Basics Of Process Fault Detection And Diagnostics

By-Rahul Dey

EE14MTECH110331

Fault Detection • Previously it was known as Fault Detection

Isolation and Recovery(FDIR).

• Fault is defined as an abnormal condition or defect at the component equipment or sub-level which may lead to failure [ISO/CD 10303-226]

• In simple words,it is a branch of control engineering ,which deals with monitoring a system,identifying when a fault has occurred and to locate the fault is known as fault detection

2Soure:wikipedia

Difference Between Detection & Diagnosis

• DetectionIt is the action or process of identifying the process of something hidden

• DiagnosisThe identification of the nature of the problem by examination of the symptoms

• Detection + Isolation = Diagnosis3

Abnormal Event Management(AEM)

• AEM involves the timely detection of an abnormal event ,diagnosing its causal origin & then taking supervisory control decision and action to bring back the process to a normal safe operating state

• An abnormal event could arise from the departure of an observed variable from the acceptable range

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Components of general fault diagnosisframework

Figure 15

Classes of failure

• Gross parameter changes in a modela) These are all the processes that cannot be

included in the modelb) All these processes are lumped to form a

single parameter,viz.gross parameterc) In gross parameter interaction along the

system boundary is also included d) Failure arises when there is disturbance in the process through the environment

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Classes of failure (contd..)• Structural changes

a) These types of failure changes the processitself

b) These types of failure changes various information between variable

c) These types of failure occurs due to hardfailure in the component

d) For tackling such a failure,the diagnosticsystem removes the model equation of thefaulty component & to change the otherequation accordingly

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Classes of Failure(contd..)• Malfunctioning Sensors & actuators

a) Serious error usually occurs with sensors &actuators due to the following reasons:1. Fixed failure2. A constant bias3. Out of range failure

b) Feedback signals,which are essential for controlof the plant.A failure in feedback component,could result the plant go to unstablility,unlessfailure is not detected quickly

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Desirable characteristics of a fault diagnostics system

• For comparing different diagnostic approach,there is a set of desirable characteristics that a diagnostics system should posses.

• With the help of the set of desirable characteristics one can compare the different diagnostics classifier.

• When a fault occurs in a process,diagnostic classifier would propose a set of fault that explains the fault

• The main aim of the diagnostic classifier would require the actual faults to be subset of the proposed faults

• Resolution of diagnostics classifier would require the proposed fault set to be minimum

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Quick detection & diagnosis

• The diagnosis system should respond quickly in detecting and diagnosing malfunction

• Quick response to 1failure diagnosis Tolerable performance

during normal oprtn

• A system that has a quick response to any failure,so it able to detect failure particularly the impulsive changes quickly,so if the system is prone to noise,it can lead to false alarming during normal operation

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Isolability

• It is the ability of the diagnostics system to distinguish between different failures

• That is the diagnostics classifier should be able to generate output that are orthogonal to the fault that has not occurred

• But again here also there is a trade-off between isolability and rejecting modelling uncertainities

Isolability 1rejection of modelling

uncertainities

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Robustness

• Robustness is the ability of the process to cope with error & disturbances during operation

• Diagnostic system should be robust to various noise and uncertainties

• It is better if the performance degrades gracefully instead failing totally and abruptly

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Novelty Identifiability• Novelty identifiability,in simple words means

that,it is the threshold decided by the diagnostic system whether the current process is functioning normally or not,if not,whether the fault is a known fault or an unknown fault

• Sufficient data may be available to model the normal behaviour of the process,but we don’t have large data available for modelling the abnormal region

• Due to the unavailability of the data from abnormal region,so it is possible that the abnormal region has been not modelled properly.

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Adaptability

• The diagnostic system should be adaptable to changes such as :

a) Changes in external input

b) Structural changes due to retrofittingc) Change in operating condition due to

disturbances

d) environmental condition,

• The diagnostic system should adaptable to changes,and should gradually develop the system as new cases and problem emerges

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Explanation facility

• The job of the diagnostic system is not only identifying faults but also providing on how the fault started & propagated to the current situation

• It is actually the ability of the diagnostic system to reason about cause and relationship in process

• The diagnostic system should justify its recommendation so that operator can act accordingly

• Also it is the job of the diagnostic system to justify why certain fault were proposed and rest were not

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Multiple Fault Identifiability

• The ability to identify multiple fault is an important but difficult task due to interacting nature of the faults

• The interaction among the non linear is synergistic that is it cant be determined by the components

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Quantitative Model-Based Methods(Introduction)

• In this approach the most frequently used FDI methods are observers,parity relations,Kalmanfilters and parameter estimation

• Most of the work on quantitative model-based approaches have been based on general input-output and state-space models

• Both of the above types of model find an important place in fault diagnosis studies

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Quantitative Model-Based Methods(Redundancy)

• Model-based FDI mainly relies on an explicit model of the monitored plant

• There are two steps in any model based FDI methods :-1. Generating inconsistencies between the actual and expected behavior, such inconsistencies are also called residuals,these are the ‘artificial signals’ reflecting potential faults 2. Choosing a decision rule for diagnosis

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Redundancy(contd..)• The check for inconsistency needs some form of

redundancy

• There are mainly two kind of redundancy :a) Hardware redundancy b) Analytical redundancy

• Hardware redundancy a) These kind of redundancy requires extra

sensors.b) It is mainly used in the control of safety-critical

system such as aircraft,nuclear power plantEX: Triple Modular Redundancy (TMR)

c) However hardware redundancy are costly toimplement,which is their main drawback

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Redundancy(contd..)• Analytical Redundancy

Also known as functional,inherent or artificial redundancy is achieved from the functional dependencies among the process variables & is usually provide by a set of algebraic or time relating relationships among the states,input and output of the systems

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DIRECT TEMPORALThis type of redundancy is accomplished by algebraic relationship among different sensor measurement

This type of redundancy is obtained from differential or difference relationship among different sensor output & actuator input

Such relation are useful in computing the value of sensor measurement from measurement of other sensors

This type of redundancy is useful for sensor and actuator fault detection

The computed value is compared with sensor data & a difference indicates a fault

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• The main characteristics of analytical redundancy in FDI is to compare the actual system behavior against system model for consistency

• Any inconsistency expressed as residuals,can be used for isolation and detection

• The residual should be close to zero when no fault occurs but show significant values when there is fault

• For the generation of the diagnostic residuals,we require an explicit mathematical model of the system

General scheme for using analytical redundancy

Types Of Models• Most of FDI methods use discrete black-box plant

models such as input-output or state space model & assume linearity of the plant

• Considering a system with 𝑚 input & 𝑘 output𝑢 𝑡 = [𝑢1 (𝑡)……… . . 𝑢𝑚(𝑡)]𝑇

𝑦 𝑡 = [𝑦1 (𝑡)……… . . 𝑦𝑘 (𝑡)]𝑇

• The basic model in state space form is𝑥 𝑡 + 1 = 𝐴𝑥 𝑡 + 𝐵𝑢 𝑡𝑦 𝑡 = 𝐶𝑥 𝑡 + 𝐷𝑢 𝑡

where 𝐴, 𝐵, 𝐶, 𝐷 are parameter matrices

• The same system can be expressed in the input-output form

𝐻 𝑧 𝑦 𝑡 = 𝐺 𝑧 𝑢 𝑡where 𝐻 𝑧 & 𝐺(𝑧) are polynomial matrices in 𝑧−122

Types Of Models(contd..)𝐻 𝑧 is diagonal

𝐻 𝑧 = 𝐼 + 𝐻1𝑧−1 + 𝐻2𝑧

−2+. . +𝐻𝑛𝑧−𝑛

𝐺 𝑧 = 𝐺0 + 𝐺1𝑧−1 + ⋯+ 𝐺𝑛𝑧

−𝑛

• Both the above model are that of ideal situation, where there is no fault, disturbance or noise

• State space model with fault𝑥 𝑡 + 1 = 𝐴𝑥 𝑡 + 𝐵𝑢 𝑡 + 𝐸𝑝 𝑡𝑦 𝑡 = 𝐶𝑥 𝑡 + 𝐷𝑢 𝑡 + 𝐸′𝑝 𝑡 + 𝑞(𝑡)

• Input output model with fault𝐻 𝑧 𝑦 𝑡 = 𝐺 𝑧 𝑢 𝑡 + 𝐻 𝑧 𝑞 𝑡 + 𝐹 𝑧 𝑝 𝑡

𝑞(𝑡) = output sensor fault𝑝(𝑡) = actuator faults & certain plant faults, disturbances as well as some input sensor faults

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Residual Generation In Dynamic System• Both of above models, state space or input output

alike can be written as𝑦 𝑡 = 𝑓 𝑢 𝑡 , 𝜔 𝑡 , 𝑥 𝑡 , 𝜃 𝑡

𝑦 𝑡 , 𝑢 𝑡 = measurable output & input𝑥 𝑡 , 𝜔 𝑡 = unmeasurable state variable &

disturbances𝜃 𝑡 = process parameter

• Process fault usually changes in the state variables and/or changes in model parameters

• Based on the process model, one can estimate the unmeasurable 𝑥 𝑡 𝑜𝑟 𝜃(𝑡) by observed 𝑢 𝑡 & 𝑦(𝑡) using state estimation or parameter estimation

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Kalman filters• The plant disturbances are random fluctuations

and only the statistical parameters of the plants are known

• FDI in such type of systems can be done by monitoring the process or prediction error

• It can be done using the optimal state estimate such as the Kalman filters

• KF is a recursive algorithm for state estimation

• The KF in s/s model is equivalent to an optimal predictor for linear stochastic system in the input output model

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Kalman Filter Equations• The system model is given as

𝑥 𝑡 + 1 = 𝐴𝑥 𝑘 + 𝐵𝑢 𝑘 + 𝑤(𝑘)𝑦 𝑡 = 𝐶𝑥 𝑘 + 𝑣 𝑘

𝑤 𝑘 &𝑣 𝑘 are process and measurement noise

• Where 𝑤 𝑘 & 𝑣(𝑘) are standard gaussian with zero mean

• 𝐶𝑜𝑣 𝑤 𝑘 = 𝐸 𝑤 𝑘 𝑤 𝑘 𝑇 = 𝑅1 𝑛×𝑛

𝐶𝑜𝑣 𝑣 𝑘 = 𝐸 𝑣 𝑘 𝑣 𝑘 𝑇 = 𝑅2 𝑚×𝑚

• Observer design 𝑥(𝑘 + 1) = 𝐴 𝑥(𝑘) + 𝐵𝑢 𝑘 + 𝐺(𝑘)𝑛×𝑚[𝑦 𝑘 − 𝑦(𝑘)]𝑚×1

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Kalman Filter Equations(contd..)• State Estimation Error

𝑒 𝑘 = 𝑥 𝑘 − 𝑥 𝑘𝑒 𝑘 + 1 = 𝑥 𝑘 + 1 − 𝑥 𝑘 + 1

then substituting 𝑥 𝑘 + 1 from model equation and 𝑥 𝑘 + 1 from observer equation,andsolving,we get

𝑒 𝑘 + 1 = 𝐴 − 𝐺𝐶 𝑒 𝑘 − 𝐺𝑣 𝑘 + 𝑤(𝑘)to design an optimal-estimator such that

of estimation error, 𝑝 𝑘 is minimized

• minimizedo find 𝐺 𝑘 , minimize 𝑐𝑜𝑣 𝑒 𝑘 + 1

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• But,𝐸 𝑒 𝑘 + 1 = 𝐸 𝐴 − 𝐺𝐶 𝑒 𝑘 − 𝐸 𝐺𝑣 𝑘 + 𝐸 𝑤 𝑘

= 𝐸 𝐴 − 𝐺𝐶 𝑒 𝑘 − 𝐺𝐸 𝑣 𝑘

= 𝐴 − 𝐺𝐶 𝐸 𝑥 𝑘 − 𝑥 𝑘

nimize 𝑃 𝑘 + 1 with the help of decision variable help of decision variable 𝐺 𝑘

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• 𝑃 𝑘 + 1 = 𝐸 𝐴 − 𝐺𝐶 𝑒 𝑘 𝑒 𝑘 𝑇 𝐴 − 𝐺𝐶 𝑇 + 𝐺𝑣 𝑘 𝑣 𝑘 𝑇𝐺𝑇 + 𝑤 𝑘 𝑤 𝑘 𝑇

= 𝐴 − 𝐺(𝑘)𝐶 𝑃 𝑘 𝐴 − 𝐺(𝑘)𝐶 𝑇 + 𝐺 𝑘 𝑅2𝐺 𝑘 𝑇 + 𝑅1

• Now let us assume that 𝐸[𝑒 𝑗 𝑒 𝑗 𝑇] is minimized for 𝑗 = 0,1. . , 𝑘by selecting 𝐺 0 , 𝐺 1 , . . 𝐺(𝑘 − 1)

• Now we need to find 𝐺 𝑘 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑃(𝑘 + 1) is minimized• 𝑃 𝑘 + 1 = 𝐴 − 𝐺(𝑘)𝐶 𝑃 𝑘 𝐴 − 𝐺(𝑘)𝐶 𝑇 + 𝐺 𝑘 𝑅2𝐺 𝑘 𝑇 + 𝑅1

• Solving by adding and subtracting terms we will get,

• 𝑃 𝑘 + 1 = 𝐴𝑃 𝑘 𝐴𝑇 + 𝑅1 − 𝐴𝑃(𝑘)𝐶𝑇 𝑅2 + 𝐶𝑃 𝑘 𝐶𝑇 −1𝐶𝑃(𝑘)𝐴𝑇

• 𝑃 𝑘 + 1 = 𝐴𝑃 𝑘 𝐴𝑇 + 𝑅1 − 𝐺 𝑘 𝐶𝑃 𝑘 𝐴𝑇

where,

• 𝐺 𝑘 = 𝐴𝑃(𝑘)𝐶𝑇[𝑅2 + 𝐶𝑃(𝑘)𝐶𝑇]−1

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Kalman Filter Summary• 𝑥 𝑘 + 1 𝑘 = 𝐴 𝑥( 𝑘 𝑘 − 1) + 𝐵𝑢 𝑘 + 𝐺(𝑘)[𝑦 𝑘 − 𝐶 𝑥( 𝑘 𝑘 − 1)]

• 𝐺 𝑘 = 𝐴𝑃( 𝑘 𝑘 − 1)𝐶𝑇[𝑅2 + 𝐶𝑃( 𝑘 𝑘 − 1)𝐶𝑇]−1

• 𝑃( 𝑘 + 1 𝑘) = 𝐴𝑃 𝑘 𝑘 − 1 𝐴𝑇 + 𝑅1 − 𝐺 𝑘 𝐶𝑃 𝑘 𝑘 − 1 𝐴𝑇

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