1 having no initial fault in the system and thereafter at time 10 s motor fault is introduced reducing by 20% from the nominal value. In that article [Samanta et al. (2009)], the variation of all sensor

data with time were plotted, from which one can conclude that something has happened at 10 s but the particular faulty component cannot be isolated by online inspection although the information is contained in those signals. To isolate the faulty component FDI analysis was done there by using the algorithm given by Ould Bouamama et al. (2003).

Fig.1: Schematics of a vehicle mounted hoisting system


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Fig.2: Bond graph model of the system in Fig.1 in integral causality

Fig. 3: Bond graph model of the system in Fig.1 in derivative causality

Table 1: Nominal values (hypothetical) of model parameters of hoisting mechanism

Symbol Value Symbol Value Mt 10,000 kg Ra 1 Ω Jt 500 kg.m2 1N.m.Amp-1 Jp 1 kg.m2 10 rad/s mh 100 kg A 1 V

Kf, Kr 1×105 N/m a 1 m Rf, Rr 0.3 N.s/m b 1.7 m

Kh 1×103 N/m c 0.3 m Rh 0.1 N.s/m r 0.2 m


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In that algorithm, to derive the ARR all the storage elements are to be brought under preferred differential causality and negative of measured quantities from detectors are imposed on the system (i.e. on 1 or 0 junction of the bond graph) as pseudo source and reactive factor in the bond corresponding to the pseudo source is ARR when expressed in symbolic form. The number of ARRs thus derived is equal to the number of sensors installed in the plant. Seven numbers of ARRs, given in Eq. (3), are obtained as the same numbers of sensors are installed in the system. Two conditions for structural observability, e.g. attainability condition and sufficient condition, need to be satisfied in the first step [Dauphin-Tanguy et al. (1999)]. Attainability condition states that, in integrally causalled model, each storage element has a causal link to each observer (sensor). Sufficient condition will be satisfied when all the storage elements can be assigned derivative causality without violating the junction causality norm, with dualisation of sensor causality permitted. It is observed from Fig. 2 & 3 that both the conditions are satisfied, and hence the system is structurally observable.

0).(...:1 hmmympmhhmhmhhn ffcbffrRefdt

dmgmARR

01

..:2 hmh

hmpmymmn edt

d

KffrffcbARR

0.1

.:3 rmr

mymn edt

d

KfbfARR

0.1

.:4 fmf

mymn edt

d

KfafARR

0..

.cos.:5

mymrrmhmmpmymhhm

a

mpmmymffmmn

fbfRebffcbfrfRecb

R

fftAfafReaf

dt

dJARR

0

..:6

hmmympmhhm

ymymmrrmmymffmn

ffcbfrfRe

fdt

dMffbRefafReARR

(3)

0..cos

:7

hmmympmhhmpmpa

mpmn ffcbffrRerf

dt

dJ

R

fftAARR

Note that the ARRs thus derived are nominal ARRs, i.e. parameter uncertainty is not considered, so designated as ARRn. The ARRs in Eq. (3) can also be obtained from the diagnostic bond graph (DBG) shown in Fig. 4, and the FSM can be obtained directly (without having ARRs) from the causal analysis on DBG [Samantaray et al. (2006)]. This was based on the use of a set of substitutions, which lead to a graph structure, called a DBG model, where sensor data from the system become inputs and residuals become outputs. Analysis of the causal paths to each residual is used to generate fault signatures. Let us consider the inverted causality in the effort sensor, De: ehm (refer Fig. 3). The flow in the bond connected to this sensor is the reactive factor of power, which is nothing but a residual. This element can then be represented as an effort source (measurements from real process), and a virtual flow sensor (Df*) is attached to measure the flow or residual (signal bond 41 in Fig. 4). Such virtual sensors have computational existence, only. The DBG for the system considered is shown in Fig. 4, and the ARRs given in eq. (3) can be derived by writing the expressions of the corresponding virtual sensors. Also, the residuals can be obtained numerically from DBG, if the symbolic form is not available due to existence of implicit causal loop. The FSM can then be obtained by direct exploitation of causal paths. As an example, consider the virtual sensor for residual Rd7 is represented in the bond number 54 in Fig. 4. The following are the causal paths to that residual.

54552924262527

545534353638

545556

eeeeeffRa

eeeeeeRh

eeeJp

From these causal paths, the components involved in the residual Rd7 are obtained as: ],,,[7 ahp RRJK .

The fault signature obtained from ARR7 of Eq. (3) is identical to the one obtained through the analysis of the causal paths. However, the later method is suitable for all the cases, even when closed form symbolic residuals cannot be derived. The fault signature matrix of the hoisting system is given in Table.2.


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Fig. 4: Diagnostic Bond graph model of the system in Fig.1

Table 2: Fault signature matrix (FSM)

Rd1 Rd2 Rd3 Rd4 Rd5 Rd6 Rd7 Ib Kf 0 0 0 1 0 0 0 1 Kr 0 0 1 0 0 0 0 1 Kh 0 1 0 0 0 0 0 1 Rh 1 0 0 0 1 1 1 1 mh 1 0 0 0 0 0 0 1 Jp 0 0 0 0 0 0 1 1 Ra 0 0 0 0 1 0 1 0 µ 0 0 0 0 1 0 1 0

The ARRs and FSM can be derived by using model builder software based on bond graphs [Ould

Bouamama et al. (2005)]. All the component faults listed in the matrix were isolable (Ib =1), except the motor fault. Motor fault results if either the armature resistance (Ra) or the gyrator modulus changes, rendering

identical signature C = [0 0 0 0 1 0 1]. However, at this stage, one can recognize a motor fault as a whole when this signature would result. As a solution, in this work, we defined a fault subspace Sp in FSM (Table 2) corresponding to identical signature where higher level fault isolation technique need to apply. We have used multi-block parallel simulation technique [Feenstra et al. (2001), Samantaray et al. (2005)] for finer fault isolation. Also, instead of using constant bilateral threshold, as in [Samanta et al. (2009)], we have used an adaptive threshold, designed in [Sié Kam and Dauphin-Tanguy (2005), Djeziri et al. (2006)], so that the residual is sensitive to fault and insensitive to modeling uncertainties and sensor noises, i.e. robust residual. In this case study, only modeling uncertainties are considered. However noises may be incorporated and then noisy signals filtered out by using low pass filter, or various data reduction methods [Blanke et al. (2003), Simani et al.

Sp


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(2003), Samantaray and Ould Bouamama (2008)] before entering to the diagnostic block. This approach of robust residual generation and associated signal processing is known as passive robustness in fault diagnosis paradigm [Puig et al. (2000), Puig and Quevedo (2002)]. Different methods to achieve active robustness are also available: unknown input observers [Viswanadham and Srichander (1987), Chen et al. (1996), Lin and Wang (2000)], robust parity equations [Patton and Chen (1991)], etc. Perfect decoupling of residuals from uncertainties arising out of parameters is not always possible and depends on the number of measurement signals and their locations [Viswanadham and Srichander (1987)]. It is only possible if there are structured uncertainties. On the contrary, passive robustness tries to accomplish robustness in the decision-making stage. In a passive approach, the effect of the parameter uncertainty is propagated to the residuals and then an adaptive threshold is used to envelop these residuals to achieve robustness [Puig et al. (2000)].

The paper is organized as follows. First, we applied the adaptive threshold designed in [Sié Kam and Dauphin-Tanguy (2005), Djeziri et al. (2006)] to achieve robustness in fault detection. Next, multi-tier parallel simulation method, given by Feenstra et al. (2001) and Samantaray et al. (2005), is used for isolation of structurally nonisolable faults (motor fault). At last, the problem of prediction of useful life of system components (prognosis) [Muller et al. (2008), Medjaher and Zerhouni (2009)] is addressed.

3.1.1. Robust fault diagnosis

Let us consider uncertainties in model parameters. C-elements: Because the actual value of the C-element is never known, it can be written as

δC)(1CΔCC nn , where Cn is the average estimated (nominal) value and nCC R is the uncertainty in estimation. When the C-element is in derivative causality, its constitutive relation may be written as:

Cn

Cnn

wC

ee

Ce

CCf /1/11

11

(4)

where, n1/C Cδ is the uncertainty in estimating the capacitance and Cw /1 may be considered as a disturbance.

Such a way of separating uncertainties is termed Linear Fractional Transformation (LFT). The bond graph representation the C-element in derivative causality with the uncertain parameter value is given in Fig. 5. Using the same principle, the DBG in Fig. 4 can be drawn in LFT form as given in Fig. 6.

Fig. 5. C-element with derivative causality transformed to LFT form

Fig. 6: LFT form model for the hoisting system

Now the ARRs from the LFT model (Fig. 6) are derived as follows:


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7...1: iforARRARR iini (5)

Where, )(1 hmmh fdt

d ,

)(/12 hmKh edt

d ,

)(/13 rmKr edt

d ,

)(/14 fmKf edt

d

,

)(5 mJ fdt

d

,)(6 ymM f

dt

d ,

)(7 pmJp fdt

d

The ARRs derived above have been separated into their nominal part (ARRin: i = 1:7) and uncertain parts 7:1, ii . Thus the corresponding residual thresholds are: iiniLiiniu RdRdRdRd , . Note that the

uncertainties in various parameters are uncorrelated. This means there is as much likelihood of them cancelling out each other as the likelihood of them adding up. Therefore, we have taken absolute values of the uncertain parts.

For simulation purpose, the integrally causalled model (Fig. 2) is converted to matlab-simulink block model, shown in Fig. 7, where Ra is modeled as step function with initial value 1 and final value 1.1 at step time 30 s so that 10 % fault in Ra is realized at that time. Again, to introduce fault in case of and Kh, we have

used switches, where clock source is connected to the control input (input 2 i.e. u2) and when the threshold criterion is fulfilled ( .102 su ) then data port would be shifted from nominal to faulty value. We have

simulated 20 % and 1% fault for and Kh, respectively.

Fig. 7: Behavioral model (plant) with scope of fault incorporation


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Fig. 8: Simulink model for adaptive threshold generation and residual evaluation

The scheme of using adaptive threshold is shown in Fig. 8, where a nonfaulty behavioral model replicating the plant inside computer would run in parallel to the plant model. The upper and lower bounds of residuals are generated within diagnostic model 2 only, where δ values corresponding to parameter uncertainty may be adjusted based on experimentation. We have assumed the following values for δ for this simulation study.

Table 3: Values of δ used in simulation

Symbols value Symbols value

Jp 1.7 Kh/1 20

Kr/1 400 mh 2

J 5×10-3 M 1×10-3

The simulation result corresponding to the normal operation upto 10 s and then introducing fault by

reducing Kh in behavioral model (plant) from 1×103 N/m to 9×102 (i.e. abrupt rope slackening) is shown in Fig. 9, which is in match with the row corresponding to Kh in FSM (Table 2), i.e. Rd2 is out. So this fault is isolated. On the contrary, the faults in µ and Ra (one at a time) are structurally nonisolable (Ib = 0) as validated from the simulation results shown in Fig. 10-11. Note that 20% fault is considered for µ and 10 % for Ra, both at 10 s. Now, these nonisolable faults can be isolated by further quantitative analysis, i.e. using second level decision procedure, addressed in the next section.


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Fig. 9: Residuals responses when Kh is faulty


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Fig. 10: Residuals responses when µ is faulty


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Fig. 11: Residuals responses when Ra is faulty

3.1.2. Second level decision procedure using parallel multi-block simulation Although, both the residuals Rd5 and Rd7 exceed threshold after the inception of fault (Ra and ) at 10 s leading to same signature, their quantitative value differ from each other. So the parameters can be estimated by quantitative analysis of the residuals as follows.

Let us write 0,,,,,,,:ARR T21 YYUUfYU,f mi , (6)

where, T321 uuuU is the input vector, T21 m dis the nominal parameter vector and

T321 yyyY is the output vector. During a fault, if the residual corresponding to the ARR given in Eq.

(6) is evaluated with nominal parameter values, then it deviates from its corresponding threshold. Then the value of the parameter associated with fault hypothesis i can be estimated, either algebraically or numerically, from

the relation:

0,,,,,,T

21 YYUUf mi , (7)


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assuming that rest of the parameters are normal. If there are more than one ARRs involving i , it can be

estimated from any one ARR, but algebraic loop may be avoided and the simplest form may be used in order to reduce the computational time [Samantaray et al. (2005)].

Once the set of estimated parameter values m 1 are obtained corresponding to m possible single-

faults, the list of fault candidates may be reduced to a smaller set by using some predefined rules, which define plausible bounds for each parameter. For example, the spring stiffness cannot be negative; wire resistance can only increased, etc. Only those estimated parameter values, which satisfy their feasibility domain, say a number

mk , are considered as relevant fault candidates k 1 . For isolating the fault, k number of fault models

kii 1,FM are simulated, where each fault model kii 1FM uses the parameter set

( kii 1,1 ). If a fault has actually occurred in the ith parameter, then iFM should replicate the faulty

plant behaviour and the output response (Y*) from it should match best with that (Y) from the plant. This response matching norm can be used to isolate the component corresponding to the ith parameter as the actual fault candidate. The parallel models are activated by a special switching mechanism when abnormality in residual(s) is detected by the first phase decision procedure. The identification procedure thus described is treated as a second phase refining decision procedure and is shown schematically in Fig. 12.

Fig. 12: Schematic diagram for isolation of structurally nonisolable fault

The actual fault candidate can be localized, if the transient response corresponding to a fault is unique from the response corresponding to all other k-1 faults, both qualitatively and quantitatively. If transient responses from many fault models are equally close to the plant response, the fault cannot be isolated, but the list of fault candidates is reduced. Fewer fault candidates help in quickly carrying out maintenance work.

Before solution of fault models corresponding to different fault hypotheses, values of state variables in non-faulty state are estimated from an observer and copied as initial values to each of the m number of fault models. The observer may be based on an open-loop observer, a Kalman filter or an extended Kalman filter [Chui and Chen (1987)], depending on the application. In case of a closed-loop observer, faults in the system influence the observer, which has been used extensively to develop observer based FDI schemes. However, in our case, the initial values chosen to run fault models are taken corresponding to a time at which the process was working without any fault. In other words, we are using estimates of state variables corresponding to the last known non-faulty state, i.e. when the residual was within the described threshold. However, note that a database of process and observer history is required for this purpose. Therefore the kind of observer used is unimportant as long as the response time of the observer is sufficiently fast. However, there will be a time-shift between the response of the fault models and the actual process response due to the time delay suffered in detection of the fault, copying of the initial states from the observer etc.

For the present application, Ra and are estimated from ARR7n as this is simplest, alternatively those can

be estimated from ARR5n. The symbolic expressions are given in Eq. (8-9).


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))((

)(cos

hmmympmhhmpm

p

mpma

ffcbfrfRerdt

dfJ

fftAR

(8)

)(2

)))((()(4)cos(cos 2

mpm

hmmympmhhmpm

pampm

ff

ffcbfrfRerdt

dfJRfftAtA

(9)

Note that in the expression of we have not considered positive sign before radical, as the gyrator modulus of motor cannot go up after any fault (physically implausible).

Now, with the available measurement of states the values of Ra and are estimated online as shown in the

estimation block in Fig. 13. The variation of estimated values with time is given in Fig. 14-15. Now out of these two estimations which one is correct? To figure out that, we need to use the parallel simulation scheme and state comparison (see Fig. 16). The time response of state comparison is shown in Fig. 17-18.

Fig. 13: Estimation block obtained from Fig. 4

Fig. 14: Online estimation of


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Fig. 15: Online estimation of Ra

Fig. 16: Multi block model of hoisting mechanism for parallel simulation


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Fig. 17: Comparison of states from FM1 (using estimated ) and Plant model

Fig. 18: Comparison of states from FM2 (using estimated Ra) and Plant model

From Fig. 17 and Fig. 18, it is observed that state output from FM2 is matching best with that from plant, since the error comparison of states are of lower order for Fig. 18. That means FM2, which uses estimated Ra, is replicating the plant model. Hence the faulty component is isolated as Ra.

3.1.3. Prognostic analysis

We have validated another single fault by changing the value of Kh from 1000 N/m (nominal) to 990 N/m (faulty) at 10 s. The residual responses are given in Fig. 10, which generates C = [0 1 0 0 0 0 0]. However, in this case, the value of Kh cannot be estimated from the present form of ARR2n (refer Eq. 3). This is because of

the multiplication factor of derivative hme , which is responsible for singularity in this form as the state

hme comes to static steady state soon after the transition period of fault is over. Although the corresponding fault

can be isolated from residual sensitivity, the parameter should be estimated continuously for fault tolerant control or reconfiguration and more importantly for prediction of life, i.e. prognosis. To avoid the singularity problem, we have proposed the estimation of rope stiffness by using integral form of ARR2n, as given in Eq. (10)..

constant.)(

01

..:2

dtfdtfrdtfdtfcb

ehmK

edt

d

KffrffcbARR

hmpmymm

h

hmh

hmpmymmn

(10)


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The constant term in Eq. (10) can be evaluated using the measured states in nonfaulty condition, i.e. when

the residuals do not cross the threshold boundary and nominal value of Kh is known in that time can be used for

the constant term evaluation. In the present case study, the constant term is evaluated as zero. The simulink

block diagram, shown in Fig. 19, uses Eq. (10) for online estimation of Kh (Khest in Fig. 19).

The parameter hnk (nominal value of rope stiffness, i.e. 1000 N/m) is considered to drift progressively with

time and settle at half of its nominal value. In the simulation, we have introduced the fault at starting point by

assuming a function:

)( 1

2

11)( )(

0 fsttr

thnhn tuektk f , where 5101 r s-1, 0ft s (starting time)

and su is the unit step function. The time variation of estimated Kh is shown in Fig. 20, from which one can

replace the rope when Kh goes beyond a preset benchmark value of Kh. The dashed line is shown as an example whereby the maximum allowable (benchmark) value of Kh is assumed to be 800 N/m, the rope life therefore is

concluded to be 4101.5 t s.

Fig. 19: Block diagram for online estimation of Kh using integral form of ARR.

Fig. 20: Online estimation of Kh


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4. Conclusions

Model based FDI of hoisting mechanism is reported in this article. The information gathered in this problem is that ARRs and FSM for a system containing in different energy domains can be algorithmically derived from a bond graph model and those can be effectively used for condition monitoring, fault disambiguation and parameter estimation. The scheme of achieving robust diagnosis is applied to ensure no misdetection and false alarm in fault detection. We have designed an adaptive threshold which can decouple the effect of parameter uncertainty and thus enhancing robustness of FDI. It is shown that the structurally nonisolable faulty components can be isolated through parallel multi-tier simulation scheme. The online estimation of parameters has been used for prognostic analysis also.

However in the whole analysis only single fault hypothesis is considered, the work can therefore be extended toward multi fault analysis, where more than one fault may be faulty at a time ( may not be simultaneous) and different types of fault (abrupt, progressive or incipient) may be associated with different components. The residual sensitivity to different parameters may be examined. The parameter estimation may be effectively used for fault accommodation either through system reconfiguration or through fault tolerant control. All the analytical results are subject to experimental validation in future.

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