Diagnosis and FTC by Prof. Blanke [pdf] - NTNU

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Fault-tolerant Control
Lecturer
Mogens Blanke
Professor II at CeSOS –
NTNU
Professor of Automatic
Control at Ørsted rsted-DTU DTU
Research interests:
•Autonomous Control Systems
•Fault-tolerant and Dependable Control
•Satellite Attitude Control
•Marine Control Systems
•System Modelling and Identification
•Diesel Engine Control
Reference for todays lecture:
M.Blanke, M. Kinnaert, J. Lunze and M.
Staroswiecki
Diagnosis and Fault-Tolerant Control
Springer, June 2003
• Chr. Rovsing/European Space
Agency 1974-75: Systems
Analyst
• DTU 76-84: Ass + Assoc Prof.
• Søren T. Lyngsø 85-90,
Head of Marine Division
• Aalborg University
1990-2000 Professor of
Automatic Control
• DTU 2000 Professor of
Automatic Control
• NTNU 2005 Professor II
Mogens Blanke – Spring 2006
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Scope of lecture – Fault-tolerant Control
• Fault-tolerant Control in Context
• Analysis based on System Structure
• Fault Diagnosis – detection and isolation
• Fault-tolerant control:
– Sensor fault, Actuator fault
• An application
Fault-tolerant design
Mogens Blanke – Spring 2006
Fault-tolerant systems (and control)
Created by the need for safety, reliability and availability.
Touches on fundamental properties of controlled systems and
the control architectures.
Draws on results from many areas in automatic control
Diagnosis
The essential means to determine whether faults have
occurred
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Structure of Plant + Controller
The standard
control problem
A system with control is a hybrid entity. The system can be
changed by events, some of which are faults. We may change
the controller and/or the system by action of a supervisor.
A Hybrid System
A bathtub is a hybrid system.
Dynamic relation between
input flow q, height h and area A( h)
are
1
ht ()
= qt ()
Ah ( )
Holds only until
the edge is reached:
ht (
) = 0 for h= hedge
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q(t)
h(t)
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A Hybrid System
Dynamic relation between
input flow q, height h and area A( h)
are
1
ht (
) = qt ( ) for 0 ≤ h< hedge
Ah ( )
ht (
) = 0 for h≥ h
edge
Relation to Robust and
Adaptive Control
q(t)
h(t)
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Fault-tolerant Control
Fault-tolerance:
Ability to maintain control
objectives, despite the
occurrence of a fault.
A degradation of control
performance may be
accepted.
Obtained through fault
accommodation or system
reconfiguration.
Standard control problem - example
Objective:
Obtain bandwidth > 0.2 rad/s, loop damping > 0.7
Mogens Blanke – Spring 2006
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kkb p t 2
ψ () s = s ψ ref () s
kb t + h kkb
1 p t
1+
+ 2
s s
ψ () s =
kkb
ψ () s
p t
2
s + ( kb t + h1) s+ kpkb t
Compare with standard form
ψ () s =
s
ω
s
ψ ref () s
2
b
2 2
+ 2ςωb
+ ωb
ref
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Fault-accommodation or reconfiguration
Change in controller
parameters or
structure to avoid the
consequences of a
fault.
Accommodation:
Input-output between
controller and plant is
unchanged.
Reconfiguration:
Input-output between
controller and plant is
changed.
Handling of sensor fault -accommodation
• Fault handling: a sensor fault (gain is 0.1) accommodated by
differentiating term in position feedback.
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Handling of fault - reconfiguration
• Fault reconfiguration: a sensor failure in inner loop.
• Switch to differentiating control when fault is diagnosed
Handling of sensor fault by output
estimation
icom
power
drive
+
Speed control
im
kt
kqη
+
Ql
1
Itot
Fault:
[0.0 1.0]
kobs
∫
-
ˆn
n
1
N ∫
+
+
nm
-
Position control
θm X
+
1
N ∫
gear
kp
ˆ
θ
-
θ
+
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θref
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Handling of sensor faults by reconfiguration
Reconfiguration: Failed sensor
measurement is replaced by an
estimate, which is used in the
feedback loop
y = − + ˆ
k (1 S( fk)) yk S( fk) y( fk)
S is a diagonal matrix S ij=0(
i ≠ j)
Skk
= 1 f ,0 otherwise
k
Condition for observer k to exist
( A,[ c1,.., ck− 1, ck+ 1,..,
cm]
) is observable
where c i is a column of C
If not fully observable, the unobservable
subsystem must at least be stable
Model-matching state-feedback
O H A B
O H A B
K
+ J H A H 2 = J
E = ? I A @ F I O I J A
4 A @ A I EC A @
+ J H A H
E = ? I A @ F I O I J A
K
. = K J O
2 = J
O
O
u
u
u
u
Plant
estimator 1
est(y) | y , y , ...
2 3
estimator 2
est(y) | y 1 , y 3 , ...
estimator 3
est(y) | y 1 , y 2 , ...
y m =[y 1 ,y 2 ,y 3 ]
y est
y est
y est
Mogens Blanke – Spring 2006
The nominal (no fault) system
in closed state feedback loop:
xt ()
= ( A- BK) xt ()
yt () = Cxt ()
after the fault occurs,
xt ()
= Axt f ()- But f ()
yt () = Cfxt ()
with new state feeedback controller:
xt ()
= ( Af - BK f f ) xt ()
yt () = C x(
t)
f
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Model-matching state-feedback (2)
Ideal if we could obtain A- BK = A f - B fK f
This is only rarely possible (requires redundant actuators).
Consider the relaxed condition:
∃K f ⊂{ Kstab} :min( A- BK) − ( A f - B fK f ) ?
If the pseudo-inverse of B exists
f
T
−1
T
BK f f = A f −( A - BK) ⇒ K f = ( B fBf ) B f ( A f − A + BK)
Example on handling of faults
Mogens Blanke – Spring 2006
• Simple means needed to handle simple control loop
faults
• Applied common sense thus far, applied systematic
common sense, however.
•More challenge to come, though
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The real thing
By courtesy from C. Thybo
A stop using the emergency brake subjects the
structure to large stresses. An x-ray
certification is required after each emergency
stop. Graceful degradation pays off - a potential
application for fault-tolerant control.
Handling of faults – Induction Motor
• Accommodation: high
performance controller that
needs several sensor inputs
is replaced (at run time) with
one that can run with
remaining sensors but with
reduced performance
Stator
current
i s
θ
i sa
i sb
ε
i sq
i sd
ρ
Rotor
flux
ω mr
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Rotor
angle
em = z ω ω p m
Stator axis
• Flux vector control - high
performance, measurement needed
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Handling of faults - 2
• Accommodation: high
performance controller that
needs several sensor inputs
is replaced (at run time) with
one that can run with
remaining sensors but with
reduced performance
ω set
+
Load
-
K
ωactual
• Scalar control - low
performance, no sensors
Induction Motor Fault-tolerant Control
• Implementation of fault-tolerant control - time to
reconfigure is crucial.
• Alternative motor control algorithms are not globally
stable - large field angle deviations are not allowed.
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• The likelihood of successful reconfiguration is crucial -
the concept of coverage - reliability issues
• Coverage = Prob.(System recovers|Fault occurs)
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Fault-tolerant steering by wire
Steering
input
system
Power
supply
Computer
Sensor
system
Drive
system
Induction
motor
• Requirements to electrical
steering system:
• Must be able to steer until
vehical is stopped despite
any single fault
Wheel
Fault-tolerant steering by wire
Steering
input
system
Power
supply
Computer
Sensor
system
Drive
system
Induction
motor
Wheel
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• Function blocks and
electrical details:
• AC motor is driven by
inverter (3-phase switching)
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Properties of possible architectures
A behaviour is the relation between variables
(input and output) of a function block:
c 1: g1( u, y)
= 0
3
example 1: y( t) − 2 u( t) u( t)
= 0
⎧dx()
t
⎪ =− 2 xt ( ) + 4 ut ( ); x(0) = xo
example 2: ⎨ dt
⎩
⎪ yt () = xt () −0.01()
ut
The
service offered
by a function block is to
provide a behaviour
between input and output.
S : g( u , y ) = 0
1 1 1 1
Properties of possible architectures
The behaviour of a system with an
architecture A is SA
With the two examples:
( upper )
S = S ∩S ∩S
A
1 2 3
All components must provide the service
to give the required service from u to y
( )
1 3
( lower ) ( ft )
( d )
SA= S1 ∩ S2 ∪S2 ∩S3
We obtain fault-tolerant behaviour if
( d )
S3
is fail-operational (can work despite any
( ft )
single fault) and S1
is fault-tolerant: we obtain
*
g(u,y) 1 1 ≈ g(u1,y 1)
but perhabs degraded
(lower performance)
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A fault-tolerant architecture (patented)
• Redundant hardware is
necessary but here is a
cheap version: split
power stage in two,
wind AC motor with
dual winding.
The FDI - FTC chain
• What FDI provides
• Fault detection
• Fault isolation
• Evaluation to get confirmed
hypothesis
• Fault estimation
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• What FTC shall do
• Assess estimated structure
• Is diagnosis unambiguous,
– determine faultaccommodation
action
• If not,
– react according to highest
severity
– accommodate both/all
possible scenarios
• Reconfigure in time
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Diagnosis and Fault-tolerant Control
Chapter 1, pp 1-23
• Faults and fault-tolerance:
• A fault is a deviation of the system structure or system
parameters from the nominal situation (actuator blocks, loss of
sensor, disconnection of system component)
• A failure is an event that prevents a system to perform a
required function.
• Parameters are alowed to vary within their specified range
System behaviour
• System behaviour is a subset of all possible combinations of
input and output signals.
Let a static system be
yt () = kut s ()
The behaviour is the set of all I/O pairs is given by
B
= { ( uy , )| y= ku s }
If the system is dynamic, B⊂ U× Y
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System behaviour: normal
• System behaviour is in
the green area.
• Point A represents an
observation that may
occur for the given
system.
• Point C represents a pair
that is not consistent
with system dynamics.
System behaviour: norman or with a fault
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• A specific fault change
the behaviour from the
green to the yellow
area.
• If a common input is
applied to the system,
the normal and faulty
cases would respond
differently:
e.g. points A and C.
The normal and the faulty system can be distinguished unless the
observation is in the intersection (point B) between the behaviours.
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Safety versus fault-tolerance
Safety versus fault-tolerance
Green region:
required
performance
Yellow region:
degraded
performance
Orange region:
unacceptable
performance
Mogens Blanke – Spring 2006
Green region:
required
performance
Yellow region:
degraded
performance
Orange region:
unacceptable
performance
Red region:
danger
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Models of dynamical systems
• Notation for nonlinear system with faults
The general, nonlinear case reads:
dx() t
= gx ( ( t), u( t), d( t), f( t)),
x(0) = x0
dt
y() t = h( x(), t u(), t d(),()) t f t
where x is state vector, u is input vector,
d is disturbance vector and f(t)
a fault vector.
Models of dynamical systems
Mogens Blanke – Spring 2006
We obtain a linar model in a point of operation
xudf , , , by Taylor expansion around x = x- x
dx dx dx ∂g∂ ∂ ∂
= + = +
g
+
g
+ g
gxudf ( , , , ) x u d+ f,
dt dt dt ∂x∂u∂d∂f x(0) = x0...
∂h ∂h ∂h ∂
= + = ( , , , ) + + + h
y y y h x u d f x u d + f
∂x ∂u ∂d ∂f
or
x = Ax + Bu + E +
xd Fxf y = Cx + Du + E +
yd Fyf In the sequel, we ommit the x notation and just write x
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Models of dynamical systems
Linearized system without faults
The general, linear case reads, for A, BCD , , ... time invariant:
dx() t
= Ax() t + Bu() t + Exd(), t x(0) = x0
dt
y() t = Cx() t + Du() t + E d()
t
Linearized system with additive faults
y
dx() t
= Ax() t + Bu() t + Fxf() t + Exd(), t x(0) = x
dt
y() t = Cx() t + Du() t + F f() t + E d()
t
y y
Ship - running example
δ
rudder
angle
b
+
+
∫
ω3
δ
steering
characteristic
wave
disturbance
ω3
f ω
ωw
ω3m
rate
measurement
∫
f ψ
0
ψ
ψm
heading
measurement
Ship steering. Input is rudder angle. Output is ship’s
heading. Measured variables are turn rate and
heading angle.
+
+
+
+
+
+
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Model of normal behavior of ship
3 ( )
c1: ω3 = b h1ω3+ h3ω3 + bδ
c2:
ψ= ω3+ ω3w
c3
: ψm= ψ
c4:
ω3m ω3+ ω3w
c5
:
d1
:
δm= ψ
=
δ
dψ
dt
δ
d2
ω3
=
dω3
dt
rudder
angle
b
+
+
∫
ω3
δ
steering
characteristic
wave
disturbance
Linearized model with sensor faults
δ
rudder
angle
ω3 = b( δ + h1ω3)
ψ= ω3+ ω3w
ψm= ψ + fψ
linearized at ω3=0
ω3m = ω3+ ω3w
+ fω
δ = δ
m
b
+
+
∫
ω3
δ
steering
characteristic
ω3
ω3
wave
disturbance
+
+
f ω
ωw
+
ωw
+
+
+
ω3m
rate
measurement
∫
f ψ
ψ
ψm
heading
measurement
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∫
+
fω +
ω3m
fψ rate
measurement
+
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+
ψ
+
ψm
heading
measurement
+
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Example on requirements to diagnosis
• Requirements for motion control
• f rate (t) t det < 1deg
• f angle (t) t det < 5 deg*s
• Obtained in theory and practice
• Accommodation of sensor faults on the fly is possible
Analysis based on structure
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Aim and scope
This is fun
Design
Sequence
Overview
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• We need a tool to cope with complexity of systems
• Desire: Determine principal properties from the
structure of a system - without detailed control theory
calculations
• Principal questions we have:
• Can we diagnose a fault ?
• Can a system recover from a fault?
• Can we monitor the system?
• Method: graph based analysis of system structure
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Digraph for linear system
Example 5. 2
d
c1: x1 = x1
dt
c2 : x1 = ax2
d
c3 : x2 = x2
dt
c : x = bx + cx + du
4 2 1 2
Bipartite graph
Graph represented as
incidence matrix:
u x1 x2 x1 x2
c1
0 1 0 1 0
c2
0 0 1 1 0
c3
0 0 1 0 1
c 1 1 1 0 1
4
Graph represented as the
incidence matrix:
u x1 x2 x1 x2
c1
0 1 0 1 0
c2
0 0 1 1 0
c3
0 0 1 0 1
c 1 1 1 0 1
4
Or drawn as a bipartite graph:
N N N
Mogens Blanke – Spring 2006
N K
? ? ? ! ? "
"A simple graph is one with no parallel edges and loops.
A bipartite graph is a simple graph in which the set of
vertices can be partitioned into two sets such that every
edge is between a vertex in C and a vertex in Z: G= ( C, Z, E)
."
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Structure Graph
Structural analysis
A graph-based technique where
principal relations between variables
express the system’s properties.
Measured and controlled quantities
in the system are related to variables
through functional relations, which
need not be explicitly stated. The
user specifies a list of these relations
that together describe the
functionality of the system
considered. A list of such variables
and functional relations constitute the
system’s structure graph.
Control input
Unknown
Constraints
Measured
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Paradigm: A Fault is violation of a Constraint
Faults
Normal operation means all
functional relations are intact for
the system. Should faults occur,
one or more functional relations
cease to be valid. In the structure
graph, one or more nodes of the
graph will disappear when a fault
occurs.
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Example 5.3: tank system
F = H= A JA H H
D
K J
+ JH
= C HEJD
O J
A L A I A I H
D
6 =
G E J
Single tank system example 5.3
Tank c 1: h(t) = q i(
t) − qo( t)
Input valve c 2 : q() i t = αu()
t
Output pipe c 3 : q o(
t) = k h( t)
Level sensor c 4 : yt ( ) = ht ( )
Control algorithm c 5 : ut ( ) =
⎧1
if y( t) < h0
⎨
⎩0
otherwise
dh() t
differential constraint: c6: h( t)
=
dt
F = H= A JA H H
D
K J
+ JH
= C HEJD
O J
A L A I A I H
G J
D
Mogens Blanke – Spring 2006
6 =
G E J
G J
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Single tank system example 5.3
c 1 : ht ()
= qi() t −qo()
t
c 2 : qi( t) = αu(
t)
c 3 : qo() t = k h() t
c 4 : yt () = ht ()
c 5 :
⎧1
if y( t) < h0
ut ( ) = ⎨
⎩0
otherwise
c6: dh() t
h( t)
=
dt
c
c
c
c
c
c
1
2
3
4
5
6
u y h h qi q
0 0 0 1 1 1
1 0 0 0 1 0
0 0 1 0 0 1
0 1 1 0 0 0
0 0 1 1 0 0
Ex. 5.3: structure graph of tank system
c
c
c
c
c
c
1
2
3
4
5
6
u y h h qi q
0 0 0 1 1 1
1 0 0 0 1 0
0 0 1 0 0 1
0 1 1 0 0 0
0 0 1 1 0 0
0
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Every column in the incidence matrix corresponds to a circle vertex.
Every row corresponds to a bar vertex in the bipartite graph.
0
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Example 5.3: controlled tank
?
?
K
G E
D
? $
G
D
O
Structure graph with controller added: c5
Example 5.3: reduction of graph
? !
? "
D
H
? #
Mogens Blanke – Spring 2006
Seperate variables Z into known (K)
and unknown (X) variables
Z = K ∪ X.
Separate C into C = Ck ∪Cx.
Delete rows with constraints between known variables
Delete columns with known variables ⇒
Decompose graph (C, Z,E) into
(C ,Q(C )) and (C ,Z)
k k x
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Matching
Definition 5.1: ( Matching)
∀e1, e2∈M : e1 ≠e2 ⇒ pC( e1) ≠ pC( e2) ∧ pZ( e1) ≠ pZ( e2)
Any two edges in M have no common node - neither in Z nor in C.
Matching on the single tank
c4→h c6→h c3 → q0
c2→qi c → zero
1
c4→h c6→h c3 → q0
c1→qi c → zero
2
c
c
c
c
c
c
1
2
3
4
5
6
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u y h h qi q
0 0 0 1 1 1
1 0 0 0 1 0
0 0 1 0 0 1
0 1 1 0 0 0
0 0 1 1 0 0
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0
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Non invertible constraints
=
N N
?
N
N N
?
> ?
N ? N
N N
Matching – the ranking algorithm
Given: incidence matrix or structure
graph
1. Mark all known variables, i = 0.
2. Find all constraints with exactly one unmarked variable.
Associate rank i with these constraints.
Mark these constraints
and the associated variable.
3. If there are unmarked constraints whose variables are
all marked, mark them and connect them with the pseudovariable
zero.
4. set i = i+1
5. if there are unmarked variables or constraints, continue
with step 2.
Result: Ranking of the constraints.
?
N
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Purpose and scope of part 2
• Matching – a graph method to find the nonlinear
parity relations that can later be used in diagnosis.
• Define maximal and complete matching
• Investigate matching on an oriented graph
• Discuss causality
• Dealing with algebraic constraints
• Dealing with loops in graphs
Complete matching
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A matching M is a subset of edges such that
no edge have common node (neither in C nor in X).
Let M be number of edges
in M, then M ≤
min
C X
A matching is complete in
X if M = X.
It is complete in C if M = C .
Figure: One incomplete and
two complete matchings in X
for the tank example.
( , )
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Differential and integral constraints
(repeat)
Differential constraint
dh
c6: h=
dt
Integral constraint
c : h( t) = h( τ) dτ + h(0)
6
t
∫
0
Obvious assumptions
(5.2) All constraints in C are compatible
The constraints provide a solution to the
behaviour of a physical system.
(5.3) All constraints in C are independent
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Design of Parity Equations (5.6.3)
Analytic redundancy relation (or parity equations) are
subgraphs of the structure graph.
They are formed from unmatched constraints using the
complete matchings of the unknown variables we make
on the reduced structure graph of a system.
SaTool - constraint editor
input
Measured
Unknown
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Create
constraints
Constraints
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SaTool – A tool for Structural Analysis
Features
• Matlab ® based (version 7.0 or later)
• Graphical representation of structure as a graph
• Easy point and click manipulation of system structure
• Matching by ranking
• Backtracing
– Recursive algorithm
Matlab ® is a trademark of the Math Works Inc. USA.
Results
Mogens Blanke – Spring 2006
• Detectability
– Which faults can be detected with the found parity relations
• Isolability
– Which faults can be isolated with the found parity relations
• Parity relations
– Which relations exists
• Controllability
– Can I reach the unknown variables from the input variables
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Investigate the effect of a structural fault
• Faults can be modeled by disabling nodes
– Right click on a node to disable it
– Perform a new analysis
– Or analyze consequences of all possible violation of
constraints
• The result shows if the
– system can be reconfigured in case of faults
– Shows which further faults can be diagnosed for the faulty
system
– Showh witch secondary faults can be isolated for the faulty
system
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• Salient
feature:
• Run
analysis
on system
with fault
• Investigate
the new
result
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SaTool – A tool for Structural Analysis
Features
• Matlab ® based (version 6.5 or later)
• Graphical representation of structure as a graph
• Easy point and click manipulation of system structure
• Matching by ranking
• Backtracing
– Recursive algorithm
Matlab ® is a trademark of the Math Works Inc. USA.
SaTool
Mogens Blanke – Spring 2006
• SaTool – a rapid prototyping tool now being tested by
colleagues and industry.
• A beta version is available (GNU open source), please
contact mb@oersted.dtu.dk
• Uploaded to participants
• Work is in progress in the area of autonomous robotics
• Also a good subject for your MSc thesis !
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Case: ship station-keeping control
• Position control is required within a specified circle to
enable investigation of an unknown object
• Ship has two shafts and a
bow thruster for control
Constraints – forces from actuators
( )
( )
( )
a1: θ1
= gpu1 pitch and shaft speed on starboard shaft
a2 : θ2
= gpu2 pitch and shaft speed on port shaft
a3 : n3 = gbu3 angular velocity on bow thruster
b1:
y1 = θ1
measurement of SB pitch and rpm
b2
: y2 = θ2
measurement of port pitch and rpm
b : y = n
measurement of BT rpm
3 3 3
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Constraints – forces from actuators
c
N =− l c ( θ ) + l c ( θ ) + l c ( n ) torque from actuators
and wind
c : T c ( n ) g ( ) total sway force
c : T c ( ) c ( ) g ( ) total surge force
1 :
z y T 1 y T
N
+ gw
( vw)
2 x b 3
2 y = b 3 +
Y
w vw
3 x = T θ1 + T θ2 +
X
w vw
Constraints – dynamics and kinematics
c4: −1⎡Tx⎤ −1
v= M ⎢ ⎥−A
( ψ ) vc
⎣Ty⎦ linear acc. of body
c5
: p= A( ψ ) v+ vc
geographical position
c6: −1
ψ = I N angular acc. of body
d1
:
d
v= v
dt
differential
d2
:
d
p= p
dt
differential
d3
:
d d
ψ = ( ψ)
dt dt
double differention
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Constraints - measurements
Variables
m1
: vm= v velocity - inertial system
m2 : h1
= ψ heading - gyro 1
m3 : h2
= ψ heading - gyro 2
m4
: p1 = p position - GPS 1
m5
: p2 = p position - GPS 2
m6
: cm = vc
sea current - log&imu
m : w = v wind - anomometer
7
m w
Seperate variables into known Kand unknown X.
Seperate known vars. further into input and measured:
K = Ki ∪ Km
Unknown : X =
⎧θ1, θ2,
n3,
⎫
⎨ ⎬
⎩Nz, Ty, Tx,
ψ , vpv , , c, vw⎭
Input : Ki = { u1, u2, u3}
Measured : Km
=
⎧y1, y2,
y3,
⎫
⎨ ⎬
⎩h1, h2,
vm, p1, p2, cm, wm⎭
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The Constraint Editor in SaTool
input
Measured
Unknown
Structure Graph
A graph constists of an
ordered set of nodes and
vertices
Let variables and
constraints be nodes
Let vertices show which
variables are used by
constraints
The structure graph is bipartioned:
a variable is
always connected to
constraint(s), a constraint
always to variable(s)
Create
constraints
Constraints
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Matching of a structure graph
Matching of a structure graph means to relate
unknown variables to known ones through
constraints (solve for the unknown variables)
A matching is complete if all unknown variables are
matched
Constraints that are not used by a complete matching
(unmatched constraints) areusedto test the
correctness of the solution
These are parity relations of the problem and are
used for diagnosis
Mogens Blanke – Spring 2006
Unmatched constraints are parity relations
• Unmatched
constraints
constitute parity
relations
• Select one (shown
by white color)
• Trace the matching
backwards (yellow)
to get an analytic
expression for the
parity relation
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Parity relations (normal operation)
Autogenerated
parity
relations
r1
= a1( u1, b1( y1))
r2
= a2( u2, b2( y2))
r3 = a3( u3, b3( y3))
r4
=
c4( c2( b3( y3), m7( wm
)), c3( b1( y1), b2( y2),...
m7( wm)), d1( m1( vm)), m3( h2), m6(
cm))
r5 = c5( m3( h2), m6( cm), d2( m5( p2)), m1( vm))
r6
=
c6( c1( b1( y1), b2( y2), b3( y3), m7(
wm
)) ,...
d3( m3( h2)))
r7 = m2( h1, m3( h2))
r = m ( p , m ( p ))
8 4 1 5 2
Auto-generated parity relations
Autogenerated
parity
relations in
”clear”
equation
language
8 2 1
Mogens Blanke – Spring 2006
r1 = gp( u1)
− y1
r2 = gp( u2)
−y2
r3 = gp( u3) − y3
r4 =
X
−1⎡c ( 1) ( 2)
( ) ⎤
T y + cT y + gw wm
−1
M ⎢ ⎥−A
( h2)
c Y
⎣ cb( y3) + gw(
wm)
⎦
d
− v
dt
r5 =
d
A( h2)
vm + cm − p2
dt
r6= ⎛− lc( 2
1 1) + ( 2)...
⎞
− y Ty lc y Ty
dh2
I ⎜ ⎟−
⎜ N
2
+ ( 3)
+ ( ) ⎟
⎝ lc x by gwwm
⎠ dt
r7 = h2 −h1
r = p −p
m m
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Properties of nominal system
a1
i
a2
i
a3
i
b1
i
b2
i
b3
i
c1
i
c2
d
f1
c3
d
In nominal mode, which faults are detectable, which are isolable ?
When is it safe to continue operation ?
Given: One or more faults/failures are present
Determine:
Is it safe to continue operation with this/these fault(s)
present ?
Are there any critical faults that can not be detected ?
m
1
d
m
2
i
m
3
Could further faults be accommodated ?
Solution: analysis of the structure graph
i
m
4
i
m
5
i
m
6
d
m
7
i
8
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Fault means violation of a constraint
• A fault/failure is a violation of a
constraint
m1
: vm= v velocity - inertial system
m2 : h1
= ψ heading - gyro 1
m3 : h2
= ψ heading - gyro 2
m4
: p1 = p position - GPS 1
m5
: p2 = p position - GPS 2
m6
: cm = vc
sea current
- log&imu
m : w ≠ v wind - anomometer fault
7
m w
Analysis of multiple faults
• SaTool offers the possibility to disable constraints
(assume a failure in the constraint) and analyze the
structure graph of the remaining system
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Properties of the faulty system
A salient feature of SaToo is the ability to analyze the faulty
system with several simultaneous faults present.
When f1 and f2 have happened, what if f3 is a fault in a1, a2, a3, ..
f1
a3
f2
m5
m6
m7
a
1
i
i
i
a
2
i
i
i
a
3
-
-
-
b
1
i
0
i
b
2
i
0
i
b
3
d
d
d
Essentials from the structure graph
c
1
d
i
d
Auto-generated parity relations for diagnosis !
Controlability (structural): are unknown variables
reachable from input variables ?
Monitorability: are constraints reachable from
measured variables ?
c
2
d
d
d
f3
c
3
d
d
d
m
1
d
0
d
m
2
i
i
i
m
3
i
0
i
m
4
i
i
i
m
5
-
0
0
m
6
d
-
d
m
7
0
d
-
#
6
6
6
Mogens Blanke – Spring 2006
Detectability: does the constraint enter into any of the
parity relations ?
Isolability: is there a unique signature in the set of
parity relations by a particular fault (violation of
particular constraint) ?
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Maritime uses - Naval and Offshore
Examples from the Danish SF300 class
Mogens Blanke – Spring 2006
Hovering with different shaft configurations
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Manual Control
Summary
Mogens Blanke – Spring 2006
• Fault-tolerant control: detect and isolate a fault and
handle it by reconfiguring the controller (or the plant)
• Parity relations are as residual generators used for
fault detection and isolation
• Parity relations can be generated through an analysis
of system structure (structural analysis)
• Other methods exist to get residual generators
• Controller redesign is done on-line or off-line
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