Deteccin de fallos basada en modelos intervalares

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Detección de fallos basada en modelos
intervalares

• Joaquim Armengol Llobet
• Universitat de Girona

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Fault concepts

• Fault deviation from the normal, acceptable,
  usual, standard behaviour. Abnormal operation of
  system, sensors or actuators.
• Causes design errors, implementation errors,
  human errors, use, wear, deterioration, damages,
  ageing
• Consequences worse performances, energy waste,
  waste of raw materials, economic losses, lower
  quality, lower production, environmental damages,
  human damages

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Example

• Internal faults
• Process Tank leakage, clogged pipe.
• Sensor Offset.
• Actuator Valve is blocked.

Controller
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Fault types

• Depending on the temporal aspects
• Abrupt fault sudden and considerable. Model
  step. Example offset.
• Incipient or evolutive fault affects slowly.
  Model ramp, exponential, parabola. Example
  drift (deriva).
• Intermitent fault. Model pulses.

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Tasks

• Fault detection. Determination of the existence
  of the fault.
• Fault diagnosis. Diagnostic set of components
  that explain the fault.
• Fault isolation kind, location, root cause,
  faulty component.
• Fault identification size, type, time of the
  fault.
• Fault tolerance. Fault handling.

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Fault detection by analytical redundancy

• The behaviour of the real process and the
  behaviour of a model of the process are compared.
• Sufficient condition
• IF real_output ? simulated_output THEN fault_alarm

System
Input
Output
Alarm
Comparison
Model
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Analytical Redundancy

• Problem Behaviours are always different due to
  the uncertainty. Uncertainty in industrial
  processes makes difficult the fault detection
  task.

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Residuals

• Sufficient condition
• IF real_output - simulated_output gt e THEN
  fault_alarm

Input
Output
System
Alarm
Comparison
Model
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Threshold

• Fixed.
• Variable
• Adaptive.

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Models expressing uncertainty

• Decreasing precision to increase accuracy.
• Imprecise models
• Qualitative models. Non-numerical values
• Signs positive, negative, increases, decreases
• Orders of magnitude big, small
• Example economy models
• If interest rates grow, unemployment increases.
• Semiqualitative models
• Fuzzy sets.
• Intervals.

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Uncertainty in models
Example
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Intervals decreasing precision to increase
accuracy

• Its 11.56

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Simulation
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Simulation of uncertain models

• Example 2 tanks.

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Fault detection by analytical redundancy

• Sufficient condition
• IF real_output ? simulated_output THEN fault_alarm

Output
Input
System
Interval
Envelope
Alarm
Comparison
Model
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Quantitative simulation Quantitative models
method

• Example
• Systematically Randomly

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Quantitative simulation Quantitative models
method

• Internal envelopes.

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Quantitative simulationRange computation

• Example difference equation model
• Goal finding for each time point t
• and s.t.
  
• 2 cases
• Parameters vary in time ? time variant ?
  independent functions.
• Constant parameters ? time invariant ?
  dependent functions.

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Quantitative simulationRange computation

• Global optimisation algorithms.
• Why global? Local optima ? Internal estimation of
  the envelope.
• Computational cost is very high.
• Exact can not be obtained at least due to
  rounding errors ? Exact envelope is not known.
• Error is not known ? Distance to the exact
  envelope is not known.
• Some of them are based on intervals.

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Interval Arithmetic

• Moore, 1966.
• Interval
• Interval operations

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Example of interval computation

• We want to compute the
• value of the function
• for
• The solution is

x
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Interval Arithmetic

• Natural extension of a real function.
• Property monotonic inclusion.
• Multi-incidences ? outer estimation.

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Modal Interval Analysis

• Interval
• Modal interval
• interval quantifier
• Canonical notation

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Modal Interval Analysis

• Modal interval extensions
• Interval Analysis. Unique semantics
• Modal Interval Analysis. Many different
  semantics
• Examples

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Modal Interval AnalysisImportant properties

• Interpretable Modal Interval extensions
• ? Semantics of external and internal estimations.
• Coercion theorems allow to obtain better
  approximations of f and f efficiently and
  taking multi-incidences into account. Example

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F algorithm

• Branch-and-Bound algorithm based on Modal
  Interval Analysis.
• Provides inner and outer approximations of the
  exact result.

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Fault detection using error-bounded envelopes
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Algorithm for FD with error-bounded envelopes

• Use of the output measurement into the iterative
  algorithm for generating error-bounded envelopes
• Outer zone ? Stop. FAULT.
• Inner zone ? Stop
• Either normal behaviour.
• Or fault can not be detected.
• Shift sliding window.
• Intermediate zone ? Iterate until outer or inner
  zone.

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CHEM TB3.2b SQualTrack
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SQualTrack Architecture
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Inner approximation Outer approximation Interval
measurement
Shows a bar when a fault is detected When the
intersection between the interval measurement and
the external band is void
Window length that has been used at each time
step to compute the band
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SQualTrack v. 2

• SqualTrack is one of the components of MISO
  (Modal Interval Solver)
• FSTAR Solver.
• QRCS Solver.
• QSI Solver.
• MINIMAX Solver.
• SQualTrack Solver.
• http//pau.herrero.googlepages.com/software