Transcript Document

Model-Based Diagnosis
of Hybrid Systems
Papers by:
Sriram Narasimhan and
Gautam Biswas
Presented by: John Ramirez
Introduction
Actuators
Plant
Discrete Signals
Supervisory
controller
Sensors
Sensor values
 Modern systems are
complex, and include
supervisory control that
switches modes of
behavior.
 The controller is a software
program and is not tightly
meshed with the
continuous plant dynamics.
Introduction
 The continuous
dynamics of the plant
are defined by
differential and
algebraic equations.

x(t )  f ( x(t ), u (t ), q(t ))
y (t )  g ( x(t ), u (t ), q(t )),t  0
q(t) is the discrete model
Fault Detection and Isolation
(FDI)
• The goal of this presentation is to briefly overview
the study of FDI in hybrid systems with supervisory
controllers.
• System faults may be component, actuator, sensor,
and controller faults. (We do not deal with the later)
• The methodology we will cover combines
qualitative and quantitative reasoning techniques to
perform parameterized fault isolation of plant
component faults.
Modeling for Diagnosis
Controller Model
Plant Model
Modeling for Diagnosis
 Controller Model



The primary model of the controller is
implemented as a finite state machine (FSM).
States of the FSM correspond to the states of the
controller, which in turn define modes of the
physical plant(q(t)).
The Transitions determine the conditions for
switching states.
Modeling for Diagnosis
Controller Model
3
t2
1
t1
t4
2
t3
5
t10
8
t5
t7
4
t6
6
t11
t9
t8
7
10
Controller Model for 3 tank system
Flow source 1
Three Tank
system
Flow source 2
= Valve
C = capacitance
R3
Tank
2
(C2)
R1
R2
R5
Tank
2
(C2)
9
R4
R = resistance
Tank
1
(C1)
R6
Modeling for Diagnosis
 Plant Model



Hybrid Bond Graph Models (HBG).
State equations and temporal causal graph (TCG)
can be systematically derived from the bond
graph representation of the system.
State equations along with the TCG constitute
our diagnosis models.
Methodology for Hybrid
Diagnosis
 Hybrid observer: follows the continuous dynamics
of the plant and identifies discrete mode changes.
 Fault detection mechanism: signals a fault when the
observer cannot compensate for differences
between observed and expected behavior.
 Fault isolation mechanism: generates candidate
faults and refines them with the hybrid model and
measurement from the system.
Methodology for Hybrid
Diagnosis
u
System
y
r
Hybrid
models
Observer ^
y
and mode
detector
Diagnosis
models
Fault
isolation
Fault
detection
Fault Hypotheses
Diagnosis System Architecture
The following information
is assumed to be available
to all modules:
-HBG
-FSA
-FSM
- A = all possible
autonomous events in the
system
-U = inputs
-Y = system outputs
-Parameters nominal
Methodology for Hybrid
Diagnosis
Algorithm 1:Diagnosis Module
MODULE DIAGNOSE(Minitial,Xinitial)
// Observe the system until a fault is detected
<StackM, Yestimated>=OBSERVER(Minitial,Xinitial);
//Convert the quantitative residuals to qualitative values
QualResidualcurrent = SIGNAL_TO_SYMBOL(Y,Yestimated);
//Back propagate across modes to identify fault candidates
BackHorizon=2;
Listcandidates=HYBRID_BACK_PROP(StackM,QualResidualcurrent,BackHorizon);
//Forward propagete across modes to isolate the fault
Listcandidates=HYBRID_FAULT_OBSERVER(Listcandidates,Yestimated);
END DIAGNOSE
Hybrid Diagnosis Problem
Piecewise linear hybrid dynamical systems
Presence of fault invalidates
Tracked Trajectory
Actual Trajectory
Fault
Occurs
Mode 1
Mode 2
T1
Mode 3
T2
Time
Line
Mode 5
Mode 4
Mode 7
Mode 6
T3
T4
T5
Roll Back to find fault hypotheses
Known Controlled Transition
Hypothesized Autonomous
Transition
Possible current
modes
Hypothesized
intermediate modes
Hypothesized
fault mode
Roll Forward to confirm fault hypotheses
T6
tracked mode trajectory
Fault
Detected
Fault Hypothesis: <mode,parameter>
Catch up to current
system mode to verify
hypotheses against
measurements
Note: Controller transitions
known
Autonomous transitions have
to be hypothesized
Fault Isolation
Background
 The type of plant model employed
determines the scheme to be employed.
 Traditional schemes for the continuous
domain use structured and directional
residual approaches.
 Extending these continuous methodologies to
hybrid systems becomes intractable.
Fault Isolation
 The approach we will follow involves
hypotheses generation and hypotheses
refinement.
 Qualitative approach for hypotheses
generation.
 Qualitative-quantitative combined approach
for hypotheses refinement.
Fault Isolation
Hypotheses Generation
 For initial hypotheses generation we have to
back propagate across modes.

The assumption that the controller model is
correct implies that the observer predicted the
correct mode sequence till the fault occurred.
Therefore, the mode in which the fault occurred
must be in the predicted trajectory of the
observer.
Hypotheses Generation
TCG generation
•Effort and flow variables are vertices
•Relation between variables as directed edges
•=implies that two variables associated with the
edge take on equal values, 1 implies direct
proportionality,-1 implies inverse proportionality.
•Edge associated with component represents
the component’s constituent relation.
Hypotheses Generation
Algorithm 2:Hybrid Back Propagation
MODULE HYBRID_BACK_PROP(StackM, QualRi, BackHorizon)
//Generate candidates in each mode in the mode trajectory.
<Mcurrent, Timecurrent>=Pop(StackM);
TCGcurrent=GET_TCG(HBG, Mcurrent)
//Back propagate in selected mode for candidates in the mode
Fcurrent=CONTINUOUS_BACK_PROP(TCGcurrent,QualRi);
Add(Listcandidates,<Mcurrent,Timecurrent,Fcurrent>);
Count=0;
//Go back in the mode horizon upto BackHorizon number of nodes
While(Count<BackHorizon)
//Select next mode in mode trajectory and calculate TCG
<Mnext, Timenext>=Pop(StackM);
TCGnext, GET_TCG(HBG, Mnext);
// Propagate qualitative deviations across modes
QualRnext=BACK_PROP_ACROSS_MODES(Mcurrent, Mnext, QualRi)
//Back propagate in selected mode for candidates in the mode
Fnext=CONTINUOUS_BACK_PROP(TCGnext, QualRnext);
Add(Listcandidates,<Mnext,Timenext,Fnext,1>);
End While
Return(Listcandidates)
END MODULE
Roll Back Process
•Qualitative Hypotheses Generation
• Back propagate through TCG in current mode to identify candidates
• Back propagate across mode transitions using transition conditions (need to account for
reset conditions, and change in plant configuration – invert qualitatively)
• Repeat same process for previous modes to identify more candidates
Transition
Fault Occurred
Fault Detected
System Autonomous Transition
- Tank 1 Pressure
- Tank 2 Pressure
- Tank 3 Pressure
Fault Isolation
Hypotheses Refinement
 First apply a qualitative forward propagation for
each hypothesized fault candidate.


To take into account mode changes, all possible modes
changes from the current mode are hypothesized.
A candidate is dropped when the predictions do not
match the observations across all of the hypothesized
modes
 Apply a quantitative parameter estimation on
remaining candidates.

This approach works within a single continuous mode.
Hybrid Diagnosis Problem
Piecewise linear hybrid dynamical systems
Presence of fault invalidates
Tracked Trajectory
Actual Trajectory
Fault
Occurs
Mode 1
Mode 2
T1
Mode 3
T2
Time
Line
Mode 5
Mode 4
Mode 7
Mode 6
T3
T4
T5
Roll Back to find fault hypotheses
Known Controlled Transition
Hypothesized Autonomous
Transition
Possible current
modes
Hypothesized
intermediate modes
Hypothesized
fault mode
Roll Forward to confirm fault hypotheses
T6
tracked mode trajectory
Fault
Detected
Fault Hypothesis: <mode,parameter>
Catch up to current
system mode to verify
hypotheses against
measurements
Note: Controller transitions
known
Autonomous transitions have
to be hypothesized
Quick Roll Forward
• Goal: Get to current mode, so parameter estimation can be
applied to refine faults and identify fault magnitude
•
•
Lemma 2: Sequence of k mode transitions in any order
drives the system to the same final model
Requires tracking of transients by progressive monitoring in
continuous regions of space. Taylor series expansion defines
qualitative fault signatures. Residual r(t) after fault can be
described as:
(t  t0 )
(t  t0 ) 2
(t  t0 ) k
k
r (t )  r (t0 )  r (t0 )
 r (t0 )
 ...  r (t0 )
 Rk (t )
1!
2!
k!
Fault signature: qualitative form of derivatives:
r(t ), r(t ),....,r (t )
k
0
0
Qualitative form of r k (t0 )   (above/ below normal),0 (normal)
0
Quick Roll Forward
•
In continuous case, mismatch implies fault hypothesis is not consistent. However, in
hybrid tracking, it may imply that we are not in the right mode. We need to identify
the current mode (roll forward)
•
All controlled transitions are known, but we have to hypothesize autonomous
transitions since observer can no longer predict them correctly
•
Use fault signatures to hypothesize mode transitions
Transition
Fault Occurred
Fault Detected
System Autonomous Transition
- Tank 1 Pressure
- Tank 2 Pressure
- Tank 3 Pressure
Parameter Estimation (Real Time)
 Derive transfer function model in current mode
with only one unknown (fault parameter)
 Initiate fault observer filter for each fault
hypothesis

least squares estimator for parameter estimation
 Test for convergence identifies true fault candidate
Least Square Estimation from
IOE
Q(t )    Q(t  1)   (t )  (t )
T
ˆy (t )   T (t )  (t  1)
eˆ(t )  y (t )  yˆ (t )
 (t )   (t  1)  Q (t )  (t )  eˆ(t )
Q is t hecovariance,  is t heforget t ingfact or,
 is a mat rixmade of u' s, y's and 0's,
 is t heparamet ervect or(g's and h's in IOE),
1
yˆ is t heest imat edout put ,eˆ is t hepredict ionerror
Parameter Estimation Example
Plot of prediction error
Quantitative Parameter
Estimation: Issues
• Deriving the simplified one unknown parameter equation
for least square estimator
• Convergence to local minima – need good initial estimates
• Need for persistent excitation in input – mitigated to some
extent by reducing it to a one parameter estimation
problem
• Measurement noise leads to biased estimates – need to
apply more sophisticated techniques: IVM methods
Observation: What is good for qualitative FDI is not
always good for quantitative identification using least
squares methods
Summary
 Model for Diagnosis

Controller Model


FSM
Plant Model

HBG
 Fault Isolation

Hypotheses Generation


TCG
Hypotheses Refinement

Parameter Estimation
Conclusion
 By having the supervisory controller model and
assuming that our model is correct, we do not have
to make the assumption that faults are detected in
the mode in which they occur, and we still are able
to avoid the intractability problem.
 Combination of qualitative + quantitative
approaches suitable for online diagnosis
 Approach different from discrete-event approaches
of Lunze and Sampath