Fault Detection and Diagnosis

Fault Detection and DiagnosisSENAKA SAMARASEKERA

Scope Fault detection : Indication that something has gone wrong in the system Fault Diagnosis

Fault isolation : determination of the exact location of the fault (FDI) Fault identification : Determination of the magnitude of the fault (FDD)

Fault Classes Additive ( Unknown inputs) vs Multiplicative (Changes in the parameters) Sensor faults, Plant faults, and Actuator faults

Faults, disturbances, and model errors Faults: changes in the system we want to detect and act accordingly Disturbances : we want to ignore (e.g. : sensor, plant and actuator noise) Model errors : discrepancies between the mathematical model used in signal processing and the actual system.

This is a nuisance effect we want to ignore

Type of faults

Detection and Isolation performance

Detection performance can be analyzed as Fault sensitivity : ability to detect a fault of reasonably small size Reaction speed : ability to detect a fault of reasonably small delay after their occurrence Robustness : ability to detect fault in midst of noise, disturbances and modelling errors

Isolation performance is much more complex to analyze. E.g.: Localization,

Both detection and isolation will depend on Properties of the plant, Size of the fault, Fault Multiplicity (How many faults occur simultaneously) Noise and other disturbance levels, and model errors.

Some time one fault might not be distinguishable from another fault. These occur due to limitations of observability of the system. Then for the given set of sensors, system model and plant the best we can do is to provide a fault class (i.e. Classify the faults in to different classes. In practice this is the most common method.)

Approaches to FDDModel free methodsCan be used in simple FDD scenarios such as detecting a directly observable fault. Cannot be used for multiple FDD hidden state FDD.

E.g. : Multiple sensors ( to identify sensor failure Limit checking : If then faultLimit checking can be extended to two techniques Trend checking : e.g. Time domain : if FOR then detect fault. Frequency domain: if FOR then detect fault. Time frequency plane: if FOR then detect fault.

Knowledge based checking : Enable to use coupling between symptoms and faults. This contains the expert knowledge gained by experience.

e.g. IF (condition 1 and condition 2) then detect fault 1 if condition 3 then detect (fault 1 and fault 2)

Approaches to FDD Model- based methods

Use an explicit mathematical model of the monitored plant.

The original plant model can be given as a continuous time differential equation or a Laplace domain transfer function

Since the monitoring is mostly done via computers we need to convert this to a discrete time representation (i.e. difference equation or z domain transfer function)

Most of the time we are able to linearize the system model around a suitable operating point.

ResidualsThe main concept behind model-based methods is

Analytical redundancy : Is the comparison of the actual behavior of the monitored plant to the behavior predicted by the fault free plant model . The plant measurement ae checked for consistency with the model.

Residual : is the difference between the actual and the computed estimate and are the outcome of he consistency checks. This is the main indicator function used in model based FDD.

Residuals Methods for generating residuals

Kalman filter Innovation (prediction error) of the Kalman filter can be used as a fault detection residual Since innovation is white relatively easy to construct statistical tests Fault isolation : One needs to run a bank of matched filters (one for each suspected fault) and find the maximum

SNR

Diagnostic observers: When the state is not directly observable we can design a state observer. Isolation is just like Kalmann

Parity relations: Rearrange the system (equation) in to subsystems (equations) via linear transformation so that a decision is taken (parity bit is fired) when a fault occur. Looking at the parity bit structure an error could be isolated

Parameter estimation: Fault free parameter values are specified for a parametric model of the system. If the parameter values deviate too far from the value a fault will be declared.

Residuals Parameter, state and measurand estimates are never perfect match for their actual counterparts. Therefore the residuals are never zero due to the presence of noise, disturbances and modeling errors. But they will be definitely large when a system fault occurs. (Why?)

Therefore we need to analyze the residuals and give them a threshold over which we would claim a fault occurrence.

If we make this threshold too low : we would get many false alarms

If we make this threshold too high : we would miss a fault detection

How to decide what is the best threshold level?

We have to have an understanding on the noise statistic to answer this question

Car engine exhaust emission control system

Most modern cars have emission control system that will keep the CO, Hydrocarbons and the NOx below regulatory thresholds. The regulations dictate that the OBD (Onboard diagnostics) should detect and isolate a fault in this system if the emissions increase to 50% of this value.

Stoichiometric ratio that minimizes emissions = 14.7 Excess oxygen sensor characteristics

Car engine exhaust emission control system

EGR valve dilutes the input mix with exhaust gas by lowering the combustion temperature leading to recued NO emissions.

Scope of the diagnostic task

Fault detection and isolation of the emission control system using Parity relations

Thresholds are determined empirically Thresholds are low pass filtered to reduce noise As a second stage of filtering its only after the fire counter

reaches a certain threshold that the fault is detected

The models are build using system identification methods. (grey box)

Residuals and counters with no faults Residuals and counters with a small EGR fault

Analytical redundancy based Fault detection

Contain two sub blocks

1. Residual generator

2. Decision maker

A single residual might be sufficient for fault detection. But for fault isolation the residuals should be enhanced, that is multiple residuals which are orthogonal to different faults should be designed.

Modelling additive faults Let

Let be actual plant inputs and be the actual plant output

Modelling additive faults Let the fault free model be

where and

Assumptions: System is time invariant. System is can be modelled using the shift parameter

E can incorporate the faults as

Where is the plant fault transfer function matrix.

We can rewrite this combining all fault response in

Where and

Modelling additive disturbances Let

Let be actual plant inputs and be the actual plant output

Modelling additive faults and disturbances

Using similar procedure as in faults we can incorporate the disturbances as

Where

Taking faults and disturbances together we can write

Example

Let

1. Find the fault free model. 2. Assume that is controlled and is a measurement. Assuming no plant disturbances or noise, find the model with input and output faults and noises.

Modelling multiplicative faults and disturbances

These can be modelled as changes in the system transfer function

These may reflect a

1. Parametric fault

2. Modelling error

To simplify analysis we will assume that these model changes are not time varying (for the time period of analysis).

In most cases rather than defining individual deviations for the scalar terms in the system matrix we can parametrize model using an underlying parameter vector .

Parameterizing model deviations be the model transfer function.

If the nominal (fault free) values of

we can approximate the deviations using the Jacobian matric

Then we can write

Parameterizing model deviations By grouping the model deviations to parametric faults and model errors,

This enable us to write the total equation as

Example (cont) :

Assume that the two poles and are the source of model uncertainty. Find and and there by give an expression to the model uncertainty.

Parameter estimation for linear models

Equation error method Output error method

Residual generation The residual generator is a linear discrete dynamic algorithm acting on the observables.

where

In other words

which leads to

Since

We can write

Parameter estimation

Example : Air cabin pressure valve control using BLDC


Modelling of the BLDC

Electrical subsystem

In order to get out we recognize that only two out of 3 windings are active at a given time which leads to

e.g.

Similarly 5 more equations can be written.

Let


Mechanical subsystem

Where where gives the flap position.

Where is the motor position and is the gear ratio.


LS Parameter estimation

Electrical subsystem

Let ,

Let be the sliding window data matrix.


LS parameter estimation

Mechanical subsystem

Neglecting dynamic friction and, assuming is known,

Let ,

Let be the sliding window data vector.

Here can be estimated using a state variable filter or simply an active differentiator


Generating residuals

Each of the residuals are decoupled from at least one variable.

,

FDI indicators for the cabin pressure valve system

Detection of faults using Residuals

How to be robust against

Additive disturbances: Slow varying in time, compared to the processing window. No quantitative information is available most of the time. Therefore the residuals should be orthogonal to these disturbances.

Multiplicative disturbances: Slow varying in time, the residual generator should adapt online for parameter changes

Noise : Fast varying in time. Statistical description of these are available. Filtering and appropriate thresholding (to balance false alarms and missed detections)

Fault sensitivity Fault sensitivity

Triggering limit : The value of the particular fault j which brings a particular residual i to

its detection threshold , provided no other faults and nuisance inputs are present.

This can be analyzed using

Assuming step fault (applying final value theorem)

Therefore

Fault sensitivity For a nominal (typically occurring, minimum) fault size for successful detection

When analyzing the detection quality of a we would look at the spread

Modelling error robustness Since the residual response for a model error only scenario is

we can define the limit error as

A higher limit error signifies lower sensitivity to model errors hence, better robustness.

Finding thresholds for residuals Considerations

1. Magnitude (distribution) of the Fault

2. Magnitude (distribution) of noise and other disturbances

If these information is available we can formulate the problem as a problem in statistical dicision theory.

Noise – zero mean with known covariance

Fault – with a deterministic value (mean)

Fault detection -> Testing for zero mean hypothesis

Fault Isolation -> Testing for multiple hypothesis

The Neyman- Pearson detector

Isolation properties of the residuals

Directional residuals : We can define a vector for each fault j such that the residual vectors are of the form

Where the is the scalar transfer function and is the fault magnitude.

Isolation properties of the residuals

Structured residuals – Each residual responds o a different subset of faults. This enables the definition of a fault signature or a fault code, by thresholding each fault.

Residuals from output error method

Residuals from equation error method

and are known as primary residuals. The two are related as

SISO residuals In the SISO case nothing much can be done with the single residual.

E.g. : Let Find and under different fault conditions. Comment on the diagnosis.

SISO residuals :First order system

SISO residuals : First order system

Continuous time state space parity equations

The derivatives of y can be found to be




Using the Laplace transform

Discrete time state space parity equations

Similar to the continuous time case we create time difference equations




Properties of residuals

Generation of Enhanced residuals

Structured residuals vs directional residuals

• Errors in residuals : Strong isolation /Weak isolation

Generation of structured residuals Generating good isolating patters of the residual vector

The equation error residual for p inputs and r outputs MIMO system can be given as

To generate structured residuals this is multiplied by matrix W

Generation of structured residuals

Let


Example : Let the system be SIMO with 2outputs

Rearranging the process equations and vectorising them



Fault detection with state observers

Let the system be

With the assumption that the model parameters are known the state observer estimates the state using the measurands as

The underline assumption is that the system is observable.

State observers

Asymptotic state error

iff is stable.

So the trick is to build observers that are stable.

Additive fault detection with state observers



Using the Laplace transform of the no-fault system

Using the Laplace transform of the fault system (no disturbance)

Using Final value theorem, assuming step fault and

Multiplicative fault detection with state observers

Fault isolation with state observers



Similar to the structured residuals. Use different observers that are insensitive to deviations of a sub set of measurands. Let

An observer bank is obtained using one output at a time

Leading to an output error residual


Kalman Filter based FDI



FDI based on Output observers Do not care about observing states. Just want to observe the faults.

FDI based on Output observers





Date post:	24-Dec-2015
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Fault Detection and Diagnosis

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