1. Lecture 1 Model-based Diagnosis of Continuous Systems
Gautam Biswas
Dept. of EECS and ISIS
Vanderbilt University
Acknowledge
Pieter Mosterman, Sriram Narasimhan, Eric Manders
Supported by DARPA SEC, NASA-IS, NASA-ALS, & NSF-ITR
Copyright © Vanderbilt University, 2006.

2. Overview Context (History) of the work
Initially extend qualitative consistency-based diagnosis to diagnosis of continuous (dynamic) systems (Lecture 1)
Extend to diagnosis of hybrid systems to accommodate more real-world applications (physical processes with supervisory (discrete) control) (Lecture 2)
Online diagnosis? What’s the use – extend to fault-adaptive control (Lecture 3)
Look at the bigger picture – fault-adaptive control, different fault profiles, prognosis, maintenance, safety, etc. … (Lecture 3)
Some related problems: distributed diagnosis, multi-level diagnosis, ….
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3. The need for fault-adaptivity
In complex systems even simple failures lead to complex cascades of events that are difficult to understand and manage.
How to
detect and isolate faults?
react to faults to mitigate their effect?
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4. Overview FACT – Fault Adaptive Control Technology
Goal: systematic model-based approaches to maintain system operations under degraded and failure conditions
Expanded goals: reliable, safe, autonomous operation for long-duration missions
Approach: Develop the technology and required tool-suite using Model-Integrated Computing approach to achieve this
Components:
Modeling Approaches – hybrid dynamic processes of the plant + reconfigurable controllers
Online monitoring of system behavior
Online fault detection, isolation, and identification
Adaptive models – update plant model after failure
Model-predictive fault-adaptive control
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5. Model-based Tools and System Development
Visual modeling tool for creating:
Hybrid bond-graph models
Timed failure propagation graph models
Controller models (including reconfiguration)
Controller
Models
Strategy
Models
Hybrid
Diagnostics
Active
Model
Modular run-time environment contains:
Hybrid observer and fault detectors
Hybrid and Discrete diagnostics modules
Controller and reconfiguration strategy model library
Controller selector and reconfiguration manager
Active controller modules
Modules can be used standalone
Can use an RTOS as the platform
Failure Propagation
Controller
Diagnostics
Selector
Fault Detector
Plant
Models
Hybrid Observer
Interface & Controllers
Reconfiguration
Manager
Run-time Platform (RTOS)
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6. FACT Components Modeling environment: Plant + controllers
hierarchical, multi-aspect
Three views
HBG view
Plant I/O view: Sensors + Actuators
Controller view (finite state machine, extend to MPC)
parameterize faults
TFPG view: specialized discrete-event diagnosis approach with time intervals
Simulation environment: simulates physical system behavior including fault scenarios
Run-time environment: implements fault detection, isolation, identification, and fault adaptive control algorithms
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7. Lecture 1 Model-based Diagnosis of Continuous Systems
Modeling of Continuous, Dynamic Systems
Fault Detection and Isolation (FDI) system
architecture
Components of FDI system
Observer
Fault Detector
Fault Isolation
Summary and Conclusions

8. Model-Based Diagnosis of Dynamic Systems
FDI Models: examples
Continuous
Discrete
Quantitative
State Estimation
Parameter Estimation
GDE
(static)
Qualitative
Fault Signatures & TCGs
(Mosterman and Biswas)
Sampath, et al.
Lunze, et al.
Values: Qualitative vs. Quantitative
computational qualitative models do not require numerical parameter values but diagnosis is less precise
run time complexity of qualitative methods may be less
Temporal Behavior: Discrete vs. Continuous
discrete methods may be computationally simpler at run time but are less precise, coarser
complete analysis of faulty behavior up front or spurious results possible
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9. Models Why do we need models?
Suppose, we were doing diagnosis of system in steady-state
Nominal behavior of system known, don’t need model to track system behavior
Once deviations recorded, use model of system to analyze cause for deviations
If system operating in dynamic regions
Need model to track behavior of system – to determine when system behavior deviates
Use model to analyze cause for deviations
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10. Model-based FDI of Continuous Systems (TRANSCEND)
Plant
(Process)
Extended Kalman filter
Statistical Hypothesis Testing + r r
Fault
Detector
Observer  ŷ
Symbol
Generator
Observer
Model rs m
Hypothesis
Generation
fh, p
Hypothesis
Refinement fr
Diagnosis
Model
State Space Eqns
Temporal Causal Graph (TCG)
Qualitative
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11. Bond Graphs Overview
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12. Modeling Approach Bond Graphs
What are Bond Graphs?
Topological, lumped parameter domain-independent
modeling scheme for physical systems
Based on concept of reticulation
Properties of system lumped into processes with distinct
parameter values
Dynamic System Behavior: function of energy
exchange between components
State of physical system – defined by distribution of
energy at any particular time
Dynamic Behavior: Current State + Energy exchange
mechanisms
(Ref: physical systems dynamics – Rosenberg and Karnopp, 1983, 2003)
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13. Example of Reticulation
Suspension system of automobile
Input : Velocity at bottom of tire
(function of road conditions) V g K B v k
V0(t) M m
Mass of car
Mass of tire
Tire stiffness
Suspension system parameters
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14. Bond Graphs Modeling language
Domain independent
Compact representation: Generic components:
Energy storage (C, I), dissipative (R), source (Se, Sf), Transformers (TF, GY)
Idealized junctions to allow connections among components: (1- series, 0 – parallel)
Energy exchange between components through mechanisms called bonds
Dynamic system behavior governed by conservation of energy and continuity of power A B e f
Two associated variables with bond:
e: effort; f : flow; such that e  f = power
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15. Modeling with Bond Graphs
Modeling language (based on small number of primitives)
Dissipative elements: R
Energy storage elements: C, I
Source elements: Se, Sf
Junctions: 0, 1
Transformer Elements: TF, GY
physical system
mechanisms R
C, I
Se, Sf
0,1
uniform network–like
representation
domain independent
forces the modeler
to make assumptions explicit
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16. Building Bond Graphs: Examples
Impacting Trains
Tank2 C2
R12
Tank1 C1 R2 R1
Sf1
Two tank system
Lumped parameter Topological Modeling
C: C1
R: R12
C: C2
Sf: Sf1 1
Compact Representation
across domains
R: R1
R: R2
Two Tank System
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17. Bond Graphs Modeling non linear behaviors
For linear systems, BG elements are constant values
Non linear systems
parameter value = f (effort, flow, external variables)
Sf: Sf1
R: R1 1
R: R12
C: C2
R: R2
Tank2 C2
R12
Tank1 C1 R2 R1
Sf1
Two tank system p1 p2
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18. Bond Graphs Building Models for Diagnosis
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19. TRANSCEND Architecturey
Plant
(Process)
Extended Kalman filter
Statistical Hypothesis Testing + r r
Fault
Detector
Observer  ŷ
Symbol
Generator
Observer
Model rs m
Hypothesis
Generation
fh, p
Hypothesis
Refinement fr
Diagnosis
Model
State Space Eqns
Temporal Causal Graph (TCG)
Qualitative
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20. Bond Graphs Different Model Forms
Deriving different model forms
State Space equations for tracking dynamic behavior
Causal models for diagnostic analysis
Central to this
Causality Information that can be derived from the
Topological Bond Graph Model
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21. Causality in Bond Graphs
To aid equation generation, use causality relations among variables
Bond graph looks upon system variables as interacting variable pairs
Cause effect relation: effort pushes, response is a flow
Indicated by causal stroke on a
bond A B e f
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22. Causality for basic multiports
Note that a lot of the causal considerations are based on
algebraic relations
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23. Causality Assignment Procedure
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24. Causality Assignment: Example
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25. Causality Assignment: Example 2
Tank2 C2
R12
Tank1 C1 R2 R1
Sf1
Sf: Sf1
C: C1
R: R1 1
R: R12
C: C2
R: R2
Two Tank System
Causality assignment important – for building
TemporalCausal Graphs (TCGs)
that are used for diagnosis from transients
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26. Generating State Space Models
Step 1: Augment bond graph by adding
Numbers to bonds
Reference power direction to each bond
A causal sense to each e,f variable of bond
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27. Generating State Space Models
Step 2: Identify state and input variables
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28. Model-based Diagnosis TRANSCEND approach
Fault Detection +
Fault Isolation (Qualitative)

29. TRANSCEND architecturey
Plant
(Process)
Extended Kalman filter
Statistical Hypothesis Testing + r r
Fault
Detector
Observer  ŷ
Symbol
Generator
Observer
Model rs m
Hypothesis
Generation
fh, p
Hypothesis
Refinement fr
Diagnosis
Model
State Space Eqns
Temporal Causal Graph (TCG)
Qualitative
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30. Qualitative Approach to Fault Isolation
Why Qualitative ?
Accuracy of models: structural + difficulty in estimating parameters
Imprecision of real world numeric models
computational issues, e.g., convergence problems
Qualitative Constraints
magnitude deviations + higher order derivatives. (currently +, 0, -)
Topological Models
Graph-based, compositional
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31. Fault Characterization
Model parameters in TCG correspond to system components.
Fault – model parameter that is deviated from its normal operating value
Abrupt Fault – instantaneous and persistent parameter value change (modeled as a step change).
Fault Characterization
Additive: sensor and actuator faults
Multiplicative: component parameter faults
multiplicative faults directly affect dynamic response of system, therefore, much harder to analyze residuals
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32. Example Two-Tank System
Rb1
Tank2 C2
R12
Tank1 C1
Rb2 f1 f3 f8 f5 e2 e7
Focus on process faults
multiplicative
Parameterized Model
Possible faults: Bond Graph component parameters: Sf, C1, C2, Rb1, Rb2, R12
Fault profile
abrupt change in parameter value: increase or decrease
Note – partial change
not complete failure
Bond Graph Model
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33. Diagnosis from Transients
Abrupt Faults cause transients in observed measurements.
Goal: Isolate fault as quickly as possible after occurrence of transient.
Two primary tasks:
Reliable detection of transient
Isolation of fault based on transient characteristics C1 C2
Rb1
R12
Rb2
f5 (flowrate through connecting pipe):
Faults: Rb1, Rb2, R12
Faults: C1, C2
Discontinuity
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34. Two-Tank System Diagnosis Model
Rb1
Tank2 C2
R12
Tank1 C1
Rb2 f1 f3 f8 f5 e2 e7
Bond Graph Model
Important Characteristics:
Algebraic and temporal – cause effect relations
Note the state loops – there are three in this TCG
Temporal Causal Graph
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35. Transient Analysis
Our approach is to analyze measurements individually.
Transient Response of residual of a signal (can be approximated by Taylor series of order k)
r(t) = r(t0) + r'(t0)(t- t0)/ 1! + r''(t0)(t- t0)2/ 2! + …… +
r(k)(t0)(t- t0)k/ k! + Rk(t),
where Rk(t) is the remainder term based on y(k+1)(t).
Signal transient due to a fault at t0 can be expressed as discontinuous magnitude change, r(t0), plus first and higher order derivative changes, r'(t0), r''(t0), ….., r(k)(t0).
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36. Qualitative Transient Analysis
As order of derivative increases accuracy of match improves
Matching the measured
magnitude and slope of
a signal to the signature
Signature:
< >
step
Measured
signal 1 2 3 4
- ? 5
Signature
Transient Signal from 2nd order system
(1st to 4th order Taylor’s series expansion
shown as dashed lines)
Tracking a signal when only the magnitude change and slope of signal are measured
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37. Fault Signature
Signal feature vector in response to a fault expressed as a sequence of qualitative derivative values at the point of failure.
Qualitative Fault Signature of order k: fault signature that includes derivatives up to order k.
Assumptions – (I) Abrupt faults, (ii) The sampling rate of the measured signals is set to be fast enough so that no qualitative change in the transient dynamics is missed. .
The Diagnosis Process
1. Generate Fault Hypotheses –
Backward Propagation
2. Predict Behaviors –
Forward Propagation
(Individual Fault Signatures)
3. Monitoring - compare signatures to observations: Progressive Monitoring & Discontinuity Detection
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38. Analysis of Fault Transients
Analyze behavior immediately after fault onset
Time point of failure occurrence important
Detection + estimation problem
Transient Dynamics
Captured as fault signatures:
Magnitude
First and higher order derivatives
Progressive Monitoring
Higher order derivatives progressively affect magnitude and slope of signal
For details of algorithms, see Mosterman & Biswas (1999)
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39. Run-time System Fault Detection
If there is no fault, then the Kalman filter will track –
Residual: Gaussian white with zero mean
To detect a fault, the generalized likelihood ratio test (LR) is used for hypothesis testing  n
Faults
System + + +
residual
Model 
Kalman Filter 
Residuals deviate from zero because of
Noise (n)
Separation of effects necessary!
Modeling errors () X
Sensor inaccuracies ()
Faults !
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40. Fault Detection and Symbol Generation
Approach: model based
Residuals: difference between ‘ideal’ and measured behavior
Fault detection: residual evaluation
Fault detection says “yes” if ‘significant’ deviation is detected
Generated symbols:
Sign of deviation
Sign of first derivative r
+ – t
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41. Fault Detection Assumptions (model based again…)
Noise is white, Gaussian N(0, )
Variance is constant (slowly changing)
Modeling errors and sensor inaccuracies are treated as noise
‘Significant’ residual deviation  Fault
Significance is determined by a statistical hypothesis test (approximate Z-test):
Mean value and variance used
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42. Z-test Fault detection Z-test Mapping the confidence level (1-) to z:
time
Variance estimation
Mean estimation
VarDelay k N2 N1
Mean: info on the transient residual:
Variance: info on the nominal dynamics
Typical values:
N1=50; N2=5
VarDelay=150
Fault detection
Z-test
Z-value distn.:
Decision making:
where N2 is the number of samples and  is the standard deviation.
The  is also estimated, but from a larger set of samples (N1 >> N2) Z
/2 z- z+
Mapping the confidence level (1-) to z:
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43. Symbol Generation Decision:
1st symbol: sign of deviation after fault detection 
2nd symbol: sign of slope after fault detection
Is difference significant? N3 r
Estimation of the ‘initial value’
Estimation of the ‘later value’
Fault detection t k
Additional Decision
If 1 & 2 have opposite
signs, then change is
discontinuous
Decision:
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44. Slope Symbol Generation Example20 40 60 80
100
120
140
160
-0.5
0.5
N3=1
N3=10
N3=20 m d +z + s -z r
/sqrt(N 3 )
detection time 20 40 60 80
100
120
140
160
0.5 1
1.5
Residual
Choice of N3
Small value  large threshold, thus long detection time
Large value  significant delay, may suppress short transients
Typical values of N3 are 5…20
Better solution: adaptive settings …
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45. Detection, Symbol Generation Complete
Onto Qualitative Fault Isolation

46. Generate Fault Hypotheses – Back Propagation
Rb1
Tank2 C2
R12
Tank1 C1
Rb2 f1 f3 f8 f5 e2 e7
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47. Prediction by Forward Propagation Fault Signatures
Qualitative Signature: magnitude + first and higher order derivative changes expressed as +,0,- values.
How to generate signatures from TCG ?
Temporal links imply integrating edges, affects derivative of variable on the effect side
Start with 0-order changes
Every integrating edge increases order by one
Rb2+ 2nd Order
signature of
e7: < 0,+,- >
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48. Progressive Monitoring track system behavior after failure1 2 3 5 4 6
System Behavior Convolutes the Predicted Transient at Time of Failure
dynamically change the signature: Justified by Taylor’s series
measured k 1 + 2 3 4 5 6 k
signature1 k
signature2 1 2
no match! 3
match!
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49. Three Tank Results
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50. Analysis of Qualitative Fault Signatures
Discriminatory Power of Qualitative Fault Signatures
Abrupt change – direction of abrupt change + direction of change immediately following abrupt change,
(+,+), (+,-), (-,+), and (-,-)
No abrupt change – first direction of change of the signal only, (0,+), and (0,-)
Problem: further +,- changes provide no discriminatory evidence because qualitative information contains no time constant information.
Ways to handle this problem:
measurement selection – end up needing many more measurements than log4[k], where k – number of fault hypotheses.
estimate fault parameters values; true fault is the one whose parameter is consistent across multiple measurements.
Use gradient descent methods for parameter estimation.
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51. Summary Modeling for Diagnosis
Lumped Parameter Modeling of Dynamic Systems
Topological Models have their advantages: can derive causal models that facilitate diagnosis
FDI of continuous systems by qualitative analysis of transients
Fault detection and Fault Isolation algorithms
Advantages and limitations of qualitative analysis
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52. References
Dean C. Karnopp, Donald L. Margolis, and Ronald C. Rosenberg,
System Dynamics: Modeling and Simulation of Mechatronic Systems, 4th Edition , John Wiley, New York, NY.
P.J. Mosterman and G. Biswas, “Diagnosis of Continuous Valued Systems in Transient Operating Regions,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 29, no. 6, pp , Nov
E.J. Manders, P.J. Mosterman, and G. Biswas, “Signal to Symbol Transformation Techniques for Robust Diagnosis in TRANSCEND,” Tenth Intl. Workshop on Principles of Diagnosis, Loch Awe, Scotland, pp , June 1999.
E. J. Manders, S. Narasimhan, G. Biswas, and P. J. Mosterman, ``A Combined Qualitative/Quantitative Approach for Fault Isolation in Continuous Dynamic Systems,’’ 4th Symposium on Fault Detection, Supervision and Safety for Technical Processes (Safeprocess 2000), Budapest, Hungary, pp , June 2000.
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53. Next Lecture 2 Hybrid Modeling and Diagnosis of Hybrid Systems
Combining Qualitative FDI with Quantitative parameter estimation methods for more informed and more refined diagnosis
Example Applications
Generation of Toolsuite
Simulation experiments
Run time system
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