Transcript FACT/SEC May 02 - Vanderbilt University

Lecture 1
Model-based Diagnosis of Continuous Systems
Gautam Biswas
Dept. of EECS and ISIS
Vanderbilt University
[email protected]
http://www.vuse.vanderbilt.edu/~biswas
Acknowledge
Pieter Mosterman, Sriram Narasimhan, Eric Manders
Supported by DARPA SEC, NASA-IS, NASA-ALS, & NSF-ITR
Copyright © Vanderbilt University, 2006.
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 faultadaptive 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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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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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 toolsuite 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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Model-based Tools and
System Development
Visual modeling tool for creating:
•Hybrid bond-graph models
•Timed failure propagation graph models
•Controller models (including reconfiguration)
Hybrid
Diagnostics
Active
Model
Failure Propagation
Diagnostics
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
Fault Detector
Hybrid Observer
Interface & Controllers
Controller
Models
Strategy
Models
Controller
Selector
Plant
Models
Reconfiguration
Manager
Run-time Platform (RTOS)
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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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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
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
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8
possible
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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Model-based FDI of Continuous
Systems (TRANSCEND)
u
y
Plant
(Process)
Statistical
Hypothesis
Testing
Extended Kalman filter
+
Observer
r
r

Fault
Detector
xˆ
ŷ
Observer
Model
Diagnosis
Model
State Space
Eqns
Symbol
Generator
rs
m
Hypothesis
Generation
Temporal Causal Graph (TCG)
10
f h, p
Hypothesis
Refinement
fr
Qualitative
7/17/2015
Bond Graphs
Overview
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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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Example of Reticulation
Suspension system of automobile
V
g
M
Mass of car
Suspension system parameters
B
K
v
Mass of tire
m
V0(t)
k
g
Tire stiffness
Input : Velocity at bottom of tire
(function of road conditions)
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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: (1series, 0 – parallel)
• Energy exchange between components through
mechanisms called bonds
A
e
f
B
Two associated variables with bond:
e: effort; f : flow; such that e  f = power
• Dynamic system behavior governed by
conservation of energy and continuity of power
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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
mechanisms
physical system
R
C, I
Se, Sf 0,1
uniform network–like
representation
domain independent
forces the modeler
to make
assumptions
explicit
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Building Bond Graphs:
Examples
Impacting Trains
Sf1
Two tank system
V1
Tank1
Engine
R12
F(t)
1
2
Snubber
R12
C2
F(t) V1
Se
1
R: R12 C: C2
0
R23
1
1
C: C1
Vg
R2
V2
1
0
V3
1
C3
R3
I1
Sf: Sf1
3
C2
C1
Lumped
parameter
Topological
Modeling
V3
Tank2
C1
R1
V2
0
R: R1
1
I2
I3
0
Compact Representation
across domains
R: R2
Two Tank System
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Bond Graphs
• Modeling non linear behaviors
– For linear systems, BG elements are constant
values
– Non linear systems
• parameter value = f (effort, flow, external variables)
Sf1
Two tank system
Sf: Sf1
Tank1
Tank2
p1
R: R12 C: C2
0
R: R1
1
0 p2
R: R2
C1
R1
R12
C2
R12  k 
R2
p1  p2
R12  k1  f12
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2
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Bond Graphs
Building Models for Diagnosis
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TRANSCEND Architecture
u
y
Plant
(Process)
Statistical
Hypothesis
Testing
Extended Kalman filter
+
Observer
r
r

Fault
Detector
xˆ
ŷ
Observer
Model
Diagnosis
Model
State Space
Eqns
Symbol
Generator
rs
m
Hypothesis
Generation
Temporal Causal Graph (TCG)
19
f h, p
Hypothesis
Refinement
fr
Qualitative
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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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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
e
f
B
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Causality for basic multiports
Note that a lot of the causal considerations are based on
algebraic relations
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Causality Assignment
Procedure
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Causality Assignment:
Example
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Causality Assignment:
Example 2
Sf1
C: C1
Tank1
Tank2
C1
C2
Sf: Sf1
0
R: R12 C: C2
1
0
R2
R12
R1
R: R1
R: R2
Two Tank System
Causality assignment important – for building
TemporalCausal Graphs (TCGs)
that are used for diagnosis from transients
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Generating State Space
Models
• Step 1: Augment bond graph by adding
1)Numbers to bonds
2)Reference power direction to each bond
3)A causal sense to each e,f variable of bond
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Generating State Space
Models
• Step 2: Identify state and input variables
x  f ( x , u)
f2  V (t )  e3  e4  V (t )  R3 . f 3  e5
L : I2

C : C5
S e : V (t )

R : R3
e5 
R : R6

27
V (t ) R3
1
 . f 2  .e5
I2
I2
I2
R3
1
1
. f 2  .e5  .E (t )
I2
I2
I2
1
1
1
.f 5
. f4 
. f6
C5
C5
C5
e
1
. f2  6
C5
C5 .R6
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Model-based Diagnosis
TRANSCEND approach
Fault Detection
+
Fault Isolation (Qualitative)
TRANSCEND architecture
u
y
Plant
(Process)
Statistical
Hypothesis
Testing
Extended Kalman filter
+
Observer
r
r

Fault
Detector
xˆ
ŷ
Observer
Model
Diagnosis
Model
State Space
Eqns
Symbol
Generator
rs
m
Hypothesis
Generation
Temporal Causal Graph (TCG)
29
f h, p
Hypothesis
Refinement
fr
Qualitative
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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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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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Example
Two-Tank System
f1
• Focus on process faults
Tank1
Tank2
– multiplicative
C1
Rb
1
• Parameterized Model
C2
R12
Rb2
Rb1
e2
f5
f3
C
R
C1
1
0
1
f8
• Fault profile
C2
5
4
– Possible faults: Bond Graph
component parameters: Sf, C1,
C2, Rb1, Rb2, R12
C
R12
2
Sf
e7
– abrupt change in parameter
value: increase or decrease
7
6
0
• Note – partial change
8
3
R
R
Rb1
Rb2
– not complete failure
Bond Graph Model
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Diagnosis from Transients
Abrupt Faults cause transients in observed measurements.
Goal: Isolate fault as quickly as possible after occurrence
of transient.
C1
C2
Two primary tasks:
Rb1
R12
1. Reliable detection of transient
2. Isolation of fault based on transient characteristics
f5 (flowrate through connecting pipe):
Faults: Rb1, Rb2, R12
Rb2
Faults: C1, C2
Rb 2
Discontinuity
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Two-Tank System
Diagnosis Model
f1
C
R
C1
Tank1
2
Tank2
C2
Rb
1
R12
f3
f3
-1
f1
f2
e7
f5
1
Rb1
1
C1 dt
e3
e6
-1
=
e2
f8
=
e4
1
e5
1
R12
f5
=
f6
=
1
f7
1
C2 dt
-1
f4
4
0
=
f8
Temporal Causal Graph
34
1
Rb 2
1
7
6
0
8
R
R
Rb1
Rb2
Bond Graph Model
e7
=
-1
C2
5
3
Rb2
Rb1
e2
1
1
Sf
C1
C
R12
e8
Important Characteristics:
Algebraic and temporal –
cause effect relations
Note the state loops – there
are three in this TCG
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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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Qualitative Transient Analysis
As order of
derivative increases
accuracy of match
improves
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
36
Matching the measured
magnitude and slope of
a signal to the signature
Measured
Signature
step
signal
0
+. . .
+-+-
1
+- . .
+-+-
-- ..
+-+--+-
3
--..
--+-
4
-?..
--+-
-+..
--+-+-.
2
5
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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 37
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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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Run-time System
Fault Detection
Faults
System

+
n
+
+ residual

Model

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
Kalman Filter
Residuals deviate from zero because of
– Noise (n)
– Modeling errors ()
–
Sensor inaccuracies ()
–
Faults
X
Separation
of effects
necessary!
!
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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:
1. Sign of deviation
2. Sign of first derivative
40
r
+–
t
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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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Z-test
Mean: info on the transient residual:
Variance: info on the nominal dynamics
Typical values:
N1=50; N2=5
VarDelay=150
k
N1
Variance
estimation
Decision making:
time
N2
VarDelay
Mean
estimation
  z  / N 2
No fault
Fault
otherwise
Fault detection
Z-test
Z-value distn.:
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)

Mapping the confidence level (1-) to z:
Z
/2 z42
z+ /2
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Symbol Generation
•
•
1st symbol:
sign of deviation after fault detection 
2nd symbol:
Is difference
sign of slope after fault detection
N3
significant?
r
Estimation of the ‘initial value’ 1
Fault detection
t
k
Decision:
k    2 k   1
Estimation of the ‘later value’  2 k 
 1
  z  

 N
3

 1
   z  

 N
3

otherwise
43
1 
symbol 
k 
1 
symbol k 
symbol 0
Additional Decision
If 1 & 2 have opposite
signs, then change is
discontinuous
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Slope Symbol Generation
Example
N3=1
0.5
1.5
detection time
Residual
1
0
0.5
0
0
20
40
60
80
100
120
-0.5
160 0
140
20
40
60
80
N3=10
100
120
140
160
20
40
60
80
N3=20
100
120
140
160
0.5
Choice of N3
•
•
•
0
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 …
44
-0.5
0
0.5
d
d+z + d
d-z+ d
+z + r /sqrt(N3)
-z+ r /sqrt(N3)
0
-0.5
0
20
40
60
80
100
120
140
160
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Detection, Symbol Generation
Complete
Onto Qualitative Fault Isolation
Generate Fault Hypotheses –
Back Propagation
f1
Tank1
Tank2
C1
Rb
1
Rb1
e2
C
R
C1
0
3
e7
1
-1
C2
5
4
f3
f8
C
R12
2
Sf
Rb2
f5
f3
1
C2
R12
f1
1
f2
1
Rb1
1
C1 dt
e3
=
e2
e6
-1
=
e4
1
R12
1
e5
f5
=
f6
=
1
1
C2 dt
f7
e7
7
6
0
-1
R
R
Rb2
e7
+
-
C2
f7 +
f8 f6 +
Rb2
e8 f5 +
=
f4
8
Rb1
=
-1
f8
1
Rb 2
e8
+
+
e7 R12
e5 +
-
e6 -
e7 -
C1 -
f3 -
e4 +
e2 +
f2 +
f4 -
Rb1
e3 -
e2
-
f5 -
Possible Faults: C2 - Rb2 + R12 - C1 - Rb1 +
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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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Progressive Monitoring
track system behavior after failure
• System Behavior Convolutes
the Predicted Transient at
Time of Failure
11 22
33 44
k
measured
0
1
2
0
+
+
3
4
5
+
+
+
+
0
-
6
+
-
55
t
6
k
-
– dynamically change the
signature: Justified by Taylor’s
series
signature1
0
1
2
0
+
+
+
+
+
-
3
+
+
-
k
0
signature2
+
-
+
no match!
match!
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Three Tank Results
F1
1
f2
1
C1dt
e2
-1
e5
-1
e1
=
=
e3
f3
1
e4
1
R12
=
f4
=
f5
1
f6
1
C2dt
e9
-1
=
e6
-1
=
e7
f7
1
e8
1
R23
=
f8
=
f9
=
1
1
C3dt
f10
e10
=
e11
1
Rb
f11
-1
Measured Variables are p1, p2 and f12
Faults Introduced Faults Identified Steps Taken
C1+
C1+
6
C2+
C2+
5
C3+
C3+
8
R12+
R12+
10
R23+
R23+, Rb+
14
Rb+
R23+, Rb+
21
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Analysis of Qualitative Fault
Signatures
Discriminatory Power of Qualitative Fault Signatures
1. Abrupt change – direction of abrupt change + direction of
change immediately following abrupt change,
(+,+), (+,-), (-,+), and (-,-)
2. 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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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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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. 554-565, Nov. 1999.
•
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. 155165, 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. 512-517, June 2000.
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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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