Supervision Systems Design Prof. Belkacem OULD BOUAMAMA Research Director Ecole Polytechnique de Lille

Supervision Systems Design

Prof. Belkacem OULD BOUAMAMA Research Director Ecole Polytechnique de Lille http://www.polytech-lille.fr/ Head of the research group “Bond Graphs” «LAGIS UMR CNRS8219» Laboratory Avenue Paul Langevin, F59655 Villeneuve d'Ascq cedex Tel : +33(0)3 28 76 73 97, GSM: +33(0)6 67 12 30 20 [email protected] http://www.mocis-lagis.fr/membres/belkacem-ould-bouamama/

http://www.polytech-lille.fr/

mailto:[email protected]

http://www.mocis-lagis.fr/membres/belkacem-ould-bouamama/

http://www.mocis-lagis.fr/membres/belkacem-ould-bouamama/

Prof. B. Ould Bouamama Polytech Lille « Supervision Systems Design»

PLAN Supervision : Introduction and definitions

Supervision software's

Synthesis of monitoring systems

Structural analysis and bipartite graph

Information redundancy for FDI

Observers for FDI

LFT Bond graphs for robust FDI

Design of supervision system.

Application to a industrial systems

Conclusions and bibliography

Prof. B. Ould Bouamama Polytech’Lille « Supervision Systems Design »

Part 1: Introduction

Bibliography

Blanke, M., Kinnaert, M., Lunze, J. and Staroswiecki, M. (Eds)(2007) Diagnosis and Fault-Tolerant Control, Berlin:Springer-Verlag.

"Automatique et statistiques pour le diagnostic". T1 et 2 sous la direction de Bernard Dubuisson, Collection IC2 Edition Hermes, 204 pages, Paris 2001.

A.K. Samantaray and B. Ould Bouamama "Model-based Process Supervision. A Bond Graph Approach" . Springer Verlag, Series: Advances in Industrial Control, 490 p. ISBN: 978-1-84800-158-9, Berlin 2008.

D. Macquin et J. Ragot : "Diagnostic des systèmes linéaires", Collection Pédagogique d'Automatique, 143 p., ISBN 2-7462-0133-X, Hermès Science Publications, Paris, 2000.

B. Ould Bouamama, M. Staroswiecki and A.K. Samantaray. « Software for Supervision System Design In Process Engineering Industry ». 6th IFAC, SAFEPROCESS, , pp. 691-695.Beijing, China.

B. Ould Bouamama, K. Medjaher, A.K. Samantary et M. Staroswiecki. "Supervision of an industrial steam generator. Part I: Bond graph modelling". Control Engineering Practice, CEP, Vol 1 14/1 pp 71-83, Vol 2. 14/1 pp 85-96, 2006.

B. Ould-Bouamama. Contrôle en ligne d'une installation de générateur de vapeur par Bond Graph. Techniques de l'Ingénieurs AG3551. 28 pages 2014

B. Ould-Bouamama. La conception intégrée pour la surveillance robuste des systemes. Approche Bond Graph. Techniques de l'Ingénieurs AG3550. 24 pages 2013

R.Merzouki, A.K.Samantaray, M.Pathak and B. Ould-Bouamama. Intelligent Mechatronic Systems: Modelling, Control and Diagnosis. Springer Verlag, ISBN: 978-1-4471-4627-8, 943 pages, 2013.

PhD Thesis, several lectures can be doownloaded at : //www.mocis-lagis.fr/membres/belkacem-ould-bouamama/

Publications and co publications in the BG and FDI domain

5

BG for M

odell

ing

Bg for

Sup

ervi

sion

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hatro

nics

BG theo

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LFT BG

Intel

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Aims

Acquire the methodological and practical knowledge on development and implementation of online monitoring systems (detection and isolation of faults)

Understanding and acquire the structural analysis methodology for integrated design of complex systems supervision

Understanding how online monitoring systems (SCADA system) can be developed and implemented

Understanding the links between maintenance, control, on-line diagnosis, reconfiguration and analysis of operating modes and criticality

6


What is a supervision : two levels FDI FTC? Supervision :

Set of tools and methods used to operate an industrial process in normal situation as well as in the presence of failures.

Supervision (IFAC): Monitoring a physical system and taking appropriate actions to maintain the operation in the case of faults.

Activities concerned with the supervision : Fault Detection and Isolation (FDI) in the diagnosis level, and the Fault Tolerant Control

(FTC) through necessary reconfiguration, whenever possible, in the fault accommodation level.

SUPERVISION

FDI : How to detect and to isolate a faults ?

FTC : How to continue to control a process ?

Supervision Graphical User Interface (GUI)

Monitoring of variables (Data

acquisition)?

Surveillance (Alarms)

Control

9

Synoptique fonction essentielle de la supervision, fournit une représentation synthétique, dynamique et

instantanée de l'ensemble des moyens de production de l'unité

permet à l'opérateur d'interagir avec le processus et de visualiser le comportement normal

Courbes: donne une représentation graphique de différentes données du processus

Historisation du procédé:• - permet la sauvegarde périodique de

grandeurs (archivage au fil de l'eau)• - permet la sauvegarde

d'événements horodatés (archivage sélectif)

• - fournit les outils de recherche dans les données archivées

• - fournit la possibilité de refaire fonctionner le synoptique avec les données archivées

• ( fonction de magnétoscope ou de replay)

• - permet de garder une trace validée de données critiques (traçabilité de données de

• production) Gestion des Alarmes

Role of GUI (IHM)

Fonction of supervision systems Management

ERP : Enterprise Resource planning : planning of resources

• integration of different business functions in a centralized computer system configured according to the client-server mode.

MRP : Manufacturing Resource Planning : planning of production

• Planning system which determines the component requirements from requests of finished products and existing suppliesPRODUCTION

Process SCADA : Supervisory Control & Data Acquisition PC & PLC Process Control/ Programmable Logic Controller

Supervisor A system that performs supervision by means of fault detection

and isolation, determination of remedial actions, and execution a corrective actions.


Supervision and Monitoring

Monitoring A continuous real time task of determining the conditions of a physical system, by recording

information recognising and indicating anomalies of the behaviour (local security)

Automatic control Control of parameters (to maintain the quality of products)

Supervision Centralize monitoring and control tasks Two parts of SCADA system

• hardware (collect of datas) • Software (control, display, monitoring)


Supervision in the hierarchy of a manufacturing company

12


Global Function of the supervision

13


Supervision softwares

Les logiciels de supervision sont une classe de programmes applicatifs dédiés à la production dont les buts sont :

- l'assistance de l'opérateur dans ses actions de commande du processus de production (interface IHM dynamique...)

- la visualisation de l'état et de l'évolution d'une installation automatisée de contrôle de processus , avec une mise en évidence des anomalies (alarmes)

- la collecte d'informations en temps réel sur des processus depuis des sites distants (machines, ateliers, usines...) et leur archivage

- l' aide à l'opérateur dans son travail (séquence d'actions/batch , recette/receipe) et dans ses décisions (propositions de paramètres, signalisation de valeurs en défaut, aide à la résolution d'un problème ...)

- fournir des données pour l'atteinte d'objectifs de production (quantité, qualité, traçabilité, sécurité...)





Wonderware InTouch Wonderware InTouch is the world’s number one Human Machine Interface (HMI) , Used in

over one-third of the world’s industrial facilities open and extensible solution that enables the rapid creation of standardized, reusable

visualization applications and deployment across an entire enterprise. Extensible library with more than 500 graphical symbols to build the system.



PANORAMA : Ergonomic HMI module for alarms and events, an operating unit of historical datas.

SIMATIC WinCC (Siemens) Supervision system with scalable features for monitoring automated processes,

provides a full SCADA functionality in Windows Totally Integrated Automation System : Engennering, Communication, Diagnosis, Safety,

Security, Robustess



DSPACE MATLAB-Simulink More used for fast prototyping based on RealTime Interface (RTI)

Simulink model

RTI

Residuals


How to select SCADA systems Simplicity, Usability

Solvers

Image processing (icons, libraries, …)

Supervision Control Surveillance Alarm processing

Archiving

Programing

Performances/Price : Price : hardware + Operating system, software, support, documentation

Supervision system Architecture

Réseau d’entreprise

Réseau d’atelier (Ethernet)

Réseau de terrain (Profibus, Modbus, Asi…)

Postes de Supervision

Automate

(PID, TOR…)

Opérateur

Terminald’atelier

Actionneurs

Capteurs


Part 2: Objectives and definitions


Definitions Safety (sûreté)

Ability of a system to dispose of its functional performance (reliability, maintainability, availability) and not to cause a danger for persons or equipment or environment

Safety is rather protection against accidental events. Security (sécurité)

The condition of being protected from or not exposed to danger. Security is rather protection against intentional damages.

Example : Aircraft security is about protecting the aircraft and it's contents from criminal activity and terrorism

(Control of documents) Aircraft safety is about protecting the people by making the aircraft less likely to be involved in a crash

(maintenance…)

Somme definitions Fault

Unpermitted deviation of at least one characteristic property or parameter of the system from acceptable / usual / standard condition

• Incipient fault (naissante): A fault where the effect develops slowly e.g. clogging of a valve). In opposite to an abrupt fault.

• Abrupt fault : A fault where the effect develops rapidly (e.g. a step function). In opposite to an incipient fault.

• Active fault- tolerant system : A fault-tolerant system where faults are explicitly detected and accommodated. Contrary to a passive fault-tolerant system.

Failure (Défaillance) Permanent interruption of a systems ability to perform a required function under specified

operating conditions – incipient failures (naissantes),– Having a transitory nature– constants– Evolving over time– catastrophic

Types of fault

Capteur de position

u Cm

m

Controller fault

Actuator faultPlant fault

Défaut capteur


Somme definitions Fault detection :

Determination of faults present in a system and time of detection

Fault diagnosis: Determination of kind, size, location, and time of occurrence of a fault. Includes fault detection, isolation and

identification

Fault isolation : Determination of kind, location, and time of detection of a fault. Follows fault detection.

Fault modeling : Determination of a mathematical model to describe a specific fault effect.

Fault-tolerance : The ability of a controlled system to maintain control objectives, despite the occurrence of a fault. A degradation of

control performance may be accepted. Fault-tolerance can be obtained through fault accommodation or through system and /or controller reconfiguration.

Fault-tolerant system : A system where a fault is accommodated with or without performance degradation, but a single fault does not

develop into a failure on subsystem or system level.

Sensor fusion Integration of correlated signals from different sensors (information sources) into a single representation or action.


Somme definitions

Fault accommodation (1) - A correcting action that prevents a certain fault to propagate into an undesired end-effect. (2) - Change in controller parameters or structure to avoid the consequences of a fault. The original control

objective is achieved although performance may degrade.

Disturbance: An unknown (and uncontrolled) input acting on a system

Perturbation: An input acting on a system which results in a temporary departure from current state

Constraint: The limitation imposed by nature (physical laws) or man. It permits the variables to take certain values in

the variable space.

Decision logic The functionality that determines which remedial action(s) to execute in case of a reported fault and which

alarm(s) shall be generated.

Detector An algorithm that performs fault detection and isolation


Somme definitions

Analytical redundancy Use of more than one not necessarily identical ways to determine a variable, where one way uses a

mathematical process model in analytical form.

Hardware redundancy Use of more than one independent instrument to accomplish a given function.

Availability: Probability that a system or equipment will operate satisfactorily and effectively at any point of time.

MTTR: Mean Time To Repair MTTR = 1/µ; µ: rate of repair

Reliability: Ability of a system to perform a required function under stated conditions, within a given scope, during a

given period of time. Measure: MTBF = Mean Time Between Failure. MTBF = 1\la; la is rate of failure [e.g. failures per year]


Somme definitions : Models Qualitative model

A system model describing the behavior with relations among system variables and parameters in heuristic terms such as causalities or if-then rules.

Qualitative equation Equations whose functional form and coefficient values are not completely specified.

Quantitative model A system model describing the behavior with relations among system variables and parameters in

analytical terms such as differential or difference equations. Residual

Fault information carrying signals, based on deviation between measurements and model based computations.

Threshold Limit value of a residual's deviation from zero, so if exceeded, a fault is declared as detected

Symptom Change of an observable quantity from normal behaviour


Introduction From 1840: automatic control (Watt regulator)

Task: improve the quality of finished products,

from 1980, new Challenge : Supervision Rôles : Provide the human operator assistance in its emergency management tasks alarm

situations to increase the reliability, availability and dependability of the process.

Apparition of integrated automation Control, diagnosis, optimization …

Integrated automation

Supervision

Monitoring

Regulation

Instrumentation

Input Outputs

FDI, FTC, aided decision tools

Monitoring the state of the process, user interface

Control, optimisation

Selection and implementation of sensors and actuators

ObservationsDecisions

level 3

level 2

level 1

level 0

Haz

ard

ous

area

Haz

ard

ous

Are

aHazardous Area

Relation between FDI et FTC Perf=F(Y1,Y2)

UNACCEPTABLE PERFORMANCES

DEGRADED PERFORMANCES

Y1

Y2

Degraded performances

RequiredPerformances

Reconfiguration

Fault

SUPERVISION in INDUSTRY

FTC LevelFault accommodationReconfiguration

List of faulty components

Corrective maintenance (after fault occurs)

Set points

Sensorsy

x

u

ur

Controllers

Actuator

Process

FDI LevelOn line Fault Detection and

isolation

Supervision system : different steps

u

Corrector Actuator Process Sensorx

y

Fault Tolerant control reconfiguration (FTC)

Maintenance service

Physical system to be monitored

DECTECTION

ISOLATION

Diagnosis(Identification of the type of fault

Offline

Graphical User Interface


FDI Purpose

Objectives : given I/O pair (u,y), find the fault f . It will be done in 3 steps :

DETECTION detect malfunctions in real time, as soon and as surely as possible : decides whether the

fault has occured or not

ISOLATION find their root cause, by isolating the system component(s) whose operation mode is not

nominal : find in which component the fault has occured

DIAGNOSIS diagnose the fault by identifying some fault model : determines the kind and severity of the

fault

FDI: Medical interpretaion

0T

37

+

-NON

OUI Clin

ical

exa

min

atio

n (D

ET

EC

TIO

N)

Dia

gn

osis

(I

SO

LATIO

N)

FDI steps in technological process supervisions

35

Alarms generation

Datas from Actual process Model

+-

DIAGNOSIS

Type of failures

Detection : Is it really a fault ?

isolation : Which component is faulty ?

Identification : What is the type of fault?

DECISION


Technical specifications

FT (Fault Tolerance) and FTC (Fault Tolerant Control) FT (Fault Tolerance)

Analysis of fault tolerance : The system is runing under faulty mode• Since the system is faulty, is it still able to achieve its objective(s) ?

Design of fault tolerance : • The goal is to propose a system (hardware architecture and sofware which will allow, if

possible, to achieve a given objective not only in normal operation, but also in faulty situations.

Control and Fault Tolerant Control Control algorithms :

• implement the solution of control problems : according to the way the system objectives are expressed

FTC algorithms• implements the solution of control problems : controls the faulty system• the system objectives have to be achieved, in spite of the occurence of a pre-specified set

of faults


Control Problem

Traditional control : two kinds of objectives control of the system , estimation of its variables

Problematic : Given • a set U of a control law ( open loop, closed loop, continuous or

discrete variables, linear or non-linear)• a set of control objective(s) O, • set of uncertain constraints C(), (dynamic models)

The solution is completely defined by the triple <O,C(), U >


FTC problem

FTC Controls the faulty system: 2 cases 1) fault adaptation, fault accommodation, controller reconfiguration

• change the control law without changing the system 2) system reconfiguration

• change both the control and the system :

The difference with Control problem

System constraints may change. UCOUCO

UCOUCO

ffnn

fn

,(,,(,

,(,,(,

:e2.Structur

:rs1.Paramete

Admissible control laws may change. rffnnn UCOUCO ,(,,(,


Passive and active fault tolerance

Passive fault tolerance Active fault tolerance

control law unchanged when faults occur

Normal modeControl law solves < O, Cn(n), Un > Faulty mode Control law also solves < O, Cf(f), Uf >

f F

specific solution for normal and faulty mode

<O,Cn(n),Un > and < O, Cf(f), Uf > f F

ROBUST TO FAULTS

Knowledge about Cf(f) and Uf must be available . FDI layer must give information.

Fault accommodation and System reconfiguration

FDI system

Systemreconfiguration

Provide estimation of Cf(f) Uf of the fault impact

solve < O, Cf(f), Uf >

Fault

solve < O, f(f), Uf >

Provide estimation of f(f), Uf of the fault impact

Fault

FDI cannot provide any estimation of the fault impact

solve < O, Cr(r), Ur >

Fault

Fault accommodation

Fault accommodation

ProcessController

FDIFault

Accommodation

Controller parameters

Ref.

Yu

Supervision

Control system

Fault Reconfiguration

FDI

New controlconfiguration

Reconfiguration

YrefNominal Controller

Process Yu

u'New Controller

Y’ref

Y’

Supervision

Con

trol syste

m


Part 3: HOW TO DESIGN SUPERVISION SYSTEMS ?


DIAGNOSTIC METHODS (2/2)

Qualitativemodels

Bond Graph

bipartite graph

Causal graphs

Observers

Identification

Analytical redundancy

Kalmanfilter

Quantitative models

Artificial Intelligence

Pattern recognition

Expert systems

Fuzzy logicLearning

FDI Methods

Data analysis

Signal processing

Based models No model based

Model-based FDI

S E N SO R S

Process actual operation

RESIDUALGENERATOR

MODEL OF THE NORMAL OPERATION ALARM GENERATION

0

Isolation Identification

ALARM INTERPRETAION

Detection


FDI based on Identification and observer

y

Modèle

Uy

Residual+-

ref

yObservateur

Uy

Residual+-

y

identification based

Observer based

No model based Pattern recognition methods

Determination of a set of classes (learning step) For each class is associated an operating mode (normal and faulty)

• Advantage Methods : statistical learning, data analysis, pattern

recognition, neuronal networks, etc. Only experimental data are exploited No complex analytical model

?

?

?

• Problems • need historical data in normal and in abnormal situations,

• every fault mode represented ???• generalisation capability ??

++++

++

+ ++++

+

+D2

Example : FDI of a valve

48

1) No model based

Pressure difference Pr = P1-P2

Flow

Q(t)

*

**

***

**

**

*

*

*

*

*

**

D1

1) Pattern recognition step (classification of different modes)

QP1 P2

2) On line surveillance step


QUALITATIVE METHODS

Use expert knowledge based on « If then else » : applying models of human thinking to physical systems Example : « If P1 increase then Q increase, else valve is blocked»

advantage of qualitative methods: No need of numerical value of parameters neither deep knowledge of the system système. Easy to be implemented

Issue Sensor faults not detected Lower and upper values of the deviation cannot be fixed precisely Combinatory problem can appear for complex systems (multivariable)

Model based : example

50

Step 1 determination of fault indicator offline)

21 2

m

m

P P P KQ

P P

Q Q

System Model

Measurement equations

21m mP KQ ARR

Generation of a fault indicator

QP1 P2

2m mP KQ residual

Numerical evaluation

Analytical model, parameters

Threshold

Residual signal


Steps in FDI system (1/4)

1. DETECTION Logic operation : We state the system is faulty or not Criteria

• No detection or too late detection Catastrophic ➽consequences for the process

• False alarms Unnecessary stops of the production unit. ➽

There are 4 hypothesis• H0 : Assumption of normal operation (Decision domain D0)• H1 : Assumption of faulty mode operation (Decision domain

D0)• Dx : No decision domain



Problematic• Given R=[r1, ….rn] fault indicators

• Two distributions are known p(Z/H0) and p(Z/H1)

• One of two hypotheses, H0 or H1 is true

What to do ?• Verify if each ri (i=1,..n) belongs to p(Z/H0) and p(Z/H1)

• 4 possibilités

H0 H1

Decide H0 OK Missed detection

Decide H1 False alarm

OK



2. ISOLATION To be able to isolate the failed components (Alarm filtering) using logic

operations Criteria

• No isolability Catastrophic consequences for the process➽• False isolability Unnecessary stops of the production unit or ➽

equipment.

3. IDENTIFICATION (DIAGNOSIS) When the fault is located, it is then necessary to identify the specific causes of

this anomaly. Are the used logic operation based on signatures identified by experts and validated through expertise and repair faults.


Technical specificationsS

pec

ific

atio

ns

Sp

ecif

icat

ion

s

Which parameters must be supervized ?

What are the non acceptable values ?Objectives

Performancesfalse alarm

missed detection

detection delay

Available data

other (cost, complexity, memory, ...)Constraints

Logic Diagnosis : Systems and faults (1)

COMPS = {comp1, comp2, comp3, comp4, comp5}

xa

b

c

d

y

z

e

f

comp1

comp2

comp3

comp4

comp5

A system is a set of interconnected components

A system is a triplet (SD, COMPS, OBS)

SD : System Description, COMPS : Set of componentsOBS: set of observations

System (2)

COMPS = {input valve, tank, output pipe, level sensor}

x = a by = bz = c de = x yf = z ( y)

xa

b

c

d

y

z

e

f

comp1

comp2

comp3

comp4

comp5

Continuous Hydraulic system

Discrete electronic system

SD


SM (or SD) is the set of all those constraints

Input valve

Tank

Output pipe )()( tlktqo

)()()(

tqtqdt

tdloi

)(1)(

0)(0)(

tqtu

tqtu

i

i

Level sensor),0(

)()()(

N

ttlty

System (4)


Examples of internal faults (1)

y b OK(comp2) is false

xa

b

c

d

y

z

e

f

comp1

comp2

comp3

comp4

comp5

Examples of internal faults (2)

Process fault : the tank is leaking

Sensor fault : noise has improperstatistical characteristics

),(

)()()(

bN

ttlty

)()()()(

tqtqtqdt

tdlloi

Actuator fault : input valve is blocked open

)(1)(

)(0)(

tqtu

tqtu

i

i

Examples of external faults (2)

min1210 )()()(2

1

tttdttqt

t

max

min

Control algorithm objective :

cannot be achieved for too large output flows

maxmin )( t

Controller

a = 2 !! (it should equal to 1)

xa

b

c

d

y

z

e

f

comp1

comp2

comp3

comp4

comp5


Diagnosis algorithm

SD is now ...

OK(input valve)

OK(tank)

OK(output pipe)

OK(level sensor)

)(1)(

0)(0)(

tqtu

tqtu

i

i

)()()(

tqtqdt

tdloi

)()( tlktqo

),0(

)()()(

N

ttlty

OK(comp1) x = a bOK(comp2) y = bOK(comp3) z = c dOK(comp4) e = x yOK(comp5) f = z ( y)

Problems

1) For some given S COMPS, how to check the consistency ofSD {OK(X)X S} OBS

2) How to find the collection of the NOGOODS

How to check the consistency

OBS (controls, measurements)

Properties that OBS should satisfy / values that OBS should haveTEST

Actual system Nominal system model

Detection

Compare actual system and nominal system

Problem statement


Two means to check consistency

Analytical Redundancy properties that OBS should satisfy if actual system healthy properties that are satisfied by the nominal system trajectories check whether they are true or not

Observers values that OBS should have if actual system healthy simulate / reconstruct the nominal system trajectories check whether they coincide with actual system trajectories


Chap.2 : ANALYTICAL REDUNDANCY

Representation

),(

),,,,(

m

p

xCy

tduxfdtdx

Parameters:

esDisturbanc:

d

PROCESS Capteurs

qp dx0

x(t)y(t)

u(t)

qm

PROCESS Capteurs

qp dx0

x(t)y(t)

u(t)

qm sp

),,(

),,,,,(

sm

pp

xCy

tduxfdtdx

Model of the faulty systemModel of the faulty system

Model of the healthy systemModel of the healthy system


State space representation

HGdDuCxy

FEdBuAxdtdx

Faults

DisturbancesLinear caseLinear case

Nonlinear caseNonlinear case

),,,(

),,,(

duxCy

duxFdtdx

HGdDuCxy

FEdBuAxdtdx

Faults

Disturbances


When the system is faulty ?

Given a system

The system works in normal regime (hypothesis H0) means : y is produced according law C and x is produced according law f and is produced according law of probability P

The system works in failure mode hypothesis H1) means : y is not produced according law C, or x is not produced according law f, or is not produced according law of probability P

),(

),,,()(

xCy

tuxftx

noise:

parameters:

input :

tmeasuremen:

state :

u

y

x

Analytical redundancy :How to generate ARRS ? Given

The ARR express the difference between information provided by the actual system and that delivered by its normal operation model

What is Residual ?

68

)(

),()1(

xCy

uxfx

)1()(1 yCx

u

y

r

( )R Eval ARR

All variables are known

: unknown variable

, : Known variables

x

u y

1( ( ))( , , )d C y

y u ARRdt

11( ( ))

( ( ), )d C y

f C y u ARRdt

11( ( ))

( ( ), )d C y

f C y u ARRdt


Analytical Redundancy Relations (ARR) and Residuals (r)

Definition ARR

• ARR is a mathematical model where all variables are known. The known variables are available from sensors, set points and control signal.

ARR : F(u,x0, y, )

Residual r• Residual is the numerical value of ARR (evaluation of ARR) R is a signal,

ARR is an expressionR= Eval (ARR)

Problematic : How to generate ARRs ? Issue : Elimination of unknown variables theory


General principle

Analytic modelmeasurement equationsorstate and measurement equations

ARR: 0Φ(u,y)

Off-lineElimination of unknown variables techniques

0 r

RESIDUALS

),yΦ(u actualactual

On-lineComputation of ARRs (actual system)

Hardware and analytical redundancy

71

R

S1 or S2

0P

.P*Q 111

dt

dCR

S2

Hardware redundancy

Detection IsolationSensors

0S

.S*Fr 1111

dt

dCR

S3 S2 S1

F2

F1

0*Q2 PR 0S*Fr 122 R

Analytical redundancy

?

LeakageS1F1 Valve R F2

r1

r2

1 1

0

1

10

1

1

0

1

1 1

0

1

10

1

1

0

1

Signature Fault Matrix (SFM)


Detectability and isolability

otherwise0

1 ijij

ARR if Es

Sij : boolean value (0,1)

Ej (j=1,m) : Fault which may affect the jth component

Fault Signature Matrix (FSM)

Ib1 Ib2 … Ibm

Mb1 Mb2 … Mbm

E1 E2… Em

ARR1 S11 S12 … S1m

ARR2 S21 S22 … S2m

………

.

.

.

.

.

.

………

.

.

.

ARRn Sn1 Sn2 … Snm

Ib : Isolability

Mb: Detectability

DEFINITION


Detectability and isolability

Detectability

A component fault Ej is detectable (Mbj=1) if at least one sij (j=1,m) of its signature vector VEij is different than zero

0),,1( Eijij Vsmjj

IsolabilityA component fault Ej is isolable (Ibj=1) if it is detectable and its signature vector VEij is different from others .

otherwise0

)(),1( 1 iVVm if I

ElEjbj

The signature vector VEj (j=1,m) of each component fault Ej is given by the column vector:

TnmjjEj sssmjV ...),1( 21


Detectability and isolability example Faults and ARRFault Signature Matrix (FSM)

Ib 0 0 0 0 1

Mb 1 1 1 1 1

F1 S1 Leak. Valve R F2

ARR1 1 1 1 1 0

ARR2 0 1 0 1 1

21

211 .

ARRARRARR

FValveRLeakSFF

Signature vectors

10

11

01

11

01

2

1

1

F

RValve

Leak

S

F

V

V

V

V

V

Hamming Distance ji SCD

C: Binary coherence vector

Sj : Signature vector of the jth component to be monitored

to isolate k failures, the distance should be equal to 2k + 1.

Hamming Distance

Hamming Distance of given example

F1 S1 Leak. Valve R F2

F1 0 1 0 1 2

S1 0 1 0 1

Leak. 0 1 2

Valve R 0 1

F2 0

Signature vectors

10

11

01

11

01

2

1

1

F

RValve

Leak

S

F

V

V

V

V

V

The Hamming distance shows the ability to isolate two faults.

1 2

1 1

1

1

,

,

,

,

1 0 0 1 2

1 0 1 1 1

1 0 1 0 0

1 0 1 1 1

F F

F S

F leak

F Valve

D

D

D

D

Hamming distance (example)


Hardware redundancy : Simplest redundancy

Hardware redundancy uses only measurement equations (therefore it can detect only sensor faults)

Example : duplex redundancy

Model : y1 = x

y2 = xStatic ARR : y1 - y2 = 0

Duplex redundancy

r

t

Max threshold

Min threshold

Alarm

Fn. normal

Low pass filter

+-

m1

m2

m1f

m2f

Alarm generator

Max threshold

Min threshold

AlarmsSensor 1

Sensor 2

Noised signal

Noised signal Low pass

filter

Process

Variable x

r1

r2

Triplex redundancy

r1

tr2

tr3

t

Residuals r1 = m1f - m2 f r2 = m1f – m3f r3 = m2f – m3f

Low pass filter

m1

m2

m1f

Thresholds

AlarmsLow pass filter

Residual generation

m2f

m3f

m3

Decision procedure

r2

r3

r1

Sensor 2Variable x

Low pass filterSensor 1

Sensor 3


Fault detection : three steps

y1

y2

Sensors

acquisitionResidual generation

r = y1 - y2

+

-

Residual evaluation

= 0 ?

yes

no


Fault detection : Problematic

y1 - y2 = 0it is not impossible (but it is not certain) that both sensors are healthy

Why is it so ???

because there might be non detectable faults

non detectable faults

y1 = x + f1

y2 = x + f2

r = y1 - y2 = f1 - f2

r = 0 even when there is a combination of faults f1 and f2

such that : f1 - f2 = 0

Example : common mode failures

Computation form Evaluation form

Redundancy with Non detectable faults

Given fault model


yes is never true

no is always true because y1 = x + 1

y2 = x + 2

we need a model of the uncertainties

Assume we know 1 [a1, b1], 2 [a2, b2], then we know

1 - 2 [a12, b12]

r = y1 - y2 = 1 - 2

Redundancy with uncertainties

y1

y2

= 0 ?Residual

Generationr


y1 = x + 1

y2 = x + 2

r = y1 - y2 = 1 - 2

Redundancy with noises

Assume we know P(1) and P(2), then we know P(1 - 2)

is r distributed according to P(1 - 2) ???

r

P(1 - 2)

r

d(1 - 2)we need a Statistical decision theory


triplex redundancyy1 = xy2 = xy3 = x

two residuals

r1 = y1 - y2 = 0r2 = y2 - y3 = 0

Remarks* any linear combination of residuals is a residual (r3 = y2 - y3)

How to isolate the fault ?

3

2

1

110

011

2

1

y

y

y

r

rThe set {r1, r2} is a residual basis in the following sense :


Fault isolation (fault model)

Triplex redundancyy1 = x + f1 x = y1 - f1

y2 = x + f2 x = y2 - f2

y3 = x + f3 x = y3 - f3

y1 - f1 = y2 - f2

y2 - f2 = y3 - f3

r1 = y1 - y2 = f1 - f2

r2 = y2 - y3 = f2 - f3


Fault isolation

r1 = y1 - y2 = f1 - f2r2 = y2 - y3 = f2 - f3

f1 f2 f3r1 1 1 0r2 0 1 1

Structured and directional residuals Structured and directional residuals

Directional residuals Directional residuals

3

2

1

321

3

2

1

.110

011

2

1

f

f

f

WWW

f

f

f

r

r


Conclusion about hardware redundancy

detect sensor faults (if detectable)

isolate sensor faults (if enough redundancy)

needs noise models for statistical decision

needs uncertainty models for set theoretic based decision

powerful approach but multiplies weight and costs

limited to sensor faults


Static Analytical redundancy


Parity Space Given linear system

Static redundancy Suppose m>n : Then, a decomposition of matrix C can be given under following form as :

Such that C1 is inversible then measurement equation y(t) can be written :

1

,

x x

y y

n m

x t A x t B u t F d t E t

y t C x t D u t F d t E t

x y

2

1

C

CC

d: fault, Ԑ: uncertainties

1 1

2 2

1 1 1

2 2 2

y ( )x( ) u ( ) ( )

y ( )y y

y y

F Et C Dt (t) d t t

F Et C D


Parity Space

Then unknown variable X is calculated from y1,

and eliminated by replacing x(t) in Y2 : we obtain an ARR

Evaluation and calculation form can be obtained

1 1

11 1 1x( ) y ( ) u( ) ( ) ( )y yt C t D t F d t E t

2 1 2 1

1 1 1 12 2 1 1 2 1 1 1 1y ( ) y ( ) ( )u( ) ( ) ( ) ( ) ( ) 0y y y y

t C C t D C D t F C F d t E C E t

2 1 2 1

1 12 2 1 1 2 1 1

1 11 1

( ) y ( ) y ( ) ( )u( )

( ) ( ) ( ) ( )y y y y

t t C C t D C D t

F C F d t E C E t


Parity space approach

Parity space approach to eliminate unknown variable x (Chow 84). : Find an orthogonal matrix W to C such that (WC=0) by multiplying measurement

equation y=CX by W :

Then The system of measurement equation is overdertermined w.r.t. to x :

• We have m-n ARR, while W has m-n linearly independent rows

)()( t WEy εW WFy d(t) WD u(t) t WEy ε WFy d(t)

WD u(t) WCx(t) Wy(t)

( )y(t) Cx(t) D u(t) Fy d(t) Ey ε t


Static Parity space Given measurement equation :

Columns of C : vector subspace of dimension R(C) : we note CR(C)

Given additional subspace to CR(C) noted Wm-R(C)

Wm-R(C) is named parity space Thus : CR(C) Wm-R(C)=Rm ( sum of vector space)

mCRCRang

nmC

nkx

mky

kGfkdHkDukxCky

)()(

)dim(

1))(dim(

1))(dim(

)()(.)()(.)(

Projection of measurement equation onto parity space

ARR: in the absence of faults and disturbances (d(k)=f(k)=0)

( ) ( ) . ( ) ( )W y k Du k W H d k Gf k

RRAkDukyW 0)()(

( ) . ( ) ( ) . ( ) ( )Wy k WC x k WDu k WH d k WGf k

=0

Calculation form Evaluation form

Forms of vector parity

)()(.)()( kGfkdHWkDukyW

Evaluation form

)()(.)( kGfkdHWkr 0)()()( kDukyWkr mesurémesuré

Calculation form


Hardware redundancy based on substitution Example : triplex redundancy

y1 = x + f1 x = y1 - f1

y2 = x + f2 x = y2 - f2

y3 = x + f3 x = y3 - f3

y1 - f1 = y2 - f2

y2 - f2 = y3 - f3

r1 = y1 - y2 = f1 - f2r2 = y2 - y3 = f2 - f3


Hardware redundancy based on parity space

ARR generation using parity space

Parity space of dimension 2. Then a basis W can be choosen WC=0 (2 vectors orthogonal to C). Among those solutions, Parmi toutes les solutions choisissons :

Projection of Y(t) onto parity space gives:

)()(.)( kGfkxCky

13)dim(,11)(dim(,13))(dim(

)(

)(

)(

)(

1

1

1

)(

)(

)(

2

2

1

2

2

1

Ckxky

kf

kf

kf

kx

ky

ky

ky

110

011W

)()()()()()(

)()()()()(

323212

21211

kfkfkykykykr

kfkfkykykr

)(

)(

)(

)(

)(

)(

110

011)()(.

)(

)()(

3

2

1

3

2

1

2

1

kf

kf

kf

ky

ky

ky

kGfkyWkr

krkr


Directional residuals r(k) can be expressed as :

Dimension of the parity space is 2. The direction of the residual vector depends on the specific direction of each fault.

)(

)(

)(

)(

)()(

3

2

1

3212

1

kf

kf

kf

WWWkr

krkr

1 2 3

1 1 0Let choose: , ,

0 1 1W W W

)()()()( 332211 kfWkfWkfWkr

r1

r2

f1

f2

f3

Example of static redundancy Given parity space

Cxy

BuAxzx

3

2

1

,0,

01

10

01

,0

1,

5.02

01.0

y

yy

yDCBA

13

22

11

212

11

5.02

1.0

xy

xy

xy

xxzx

uxzx

1.0

1

z 5.0

2

z

u

y2

y1 y2

x1 x2

y3

To eliminate x, one find W such that : Wy = WCx = 0

00 WCWcxWy ),)()dim(

123)()(

)(

mWRangW

CRangmWRang

mCRang


Example of static redundancy Residuals are :

While dim(W)=1x3, then W = (a b c) All vectors under form : W= [a 0 -a] cancels WC

One find thus the hardware redundancy:

0Wyr

31

3

2

1

0.0 yyr

y

y

y

aaWyr


Conclusion about hardware redundancy There is a static redundancy if one can find :

A set of vectors W orthogonal to C such that : WC = 0 • Row vectors of W define parity space :• Projection of measurement equation onto parity space gives :

– Static ARR: W.Y = W.C.X = 0

Hardware redundancy concerns only sensor FDI

Widely used in industry


A bit more complex

Analytical redundancy (dynamic)


State space model

)()()(

)()()(

tDutCxty

tButAxtx

)()()(

)()()1(

tDutCxty

tButAxtx

Continuous time Discrete time

If there exists W such that WC = 0then static redundancy relations can be found

Dynamic Analytical Redundancy


)()()(

)()()(

tDutCxty

tButAxtx

)()()( tuDtxCty

)()()()( tuDtCButCAxty

)(

)(0)(

)(

)(

tu

tu

DCB

Dtx

CA

C

ty

ty

Dynamical Analytical redundancy (continuous)

Differenciation of y


( 1) ( ) ( )

( ) ( ) ( )

x t Ax t Bu t

y t Cx t Du t

( 1) ( 1) ( 1)y t Cx t Du t

( 1) ( ) ( ) ( 1)y t CAx t CBu t Du t

( ) 0 ( )( )

( 1) ( 1)

y t C D u tx t

y t CA CB D u t

Dynamical Analytical redundancy (Discrete)

Differenciation of y


)(

)(0)(

)(

)(

tu

tu

DCB

Dtx

CA

C

ty

ty

If there exists W such that

021

CA

CWW

W

then 0

)(

)(0

)(

)(21

tu

tu

DCB

D

ty

tyWW

Analytical redundancy (dynamic)


)(

...

)()(

...

0.........

......

0...0

)(......

)()(

)()1()()( tu

tutu

DCBBCA

DCB

D

tx

CA

CAC

y

tyty

pppp

Observability matrix OBS(A, C, p)

Toeplitz matrixT(A, B, C, D, p)

Analytical redundancy (general)

Dérivation de y

Dérivation de y(n)

)()()(

)()()(

tDutCxty

tButAxtx

)()()()( tuDtCButCAxty


Expressions of dynamical ARRs

)(

...

)()(

...

0.........

......

0...0

)(......

)()(

)()1()()( tu

tutu

DCBBCA

DCB

D

tx

CA

CAC

y

tyty

pppp

)()( ).,,,,()().,,( pp upDCBATtxpCAOBSy

0),,,,(. )()( pp upDCBATWyW

If there exists W such that 0),,(. PCAOBSW

)()( ).,,,,()().,,( pp upDCBAWTtxPCAWOBSyW

ARRs are :

Rows of W are a basis of Ker(OBS), define the parity spaceParity space dimension is number of sensors

RESUME REDONDANCE DYNAMIQUE

Given the system

At time K+1

Using (1) we have

Then:

generalizing until the order p

)()()(

)()()1(

kDukCxky

kBukAxkx

(1)

(2)

)1()1()1( kDukCxky (3)

(4))1()()()1( kDukCBukCAxky

))1(

)(0)(

)1(

)(

ku

ku

DCB

Dkx

CA

C

ky

ky

)(

...

)1()(

...

0.........

......

0...0

)(...

)(

...

)1()(

)1()( pku

kuku

DCBBCA

DCB

D

kx

CA

CAC

pky

kyky

pp

),().,,,,()().,,(),( kpupDCBATkxpCAOBSpky


fFtDutCxty

fFtButAxtx

y

x

)()()(

)()()(

0),,,,(),,,,( )()()( pyx

pp fpFCFAWTupDCBAWTyW

)()()( ),,,,(),,,,( pyx

pp fpFCFAWTupDCBAWTyWr


= 0 when no fault

0 when fault is present

Fault detection


Cayley-Hamilton Theorem Consequence of Cayley-Hamilton Theorem

It exists order p such that rank of OBS(A,C,p) matrix is smaller than the number of rows : thus we can find a matrix W such that :

W.OBS(A,C,p) = 0

Additional space to OBS, defined by W, is named « Parity space ». By projection of measurement equation (3) onto this space, we obtain: Dynamic ARR : The

residual is

0),(),,,,(),( pkupDCBAWTpkyW

0),(),,,,(),()( pkupDCBAWTpkyWkr( ) ( 1) ( )

dim( ) ( ( ), ( 1))

rank W m p rank T

W rank W m p

Application

( 1) ( ) ( )

( ( )

x k Ax K Bu k

y k Cx k

1

2

0.1 0 1 1 0 0, , , ,

2 0.5 0 0 1 0

yA B C I y D

y

)k,p(u).p,D,C,B,A(T)k(x).p,C,A(OBS)p,k(y

2

( ) 0 0 ( )

( 1) 0 . ( 1)

( 2) 0 ( 2)

y k C D u k

y k CA CB D u k

y k CA CAB CB u k

1( ) ( 1) ( ) 2*(1 1) 2 2

dim( ) ( ( ), ( 1)) (2,4)

rank W m p rank T

W rank W m p

Calcul W : derivation first order : ),1().1,,,,()().1,,()1,( 1 kuDCBATkxCAOBSky

Derivation up to second order

00

01

00

00

5.02

01.0

10

01

,25.02,1

001.011

2 TCA

COBSA

CB

D

D

Application

00

5.02

01.0

10

01

0(.). 1

dcbaOBSW Find two linearly

independent vectors W

0.1 2 02 equations 4 unknowns

0.5 0

a c d

b d

20142et 0

010011.0,0,0

1205.0020,5.01,0

3

2

1

Wdc

Wcadetb

Wdcdbdeta

We fix arbitrarily 2 unknowns

Residuals expressions are then :

pkupDCBAWTpkyWkr ,(),,,,(),()(

W3 is linear combination of W1 and W2

3 2 1

1

2

0.5 2 ,W W W

WW

W

Application

)1,()1,,,,()1,()( 1 kuDCBAWTkyWkr

)1(

)(.

00

01

00

00

.2014

01001

)1(

)1(

)(

)(

.2014

01001

)(

)()(

2

1

2

1

2

1

ku

ku

ky

ky

ky

ky

kr

krkr

221

11

22212

111

11111

24)()1(2)()(4)(

1010)()(10)1(10)()(

yyzyzzrkykykykr

uzyyzzrkukykykr

121

22

11

212

11

5.0

2,

1.0

5.02

1.0

yz

yz

uy

xy

xy

xxzx

uxzx

If r=0, we obtain initial model

Second order residual

Matrices OBS and T will be :

We obtain after claculation

Analysis 2nd order residual (cf r4) is sensible only to Y2 (Good for isolation) If the order is increased, are obtained the same ARRS but time shifted RRAs (filtered)

0

0

00

22

CBCAB

DCB

D

TCAB

CAC

OBS2( ) ( 1) ( ) 2*(2 1) 2 4

dim( ) ( ( ), ( 1)) (4,6)

rank W m p rank T

W rank W m p

221

11

2

111

11

24)(

1010)(

yyzyzzr

uzyyzzr

1st order residual (obtained before)

uzyyzyzr

zuzyzyzr

yzyzyzr

zuzyyzr

222

12

24

21

11

23

21

22

12

2

111

11

402012

)(1010

24

)(1010

2nd order Residual


Conclusions

detects any fault (if detectable)

isolates any fault (if enough redundancy)

estimates the unknown variable with several estimation versions

needs noise models for statistical decision

needs uncertainty models for set theoretic based decision


CHAP3:Structural Analysis

Structural analysis Motivations Structural description Structural properties Matching Causal interpretation of matchings Subystems characterization System decomposition Conclusion


Motivations Complex systems : hundreds of variables and equations

Many different configurations

Many different kinds of models (qualitative, quantitative, static, dynamic, rules, look-up tables, …)

Description of physical plants as interconnected subsystems

Analytic models not available

The structural description of a system expresses only the links between the variables and the constraints

Structural analysis Analysis of the structural properties of the models, i.e. properties that are independent on the actual values of the parameter.

Graphs : some definitions

118

A graph is an ordered pair G = (V, E) which consists of a set V of vertices or nodes together with a set E of edges or lines A graph is used to specify relationships among a collection of items. The are Simple (undirected graphs) and oriented (directed) graphs

Examples social networks, in which nodes are people or groups of people, and edges represent some

kind of social interaction Communication networks : computers are nodes, and the edges represent direct links along

which messages can be transmitted.

A

B

C D

A

B

C D

Undirected (simple) Graph Directed (oriented) Graph (A

points to B but not vice versa

Digraph: definitions Given the state equation

The digraph ? [Blanke and al. 2003]

Graph whose set of vertices corresponds to the set of inputs ui, output yj and state variables xk

Edges are defined as :

• An edge exists from vertex xk (respectively from vertex ul ) to vertex xj if and only if the state variable xk (respectively the input variable ul ) really occurs in the function F (i.e. vertex ui ) in the function

• An edge exists from vertex xk to vertex yj if and only if the state variable xk really occurs in the function g

Physical means Digraph is a structural abstraction of the behaviour model where

• Edges represent mutual influence between variables :

• The time evolution of the derivative xi depends to the time evolution of xk

g f C

yu x Z

uxgy

uxFx

),,(

),,(

Directed graph representation

g f C

yu x Z

uxgy

uxFx

),,(

),,(

1 1

2 2

1

2

( ) ( )0 0( )

( ) ( )

( )( ) 0

( )

0 0, , 0

x t x tau t

x t x tb c d

x ty t e

x t

aA B C e

b c d

Edge represents mutual influence between variables (x1 influences y

Means : the time evolution of the derivative

depends to the time evolution of x2

2x

Directed graph representation

ux2

x1

y


Structural description Behaviour model of a system : a pair (C, Z)

Z = {z1, z2,...zN } is a set of variables and parameters, C = {c1, c2,...cM } is a set of constraints

Variables quantitative, qualitative, fuzzy

Constraints algebraic and differential equations, difference equations, rules, etc.

time continuous, discrete

SensorController

Structure of controlled system

1 2 ....c p m nC C C C c c c

ProcessX YUYref

1 2 .... mZ X U Y z z z

U, subset of control variables

Y, subset of measured variables

X, subset of unknown variables

-

+

C : set of constraints

Cc CpCm

Z : set of variables

Structure = binary relation

S : C x Z {0, 1}

(ci, zj) S(ci, zj)

S=(C,Z)

: Known variables

X:Known variable

Z K X

K U Y

: Control constraints

: Structural constraints

: Measurement constraints

c

P

P

C

C

C


Bipartite graph

A graph is bipartite if its vertices can be partitioned into two disjoint subsets C and Z such that each edge has one endpoint in C and the other one in Z. Bi-partite graph : links between variables and constraints


Definition

The structural model of the system (C,Z) is a bipartite graphe (C,Z,A) , Where A is a set of edges defined as follows :

Example

, if the variable appearsin the constraints i j ic z A c

1 2 1 2

( , )

C : 0, : 0

S S C Z

C C C U Ri C y i

Z i u

C1

C2

i

y

u

C Z

Example bipartite graph (1)

1

2

3

4

5

c : 0

c : 0

c : 0

c : ( ) 0

c : 0

R

L

C

m C

e R L C

u Ri

diu L

dtdu

i Cdt

u F u

u u u u

mzzzZ ....:Variables 21

: Known variables (measured)

: Unknown variables

m e R L C

m e

R L C

Z u u u u u i K X

K u u

X u u u i

ueuC C0

uR i uL

R0

um

L0

1 2Constraints : .... nC c c c

0 0 0Parameters: = R L C

( , , )S S C Z

Remark !In some papers are introduced 2 additional constraints (differential) and corresponding variables to express just the derivative of variable:

6 1 7 2: , : cdudic z c z

dt dt

Example : bipartite graph (2)

m e R L CZ u u u u u i

1

2

3

4

5

Constraints

c : 0

c : 0

c : 0

c : ( ) 0

c : 0

R

L

C

m C

e R L C

u Ri

diu L

dtdu

C idt

u F u

u u u u

( ) 6

( ) 5

card Z Z

card C C

K=known variables X=Unknown variables

C

um

ue

uL

uC

uR

i

c1

c2

c3

c4

c5

Z

Cardinal = size (dimension) of a vector

Example : bipartite graph (3)

Cum

ue

uL

uC

uR

i

z1

z2

c1

c2

c3

c4

c5

c6

c7

Z

0:c

:c

0:c

0)(:c

0:c

0:c

0:c

sConstraint

27

16

5

4

3

2

1

dt

duz

dt

diz

uuuu

uFu

idt

duC

dt

diLu

Riu

C

CLRe

Cm

C

L

R

21 zziuuuuuZ CLRem

The differential constraints could be added

Differential constraints and variables

Incidence matrix A bipartite graph can be represented by an adjacency matrix (named incidence matrix). This is a Boolean

matrix where each row corresponds to a constraint ci and each column to a variable zj. A “1” at position (i, j) indicates that there is an edge connecting the constraint ci and the variable zj.

F/Z uR uL uC i um ue

c1 1 0 0 1 0 0c2 0 1 0 1 0 0c3 0 0 1 1 0 0c4 0 0 1 0 1 0c5 1 1 1 0 0 1

Variables Z

UnKnown variables Known variables

Co

nst

rain

ts C

The incidence matrix B is the matrix whose rows and column represent the set of constraints or variables, respectively. Every edge (ci, zj) is represented by « 1 » in the intersection of ci and zj.

1 if

otherwise 0

ij j i

j

b z c

z

Subsystem : definition

Definition 1. The Structure of a system is a bipartite graph G(C, Z, A) , where A is a set of

edges such that :

(c, z) C Z, a = (c, z) A the variable z appears in the constraint c

Definition 2. The structure of a constraint c is a subset of variables Z(c) such that : z

Z(c), (c, z) A

Definition 3. A subsystem is a pair (, Z()) where is a subsystem of C and Z() = c

Z(c).

Example of a subsystem

A subsystem is a pair (, Z()) where is a subset of C and Z() = c , Z(c).

C/Z uR uL uC i um ue

c1 1 0 0 1 0 0c2 0 1 0 1 0 0c3 0 0 1 1 0 0c4 0 0 1 0 1 0c5 1 1 1 0 0 1

C/Z uR uL i

c1 1 0 1c2 0 1 1

Subsystem (R,L)


Differential and algebraic equations

Are used three kinds of equations: Differential Algebraic Measure

Used variables are

)()(

),,(0

),,(

),,()(

txdtd

ztx

uxxh

uxxgy

uxxFtx

iii

da

da

dad

yuxxxZ dda }

sconstraint aldifferenti:

dtd

Fhgdtd

C

Hydraulic example

Tank dx(t)/dt - qi(t) + qo(t) = 0

Input valve c2: qi(t) - αu(t) = 0

Output pipe c3: q0(t) - kv(x(t)) = 0

Level sensor 1 c4: y1(t) - x(t) = 0

Level sensor 2 c5: y2(t) - x(t) = 0

Output flow sensor c6: y3(t) - qo (t) = 0

Control algorithm c7: u(t) = 1 if lmin y1(t) lmax

u(t) = 0 else

U(t)

y1 y2

y3

qi

q0

LC

x=volume

Bipartite graph and incidence matrix

c1: dx(t)/dt - qi(t) - qo(t) = 0

c2: qi(t) - αu(t) = 0

c3: q0(t) - kv(x(t)) = 0

c4: y1(t) - x(t) = 0

c5: y2(t) - x(t) = 0

c6: y3(t) - qo (t) = 0

c7: u(t) = 1 if lmin x(t) lmax

u(t) = 0 else

c1

c2

c3

c4

c5

c6

c7

x(t)

qi(t)

qo(t)

u(t)

y1(t)

y2(t)

y3(t)

Unknown variables

Known variables

Ci(i=1-7) x qi qo u y1 y2 y3

C1 Tank 1 1 1 0 0 0 0

C2 Valve 0 1 0 1 0 0 0

C3 Pipe 1 0 1 0 0 0 0

C4 LI1 1 0 0 0 1 0 0C5 LI2 1 0 0 0 0 1 1

C6 FI 0 0 1 0 0 0 1

C7 LC 0 0 0 1 1 0 0

State space model and digraph

1 1

2 2

3 3

f: ( ) ( ) ( ) 0

: ( ) ( ) 0

: ( ) ( ) 0

: ( ) ( ) 0

v

v

x t K x t u t

g y t x t

g y t x t

g y t K x t

U(t)

y1 y2

y3qi

q0

LC

x=volume

u

y3

y2

y1

x

Bipartie graph representation

Digraph representation

Subsystems A subsystem :

is a pair (Ci, ,Q(Ci) where Q(Ci) is the set of variables constrained by constraints Ci Q(Ci) consists of 2 parts

Qc(Ci): correspond to known variables and Qx(Ci): correspond aux unknown variables

Example : Hydraulic system

U(t)

y1 y2

y3qi

q0

LC

1, , ,i oTank COMP C x q q

C1 Q(C1)

Dulmage-Mendelsohn decomposition The number of solutions for Qx(Ci) obtained from Qc(Ci) characterize each subsystem

Any system can be uniquely decomposed into 3 subsystems : Over-constrained (C+,X+) Just-constrained (C0,X0) Under-constrained (C-,X-)

Only the over-constrained subsystem is monitorable

C/Z x X-{x} y1 y2

f1 1 0 1 0

f2 1 0 0 1

c1 : F1(y1, x) = 0

c2: F2 (y2, x) = 0

Example of overdetermined system

x=(F2)-1 (y2)

x=(F1)-1 (y1)


Under determined subsystem

(C, Q(C)) is under determined if, For each value of known variable Qc(C), the set of

unknown variables Qx(C) verifying the constraints C has a cardinal higher than one. : card(C)<card(Qx(C)) (number of equations less than number of variables)

Causes :• not enough equations to determine x• variables Qx(C) cannot be calculated from known variables

Qc(C) and constraints C. • Result of insufficient modeling of the system, or non

observability of certain variables.


Just and over determined subsystems

(C, Q(C)) is just determined if : card(C)=card(Qx(C))

• The unknown variables Qx(C) can be calculated uniquely from known variables Qc(C) and constraints C.

(C, Q(C)) is over determined : card(C)>card(Qx(C)) Causes

• Variables Qx(C) can be calculated in different ways from the known variables Qc (C) and the constraints C

• Each subset Ci C provides a different way to calculate Qx (C). Since the results of these calculations are identical (they are the same physical variables), there are some analytical redundancy

Examples (1/2)Z={X} U {K}X={u, i}, K={y1,}

C1: u-Ri=0C2: y1-u=0i

R

uy1

y1

0

C2(y1,U)=0 11 0

Subsystem : C1(i,u)=0 )(CQ)(CQ)Q(C CX 111

(C1, Q(C1)) is under determined

Card(C1)=1<Card(Qx (C1)=2.

)(CQC 1)(CQX 1

1 1C1(i,u)=0

u i

(C2, Q(C2)) is juste determined : Card(C2)=1=Card(Qx (C2)

(C, Q(C)) is juste détermined: Card(C)=2=Card(Qx (C)=2

Example (2/2)

1 1

y1

0

C2(y1,u)=0 11 0

C1(i,,u)=0

u i y2

0

C3(i,y2)=0 00 1 1

0 (C, Q(C)) is over determined: Card(C)=3>Card(Qx (C)=2

iR

uy1y2

Z=XUK

X={u, i}, K={y1, y2,}

C1: U-Ri=0

C2: y1-u=0

C3: y2-i=0

Example : Incidence matrix

C/Z u i

C1(i,u)=01 1

y1

0

C2(y1,u)=0 11 0

y2

0

C3(u,y2)=0 01 0 1

0

y2x={u, i}K={}C1: U-Ri=0

x={u, i}K={y1}

C1: U-Ri=0

C2: y1 –U=0

x={u, i}K={y1 ,y2,}

C1: U-Ri=0

C2: y1 –U=0

C3: y2-U=0i

R

uy1


Matching and ARRs

Definition of a matching

Consider the graph G(Cx, X, Ax), restriction of the structural graph of the system where Cx : Constraints related to unknown variables X Ax : set of edges linking Cx to X.

Let a AX, We note X(a) the end of a in X and CX(a) extremity of a in CX. The edge can be written as : a = (Cx(a), X(a))

XCA A={a1, a2, …an)

X={x1, x2, …xn)

C={c1, c2, …cn)

XC(x) X(a)

Cx(a)a


Matching : Definition (1/2) G(Cx, X, A) is a matching on G(Cx, X, Ax) if and only if

1) A Ax 2) a1, a2 A a1 a2 Cx(a1) Cx(a2) X(a1) X(a2)

Interpretation A matching is : a set of pairs (ci,xi) s.t. the variable xi can be computed by

solving the constraint ci, under the hypothesis that all other variables are known

XC(x)

X(a1)Cx(a1)

XC(x)

X(a2)

Cx(a2)

a1

a2

Matching : Definition (2/2)

145

Definition A mathing is a subset of edges such that any two edges have non common node (neither

in C nor in Z) Differents matchins can be defined on a bi-partite graph

C1(i,,u)=0

C2(y1,u)=0

C3(i,y2)=0

C1

C2

C3

i

u

y1

y2

C1

C2

C3

i

u

y1

y2

Different matchings of unknown variables

Maximal matching A maximal matching on G(Cx, X, Ax) is a matching G(Cx, X, A) s.t.:

A' A, A' A G(Cx, X, A') is not a matching.

What is it ? A maximal matching is a matching such that no edge can be added without violating the no

common node property

C1

C2

C3

i

u

y1

y2

This matching is not maximal w.r.t X

(C3,u) can be added C1

C2

C3

i

u

y1

y2This matching is maximal w.r.t X :Any matching can be added

Complete and incomplete matching

147

A matching β is complete w.r.t to C (set of constraints ) respectively to X (set of variables) if : x X, c C such that (c,x) β : complete w.r.t. C c C, x X such that (c,x) β : complete w.r.t. X

C1

C2

C3

i

u

y1

y2

This matching is incomplete w.r.t. to C(C3 is not matched) but complete w.r.t. to X

C1(i,,u)=0

C2(y1,u)=0

C3(u,y2)=0

C1 i

u

C1(i,,u)=0

This matching is complete w.r.t. to CBut incomplete w.r.t. to X

X (unknown variables)

K (known variables while measured)

CX


Matching and the incidence matrix 1/2

Select at most one "1" in each row and in each column

Each selected "1" represents an edge of the matching

No other edge should contain the same variable : it is the only one in the row

No other edge should contain the same constraint : it is the only one in the column.

Matching and the incidence matrix 2/2

C1

C2

C3

i

u

y1

y2

y2C/Z u i y1

C2(y1,u)=0

C1(u,i)=0

C3(u,y2)=0

0

0

0

0

1

1

0

1

1

101

y2

y2C/Z u i y1

C2(y1,u)=0

C1(u,i)=0

C3(u,y2)=0

0

0

0

0

1

1

0

1

1

101

y2C1

C2

C3

i

u

y1

y2


Causal interpretation of matchings Causal graph ?

The oriented bipartite graph which results from a causality assignment is named Causal graph Algebraic constraints

At least one variable can be matched in a given constraint Non invertible algebraic constraints Consider C(x1,x2)=0

C

x1x2

Possible matching

12 1( ) cannot be calculatedx C x

x1 x2

C

Impossible matching

calculated be cannot )( 21

1 xCx

C/Z x1 x2

C 1 11

C/Z x1 x2

C 1 1x


Oriented graph associated with a matching Causal and acausal constraint

u-Ri=0 : acausal constraint have not a direction. The variables have the same status: the graph is non oriented

U = Ri : causal constraint : i is known, u is calculated. Here the matching is chosen. The matched constraint is associated with one matched variable and with some non matched one

0

ui

C

C: u-Ri=0

Non matched constraint

u

i

C: U=RI

Matched constraint

Oriented graph


Oriented graph associated with a matching Matched constraints

the output is computed : the inputs are supposed to be known. The edges adjacent to a matched constraints are oriented

C/Z x x1 x2 x3

C1 1 1 1

C2 x x x

C3 x x x

C4 x x x

1

1

1

1

C-1(x1,x2,x3)

x1

x2

x3

x


Oriented graph associated with a matching Non-matched constraints

all the edges adjacent to a non-matched constraint are inputs. The relation C is redundant. All variables are inputs

C/Z x1 x2 x3

C1 1 1 1

C2

C3

C4

x1

x2

x3

c1

Maximal matching w.r.t. to XBut incomplete w.r.t. to C

C1 is redundant (is not used to eliminate X)

1

1

1


Structural propertiesDiagnosability conditions


Structural observability

Under derivative causality, the system is structurally observable if and only if : 1. All the unknown variables are reachable from the known ones (measure)

2. the over constrained and just-constrained subsystems are causal (no differential loop)

3. the under-constrained subsystems is empty


Over and just constrained system

The system is over-constrained if There is a causal matching which is complete w.r.t. all the unknown variables but not w.r.t. all the constraints.

• The unknown variables can be expressed (in several ways) as functions of the known variables.

• The subsystem is observable and redundant

The system is just-constrained if : There is a causal matching which is complete w.r.t. all the unknown variables

and all the constraints.• The unknown variables can be expressed as functions of the

known variables.• The subsystem is observable


Under-constrained system

The system is under-constrained if There is no causal matching which is complete w.r.t. the unknown variables.

• The subsystem is not observable, and not monitorable.

Structural monitorabilityThe conditions for a fault to be monitoable are :

1. the subsustem is observable 2. the fault belongs to the structurally observable over constrained part of the

subsystemm to be monitored

Under and juste constrained system

C1 i

u

C1: u-Ri=0

iR

1:C u Ri

No solution

C1: u-Ri=0C2: y1-u=0

iR

uy1

2 1

1

1

:

: 0

C u y

yC u Ri i

R

One solution

(non redundancy)

C1

C2

i

u

y1

u i

C1

y1

C21

u yC1 1

yi

R

❸ Oriented graph

Oriented graph All constraints are used: there is no a redundancy

❷ Bipartite graph

❶ System ❶ System

❷ Bipartite graph

❸ Oriented graph

Over constrained system (matching 1)

x={u, i}, K={y1 ,y2,}

C1: U-Ri=0, C2: y1 –U=0, C3: y2-U=0

C1

C2

C3

i

u

y1

y2

Maximal matching w.r.t. to X

Incomplète matching w.r.t. to C

y1

C2C1

1y

iR

C3

0 edge

0: 12 yyARR

y2

y2

iR

uy1

2 1 1: 0C y u y u 1

1: 0

u yC u Ri i

R R 1

2 1

3 2

: 0

: 0

: 0

C u Ri

C y u

C y u

1y u

3 2 2 1:C y u y y

❶ System

❷ Bipartite graph and incidence matrix ❸ Oriented graph and ARR

Over constrained system (matching 2)

y1

C3C1

2y

iR

C2

0 edge

y2

1

2 1

3 2

: 0

: 0

: 0

C u Ri

C y u

C y u

2y u

C1

C2

C3

i

u

y1

y2

y2C/Z u i y1

C2(y1,u)=0

C1(u,i)=0

C3(u,y2)=0

0

0

0

0

1

1

0

1

1

101

y2

1 2: 0ARR y y

3 2 2: 0C y u y u 2

1: 0

yC u Ri i

R

2 1 1 2:C y u y y

Exercise

y2

iR

uy1

❷ Constraints

❸ Bipartite graph and incidence matrix

❶ System

❹ Oriented graph and ARR

Alternated chain

What is alternated chains ? A path between two nodes (variables or constraints) alternates always successively variables

and constraints nodes : this path is said alternated chain Lenth of alternated chain ?

Number of constraints accrosed along the path

Reachability A variable x1 is reachable from variable x2 if there exists an alternated chain from x1 to x2

Example

C21

u y C11

yi

R

Number of constraints : 2

Number of variables : 3

Lenth of alternated chain : 2

The variable i is reachable from y1The path between i and y1 is : y1→C1 →u →C1 →iy1

Nodes

Hydraulic example : differential constraint

R

y

V

1

2

3

4

C : ( ) ( ) 0

C : ( ) 0

C :

C :

i o

o

V q t q t

q t RV

y bV

dz V

dt

C1

C2

C3

V

y

qi

qo

z

C4

y

C3

V

C2

qo

C4

z

qi

C1

Zero

Zero edge

)(tqo

)(tqi

ytqK

zVtqX

i

o

),(

,),(

Maximal matching w.r.t. to X

Incomplète matching w.r.t. to C

Graphe bipartite

yV

b

/dy bz V

dt

0 ( )y

q t RV Rb

1i

dy RyRRA q

b dt b


Differential constraints

Differential constraints can always be represented under the form: x2 = dx1/ dt

Derivative and integral causality Derivative causality

Integral causality

dtdx

x 12 1x 2x

)0(121 xdtxx 2x

)0(1x1x Initial conditions must be known


Loops

Definitions In the oriented graph, loops are a special subset of constraints, which have to be solved

simultaneously, because the output signals of some constraints in the loop are the inputs are some others in the same loop : the number of matched variables is equal to the number of constraints (length of the loop).

Algebraic loop

C/Z x1 x2

C1 1 1

C2 1 11

1

C3

V

C2

qo

x2

C1

x1

C2

Differential loop: example

V

C2C4qi

C1

zq0

RV

Vdtd

z

RVtq

tqtqV

o

oi

:C

)(:C

)()(:C

4

2

1

)(tqo

)(tqi

)(

,),(

tqK

zVtqX

i

o

Differential loop

dV

dt

1) Using derivative causality : there is no solution

2) Using integral causality : there is one solution if initial condition is known

1

2

C : ( ) ( ) (0)

C : ( )i o

o

V q t q t dt V

q t RV

V

C2qi

C1

q0

q0

(0)V


Differential loop How to broke the loop

Adding a sensor A matching without any differential loop is called a causal matching

V

C2

C4qi

C1

zq0

C3

y

Vdt

dz

bVy

RVtq

tqtqV

o

oi

:C

:C

)(:C

)()(:C

4

3

2

1

Example just-constrained system

Vdtd

z

bVy

RVtq

tqtqV

o

oi

:C

:C

)(:C

)()(:C

4

3

2

1

V

C2

C4

C1

z

q0

C3

y

qi

C/Z z=dV/dt V qi qo y

C1 1 0 1 1 0

C2 0 1 0 1 0

C3 0 1 0 0 1

C4 1 1 0 0 01

1

1

1

All unknown variables matched

All

con

stra

ints

are

mat

ched

y K

VqqV x oi

known)(

unknown)(

Suppose input flow qi is unknown

Example Over-constrained system

)(:C

:C

:C

)(:C

)()(:C

i5

4

3

2

1

uFq

Vdtd

z

RVy

aVtq

tqtqV

o

oi

V

C2

C4

C1

z

q0

C3

yu

C5

qi

C/Z z=dV/dt V qi qo y u

C1 1 0 1 1 0 0

C2 0 1 0 1 0 0

C3 0 1 0 0 1 0

C4 1 1 0 0 0 0

C5 0 0 1 0 0 1

1

1

1

1

All unknown variables matched

C1 i

s n

ot m

atch

ed

uy K

VqqV x oi

known)(

unknown)(

Redundancy

What is happened in integral causality?

)(:C

:C

)(:C

)0()()(:C

i5

3

2

1

uFq

bVy

RVtq

VdttqtqV

o

oi

V

C2

C1

q0

C3

yu

qi

V(0)

C5

C/Z V(0) V qi qo y u

C1 1 0 1 1 0 0

C2 0 1 0 1 0 0

C3 0 1 0 0 1 0

C5 0 0 1 0 0 1

1

1

1

X :All unknown variables matched

C :

All

con

stra

ints

are

mat

ched

uy K

VVqqV x oi

known)(

)0(unknown)(

The system is now just-determined : the matching iscomplete w.r.t to X and C.

1

Example under-constrained system

)(:C

:C

)(:C

)()(:C

i5

4

2

1

uFq

Vdtd

z

RVtq

tqtqV

o

oi

V

C2

C4

C1

z

q0

u

qi

C5

C/Z z=dV/dt V qi qo u

C1 1 0 1 1 0

C2 0 1 0 1 0

C4 1 1 0 0 0

C5 0 0 1 0 1

1

u K

VqqV x oi

known)(

unknown)(

1

1

1

The system is not observableThere is a differential loop


Conclusions (1/2) Structural analysis based on bipartite graphs is easy to understand, easy to apply,

Shows the relation between constraints and components,

Allows to : identify the monitorable part of the system, i.e. the subset of the system components whose faults can be detected and isolated,

Advantages Easy to implement and suited for complex systems Allows to determine the FDI/FTC possibilities No a priori knowledge of the model equations is necessary

Lack Structural analysis produces only structural properties


Conclusiosn (2/2) :What we can do with structural analysis ?

can the system be observed ? can all the system variables be computed from the knowledge of the sensors outputs can the system be controlled ?

can the system be monitored ? can the malfunction of the system components be detected and isolated

can the system be reconfigured ? can the system achieve some objective in spite of the malfunction of some components

Actual properties are only potential when structural properties are satisfied.

They can certainly not be true when structural properties are not satisfied.

Structural properties are properties which hold for actual systems almost everywhere in the space of their independent parameters


Chapter 3 : Observer-based approaches


Introduction Principle of FDI methods observer based

Reconstruction of the output from sensor and comparison of this estimation with the real output

In function of the system:• deterministe case : estimation with observers• Stochastic case : Kalman filter

Observer ? Is a state reconstructor that from measured variables preform estimation of state vector Software sensor !


What is observer ? Given

How to reconstruct based on output error

Processu x

C y

pmn uyx

tCxy

tButAxtx

,,

,)(

)()()(

0ˆ)0(ˆ

)(ˆˆ

)(ˆ)()()(ˆ)(ˆ

xx

txCy

txCtyKtButxAtx

0ˆ)0(ˆ

)(ˆˆ

)(

)()(ˆ)(ˆ

xx

txCy

ty

tuKBtxKCAtx

Simulation of the observer

)(ˆˆ

)(

)()(ˆ)(ˆ

txCy

ty

tuKBtxKCAtx

Cx

x

0x

y

)(tu

A-KC

KB


Observer and process

A

Cxx

0x

y)(tu

+

B

PROCESS

x

y

B

Kx

AA

C

xAˆ

)(tBu

)ˆ( yyK +

-

+

+

OBSERVERxCˆ

y

Convergence (1/2) Convergence conditions

pmn uyxtCxy

tButAxtx

,,,

)(

)()()(

)(ˆˆ

)(ˆ)()()(ˆ)(ˆ

txCy

txCtyKtButxAtx

)ˆ()ˆ()ˆ(ˆˆ~ˆ~

xxKCxxAxCyKBuxABuAxxxx

xxx

Convergence (2/2)

Erreur d’estimation

xKCAxxKCxxAxdt

xxd ~))ˆ()ˆ(~)ˆ(

001 ˆ)(~ xxKCApIpx )(.

)(tKCA

dttd

e s’annule exponentiellement si (A-KC) est asymptotiquement stable i.e. valeurs propres (modes) sont à partie réelles négatives :

Comment ? : Bien choisir K


Remarks

Conclusion The reconstruction error is not zero because

• The IC of the observer is choosen arbitraly and IC of the process are unknowns

How to cacal the error: We can act only on K: then choose K to stabilize the matrix A-KC ensuring convergence to zero the error

• Used Techniques: Poles Placement used to set the speed of convergence by adjusting the coefficient K (see the instructions on Matlab place and acker


Idea of diagnosis based observer

Estimation error cannot be generated (the state is not measured)

But : error of the recontructor can be calculated while Y is measured mesurée

Scheme :

Residual

Process

Observer

Compare

u

y

y

ˆy y

ˆx x

ˆy y

How to generate residuals ? 1. Par simulation

)(ˆˆ

)(

)()(ˆ)(ˆ

txCy

ty

tuKBtxKCAtx

Cx x

0xy

)(tu

A-KC

KB

Sensory +

-Residual

+

+y

process

Calculation of residual using z transform

1

Estimated y

ˆ ˆ ( ) .y Cx C zI A KC Bu Ky

)ˆ(ˆˆ xCyKBuxAxz

KyBuKCxAzIxz

KyBuxKCxAxz

)(ˆ

ˆˆˆ

KyBuKCAzIx .)(ˆ 1

BuKyKCAzICyyyzr 1(ˆ)(Residual

Calcul du résidu en p

(2)

0)0(0)(

)()()(xxt

tCxy

tButAxtx

)()(

)()( 0

pCXpY

xpBUApIpXL

(1) 01 )()( xpBUApICpY

?)(ˆ pY de Calcul

)(ˆˆ

)(ˆ)()()(ˆ)(ˆ

txCy

txCtyKtButxAtx

01 ˆ)()()(ˆ xpKYpBUKCApICpY

Using P transform

01

0111 ˆ)())(.()(

)()(~

)(ˆ)(

xpBUKCApICxpBUApICKKCApICI

prpYpYpY

(1)-(2) : Rsidual

01

011 ˆ)()().()( xKCApICxApICKKCApICIpr

Aprés quelques simplifications

111111)( VPUVPIUPPUVPLemme d’inversion de matrice :

00 xx ˆ)( 1 KCApICprResidual


Convergence and sensitivity to the noise

Analysis of r(p) 1. The reconstruction error of the output depends on the estimation error of

the CI

2. Dilemma between : convergence of the observer and the residue sensitivity to noise

• Choose the gain K so that the error converges rapidly (by imposing the eigenvalues of the matrix very low) : But if K becomes too sensitive to random noise residue


Example

Simple monovariable case

Convergence de l’erreur

)ˆ(42)ˆ()ˆ(2~ˆ)ˆ(xxKxxKCxxxxx

dt

xxd

)42()( 0

Kpp

)(.42)(

tKdt

td xxt ˆ)(

0)0(,0)(4

)()(2)(xxt

txy

tutxtx

0ˆ)0(ˆ,0

)(ˆ4ˆ

)(ˆ4)()()(ˆ2)(ˆxxt

txy

txtyKtutxtx

24

.)( 0

k

et t

5,00240 kk

Stability conditions


Simulation

0ˆ)0(ˆ,0

)(ˆ4ˆ

)(ˆ4)()()(ˆ2)(ˆxxt

txy

txtyKtutxtx

BU

PROCESS

dx/dtx

Ax

Xestimé

yestimé

Résidu

dxest/dt

OBSERVATEUR

U

1s

1s

A

A

C

K

B

C

Capteur

SIMULATION


Generalized Luenberger Observer

Given:

1. We want to estimate the output y(t) Is used observer of gain K

X(t) : state,u(t) : inputd(t) : faultse(t) : distubancess or noises

sappropriée dimensions de matrice : ,,,,,

,,,

)0(

)()()()(

)()()()()(

0

EFDCBA

uyx

xx

teEtdFtDutCxy

teEtdFtButAxtx

pmn

yy

xx

(1)

0ˆ)0(ˆ

)()(ˆˆ

)()(ˆ)()()(ˆ)(ˆ

xx

tDutxCy

tDutxCtyKtButxAtx

(2)


Erreurs estimation

2. Dynamic equations of the error estimation

(1)- (2)

3. Laplace trasnform of output error

)3(~)0(~

)()()(~.ˆ~

)()()(~~ˆ

0

xx

teEtdFtxCyyy

teEtdFtxKCAdt

xd

dt

xxd

yy

xx

)4()(~).()().()().()(~0 pxpGpepGpdpGpy ed 1

0

1

1

)(

)(

)(

KCApICpG

EKEEKCApICpG

FKFFKCApICpG

yyxe

yyxd


Remarks about the residual

1. Le résidue is sensitive to fault d(p), to disturbances and noises e(p), but also to the IC. Observation converge to 0 for t, we can neglect transitory due of CI.

2. If d=0, e=0, we have the expression obtained previously..

3. The gain K of the observer affects similarly d and e: So it is difficult to generate a residual sensitive to faults but not to disturbances

4. Analysis of matrices G indicates whether components are to be isolated from other

)4()(~).()().()().()(~0 pxpGpepGpdpGpy ed

Different influences to the residue

1. Influence of the noiseLet e(t) noise realization of a Esp (e (t) = 0 random variable ²

Find the residue in frequential • Using the above equations the terms of reconstruction errors are obtained

(assuming D = 1 Ey = 0)

0)0(

)()()(

)()()(

xx

teEtDutCxy

tButAxtx

y

0)0(

)()()(ˆˆ

))(ˆ()()()(ˆ

xx

teEtDutxCy

txCyKtButAxtx

y

Observer

)()(~)(~ tKetxKCAtx Fréquentiel )(ˆ)(~

001 pKexxKCApIpx

)()(~)()(~ tetxCtrty

)(ˆ)()(~ 100

1 peKKCApICIxxKCApICprpy

Influence of the noise to the residue

Négligeons d’abord l’influence des CI

Etude de l’influence du point de vue fréquentiel de e sur r(p)

Reduction of the noise e(jω) and r(jω) : Find a gain K, by placing the cut-off frequency of the filter such as the influence of noise is reduced

)(.)( 1 peKKCApICIpr

)(e )(r 1()( KCApICIpr

Calcul du seuil d’alarmes du résidu

Soit données les hypotheses statistiques du bruit :

Consider the estimator

0

( ( ) 0

( ( ))

Esp e t

Var e t V

)()(~)(~ tKetxKCAtx if Esp(e(t)) 0 ( ( )) 0Esp x t

If average noise e is null it is the same for the estimator

)()(ˆ

)()(ˆ)(ˆ

)()(

)()()(

tetxCy

tButxAtx

tetCxy

tButAxtx

Estimator

ˆ ( ) ( ) ( )x x x t A KC x t Ke t

Calculation of the alarm threshold of the residue

Equation variance propagation

Application to the error estimation

( ) ( ) ( )x t Ax t BW t TW

Txxx BtBVAtVtAVtV )()()()(

0)()()(~)(~

)()(~)(~VteVar

tetxCty

tKetxKCAtx

où

0~~

0~~~

).()(

))(()()(

VCtCVtV

KKVKCAtVtVKCAtVT

xy

TTxxx

Calculation of the alarm threshold of the residue

Threshold in stationary regim Determine a threshold in the decision process of the presence of faults based on the variance of y beyond which the

residue will be considered null (there is really an alarm)

0~~ ).()( VCtCVtV Txy

K

V 0 TT

xx

x

KKVKCAtVtVKCA

tV

0~~

~

))(()(0

0)(

:restationnaiEtat

Détermination of variance of the residual

t

Threshold

0

ALARM

NORMAL

: Residual y(t)

yV xV


2. Influence d’une erreur de modélisation

Problematic In practice there is always a modeling error Observer built from the model, then the reconstructed output is sensitive to

modeling errors Diagnosis is based on the difference between real and reconstructed output

• Difficult to separate due to modeling errors and those due to faults

Goal Build an observer sensitive to faults and insensitive to modeling errors


DéveloppementLet uncertain state model : consider error only on A

Estimation of the state

Cet observateur doit alors détecter, au travers de l’erreur de reconstruction de la sortie, la perturbation du système A

0)0(

)(

)()()()(

xx

tCxy

tButxAAtx δ Traduit l’apparition d’une perturbation A sur le système

0ˆ)0(ˆ

)(ˆ

)(ˆ)()()(ˆ)(ˆ

xx

txCy

txCtyKtButxAtx Représente un observateur calé sur le système nominal


Error hypothesis

Assumptions about the error Bounded : i.e slight inaccuracy of the model coefficients

Problem to solve : générate residuals 1. less sensitive to A 2. with a maximum sensitivity to faults


Influence of parameter uncertainties 1. Influence of variations of A to the residues

Error estimation (from previous equations) :

Frequential domain

)(ˆ)(~

)(ˆ)(~

100

1

100

1

pAxKCApICxxKCApICpy

pAxKCApIxxKCApIpx

)(ˆ)()(~ txtxtx

00 ˆ)0(~

)()(~)()(~

xxx

tAxtxKCAdt

txd

The reconstruction error is sensitive to inaccuracies A and to the state x(t) (not eliminated here)

Influence of input and A Influence of input u to the resdiue

For IC=0, and replacing x(p) by its expression we have :

Then residue depends on u and A • We exploit this property to distinguish the influences to the residue of faults and

uncertainties• How ? : • While A is unknown , the error estimation is expressed in terms of what is applied

(i.e. u) for (A )• we calculate the threshold for max A

)(max~ 11 tuBKCAIjAKCAIjCyA

u(p).)()(~ 11 BKCApIAKCApICprpy

Decision Scheme of the decision procedure

1. If the residual value is below the threshold then diagnosis is reserved because the error may be due to uncertainties

2. Beyond this threshold amplitude of the residue indicates the presence of a fault different from model errors

U (bornée)Upper bound of the construction error (residue)

)max( A

seuil : ~y

t0

ALARM

NORMAL

(t)y~ Résidu

~y

~y


Unknown Input Observers (UIO) Problematic

Models where the output of the actuators is not measured Evaluation of RRAs requires knowledge measures and inputs So: is used unknown input observers (UIO: Unknown Input Observers)

Principle Let a system with known inputs u(t) And unknown inputs )(tu


Observateur à entrée inconnue Let system with UI

Consider then the following observer :

The error estimation will be :

( ) ( ) ( ) ( )

( )

known, : unknown

x t Ax t Bu t Fu t

y Cx t

u : u

)()()(ˆ

)()()()(

tEytztx

tKytGutNxtz

)()()()(~)()()()()()()(ˆ)()(~

tztxECItx

tECxtztxtEytztxtxtxtx

Let intermediate variable

ˆ( ) ( )z t Tx t


Differentiating and substituting x (t) and z (t), then::

)()()()()()()()(~ tKytGutNztuFtButAxECItx

)()()()()()(~)(~ tuPFtuGPBtxKCNPPAtxNtx

Let : P = I+EC


The reconstruction error of the state of the UIO

While the input is unknown, we try to have :

This reconstruction tends then asymptotically to zero iff :

)()()()()()(~)(~ tuPFtuGPBtxKCNPPAtxNtx

)(~~)(~)(~

txCy

txNtx

stable N

PF

PBG

NPPAKC

ECIP

0

stable N

NEKL

KCPAN

PBG

CCFFIP 1)(

1)(0 CFFEECFF


Calculation of UIO Procedure to calculate the UIO

Calculate the generalized inverse of CF Deduct P and G We fix the poles of N and then we deduce K and N L is calculated

The unknown input is not involved in the expression of residue.


Estimation of UI Initial equation of the system :

If (CF)-1 exists we will have :

inconnu

Connu

:

)(

)()()()(

u

u:

tCxy

tuFtButAxtx

)()(

)()( 1 tCButCAx

dttdy

CFtu


Different UIO schemes SOS : Simplified Observer Scheme

Only one UIO Allows to detect faults. No isolation possibilities

DOS : Dedicated Observer Scheme Bank of UIO Each observer is sensitive to one fault (diagonal structure)

D.O.S w.r.t. actuators

Actuators System Sensorsu y

UIO 1

u1umue1

emuUIO mu

Diagonal structure w.r.t. actuator faults

D.O.S w.r.t. sensors


UIO 1u1

umue1

emuUIO mu

Diagonal structure w.r.t. sensor faults

G.O.S w.r.t. actuators


UIO 1

u1umue1

emuUIO mu

Each residual is affected by all faults except for one sensor fault


BOND GRAPH FOR ROBUST FDI

Chap.5/214


PLAN 1) Motivations et positionnement

2) Problématique des méthodes à base de modèles

3) Bond graph et le diagnostic

4) Conception d’un système de supervision

5) Outil logiciel pour la conception de systèmes de supervision

6) Application a un générateur de vapeur


Contexte Résultats de recherche depuis 12 ans

B. Ould Bouamama and A.K. Samantaray. "Model-based Process Supervision. A Bond Graph Approach" . Springer Verlag, To be published on 2007, Berlin.

Thoma J.U. et B. Ould Bouamama. "Modeling and Simulation in Thermal and Chemical Engineering". A Bond Graph Approach. Springer Verlag, 219 pages, Berlin 2000.

More : Web : http://sfsd.polytech-lille.net/BelkacemOuldBouamama

Applications Projet Européens (CHEM, damadics) supervision de procédés chimiques et pétrochimiques, raffinerie de sucre , .. Projet nationaux : EDF Filtrage d’alarmes Projet régional : supervision de procédés non stationnaires

Outils logiciels développés Model Builder « FDIPAD » Génération de modèles et d’indicateurs de fautes formels à partir des PIDs Analyse de la surveillabilité : placement de capteurs Génération de S-function ou code C pour la simulation

La supervision aujourd’hui dans l’industrie

Integrated design for supervision

P&ID

Generate a dynamic and formal models

Generate a formal and robust ARRS

Optimal sensor placement

Diagnosability results

New sensor architecture

Process

Online implementation

Online implementation

Data from sensors

Sensors

Technical specifications

Diagnosability analysis

ARRs Uncertain Parameters

Conception intégrée de systèmes pilotés : Démarche

Thème 1

Propriétés formelles et comportementales

,...,,,,,,

),(

),,(

SeICRyux

xCy

uxFx

Dynamique Modèle

Indicateurs de fautes formels

Dimension-nement

Synthèse de lois de commande

Thème 2

Placement de Capteurs et actionneurs

Propriétés structurelles et causales

Commandabilité,

Observabilité

Surveillabilité,

Reconfigurabilité

Simplification de modèles

Thème 2

Thème 3Informatisation

Test en ligne


Pourquoi les BGs pour la conception intégrée ? Graphes et Bond Graphs : quelles différences ?

)(

),(

xCy

uxFxModèle

SfSeDfDe

GYTF

JICRS

ASG

C

,,

),(

219


Génération automatique des modèles

220


Why Graphical Approach for integrated design?

Graphical methods that are based essentially on structural models Graph structures independent of the numerical values of the syst. parameters. Structural properties are independent of the values of the system Structural description of a system expresses only the links between the variables

and the constraints Visualization of the system topology

Many different kinds of models linear, non linear can be used (qualitative, quantitative, static, dynamic, rules, look-up tables, …)

Lack Structural analysis produces only structural properties

221


State of art

222

BOND GRAPH For MODELLING (1959)

Control (Vergé, Gawtrop, Dauphin, Sueur, Rahmani..) 1991

Diagnosis Sizing

Qualitative approach (1993)Linkens, Mosterman, Kohda, ..

Quantitative approche (1995)

Coupled BG (Ould Bouamama 198)

Robust Diagnosis Extension to coupled BG Automated Diagnosis Design of supervision system

Opend loop system Linear Systems Sensor and actuator Faults

Monoenergy Bond Graph (Tagina 95)

Hybrid Bond Graph (Biswas, Mosterman (USA)


Model based approach : Issues

MODELLING Modelling step is most important in FDI design

obtaining the model is a difficult task

The constraints are not deduced in a systematic way

It is not trivial in the real systems to write the model under a "beautiful" form x=f(x,u,θ).

RESIDUAL GENERATION Eliminate the unknowns : analytic redundancy approach

• Existing methodology : parity space for linear, elimination theory (constraints under

polynomial forms)

Variables to be considered : all quantities constrained by the system components (process,

actuators, sensors, algorithms)

How to generate directly from the process ARRs and models : Bond graph tool well suited

because of its causal and structural properties.

DEFINITION, REPRESENTATION

DEFINITION

REPRESENTATION

P = e.f

e

f

1 2

Mechanical power :

Notion de causalités

f

e

f

e

f

e

A BfA Bf

e

f

A B

e

f

AA BB

Electrical

DOMAIN

Mechanical (rotation)

Hydraulic

Chemical

Thermal

Economic

Mechanical (translation)

POWER VARIABLES FOR SEVERAL DOMAINS

VOLTAGE

u [V]

CURRENT

i [A]

FORCE

F [N]

VELOCITY

v [m/s]

FLOW (f)EFFORT (e)

TORQUE

[Nm]

ANGULAR VELOCITY

[rad/s]

UNIT PRICE

Pu [$/unit]

FLOW OF ORDERS

fc [unit/period]

CHEM. POTENTIAL

[J/mole]

MOLAR FLOW

PRESSURE

P [pa]

VOLUME FLOW

/s][m3V

[mole/s]n

TEMPERATURE

T [K]

ENTROPY FLOW

[J/(K.s]]S

T2

On-Off

Vo

QO

PI

T1

Tank2

0

C:C1

De2

6

Tank1

0

C:C1

De1

2

Pump

MSf1

1

T2

On-Off

Valve1

1

R:R1

4

3 5

Valve 2

1

R:R1

Se17

8

9

PIu1

On-off

USER

u3

PI

T1

Vo

QO

Out

flow

to c

onsu

mer

Specialized software for Bond graph modelling


3) Bond graph and diagnostic :determinsit and robust case

230

Prof. B. Ould Bouamama Polytech Lille « Supervision Systems Design» 231

Bipartite graphs and Bond graphs The structural model of the system (C,Z) is a bipartite graphe (C,Z,A)

The constraints C from the bond graph model consist of structural Cs, behavioral Cb and measurement equations Cm:

The structural constraints are deduced from the set of junction equations which represent the mass and energy conservation laws. The number of junction equations is then equal to the number of equations in 0-junction

(common effort), 1-junction (common flow) and 2-ports elements (transformer TF, gyrator GY):

s b mC C C C

,

0 1s J J TF GYC C C C C

232

Behavior equations (Cb) describe the physical phenomena occurred in passive BG elements (Resistive R , Capacitive C and Inertial I):

Measurement (Cm) equations represent the sensor equations

De and Df are effort and flow detectors respectively. The set of variables

The set of variables Z consists of known (K) and unknown (X) variables. The known variable set K contains the effort (Se) and flow (Sf) source variables :

Unknown variables X are the pair of conjugated power variables (flow and effort):

b C I R

C C C C

m De DfC C C

MSe MSf Se Sf De DfK

1 1 2 2, , ,..., ,

n nX e f e f e f

Cardinality from BG model Consider the jth junction structure (JS) where occur several phenomena represented by set of n bond graph elements E : E1, …Em To this junctions are connected m sensors : S1, …Sm

This junction is completely defined by one structural equation (energy conservation) , n behavioral equations (how this energy is transformed) and m measurement equations.

E I C R De Df Se Sf

1

1

( )

( ) 1

( )

nj

b b ii

s

mj

m m jj

card C C E

card C

card C C S

The cardinal of unknown variables The number of unknown variables in 0-junction is equal to the set of flow variables plus the

common effort variable which links all elements Similarly on the 1-junction, the number of unknown variables is the sum of effort variables

labeling the components bond graph plus the common flow variable General case, the unknown variables cardinal can be written by the relation:

For global system Consider now the global bond graph model of the system to be monitored which consists of

junctions . The cardinal of the unknown variables and the cardinal of constraints can be given through the following relations:

1

( ) 1n

j j

iicard X X E

jX

( ) 2*

( ) 2*b m

b

card C N N G

card X N G

: Number of Tf and GY while there are 2 equations for GY and TFG

: Number of Physical components (R,I,C)

: Number of sensors (De,Df)

: Number of Gy and TF

:Number of Junctions (0 and 1)

b

m

N

N

G


ARRs generation from Bond Graphs ARR is a constraint calculated from over determined subsystem where all variables are known:

In a bond graph representation ARR is

0F(K)=

( , , , , , , ) 0

A A M M

F De Df Se Sf MSe MSf r

R L R J k

Covering causal path

Définion (Causal path) A causal path between two ports is an alternation of bonds and basic bond graph elements

(named nodes) such that (i) all nodes have a correct and complete causality, and (ii) two bonds of the path have in the same node opposite causal stroke direction.

Simple direct Causal path : covered following only one variable (effort or flow).

Indirect causal path : one element (R,C, I) should be crossed along the path

Mixad causal path : it comprises a gyrator (GY) imposing the change of followed variable

236

e1 0 1

f

e0 1 0

f

Passive element (R, C, I

f

e f

f

e1 GY

f

e

f

Causal path and causality

E CiC UC

i

F CiC UC

dt

dECiC .Se:E

UCiC

idtC

UC.1Sf:i UC

i

UC iCUC

i

C

0Se:E

iC

Derivative causality

0

C

Sf: i

Integral causality

How causal path can help for simulation !

E

R1

g

C

iUc

UR

1

R:R1

C:C1Se:E

E

UR

Uc

ie ic

ir

1

1R

For R elemntUR

irURR:R1 1

1

RUi Rr

For C elementUc

ic

C:C1 1

1

C

dtiC

U cc 1

1ic

For 1 junction ❶EUc

UR

+

-

EUc

UR

Df:i

Df:i

Df:i

Dualised sensors

I

Se

Df

R

1

SSf

R L

i A

R L

i A

R L

Se: ui A

RL circuit

I

Se

Df

R

1

Bond graph model in integral causality

For control and simulation

Bond graph model in derivative causality with dualised sensor why ?

Initial Conditions no knowns Df : as source of information

De

I

Se

Df

C

R

1 0 SSe

SSf

I

Se

Df

C

R

1 0

SSf

Pas de conflit de causalité,

Système sur-déterminéConflit de causalité,

Système sous-Déterminé

?

Example a DC motor

ELECTRICAL PART ua

ia

MECHANICALPART

w LOAD


Systematic State equations generation

242

wua

ia

(J,f)RaLa

im

m

MSe:Ua ia

ua 1L

w

I:J

w

R:f

Se:-L

f

J

1

R:Ra

I:La

uM

ia

uRa

uLa

ia

MGY

:K

w

Df:mDf:im

Prof. B. Ould Bouamama Polytech Lille « Supervision Systems Design»243

Automated Control analysis


Algorithme de génération des RRAs à partir du modèle BG

244

❶ Put the BG model in derivative causality dualising sensors

MSe:Ua ia

ua 1L

w

I:J

w

R:f

Se:-L

f

J

1

R:Ra

I:La

uM

ia

uRa

uLa

ia

MGY

:K

w

SSf:mSSf:im


Structural analysis

Cardinal of constraints Cardinal of Unknown variables

245

1

2

1

1

1

2

: 0,

: 0

: 0

: 0

: 0,

: 0

: 0

: 0

:

:

J A A R L e R I e

RA R A R

LLA I A

GY e e

J M L R I e R I e

GY e e

RM R M R

IJM I M

m m

m m

C U U U U i i i i

C U R i

diC U L

dtC Ki

C

C U K

C R

dC J

dtC i i

C

, , , , , ,

, ,

R R L L e e e e I I R R

m m A L

X U i U i U i

K i U

While: et R I e e I Ri i i i

( ) 2 4 2 8card X

( ) 2*b

card X N G ( ) 2*

2 4 2 2 10b m

card C N N G


Incidence matrix and Bipartie graph of the Dc motor

246

2

1

2

1

2

1

1

1

m

m

d

d

JM

RM

LA

RA

GY

GY

MJ

AJ

C

C

C

C

C

C

C

C

C

C

C

C

1

2

R

L

e

A

A

e

J

R

L

m

m

U

U

U

U

i

z

z

i

C/Z Unknown Variables X

Known Variables K

UR UL Ue i e

J

R

UA L

mi m

CJ1A

1 1 1 0 0 0 0 0 1 0 0 0

CJ1M

0 0 0 0 1 1 1 0 0 1 0 0

CGY1

0 0 ❶ 0 0 0 0 1 0 0 0 0

CGY2

0 0 0 1 ❶ 0 0 0 0 0 0 0

CRA

❶ 0 0 1 0 0 0 0 0 0 0 0

CLA

0 ❶ 0 1 0 0 0 0 0 0 0 0

CRM

0 0 0 0 0 0 ❶ 1 0 0 0 0

CJM

0 0 0 0 0 ❶ 0 1 0 1 0 0

Cm1

0 0 0 ❶ 0 0 0 0 0 0 1 0

Cm2

0 0 0 0 0 0 0 ❶ 0 0 0 1

❷ The structure junction (conservative law equation) associated with at least one sensor represents the candidate

247

1

1

0, "0" Junction

ARRs Candidates

0, "1" Junction

n

ii

n

ii

f

e

MSe:Ua

ia

ua1

L

I:J

R:f

Se:-L

f

J

1

R:Ra

I:La

uM

ia

uRa

uLaia

MGY:K

SSf:mSSf:im

11

n

A RA La Mi

U U U U RRA

21

n

f j Li

RRA

❸ The unknown variables are eliminated using covering causal paths from unknwn to known variables (measured and control signal)

MSe:Ua

ua1

L

w

I:J

w

R:RM

Se:-L

f

J

1

R:RA

I:La

uM

ia

uRa

uLaia

MGY:K

w

SSf:mSSf:im

?AU

?RAU 1 (.) :RA RA A mU i SSf i RA A mU R i

?LAU 1 ( ) :LA LA LA A mU U i SSf i

?MU 1 ( ) :M GY M mU U SSf

mLA A

diU L

dt

M mU K

:A AU MSe U :A AU MSe U

11

n

A RA La Mi

U U U U RRA

Oriented graph

1 me A m A m

diRRA MS R i L K

dt

mi:SSf

1mCAi

RAC

LAC

LAU

2GYC

m:SSf 2mC 1GYC

AU

MU

Ai

JMC

RAC R

AU:MSe

: LSe

AJC 1

MJC 1

J

RAU

L

2m

L M m M m

dRRA R J K

dt

Decision procedure: monitorability analysis

Ri/fautes L Re Ua Im Wm Jm Rm

R1 1 1 1 1 1 0 0

R2 0 0 0 1 1 1 1

Decision procedure: monitorability analysis

Ri/fautes R1 R2 Mb Ib

f1 Se:Ua 1 0 1 0

f2 Df:im 1 1 1 0

f3 Partie élec. 1 0 1 0

f4 GY 1 1 1 0

f5 Df:ωm 1 1 1 0

f6 Partie méca. 0 1 1 1

fi/fj f1 f2 f3 f4 f5 f6

f1 0 1 0 1 1 2

f2 0 1 0 0 1

f3 0 1 1 2

f4 0 0 1

f5 0 1

f6 0

Hamming Distance

Informatisation FDIPAD

Robustness problem

How to fix threshold ?

(a)

(b)

Défaut sur capteur du

courant égal à 15% de sa

valeur nominale

Fonctionnement normal

21

1 3

N

ii

S x x SN

Seuil simple: 3*std

What about parameter uncertainties ?

False alam because of parameter uncertainties !!!!

introduction of 5% of nominal value of RM

Linear Fractional Transformation

Any rational expression can be written under LFT form

256

LFT Representation

Transfert Function

LFT Représentation State space representation

1

1 2

1 11 12

2 21 22

1 1

M: Augmented uncertainties matrix

,..., , ,...,nm q n q

x Ax B w B u

z C x D w D u

y C x D w D u

I I

LFT Modelling

RR fRe

incRR

RnRincRnR

RnRRnR

eee

fRefRe

fRfRe

n

nn

n

et

Physical system Modele bloc diagramme Mathematical model

RfR eR

RfR eR

δR

eR

einc

++

Rn

fR eRn

nRn RRR fR eR RnRnR fRRe

LFT modelling

Rn

fRn eRneR

einc

++

δR

R:RfR

eR

RfR eR

1 0 R:Rn

De*:zRMSe*:wR

-δR

eRn

f1=fRn

eRn

eRn

einc

fR eR

zRwR

-δR

0

1

n

n

R

R

Example

R L

i A

R L

i A

R L

Se: ui A

Se: u1 4

1

R:Rn

De*:z R

MSe:wR 25

9

0Rδ- 6

Df: i

I:Ln

3

10

0

MSf

:wL

7Lδ-

Df*

:z L

8

R:RR:RR:R

2222

I:L

3

ARR generation : determinist (1/1)

I:L

3

1

Se:

u

R:RR:RR:R

1111

2222

4444

SSf: iDf: i

0:Φ 231J1 SSfeeee ?,, 231 eeeX

?1e See 11- Se

?2e ie R2SSf- 2-R-2

)/)(()(:ARR1 dtidiSe LR

?3e SSf- 3- L- 3

dt

die L3

R L

i A

R L

i A

R L

Se: ui A

MSe:wL

R:Rn

I:Ln

De*

:z L

De*:z R

Se: u

SSf: i

MSe:wR

1

2

3

5

4

7

8

9

10

0

1

0

Rδ-

Lδ-

0:Φ 75231J1 SSfeeeeee ?,,,, 75231 eeeeeX

?1e See 11- Se

?2e ienR2SSf - 2- 9- Rn - 9- 2

LRLR wwdtidiSenn

)/)(()(:ARR1

?3e SSf - 3 - 10- Ln- 10- 3

dt

die

nL3

6

?5e Rwe 55- MSe:wR

?7e Lwe 77- MSe:wL

MSe:wL

R:Rn

I:Ln

De*

:z L

De*:z R

Se: u

SSf: i

MSe:wR

1

2

3

5

4

7

8

9

10

0

1

0

Rδ-

Lδ-

6

LRLR wwdtSSfdSSfSenn

)/)(()(:ARR1

LR

LR

ww

dtSSfdSSfSenn

a

)/)(()(r :ARR1

ara

OUR DC MOTOR

263

Robust ARR From BG DC motor

Uncertain ARRs

264

R(t)

(t)

adaptive thresholds

Simulation results

265

Residuals in normal operation

Simulation results

266

Réaction des deux résidus robustes suite à une variation des paramètres RA et RM d'une valeur supérieure à leur incertitude relative

Simulation results

267

Réaction des deux résidus robustes suite à une variation des paramètres RA et RM d'une valeur égale à leur incertitude relative

Fault detectability index DI

The fault detectability index DI is the difference in absolute value between the effort (or flow) provided by

faults and those granted by all the uncertainties.

268


CONCLUSIONS The interest of the presented approach :

consists in the use of only one representation (bond graph modelling) for ARRs and dynamics models generation in symbolic format.

the industrial designer can easily (because of integration of the functional tool as interface with the human operator) build the thermofluid dynamic model and ARRs

Propose to the user a sensor placement to satisfy a given technical specification To add a new component in the data base in a generic way

What are the limits in model based supervision ? The performances depend on the accuracy of the model Processes are no stationary : the models change

There is not “the” method for supervision… but integration of tools is needed

Real time applications are not yet used in industry : maintenance of implemented algorithms is difficult.


APPLICATION to A steam generator Installation

Steps of performing a supervisory system

Failure Modes Analysis, Effects and Criticality Analysis,(AMDEC)

Offline monitorability and reconfigurability analysis

conditions

List of pertinent equipments

Elaboration of the supervision system

Results of monitorability and reconfigurability analysis

Sen

sor

Pla

cem

ent

Online test of the supervision system

Algorithms

Online

Ofline

Different steps for on line diagnosis system design

Sensors

Isolation decision procedure

On line FDI

Measurements for FDI and

control


Decision making tool for

supervision (FDI and FTC

levels)

Logic decision procedures

Dynamic model

Model Validation

Ofline diagnosability

analysis

Diagnosis algorithms generation

Measurements for monitoring

ARRs

Pro

cess

del

ay s

yste

m

FIR

10PR11

PIR

16

TR17

PC2

PR14

PR15

TR38PR

38

TR29PR

31

V1

V6

User

PR13

PR12

ZC1

V2

V11

BOILER

LIR

9LIR

8

LG1

TR5

PC1

PIR

7

TR6

Q4

Thermal resistor

LC1

V10

60kW

FIR

3

P2

P1

V9

STORAGE TANK

TIR2

LIR

1

LG3

STEAM FLOW

FEED WATER

CONDENSER HEAT-EXCHANGER

V8

Condensate

V4

V5

LG2

LC2

Aero-refrigerator

TIR26

Environment FIR

23

FIR

24

TIR27

TIR21

Cooling water

P3

P4

TIR22

TC5

PR27

TIR20

LIR19

LIR18

V3

TIR25

Steam generator : P &IDiagram

General views

274

GUI

Data acquisition system

Prof. B. Ould Bouamama Polytech Lille « Supervision Systems Design» 275


Architecture of the supervisions system

276

Processus

Dat

a ac

qu

isit

ion

Server

Client

Variable serveur

Variable client

Evaluation des résidus et generation des alarmes

Procedures I/OGUI

(Dspace or Panorama)

General Informations Number of sensors 28

10 Pressure sensors, 12 Temperature sensors, 5 Level sensors, 4 Flow sensors, 1 Power sensor Number of actuators 8

1 Pump (switching level control in the boiler) 1 Thermal resistor (switching pressure control in the boiler) 1 Valve (Continuous pressure control in the condenser) 1 Valve (Continuous valve position) 3 discharge valves (switching level control in the condenser) 1 Three way-valve (continuous cooling water temperature control )

Number of equipment units 1 storage tank of 0.4 m3 , 4 Pumps, 1 Boiler of 0.175 m3 , 5 controlled valves, 1 Controlled three-way-

valve 1 Condenser coupled with an exchanger, 1 Aero-refrigerator, 1 Thermal resistor of 60 KW, 1 PC-based

digital control system, 1 process delay system Automation System:

Conventional instrumentation• The used technology is the 4-20 mA

Control system • Two types of digital controllers are used: « On-off » and PI• Controlled parameters:

– Boiler pressure, boiler level, condenser level, condenser pressure, Steam flow valve position and Cooling water temperature.


Failure scenarios Plant faults

• Water leak in the boiler by opening valve V11• Thermal insulation fault taking off the calorifuge sheet• Pressure leak in the steam flow system by opening valve V3• Water leak in the storage tank by opening valve V10• Steam pipe blocked out by closing the manual valve V13

Actuator faults• Any valve can be blocked open or closed • Pump fault by switching off the power supply• The actuator control signals can be modified• Failure Discharge valves leak by opening valve V8 et V9

Sensor abrupt faults• Any sensor can be temporary disconnected• The sensor signals can be modified

Reconfigurability Degraded mode: one or two discharge valves in running Use of one or two controlled valves in the steam flow system The long loop of the heat-exchanger in fault mode: degraded mode, only the short loop is in running mode Feeding pumps are redundant Sensor system can be reconfigured

General Informations

Prof. B. Ould Bouamama Polytech Lille « Supervision Systems Design» 279/13

Modelling hypothesis

For the feeding circuit the liquid is incompressible.

I n the steam boiler, water and steam are in thermodynamic equilibrium, This is justified by the fact that we have a

good homogenous mixture of the emulsion water-steam. The mixture is at uniform pressure, which means that we neglect surface tension of the steam bubbles.

The boiler has a thermal capacity and is subject to heat losses towards the environment

All variables are described by lumped parameters.


280\93

WORD BOND GRAPH OF THE INSTALLATION

Condenser

Cooling circuit

TP,

mH ,

TP,mH ,

Condenser-Heat exchanger

mH ,

TP,Boiler

mH ,

TP,Steam

expansionmH ,

TP,

Feed water circuit

TP, mH ,

ReceiverTP,

mH ,

Discharge valves

TP,

mH ,

Voltage source

i U

Q

Thermal resistor

T

Bond graph model

281

0t2

De:T6

1

1SSf:Q4

RS

SSe:T5

: RSMSf w

SSe:T5

: RaMSf w 2: RtMSf w

2: CtMSf w

R:Ra

Se:-Ta

2:

hCMSf w

0h2

C:CB

RP

cdm BP De:L9

To th

e st

eam

expa

nsio

n sy

stem

:pRMSe w

11

R:RC

Tank

R:Rp

2 1 2Pb k P k

1

R:Rz11

: RzMSe w

R:RT1

1:

hCMSf w

0h1

0t1

C:CR

1

R:RC

From

cond

ense

r

RT RH

RP Rm

DP Dm

DT

DH

Pm

SSe:T2

1:

tCMSf w

Feed water circuit Boiler

Thermal resistor

1

R:RT2

SSf:F10

AQAT

Vm Bm

BH BT

BP

cdH

Vm

VH

RT

BP RP

21/bMTFGu

Gi

PQ

PP

cdP

: GSe u

Volta

ge s

ourc

e 0t3

AQ

GQ

VH

cdm

PH

1bPC

7P

LC

PP Pm

2b

1: RtMSf w

Dynamic simulation using Bond graph and Matlab Simulink

Modular Approach using library models

Model Validation

yr(t)

ym(t)

+

< adm?

u(t)

No

Validated model

Real system

Model

Sensors (Acquisition card)

-

yes


ARRs generation

Diagnosability analysis : Fault Signature matrix

287

23 RRAs générées

Mod

èle

bon

d g

rap

h s

ous

form

e ic

one

mét

ier

Bibliothèque de modèles

Matrice de surveillabilité


Control algorithm based on Panorama software


Variable definition based on Panorama software

Diagnosis Decision procedure based on Panorama software


Diagnosis Decision procedures based on Panorama software


Determination of thresholds


CONCLUSIONS

The interest of the presented approach : consists in the use of only one representation (bond graph modelling) for ARRs and dynamics

models generation in symbolic format. the industrial designer can easily (because of integration of the functional tool as interface

with the human operator) build the thermofluid dynamic model and ARRs Propose to the user a sensor placement to satisfy a given technical specification To add a new component in the data base in a generic way

What are the limits in model based supervision ? The performances depend on the accuracy of the model Processes are no stationary : the models change

There is not “the” method for supervision… but integration of tools is needed

Real time applications are not yet used in industry : maintenance of implemented algorithms is difficult.

Documents

Supervision Systems Design Prof. Belkacem OULD BOUAMAMA Research Director Ecole Polytechnique de Lille