Basic Structures in Metrology -...

PP1022.Nr.1ETH

IMRT

Measurement

Science

and

Technology

Basic Structures in Metrology –A Unified Approach Meeting Measurement Demands

Tutorial

Karl H. RuhmMeasurement and Control Laboratory, ETH Zurich, Switzerland

[email protected]

03. 09. 2008Version 01; 17.09.2008

www.mmm.ethz.ch/dok01/d0000866.pdf

PP1022.Nr.2ETH

IMRT

Measurement

Science

and

Technology

Administrative Matters

PP1022.Nr.3ETH

IMRT

Measurement

Science

and

Technology

•

What is Covered?-

Overview on Fundamental Topics

- Overview on the Interrelation of these Topics (Structure)

- Theory and Practice

- Laws and Examples

- Block Diagrams (Signal Effect Diagrams)

•

What is not Covered?-

Recipes for Today and Everybody

- "Measurement Uncertainty" (GUM)

- "How to Measure …

?"

- Sensors and Instrumentation


PP1022.Nr.4ETH

IMRT

Measurement

Science

and

Technology

•

Prerequisites-

Little Knowledge on Measurement and Control

-

Little Knowledge on Linear Algebra

•

Constraints-

Time for Presenting

-

Time for Reading and Doing

•

Further Reading-

Measurement Science ???

-

Signal and System Theory -

Stochastics and Statistics

-

Modelling and Identification-

Control and Optimisation


PP1022.Nr.5ETH

IMRT

Measurement

Science

and

Technology

Procedure

"Top-Down"Structures

SystematicsCommon TheoryGeneralisation

and

"Bottom –

Up"Questions from practice

Concrete ExamplesBreadth of the field


PP1022.Nr.6ETH

IMRT

Measurement

Science

and

Technology

Methods of Description

The Word(verbal)

The Mathematical Representation(analytical)

The Pictorial Representation(graphical)


PP1022.Nr.7ETH

IMRT

Measurement

Science

and

Technology

Consistency of Metrological Language

There is still an insufficient familiarity and agreement about terminology in Metrology.

Language is and will remain a dynamic process, which even normative committees obey.

We keep waiting for a convergence to a common language in the field.


PP1022.Nr.8ETH

IMRT

Measurement

Science

and

Technology

Examples

Please,take the examples as didactical examples for concepts and strategies

and not for the technological state of the art.


PP1022.Nr.9ETH

IMRT

Measurement

Science

and

Technology

Introduction

I Metrology

II Processes and Quantities

III Process and Measurement Process –Model-Based Measurement

IV

Ideal / Nonideal Measurement

Metrology and Control –

Relations?

Content


PP1022.Nr.10ETH

IMRT

Measurement

Science

and

Technology

Introduction

PP1022.Nr.11ETH

IMRT

Measurement

Science

and

Technology

Introduction

B08

34

Metrology

mon

itorin

g

measurem

ent

sensorics

anal

ytic

s

test

inspection

calibration

diag

nost

ics

observa

tion

detection

data acquisition

metrology

pattern recognitio

n

cont

rol

surveying

identification

certification

fusion

filtering

Variety of Terms

PP1022.Nr.12ETH

IMRT

Measurement

Science

and

Technology

Introduction

Variety of Concernment

B09

50

Metrology=

MeasurementScience

andTechnology

controlandsystemexperts

sensor andinstrumentation market

Developersof sensors

and instrumentationauthorities

sensor andinstrumentation

operator

processoperator

recipient of measurement results

measurementpersonal

PP1022.Nr.13ETH

IMRT

Measurement

Science

and

Technology

Introduction

Variety of Fundamentals

B07

32

SENSORICS

Chemistry• Physical Chemistry• Electro-Chemistry• Quantum Chemistry

Biology• Biochemistry• Molekular Biology• Biomedicine

Physics• Mechanics• Thermodynamics• Fluid Dynamics

MaterialsScience

Electronics• Mechatronics• Quantum Elektronics• Electromagnetic Compatibility

Metrology• Signal-, System-Theory• Statistics, Error Theory• Neural Network• Fuzzy Logic

PP1022.Nr.14ETH

IMRT

Measurement

Science

and

Technology

Introduction

Synthesis

Three Levels of

Main Topics in Metrology

PP1022.Nr.15ETH

IMRT

Measurement

Science

and

Technology

Introduction

Synthesis

• Propaedeutic foundations in metrology

• Common foundations in metrology

• Sensors, instrumentation and applications in metrology

PP1022.Nr.16ETH

IMRT

Measurement

Science

and

Technology

Introduction

B09

45

CO

MM

ON

FO

UN

DA

TIO

NS

OF

ME

TR

OLO

GY

Signals,Systems

Statistics,Stochastics

LinearAlgebra Physics Chemistry Biology

Identifi-cation

ComputerSciences

GeometryAnalysis MaterialsModelling,Simulation

PR

OP

AE

DE

UT

ICF

OU

ND

AT

ION

SO

F M

ET

RO

LOG

Y

PR

AC

TIC

AL

ME

TR

OLO

GY

Measurementof electricalproperties

Measurement ofchemical properties(AnalyticalChemistry)

InstrumentalPrinciples

Sensor Designand Integration

MetrologyStandards

e.g.Measurementof pH value

e.g.Measurement

of voltage

TheoreticalFoundations of Metrology

ApplicationalFoundations of Metrology

Organisational andOperational

Foundations of Metrology

InstrumentalFoundations of Metrology

..................................

e.g.inverse

functions

e.g.Measurementof O content

e.g.Measurement

of distance

Measurementof geodetic properties(Geodesy)

Measurementof biological

properties(AnalyticalBiology)

Measurementof nonphysical

properties

e.g.quality

assessment

e.g.tomography

e.g.sensor fusing

e.g.gauges

e.g.coordinatetransform

e.g.transfermatrix

e.g.uniform

distribution

e.g.Shannontheorem

e.g.numerical

errors

e.g.order

reduction

e.g.regressionanalysis

e.g.propertiesof silicon

e.g.thermal

expansion

e.g.chemicalactivity

e.g.oxygenbalance

Metrology -Measurement Science and Technology

2

Mathematics Natural Sciences Engineering Sciences

Control

e.g.robust-ness

Measurement Quantities and Parameters Measurement Procedures

PP1022.Nr.17ETH

IMRT

Measurement

Science

and

Technology

Introduction

Measurement

means

information acquisition

information processing

information transfer

information presentation

means

applied and ambitiousInformatics

PP1022.Nr.18ETH

IMRT

Measurement

Science

and

Technology

Introduction

Preliminary Statement:

Any Tool,

which helps to

extract and provide information

concerning a

Process P

of interest, is part of a

Measurement Process M.

PP1022.Nr.19ETH

IMRT

Measurement

Science

and

Technology

Introduction

Data↓

Information↓

Knowledge↓

Understanding↓

Sagacity↓

WisdomHoward Gardner

PP1022.Nr.20ETH

IMRT

Measurement

Science

and

Technology

I. Metrology

PP1022.Nr.21ETH

IMRT

Measurement

Science

and

Technology

The Term "Metrology"

Metrologyis a synonym for

Measurement Science and Technology

B13

42

metrology

measurementtechnology

measurementscience

(Supplement

→ Module "What is Metrology? –

Survey")

basic theoreticalorientation

instrumental and organisationalorientation

Metrology

http://www.mmm.ethz.ch/dok01/d0000813.pdf

PP1022.Nr.22ETH

IMRT

Measurement

Science

and

Technology

ButScience and Technology are not easily separable,

one can not exist without the other;Metrology is an entity.

However, we sense an

inherent, tiresome and undeniable"gap"

betweenscience and

technology

Metrology has to bridge this "gap".

(Supplement


Survey")

Metrology


PP1022.Nr.23ETH

IMRT

Measurement

Science

and

Technology

Metrology

Sub-Topics

like

Geodetic Metrology (Geomatics)Legal Metrology

Chemical Metrology (Chemometrics)Clinical Metrology

Economical Metrology (Econometrics)Financial Metrology

and so on

(Supplement


Survey")


PP1022.Nr.24ETH

IMRT

Measurement

Science

and

Technology

Metrology

Measurement Science

(Supplement


Survey")

•

basic principles and structures of measurementon the base of system theory

•

quantities intended to be measured and accessible quantitieson the base of process and measurement process models (sensor-fusion)

•

suitable physical effects on the base of natural sciences•

causality and determinism

•

principle of forward and backward analysis (mapping and reconstruction) on the base of stochastics and statistics

•

data processing on the base of signal and information theory (data-fusion, data-mining)

•

performance, verification and quality assurance

Methods in measurement science develop slowly and steadily.


PP1022.Nr.25ETH

IMRT

Measurement

Science

and

Technology

Metrology

Measurement Technology

(Supplement


Survey")

Methods in measurement technology develop rapidly and unsteadily.

•

Technological tools ashardware (instrumentation) and software (algorithms) are objective-

and application-oriented.

•

Technological tools presuppose superior and application-independent concepts, strategies and structures too,which we learn from "Measurement Science".

•

Measurement procedures at space-

and time-dependent processes may take place"on line" and "real time" or "off-line" and "non-real time".

•

The instrumentation may influence the process under observation undesirably.

•

Measurement errors arise due to nonideal technological tools.


PP1022.Nr.26ETH

IMRT

Measurement

Science

and

Technology

Metrology

Causality and Chance in Metrology

(Supplement


Survey")

•

Whether a relation complies with the conditions of causality is one of the most important questions in certain border fields of metrology application.

•

Many influences have impact on measurement results in an unwanted, yet causal and deterministic way (forward disturbance, mapping).

•

If the mechanism of effect (structure) and the amount of effect (parameter)of the deterministic influences are unknown,which is usually the case for different reasons,then we are aware of them as a random outcome in the measurement resultand interpret them accordingly (→ next Example).

•

Thus, an inference (backward evaluation, back-mapping, reconstruction) on the measurand of interest is not possible in a straightforward way.

•

Thus a measurement uncertainty remains within the result, which must be evaluated, quantified and reported.


PP1022.Nr.27ETH

IMRT

Measurement

Science

and

Technology

Metrology

Causality and Chance in Metrology

(Supplement


Survey")

Example: Quantising process with sine input signal –deterministic or random errors (quantizing noise)?

0 1 2 3 4 5 6 7 8

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time t [s]

sign

als

u [V

]

quantexample1.m; B1295.eps

original signal u(t)error signal e

q(t)

quantised signal uq(t)


PP1022.Nr.28ETH

IMRT

Measurement

Science

and

Technology

Metrology

Education in Metrology

(Supplement


Survey")

•

Despite the enormousscientific, technical, economical and social relevance of metrology,educational activities concerning Measurement Scienceremain rudimentary.

•

The common saying "everybody is able to measure"and the promises of euphoric sensor advertisements like"plug and play"suggest a "simplicity of use", which is seldom true in practise.

•

Experts know: "The only certainty in measurement is its uncertainty"and concerning sensors the slogan "plug and pray"fits much better.


PP1022.Nr.29ETH

IMRT

Measurement

Science

and

Technology

Measurement

takes place by arbitrary many, interrelatedSensors →→→ Sensor Fusion

and by arbitrary extensiveData Processing →→→ Data Fusion

This concerns different Levels of Relevance:

physical principalsinstrumental methods

signal and system theoretic concepts operational aspects

economical relevance

Metrology

PP1022.Nr.30ETH

IMRT

Measurement

Science

and

Technology

Metrology

Questions Before Any Measurement:

Different Tasks, Requirements and Opinionsin Sensor Nets and Data Nets

• What are the quantities intended to be measured?

• Which quantities are accessible?

• Which are the reference quantities?

• At which locations and at which time?

• By which instrumentation?

• How will data processing, evaluation and validation look like?

• Who will decide?

PP1022.Nr.31ETH

IMRT

Measurement

Science

and

Technology

Metrology

What are we looking for?

•

quantities→ defined, measurable, accessible, traceable (SI)(data acquisition, raw data, data collection ↔ Sensor Fusion)

•

objects, features•

parameters

•

characteristic values, characteristic functions• images, patterns•

structures

•

events•

tracks, trajectories

• trends•

causal relations→ combine, calculate, estimate, observe, forecast(data reconstruction, data analysis, data-mining ↔

Data Fusion)

1.

2.

PP1022.Nr.32ETH

IMRT

Measurement

Science

and

Technology

Metrology

Is there any systematic concept?

1.with Data Acquisition (Sensorics)?

2.with Data processing (Informatics)?

yes!

amongst others

Common Structures forProcesses and Measurement Processes

Signal and System Theory, Stochastics and Statistics

PP1022.Nr.33ETH

IMRT

Measurement

Science

and

Technology

Metrology

The measurement quantity (measurand)has to be defined!

Example 1Level measurement by measurement of the hydrostatic pressure p.

Measurable and Immeasurable Quantities?

B1277

ρ g

Σ+

+

h(t)

p (t)g

Δ

ρ

p (t)g

h(t)p(t) =

g

ρgh(t) + p (t)g

process P

acquirable process quantity p (t)

interesting, not acquirable process quantity h(t)

g

acquirable process quantity p(t) = p (t) + p (t)gh

uninteresting, not acquirable process quantity p (t)h

sub-process SP

PP1022.Nr.34ETH

IMRT

Measurement

Science

and

Technology

Metrology

The measurement quantity (measurand)has to be defined!

Example 2Temperature influence on the measurand "Length" l.

Interesting Quantities?

1 0 1 0l l (1 ( )) [m]= +α ϑ −ϑ

B13

14

–

+Σ

Πϑ1

ϑ0

Δϑ

0

1

α

1

+ Σ+

l

l

0l

Σ

process P

PP1022.Nr.35ETH

IMRT

Measurement

Science

and

Technology

Metrology

Σ

÷

Π

Π

m

B08

91

Θ

p

m

x

p

ϕ

ps

+

–

p

p

pRR

÷p

xa

a

a

wv

wv

w

wv

wv

process P ("humid air")vapor pressure line

a

wv

The measurement quantity has to be defined!

Example 3Definition of the physical quantity "Humidity".

Interesting Quantities?

1 1

W waterWD water vapours saturated

x absolutehumidityrelativehumidity

p bar pressureC temperature

K absolute temperatureR Jkg K gasconstantm kg mass

L

Qu

air

antities:

Indices:

− −

−ϕ −

ϑ °Θ

PP1022.Nr.36ETH

IMRT

Measurement

Science

and

Technology

Metrology

The measurement process M has to be defined!

↓

Exampleof extreme different measurement processes

•

Thermometer

•

Magnetic Resonance Tomograph

°C

B01

01

PP1022.Nr.37ETH

IMRT

Measurement

Science

and

Technology

Metrology

Quantities of interest are measured at the process P, at the input and at the output as well.

process P

measurement process M

measurementquantitiesof interest

estimatingmeasurement

results

process domaininstrumentation domain

B11

24

u(t) y(t)

y(t)ˆ

The measurement process M has to be defined!

PP1022.Nr.38ETH

IMRT

Measurement

Science

and

Technology

MetrologyThe measurement process M has to be defined!

Example(non-dynamic case)

°Cϑ

ϑ ϑHg

V h

B01

00

ˆ

d

30 Hg 1 0 1

22

01

V V [m ] aus V V (1 );4h V [m];

ddˆ h [°C]4 V

with 3 [ C ] cubic elongation value (see Physics)−

Δ =β Δϑ = +βϑ

Δ = ΔππΔϑ= Δ

ββ= α °

B01

04

1 d π24β V

0d π2

4 β V0

HgΔϑ [K] Δh [m] Δϑ

Δϑ

ΔV [m ]3

Δϑ

[K]

[K]

[K]

sensor process S(glas capillary with mercury)

reconstruction R(scale)



Signal Effect Diagram (Model)

Physical Relations (Model)

PP1022.Nr.39ETH

IMRT

Measurement

Science

and

Technology

Metrology

Specification of the task

•

Measurement quantities y(t)

•

resulting quantities

Relation between measurement quantities and resulting quantities:ideal,

perfect, nominal, unreal, vision, aim, thought experiment, wishful thinking

Ideal Measurement:

; → Fundamental Law of Metrology

ˆ(t)y

ˆ(t) (t)=y I y

19.3 °Cϑ

ϑB00

28

ˆ

ˆ(t) (t)=y I y

y(t) = y(t)B

1283

y(t)

D = IM

ˆ

u(t)

idealmeasurement process M


nom

process P

PP1022.Nr.40ETH

IMRT

Measurement

Science

and

Technology

Metrology

ˆ(t) (t)≠y I y

Specification of the task•

measurement quantities y(t)

•


Relation between measurement quantities and resulting quantities:nonideal, real, imperfect, facts, estimates, uncertainty

Nonideal Measurement: in particular

ˆ(t)y19.3 °C

ϑ

ϑB00

28

ˆ

ˆ(t) { (t)}=y f yin general:

ˆ(t) { (t)}=y f y

y(t) = y(t)B

1284

y(t)

ˆ

u(t)

/f(y(t)

non-idealmeasurement process M


process P

PP1022.Nr.41ETH

IMRT

Measurement

Science

and

Technology

Metrology

Result

•

Measurement results never equalmeasurement quantities y(t) intended to be measured

•

Measurement deviations, measurement errors ey

(t)

•

Measurement uncertainties uy

(t)

ˆ(t)y

PP1022.Nr.42ETH

IMRT

Measurement

Science

and

Technology

II. Processes and Quantities

PP1022.Nr.43ETH

IMRT

Measurement

Science

and

Technology

Processes and Quantities

• Elements of the Signal Effect Diagram

• Group Notations of Signals

• Processes and Systems

• Quantities and Signals

• Basic Structures of State Space Representation

• Inversion of Interconnected Systems

PP1022.Nr.44ETH

IMRT

Measurement

Science

and

Technology


Elements of the Signal Effect Diagram

PP1022.Nr.45ETH

IMRT

Measurement

Science

and

Technology



Supplement→ Module "Elements of the Signal Effect Diagram"

• Signal line with arrow

• Signal junction dot on signal lines

• Signal connection between signal lines

• Functional block between signal lines

• Connection of Functional Blocks


PP1022.Nr.46ETH

IMRT

Measurement

Science

and

Technology



Signal Linescalar signal vector signal

x(t)

(1)

x(t)

(N) B01

39

1

2

T1 2 n N

n

N

x (t)x (t)

x(t) (t) x (t) x (t) x (t) x (t)x (t)

x (t)

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= = ⎡ ⎤⎢ ⎥ ⎣ ⎦⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

x … …

x(t)

x(t)

x(t)

x(t)

B07

46


Signal Junction


PP1022.Nr.47ETH

IMRT

Measurement

Science

and

Technology



Signal Connection


linear connection nonlinear connection

scalar addition vector addition

B01

79 (N)

x (t)1

2

x (t)1

2

±

+Σ Σ

(1)

x(t) = x (t) ± x (t)1 2 x(t) = x (t) + x (t)1 2

x (t) x (t)

multiplication division

B01

80

1

2

Π

1

2

..x(t) = x (t) x (t)1 2

. x(t) =

x (t)

x (t)

x (t)

x (t)

x (t)

x (t)

1

2


PP1022.Nr.48ETH

IMRT

Measurement

Science

and

Technology



Functional Block


linear system non linear system

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

B00

85

Connection of Functional Blocks

series connection parallel connection feedback connection

Σ Σ

B00

66

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


PP1022.Nr.49ETH

IMRT

Measurement

Science

and

Technology


Cause and Effect Diagram(Fishbone Diagram, Ishikawa-Diagram)

Supplement→ Module "Cause and Effect Diagram"

B12

18

f(cause)cause effect

cause effect

or

or

effect = f(cause)

B12

19

causes effect

Advantages•

quick and comprehensive overview

•

information aboutqualitative influences

Disadvantages•

type of signal connections not shown

•

weighting of influences not possible•

expansion to multivariable systemsnot possible


PP1022.Nr.50ETH

IMRT

Measurement

Science

and

Technology


Group-Notation of Quantities

→

Notation of State Space Description

Supplement→ Module "Group-Notation of Quantities"

x(t)

u(t) y(t)

B11

54

x(0) initial valuesof state quantities

process P

output quantitiesinput quantities

state quantities


PP1022.Nr.51ETH

IMRT

Measurement

Science

and

Technology


Group-Notation of Quantitiesu(t)

Input Quantities

v(t)

Disturbing Quantitiesw(t)

Reference Quantities, Additional Input Quantities

x(t)

Inner Quantities (State Quantities)x(0)

Initial Values of Inner Quantities (State Quantities)

y(t)

Output Quantities,Acquirable by Measurement Directly

z(t) Output Quantities,Acquirable by Measurement Indirectly Only

Supplement→ Module "Group-Notation of Quantities"

x(t)

u(t)

v(t)

w(t)

y(t)

z(t)

x(0)

B09

49

process P


PP1022.Nr.52ETH

IMRT

Measurement

Science

and

Technology


Process and System

PP1022.Nr.53ETH

IMRT

Measurement

Science

and

Technology


Supplement→ Module "Process and System"

Two Terms:Process and System

A process (element, object, plant)is a natural, artificial or combined arrangement of objects in the physical world, which are related by the cause and effect principle temporally and locally:

real world, reality

A system (model, map)is an artificial, human thought environment containing ideas about a process of interest; it is an abstraction in a virtual world. And it is artificial insofar that it contains only those aspects of the reality of a process deliberately, which are characteristic for a certain problem formulation and which are accessible by common and individual knowledge.

virtual world, imagination, system thinking


PP1022.Nr.54ETH

IMRT

Measurement

Science

and

Technology



System =

ModelExamples of different representations: Spring Balance

m

B08

54

h

f

c, κ

h(t)

c

κ

m

B09

08

f(t)

2

2 1

1

1 1

2

mh(t) ch(t) h(t) ma(t) c v(t) h(t) f(t) Nor

c c 1h(t) h(t) h(t) a(t) v(t) h(t) f(t) msm m m m m

mitm kg Ns m mass f N forcec Nsm damping value h m elevation

Nm spring value h v ms velocityh a ms a

−

−

−

− −

−

+ + κ = + + κ = ⎡ ⎤⎣ ⎦

κ κ ⎡ ⎤+ + = + + = ⎣ ⎦

κ == cceleration

1

2

01

0

0 1 0v(t) h(t) msf(t)c 1a(t) v(t) msm m m

with the initial valuesmhh(0)

vv(0) msand the output equation

h(t) 1 0v(t) 0 1a(t) c

m m

−

−

−

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +κ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− − ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤⎡ ⎤= ⎢ ⎥⎢ ⎥⎢ ⎥

⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡⎡ ⎤⎢ ⎥ =⎢ ⎥⎢ ⎥ κ⎣ ⎦ − −

⎣

1

2

m0h(t)

0 f(t) msv(t)

1 msm

−

−

⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎡ ⎤

+ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎢ ⎥⎣ ⎦⎢ ⎥ ⎢ ⎥⎦ ⎣ ⎦

Σf(t)

B11

19

+

––

2 1

h(0)v(0)

h(t)v(t)a(t)

c

κ

m

m

1

u(t) x (t).m x (t) 22 x (t)1

.x (t)

1

h(t)

v(t) = y(t)

a(t)

1x (0)2 x (0)

a0

b 0

a1

Zeit t

Aus

schl

ag h

T = 1 / s

h(t)

Δt = 1 / fPd

h0

h

d

Enveloppe

h hk k+1

B09

09


PP1022.Nr.55ETH

IMRT

Measurement

Science

and

Technology



System =

Model

•

qualitative models-

thought models, ideas

-

verbal descriptions-

believes, prejudices

-

experiences, estimates

•

quantitative models-

all types of physical laws

-

all kinds of quantitative pictures and drawings-

all kinds of structure, effect, cause and effect diagrams

-

all types of sets of mathematical equations-

all types of sets of stochastic and statistical equations


PP1022.Nr.56ETH

IMRT

Measurement

Science

and

Technology


Quantity and Signal

PP1022.Nr.57ETH

IMRT

Measurement

Science

and

Technology



Two Terms:Quantity and Signal

A physical quantity (event, entity)represents the dependencies (effect, relation, interrelation, interaction, correlation, correspondence) concerning a process of the physical world. Quantities are well defined by agreed international declarations, however, physical processes only indirectly by the description of physical quantities and physical laws involved.

real world, reality

A signal (model)is an artificial, human thought environment containing ideas about a quantity of interest; it is an abstraction (model) in a virtual world. And it is artificial insofar that it contains only those aspects of the reality of a quantity deliberately, which are characteristic for a certain problem formulation and which are accessible by common and individual knowledge.

virtual world, imagination, system thinking


PP1022.Nr.58ETH

IMRT

Measurement

Science

and

Technology



Process, System and Quantity, Signal

B10

25

y(t)u(t)

y(t)u(t) ˆˆ

real world, reality

virtual world, imagination, idea, map, model

process

quantities,events

systemsignals

transfer

simulation

estim

atio

n

mod

ellin

gm

easu

rem

ent

The relations between process and system or quantities and signals respectively are manifold.

The most important ones are

transfer and simulationas well as

modelling and estimation


PP1022.Nr.59ETH

IMRT

Measurement

Science

and

Technology



Process, System and Quantity, Signal

From the point of view of mathematics

a system is an operator op between signals


PP1022.Nr.60ETH

IMRT

Measurement

Science

and

Technology


Characteristic Values, Characteristic Functions

There are virtual quantities(characteristic values and characteristic functions).

"Virtual" means that there is no correspondence of these quantities in the real physical world. Virtual quantities are men-defined quantities, for example:

distributions, mean values, correlations, convolutions, spectra,

intervals, bounds, errors, uncertainties, efficiencies and so on

This does not change the question about system properties and their transfer behaviour:How do characteristic values and characteristic functions of the

input

effectcharacteristic values and characteristic functions of the output?

B08

65

output signals y(t),their properties

and their transformations

(e.g. distributions, mean values,correlations, spectra, intervals,bounds, errors, uncertainties)

input signals u(t),their properties

and their transformations

(e.g. distributions, mean values,correlations, spectra, intervals,bounds, errors, uncertainties)

system



PP1022.Nr.61ETH

IMRT

Measurement

Science

and

Technology


Characteristic Values, Characteristic Functions

Example 1Pressure p [bar] as a physical quantity of interest.

deterministic signal trajectory described• by a deterministic function p(t), if it changes with time t harmonically for example,• by a numerical value p(tobs

), if only the observation time instant tobs

is of interest.

B11

99

p(t ) [bar] p(t) = p + p cos2πft [bar]0obs

system

signal

p(t) [bar]

functionof the signal

valueof the signal

bymeasurement and data processing

one gets for example

deterministic



PP1022.Nr.62ETH

IMRT

Measurement

Science

and

Technology

Processes and QuantitiesCharacteristic Values, Characteristic Functions

Example 2Pressure p [bar] as a physical quantity of interest.

random signal trajectory is described• by a characteristic function, pd(p) (probability density function) for example, of the

stationary pressure trajectory,• by a characteristic value μp

(tobs

) (arithmetic mean value), if only the observation time instant tobs

is of interest.

B12

02

μ (t ) [bar]obsp

p p ed

p

p p

p( )

( )

=−

−

1

2 2

12

2

2

πσ

μ

σ [bar ]-1

signal

pressure p(t) [bar]

characteristic functionof the signal

characteristic valueof the signal

bymeasurement and data processing

one gets for example

random

arithmeticmean value

probability density function

system



PP1022.Nr.63ETH

IMRT

Measurement

Science

and

Technology


Task Definitions Concerning a System


given: given: searched: scope of functions

1. input signals u system structure,system parameter p

→ output signals y function, transformation, control, convolution, simulation, forecast, measurement

2. output signals y system structure,system parameter p

→ input signals u reconstruction, inversion, deconvolution, infer, diagnosis estimation, decoding

3. input signals u output signals y → system structure,system parameter p

structure identification, parameter identification, calibration, test


PP1022.Nr.64ETH

IMRT

Measurement

Science

and

Technology


The Term «Transfer»

Concerning A System

PP1022.Nr.65ETH

IMRT

Measurement

Science

and

Technology


Colloquially we say thata signal, which acts via the input of the systemaccording to the principle of cause and effect

on the output of the system,is "transferred" through this system.

However,the output signal is of a totally different physical character than the input signal.

There is indeedno transfer, no transmittance, no flow, and so on;

there is onlyimpact, influence, effect, stimulation, animation, excitation, injection, generation,

forcing, loading, drivingon a system,

and the system responds accordingly to those activities at the inputby corresponding output signals.

Nevertheless, we speak in System Theory about the

«Transfer Response Behaviour»

of a System

Supplement→ Module "The Term «Transfer»

Concerning Systems"

«Transfer»


PP1022.Nr.66ETH

IMRT

Measurement

Science

and

Technology



Concerning Systems"

«Transfer Response Behaviour of a System»

described by

NoteWatch out for the "units" of these definitions!

transfer response equation y(t) = f(u(t),t)

transfer response diagram

transfer response value gy,u

[{y u–1}]

transfer response value matrix Gy,u

transfer response function gy,u

(t) (temporal) and gy,u

(s) (spectral)

transfer response function matrix Gy,u

(t) (temporal) and Gy,u

(s) (spectral)


PP1022.Nr.67ETH

IMRT

Measurement

Science

and

Technology



Concerning Systems"

Triple of Tasks for a System with Inputs and Outputs

•

determine the output signals, input signals and transfer response functions given•

determine the input signals, output signals and transfer response functions given

•

determine the transfer response functions, output signals and input signals given

MappingMathematics, always more precise than other disciplines, speaks about mapping:

"An input signal u is mapped by means of a system (operator op) on an output signal y"

mapping

transfersystem

systemu y

B12

11

u yy = f(u)


PP1022.Nr.68ETH

IMRT

Measurement

Science

and

Technology


Basic Structures of State Space Representation

PP1022.Nr.69ETH

IMRT

Measurement

Science

and

Technology


Classification of different structural system properties by subsystems of equations:for example for the system properties "dynamic" and "non-dynamic".

•

Subsystem of Equations I–

description of the relations between the inner quantities x(t) themselves, which are needed for the characterisation of the dynamic behaviour (delay, dead time, rate, etc.)

–

effects of the input quantities u(t) on the inner quantities x(t)•

Subsystem of Equations II

–

description of the relations between the inner quantities x(t) themselves, which are needed for the characterisation of the nondynamic behaviour

–

effects of the input quantities u(t) and all inner quantities x(t) on the output quantities y(t)

Supplement→ Module "Basic Structures of State Space Representation"

u(t)

x(t)

y(t)

B00

34

effectsof inner quantitieson environment

non dynamic relationsbetween inner quantities

dynamic relationsbetween inner quantities

effectsof outer quantities on

dynamic relations

effectsof outer quantities onnon dynamic relations

I

II

differential equations

algebraicequations

system


PP1022.Nr.70ETH

IMRT

Measurement

Science

and

Technology


Basic structure of a linear, dynamic System

GivenModel of a linear time invariant process (LTI) of first order

Differential equation


1

single input, single output (SISO)

x(t) ax(t) bu(t) [{x}s ]ormultiple input, multiple output (MIMO)

(t) (t) (t)

−= +

= +x Ax Bu

Σ(M) (N) (N) (N)

(N)

u(t) x(t) x(t)

A

B.

x(0)

B00

88

differentialequation

solution

NoteA linear differential equation of (N)th

ordercan be written as

(N) differential equations of (1)st

order


PP1022.Nr.71ETH

IMRT

Measurement

Science

and

Technology


Systematic structure of a non-linear, dynamic, multiple-input, multiple-output system


B05

18u(t) y(t)

g(x(t),u(t),t)

x(t)x(t)

x(0).

(M)(N)(N)

(P)

f(x(t),u(t),t)

Systematic structure of a linear, dynamic, multiple-input, multiple-output system

(P)

u(t) y(t)

B00

87

(P)

(P)

(M)Σ

dynamicsub-system

non-dynamic

sub-system

system


PP1022.Nr.72ETH

IMRT

Measurement

Science

and

Technology


Systematic structure of a linear, dynamic, time-invariant, multivariate system


0

(t) (t) (t) (dynamic relations at the input summator)(0) (initial values)(t) (t) (t) (nondynamic relations at the output summator)

= +=

= +

x A x Bux xy Cx Du

(t) (t) (t)(t) (t) (t)

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤= =⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

A Bx x xPC Dy u u

compact

(N)

B00

80

C

A

(P)(N)

.x(t)

Bu(t)

(M)

D

Σ Σ

x(0)

x(t) y(t)

A BC D

u(t)

(M) (P)

y(t)

u(t)

(M) (P)

y(t)P

dynamic system

dynamic subsystem

non-dynamic subsystem


PP1022.Nr.73ETH

IMRT

Measurement

Science

and

Technology


State space representation

less useful for

nonlinear systems

dynamic systems with lumped parameters (time and space dependent

systems)


useful for

all

mono-

and multivariate systems

dynamic and non-dynamic systems

deterministic, linear, time-invariant, time-variant, time-continuous, time-discrete systems


PP1022.Nr.74ETH

IMRT

Measurement

Science

and

Technology


The Complexity of Processes

is described by the

Structure of Mathematical Relations

within

Signal and System Theory

and

Stochastics and Statistics



PP1022.Nr.75ETH

IMRT

Measurement

Science

and

Technology


Inversion of Interconnected Systems

Supplement→ Module "Inversion of Interconnected Systems"


PP1022.Nr.76ETH

IMRT

Measurement

Science

and

Technology



Reconstruction by Inversion

u y y u

B11

36

S Rmapping,coding,

transformation,transfer

reconstruction,decoding,

back-transformation,inversion

As soon as the response (effect) y of a system S is known,an infer delivers an estimate of excitation (cause) u

by a reconstruction process R(inversion of the model of the system).


PP1022.Nr.77ETH

IMRT

Measurement

Science

and

Technology



Connection of Functional Blocks

series connection parallel connection feedback connection

Σ Σ

B00

66

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


PP1022.Nr.78ETH

IMRT

Measurement

Science

and

Technology



Inversion of Connected Systems

QuestionGiven two subsystems

with the steady state transfer response values g1

and g2

.How does the inversion procedure of the three possible connections look like?

Result

1 11 1 1series connection: g g g [{yu }] g [{uy }]2 1 g g2 11/ g1 1 11parallel connection: g g g [{yu }] g [{uy }]1 2 1 g / g2 1

g 11 1 1fffeedback connection: g [{yu }] g g [{uy }]fb1

1inverse{g} ori

g g gff fb ff

gin l ga g

→

→

→

− − −= =

− − −= + =+

−=

− − −= = ++

"ff" feed-forward; "fb" feed-back


PP1022.Nr.79ETH

IMRT

Measurement

Science

and

Technology



QuestionGiven two subsystems

with the steady state transfer response values g1

and g2

.How does the inversion procedure of the three possible connections look like?

Result

Σ Σ

g1

g1 g1

g1

1

1

g2

g2

g2

g2

1u = uB

0922

u u

Σ

gff

gfb

uuΣ

gff

gfb

1

y

y

y = y2 u

y

2

1

y

y

y

u

u

u

y = u1 21

2

1

ff ff

fbfb u

series connection

parallel connection

feedback connection

series connection

feedback connection

parallel connection

function reconstruction


PP1022.Nr.80ETH

IMRT

Measurement

Science

and

Technology



SymmetryNotice the vertical symmetry line in the signal effect diagram!

Correspondences•

The inverse of a series connection remains a series connection.

•

The inverse of a parallel connection is a feedback connection.

•

The inverse of a feedback connection is a parallel connection.


PP1022.Nr.81ETH

IMRT

Measurement

Science

and

Technology



Multivariable SystemLinear multivariable systems are systematic extensions of simple

monovariable systems.

All individual transfer response values

are subsumed in the transfer response value matrixes G

or G–1.

ExampleSensor Fusion and Data Fusion

Σ Σ

Σ

g11

g21

g12

g22

++

++ +

+

–

–

u1

u2

u1

u2

ˆ

ˆ

y1

y2B09

24

g111

21g

12g

22g1Σ

sensor process S(Sensor Fusion)

reconstruction process R(Data Fusion)

p m p m1

y ,u y ,ug or g −


PP1022.Nr.82ETH

IMRT

Measurement

Science

and

Technology pdisturbed p u ,p g gu p

u 0 gΔϑ

Δϑ Δϑ

⎡ ⎤⎡ ⎤ ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥Δϑ⎣ ⎦ ⎣ ⎦⎣ ⎦

ExamplePressure sensor disturbed by temperature (disturbance model)

p; Δϑ

up

; uΔϑ

Σ

pp

ΔϑΔϑ

Δϑ

pu

u

B12

82

++

g

g

p

u ,Δϑ

gΔϑ

pu

pu

Δϑu

Δϑ

p

Op{..}

p

p

Δϑ

Op{..}y = uS y

S


sensor process S(sensor fusion)

dist

dist

process P

Supplement→ Module "Disturbance Sensitivity of a Sensor"



PP1022.Nr.83ETH

IMRT

Measurement

Science

and

Technology pdisturbed p u ,p g gu p

u 0 gΔϑ

Δϑ Δϑ

⎡ ⎤⎡ ⎤ ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥Δϑ⎣ ⎦ ⎣ ⎦⎣ ⎦

p

disturbed

u ,

corr pp p

g1 1p ug g g

ˆ u10g

Δϑ

Δϑ

Δϑ

Δϑ

⎡ ⎤−⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥Δϑ ⎣ ⎦⎣ ⎦

⎢ ⎥⎣ ⎦

p; Δϑ

up

; uΔϑ

corrˆ

pΔϑ

Σ Σ

pp

ΔϑΔϑ

Δϑ

Δϑ

ppu

u

B09

21

+ ++ –

ˆ

ˆg

g

g

p

Δϑ

u ,Δϑ gu ,Δϑ

1

1gΔϑ

gp

pu

Δϑ

ppu

Δϑu

Δϑ

p

Op {..}Op{..}–1

p p

pup

Δϑ

Op {..}Op{..}–1y = uM y = yMy = u S R ˆ


sensor process S(sensor fusion)

reconstruction process R(data fusion)

dist

dist

corr

corr


symmetry axis

sensor process Sreconstruction

process R

process P

Supplement→ Module "Disturbance Sensitivity of a Sensor"

Processes and QuantitiesExample

Pressure sensor disturbed by temperature (reconstruction design)


PP1022.Nr.84ETH

IMRT

Measurement

Science

and

Technology



ConstraintsThe number (P) of the output quantities y

must equal the number (M) of the input quantities uso that the

(square (P)x(M)) transfer response value matrix Gcan be inverted.

ExampleMeasurand u of interest

is not reconstructable fromerroneous sensor output y.

(P) (M)≠

u

vy y

u

v

ˆ

ˆB

1137

S R

?


PP1022.Nr.85ETH

IMRT

Measurement

Science

and

Technology



ConstraintsThe inversion of integrating subsystems (time delay systems)

is done bydifferentiating subsystems,

in theory at least.

Difficulties arise, when high frequency content of the signals is involved.Noise content will be amplified

Counter-ExampleReconstruction process with a differentiator

for a temperature sensorwith slow changing, "noise-free", since filtered, measurement quantity.

Inverse Dynamics


PP1022.Nr.86ETH

IMRT

Measurement

Science

and

Technology


Supplement→ Example "Dynamic Reconstruction within the Measurement Process"

Inverse DynamicsReconstruction process for a sensor

with slow changing, "noise-free" measurement quantity.

Sensor model of first order (gS b [ySy–1]; TS 1/a [s]; gS [ySy–1])

time domain model frequency domain model1

S S SS S S SS

S S S S S1x (s) g (s)u (s)dT x (t) x (t) g u (t) [{y }]

dtg u (s) [{y }Hz ]

1 T s−= =

++ =

time domain inverse frequency domain inverseS

S SS

S 1R S S

S S S

1 Ty(s) g (s)y (s)1 d Ty(t) ( y (t) y (t)) [{y}]T d

( s) y (s) [{y}Hzg

]Tt g

−= == ++

Σ

y(t) y(t)

y(t)

B00

16

T

d/dts g

y (t)u (t)S

S

S

S

TS

Sˆ

u(t)

y(s) y(s)

y(s)y (s)u (s)S Sˆ

u(s)

++

1

g TSΣ ...dt1/s

TS

1

x (t)S

x (s)S

x (t)S

s x (s)S

+

_

.

process P

reconstruction process Rsensor process S




PP1022.Nr.87ETH

IMRT

Measurement

Science

and

Technology

Supplement

→ Module "Reconstruction Process –

Survey"

B09

72

reconstructionprocess

R

directlaw

CAUSEEFFECT

measurementquantities

sensorquantities

inverslaw

estimation of measurement

quantities=


estimationof the

CAUSE

sensorprocess

S


Intermediate Results

Sensor Process SReconstruction Process R

Two Directions of Viewingwithin the measurement process


forward:

Cause → Effect:

«Sensorics»backward:

Effect → Cause:

«Reconstruction»


PP1022.Nr.88ETH

IMRT

Measurement

Science

and

Technology

Intermediate Results

Non-ideal sensor processes must not lead to

measurement errors and uncertainties necessarily

or

Only if the transfer response of the

reconstruction process R does not equal the inverse transfer response of the

sensor process S, only then

measurement errors and uncertainties arise.


PP1022.Nr.89ETH

IMRT

Measurement

Science

and

Technology


Nonideal Transfer Response Behaviour of theSensor Process(Non-Linearity)

ExampleFlow Measurement

Nonideal sensor: nevertheless, ideal measurement is possible in principle.

ˆ

V(t)*

*

y(t)

Δp

y (t)S y(t)y(t)f (y)S f (y )R S

V(t)*

B08

36

ˆ

V(t)u (t)

reconstructionprocess R

process domaininstumentation domain

sensorprocess S


process P

PP1022.Nr.90ETH

IMRT

Measurement

Science

and

Technology


Intermediate Result

The model of the reconstruction process Ris the inverse model of the sensor process S

that means

without model of the sensor processthere is no inversion in the reconstruction process

that means

No Exception!Measurement is Always

Model-Based Measurement

PP1022.Nr.91ETH

IMRT

Measurement

Science

and

Technology

III. Process and Measurement Process – Model-Based Measurement

PP1022.Nr.92ETH

IMRT

Measurement

Science

and

Technology


Identification (Calibration) Process

Reconstruction Process

Observer

Kalman Filter

Model-Based Measurementconcerning

Signals

and

Systems

Main Examples

PP1022.Nr.93ETH

IMRT

Measurement

Science

and

Technology

Identification (Calibration)of processes and measurement processes

B11

30

u(t)

–

+

e (t)yΣ

y(t)

y(t)

u (t) y (t)

–

+ e (t)y

Σ

refref

u (t) y (t)M M

process P

non idealmeasurement

process M


measurementresults


measurementerrors

measurement mode

calibration mode

generating processof reference quantities

< unknown

< known

< known

< known< known< known

< unknown

measurmenterrors

calibration process C


PP1022.Nr.94ETH

IMRT

Measurement

Science

and

Technology



ExampleHarmonic test signal for frequency response determination.

Acceleration Sensor Calibration(principle of "Orthogonal Cross Correlations Function"; "Impedance Spectroscopy")

B07

89

u(t) = u cos(2πft + ϕ ) y(t) = y cos(2πft + ϕ )

Π Mτ y(t) u(t+τ)

y,uR (τ)

y0

^^

0ux(t) = x cos2πft

^

.

process of interest


driving signaltest signal

correlator C

sub-process

response signal

funktion generator FG

process P

PP1022.Nr.95ETH

IMRT

Measurement

Science

and

Technology



ExampleHarmonic test signal for frequency response determination.

Acceleration Sensor Calibration(principle of "Orthogonal Cross Correlations Function"; "Impedance Spectroscopy")

Signal Analyser:

f

M

M

x(t)B

0286

arctan–––sxxc

sx xc

sx (f)

x (f)c

+2 2x(f)

0sin2πft

cos2πft

x(t)sin2πft

x(t)cos2πft

x(t)

x(t)

ϕ (f)

Π

Π

functiongenerator FG

signal analyser SA(Fourier-decomposition)

PP1022.Nr.96ETH

IMRT

Measurement

Science

and

Technology


Supplement→ Module "Reconstruction Process –

Survey"

Reconstruction ProcessThe original quantities must be reconstructed or estimated

from the mapping output quantities uS

(t) and yS

(t)of the sensor process S.

The term «reconstruction»

involves also the estimation of immeasurable quantities z(t) of the process P

by measurement of input and output quantities u(t) and y(t).

B11

33

u(t)

y(t)

y(t)

u(t)u(t)

y(t) ˆ

ˆu (t)

y (t)S

S

?

?


process Pprocess domain

instrumentation domain

sensorprocess S


B11

34

u(t)

y(t)

y(t)

u(t)u(t)

y(t) ˆ

ˆu (t)

y (t)S

S

z(t)ˆ

z(t)

?

??

measurment process M



sensorprocess S

reconstructionprocess RS

reconstructionprocess RP

not measurable

ˆ(t) (t)ˆ(t) (t)

==

y Iyu Iu


PP1022.Nr.97ETH

IMRT

Measurement

Science

and

Technology



Survey"

Reconstruction ProcessThe term «reconstruction»

involves also the estimation of

immeasurable quantities z(t) of the process Pby measurement of input and output quantities u(t) and y(t).

B09

77

Op {...}SP

Op {...} Op {...} Op {...}SPS S

-1 -1

Σ+

-Σ

+

-

y(t)ˆ

e (t)y e (t)z

z(t)

z(t)

y (t)S

measurementerrors

measurementresults

measurement quantities of interest


sensorprocess S


process P



estimation ofmeasurement

quantitiesof interest

reconstructionerrors

measurablequantities y(t)

sub-process SP

S SP


PP1022.Nr.98ETH

IMRT

Measurement

Science

and

Technology



Survey"

Reconstruction of Non-Acquirable Process Quantities

ExampleLevel measurement by pressure measurement

ˆ

Σ+

B12

80

p (t)+p (t)ˆ

–

p (t)

p (t)ˆˆ

g

g 1ρ g

ρ g

Σ+

+

p(t) =

h(t)

p (t)g

h h

ˆ

Δ

ρ

p (t)g

h(t)

p(t)

p (t)g

ˆ

p(t) =

g

ρgh(t) + p (t)g

sub-process SP

inverse model ofsub-process SP


process domain


process P


acquirable process quantity p (t)

estimate of theprocess quantity

of interest

interesting, but not acquirable process quantity h(t)

sensorprocess S


g

acquirable process quantity p(t) = p (t) + p (t)gh

uninteresting, not acquirable process quantity p (t)h

S SP

resultingquantity

h(t)ˆ

fluid pressuresensor SF

gas pressuresensor SG


PP1022.Nr.99ETH

IMRT

Measurement

Science

and

Technology


Supplement→ Example "Measurement of the Terrestrial Circumference"

Reconstruction of Non-Acquirable Process Quantities

ExampleMeasurement of the Terrestrial Circumference –

Eratosthenes (276 –

194 BC)

c

S

N

1

α1,2

α1,2

1,2c

2

B14

04

α2α1

sun light

earth

mer

idian

ellip

se

tropic

21,2

1, 1 [unit of circuc c lar length360 50

c ]α

= ≈

11,2 1,

22

,

360 50 [unit of circular lc c ength]c = ≈α

ΠB

1403

αΣ

+

c

1360

1,2α1

α2

c1,2

α1,2

c360o

o

ˆ

c

c1,2

α1,2

..



process P

sensorprocess S



PP1022.Nr.100ETH

IMRT

Measurement

Science

and

Technology



Survey"

Reconstruction ProcessThe term «reconstruction»

involves also the estimation of

immeasurable quantities z(t) of the process Pby the measurement of the input and output quantities u(t) and y(t).

B09

78

Op {...}SP

Op {...} Op {...} Op {...}SPS S

1

Σ+

Σ+

ˆy(t)ˆ

zye (t) e (t)

z(t)

y(t)

z(t)

y (t)S

u (t)OLO

y (t)OLO

measurementerror

quantities

process domain


sensorprocess S


process P

sub-process model SPM(open-loop observer OLO)


erroneousestimation of


observererror

quantities

measurable quantities

sub-process SP

immeasurable quantities of interest


PP1022.Nr.101ETH

IMRT

Measurement

Science

and

Technology


Supplement→ Module "Observer -

Survey"

ObserverThere are several types of observers,

two main types:

Open-Loop Observerand

Closed-Loop Observer

More complex observers serve further practical needs, for example

Minimal ObserverUnknown Input Observer (UIO)

Nonlinear Observer

Without exception these ask for a deeper understanding of the theoretical prerequisites. Their design is always demanding.


PP1022.Nr.102ETH

IMRT

Measurement

Science

and

Technology



Survey"

Basic Structure of an Observerx(t)(N)

u(t) y(t)

x (t)

B14

10

(M) (P)

(N)

obs

(N)

x (0)obs

process P

observer OB

idealmeasure-

ment

idealmeasure-

ment

The quality of the observation is indicated by theobservation error vector eobs

(t),which asymptotically approaches zero with time t under certain prerequisites.

B05

26

u(t) y(t)

(M) (P)

x(t)(N)

Σ

(N)x (t)

(N)x(t)

e (t)obs

(N)

+

–

x (t) =f(u(t),y(t),t)

obs

obs

(N)x (0)obs

process P

observer OB

notmeasurable

measurable

idealmeasure-

ment

idealmeasure-

ment


PP1022.Nr.103ETH

IMRT

Measurement

Science

and

Technology



Survey"

Observer Structures

Simulator

x (t)sim

(N)

x(t)(N)

u(t) y(t)

B14

18

(M) (P)

(M)

u(t)y (t)

(P)sim

x (0)(N)

sim

process P

simulator SIM(process model PM)

If the input quantities u(t), to be delivered to the process model PM,can not be acquired by measurement,

but have to be created artificially by a function generator FG,we call this Simulation.

And if, additionally, these input variables are random variables,we talk about Monte Carlo Simulation.


PP1022.Nr.104ETH

IMRT

Measurement

Science

and

Technology


Supplement→ Module "Observer –

Survey"

Observer Structures

Open-Loop ObserverThe input quantity vector u(t) is ideally measured by the measurement process MU

and is led to the process model PM, which works in parallel to process P.

The internal quantities xobs

(t) of interest and the output quantities yobs

(t)are delivered by the implemented model.

B00

23

u(t)C

A

B

.

Σ

C

A

B

.y (t)

x (t)

Σ

x(t)x(t)

x(t)

y(t)

obs

obs

x (0)obs

x (t)obs x (t)obs

(M) (N) (N)

(N)

(N)

(P)

(P)(N) (N)

Σe (t)x obs

(N)+

_

immeasurable

measurable

process P

process model PM

idealmeasurement MU




PP1022.Nr.105ETH

IMRT

Measurement

Science

and

Technology


Supplement→ Module "Observer –

Survey"

Observer Structures


A feedback path (observer controller OC) is implemented. The concept is the same as in control theory: The estimated actual values yobs

(t) are compared with the measured "set point" values y(t). Deviation signals arise.These error signals are used to influence the process model PM via the input summator

(balance

point).The influence is weighted by matrix L

(control law), still to be

determined.

u(t)C

A

B

.

.

y(t)

B00

24

C

A

B

+

–L

Σ

Σy

Σ(M) (P)(N) (N)

(N) (N)

D

(M)

+

+Σ

D

(N)

(P)

(N)

++Σ

Σ(N)(P)

e (t)x

+

–

(N)

(N)

y (t)

x(t)

x(t) x(t)

x (0)

x (t) x (t)

x (t)

e (t)obs obs

obs

obs

obs

obs

obs

process P

process model PM

observer controler OC

immeasurable

measurable

observer OBS

process domaininstrumentation domainideal measurement MU

measurement proess M

ideal measurement MY


PP1022.Nr.106ETH

IMRT

Measurement

Science

and

Technology



Survey"

Applications of Observer Structures

Open-Loop Observer

B09

78

Op {...}SP

Op {...} Op {...} Op {...}SPS S

1

Σ+

Σ+

ˆy(t)ˆ

zye (t) e (t)

z(t)

y(t)

z(t)

y (t)S

u (t)OLO

y (t)OLO

measurementerror

quantities

process domain


sensorprocess S


process P

sub-process model SPM(open-loop observer OLO)


erroneousestimation of


observererror

quantities

measurable quantities

sub-process SP

immeasurable quantities of interest


PP1022.Nr.107ETH

IMRT

Measurement

Science

and

Technology


Supplement→ Exercise "Heat Flow Measurement by an Open-Loop Observer"

Open-Loop Observer

ExampleHeat-Flow Measurement by an Open-Loop Observer

B01

24

V ;* ϑ ϑV ;*e a

temperaturesensor

volume flowsensor

temperaturesensor


PP1022.Nr.108ETH

IMRT

Measurement

Science

and

Technology



B00

81

V*

ϑ

ϑ

Q*

Q*

V

ϑ

ϑ

*

e

a

e

a

ˆ

ˆ

ˆ

ˆ

sensorprocess S



(open-loop observer)

process P

immeasurable

estimate


Open-Loop ObserverExample

Heat-Flow Measurement by an Open-Loop Observer


PP1022.Nr.109ETH

IMRT

Measurement

Science

and

Technology



Open-Loop Observer

B00

26

ϑ

ϑΔϑ

Δϑϑ

ϑ

V

m

c Q

V

cQ

V

m

kVpu

u

u

pk–1 p

a

e

aϑ

eϑ

gaϑ

geϑ g

eϑ

gaϑ

g p g p

–1

–1

–1*

ρ **

*

*

e

a

+

–

+

–

*ρ

*

*

Δ Δ ΔΔ Δ

Σ

Σ

ˆ

ˆ

ˆ

ˆ ˆ ˆ

ˆˆ

ˆ

ˆ

ΠΠ

Π

Π

process P

sensorpre-process SP

sensorprocess S

reconstruction process RSof

sensor process S andsensor pre-process SP

reconstruction process RP (open loop observer)




PP1022.Nr.110ETH

IMRT

Measurement

Science

and

Technology


Open-Loop ObserverExample: Reconstruction of not accessible quantities

by an open-loop observer(residual oxygen measurement (λ) in the exhaust gas flow)

B12

81

ˆ

Δp

O

2O

2

M{Δp (t)}O2

μ (t)Δp2O

ˆ

process domain


process P (internal combustion engine)


oxygen partial pressure difference Δp (t)

air-fuel-ratio λ(t)

lambdasensor S

sub-process SPreconstructionprocess R

averaged oxygen partial pressure difference μ (t)

estimation of the air-fuel-ratio

λ(t)

(open loopobserver)

sub-process SP

figure: Bosch

figure: Bosch

PP1022.Nr.111ETH

IMRT

Measurement

Science

and

Technology



Survey"

Applications of Observer Structures


An observer is mandatory for any state variable control in the field of control engineering. The observer is included automatically

in the controlled process CP by a control system design.

Σy(t)y (t) =Py (t) = u (t)CC= u (t) Px (t)nom +

_

B14

19

e (t)xx(t)

x (t)obs

x (0)obscontrol process C

process P

controlled process CP

observer OB


PP1022.Nr.112ETH

IMRT

Measurement

Science

and

Technology

IV. Ideal / Nonideal Measurement Process

PP1022.Nr.113ETH

IMRT

Measurement

Science

and

Technology

Ideal / Nonideal Measurement Process

• Tasks of the Measurement Process

• Models of the Error Process

• Properties of the Measurement Error

• Ideal Measurement Process –

Nominal Measurement Process

• Nonideal Measurement Process

• Calibration Process for Error Identification

• Qualification Process for Uncertainty Declaration

PP1022.Nr.114ETH

IMRT

Measurement

Science

and

Technology

Tasks of the Measurement Process


PP1022.Nr.115ETH

IMRT

Measurement

Science

and

Technology


Aim ofmeasuring procedure and data analysis:

Fulfilment of theFundamental Law of Metrology

y

!

!

!

or

o

y 1y

y 1 y

y y 0

r

ore 0

=

= ⋅

− =

=

process P

measurement process M(non-ideal)

measurement quantitiesof interest


results

process domaininstrumentation domain measurement

errors

B11

63

u(t) y(t) = y (t)

y(t)ˆ

Σ+

_e (t)y

0ˆ

(P )P

(P )M

PP1022.Nr.116ETH

IMRT

Measurement

Science

and

Technology

•

sensor process S (mapping)

•

reconstruction process R (inversion)

B11

26

u(t) y(t)

y(t)ˆy (t)S

Op {...} Op {...}1

S S

Σ+

ye (t)process P


measurement quantitiesof interest

erroneousestimating

measurementresults


sensorprocess S


sensor quantities

errorquantities

B03

98

7.52 bar

2.71Volt

p/bar

u /Vp

p/barˆ

pressuresensor


Supplement

→ Module "Sensor Process"


PP1022.Nr.117ETH

IMRT

Measurement

Science

and

Technology


Supplement

→ Examples "Interface «Process –

Sensor Process»"

InterfaceProcess –

Sensor Process

• difficult accessibility of measurement location: "observability"

• minimal physical measurement effect: "resolution"

• extreme environmental conditions: "disturbance effects", "cross-

sensitivity effects", "life-time", "stationarity"

• minimal physical signal power of information source: "loadability", "back-

loading", "impedance"

• measurement at living objects: medicine, neurology, biomedicine, biology, biotechnology

• measurement by means of a living object: "biosensor"

• unknown process model at the interface location: "observability", "reconstructability"


PP1022.Nr.118ETH

IMRT

Measurement

Science

and

Technology


Supplement

→ Module "Sensor Pre-Process"

Sensor Pre-Processsuitable measurement principles are missing, which can be

used by known sensors directly:

"indirect measurement",

"observability"

yS

yB

1027

ySP

process P

sensor pre-process SP sensor process S

process domaininstrumentationdomain

Example :orifice plate in a conduit pipe

for flow measurementby p measurementΔ −

B00

42

sensor pre-process SP

resistanceforce strech voltagepressure

pressure pressure

sensor process S

Example: Thin Film

Pressure Sensor

The principles of the pre-processes outnumber the principles of the sensors by far.


PP1022.Nr.119ETH

IMRT

Measurement

Science

and

Technology

Models of the Error Process


PP1022.Nr.120ETH

IMRT

Measurement

Science

and

Technology

Two models of error analysisVersion 1.

Modelling:

Qualitative summarising of error quantitieson the ideal measurement results

Result:

Rough impressions of possible influences

error equation v.1ˆ(t) (t) (t)= + yy y e

Question: Where do errors come from?(The cause-and-effect diagram is not suitable for quantitative considerations)

B12

19

causes effect


PP1022.Nr.121ETH

IMRT

Measurement

Science

and

Technology

Two models of error analysis

Version 1.Modelling: Linear superposition (quantitative) of error quantities

on the ideal measurement results (so-called measurement noise)

Result: Erroneous estimating measurement results(usual error model)


Question: Where do errors come from?

B12

67

u(t) +

+

e (t)y

Σy(t)y(t)y(t)

idealmeasurement

process M


erroneous,estimating

measurementresults

measurementerror quantities

process P

0


PP1022.Nr.122ETH

IMRT

Measurement

Science

and

Technology


Version 2.(Error model of signal and system theory)

Modelling, step 1: Disturbance quantities and non-ideal transfer response

Result: Erroneous measurement results and

error quantities ey

(t)

error equation v.2ˆ(t) (t) (t)= −ye y y

ˆ(t)y

B11

25

u(t)

–

+

e (t)yΣ

y(t)

y(t)

v (t)M

u (t)M

process P

non idealmeasurement process M

measurementquantities ofinterest

erroneous, estimatingmeasurement

results

process domaininstrumentation domain measurement

errors


PP1022.Nr.123ETH

IMRT

Measurement

Science

and

Technology

ˆ(t)y


Combination of version 1. + 2.Modelling, step 2: Deviation Model

(separation of ideal and non-ideal transfer responses)Result: Erroneous measurement results and

error quantities ey

(t)

Druck p / bar

0 2 4 6 8 10

Str

om i

/ mA

0

5

10

15

20

nichtidealer Sensoridealer Sensor

B1266


error equation v.2ˆ(t) (t) (t)= −ye y y

B12

78

u(t)

++

e (t)y

Σˆy(t) y (t) = y(t)

y(t)

Iu (t)M

v (t)M Δ

Σe (t)y

M

idealmeasurement

process M


erroneaous,estimating

measurementresults

measure-ment

errors

process P

0

deviationmodel ΔM


non-ideal measurement process M


PP1022.Nr.124ETH

IMRT

Measurement

Science

and

Technology

Detailed model in the state space representation (time domain)Modelling, step 3:

Only disturbance quantities and dynamic transfer response

sensor process S

Two vectors of input quantities•

measurement quantities u(t)

•

disturbance quantities v(t)

One vector of output quantities•

results of sensors y(t)

disturbingquantities

input quantities

outputquantities

Σ ΣA

B

C

D

E

F

u(t)

y(t)

v(t)

B12

79

(M)

(M )

(P)

x(t)x(t).

(N)(N)

x(0)


PP1022.Nr.125ETH

IMRT

Measurement

Science

and

Technology

Deviation model in the state space representation(separation of ideal and non-ideal transfer responses)

Modelling, step 4: Only disturbance quantities and dynamic transfer response

SS Sˆ (t) (t) (t)

error equation v.1

= + yy y e

ΣA

B C

y (t)

(P) = (M)

x (t)x (t).

(N )(N )

x (0)

ˆSy(t)

(M)D

u (t) y (t)

(P) = (M)

S S

e (t)dyn

Σ ΣA

CEv (t)

B12

68

(M )

x (t)x (t).

(N )(N )

x (0)

F

e (t)distS

(P) = (M)

(P) = (M)

u

v

u

v

v v

v

u u

u

v v

u u

ΣΣ

error process E


erroneoussensor

quantities


nominal sensor process S 0idealsensorquantities

measurement errorsdue todynamic behaviour

measurement errorsdue todisturbance quantitiesand dynamic behaviour

sensor process S


PP1022.Nr.126ETH

IMRT

Measurement

Science

and

Technology

ExtensionInfluence Structures of Disturbing Quantities


Linear SuperpositionThe amount of the disturbing effect is independent from the momentary value of the measurement quantity y. The static transfer response characteristic will be shifted in parallel, the form of

the

characteristic remains unchanged, a zero shift arises (offset, bias)

B04

02ΣM

ΔvM

y = yM

gy ,vM M

gy ,uM

y = uM

ˆ

ΣvM0

+

+

–

+

non idealmeasuring process M

B08

43

ΔvM

u [{u }]MM

y [{y }]MM

nominal characteristic at nominal disturbing quantity vM0

Linear transfer response

equation

M M M M 0M y ,u y ,v MM Mv vˆy y g g ( ) [ ]u {y}= = + −

Supplement

→ Module "Disturbing Quantities at a Measurement Process"


PP1022.Nr.127ETH

IMRT

Measurement

Science

and

Technology

ExtensionInfluence Structures of Disturbing Quantities


Nonlinear SuperpositionIf the superposition of the disturbing effect occurs nonlinearly, for example multiplicatively (modulating), the static transfer response characteristic of the measurement path is revolved or even deformed. The resulting

measurement error depends on the momentary measurement value y =

uM

for a certain value of the disturbing quantity vM

.

Nonlinear transfer response

equation

B04

08ΠM

vM

y = yM

gy ,vM M

gy ,uM

y = uM

ˆ

non idealmeasuring process M

y [{y }]

B08

44

MM

vM

u [{u }]MM

nominal charakteristic atnominal disturbance quantity vM0

M M M MM y ,u y , MMv v uˆy y g g [{y}]= =

Supplement

→ Module "Disturbing Quantities at a Measurement Process"


PP1022.Nr.128ETH

IMRT

Measurement

Science

and

Technology

ExtensionBack-loading model of the measurement process

Statements


•

"Loading" is a general phenomenon

•

Processes of interest and their quantities can beinfluenced by measurement processes.

•

In the field of natural sciences and technology causes of loading influences are of physical nature.

•

Withdrawal of information goes with withdrawal of energy.

PP1022.Nr.129ETH

IMRT

Measurement

Science

and

Technology

ExampleBack-loading model of the thermal measurement process

IC

P(t) [W]

ϑ (t)1 ϑ (t)2

ϑ (t)3

R(t) [Ω]

B07

45

S

3

1

2


PP1022.Nr.130ETH

IMRT

Measurement

Science

and

Technology

General ExamplesBack-Loading


u (t)C

u(t)

i(t)

Δu

R0

C

Δu

R

C

Δu

R

C

Δu

R1 2 3 4

1 2 3 4

1 2 3 4

B13

54

Quelle Senke

ϑ (t)

Cϑ(t)

Q(t)

Δu

R0

C

Δu

R

C

Δu

R

C

Δu

R1 2 3 4

1 2 3 4

1 2 3 4

B13

55

*Quelle Senke

1. Current through RC-Circuits

2. Heat Flow through Layers of Walls

PP1022.Nr.131ETH

IMRT

Measurement

Science

and

Technology


GeneralisationExpanded Series Connection → Series nested Feedback-Loops

B00

03

PP1022.Nr.132ETH

IMRT

Measurement

Science

and

Technology


Examplesimple electrical quadrupole

voltage divider

B06

48

u(t)

i(t)

u (t)1

i (t)1

Prozess

Σu(t) u (t)

R

i (t)

1

1

Δu (t)R

B06

68

R1

1

i(t)Σ

i (t)R1

–+

+

+

B06

67

R

i (t)R

u(t)

RΔu (t)

u (t)1

i(t)R

Ri (t)

i (t)11

1

B04

35

0 10.80.60.40

1.0

0.8

0.6

0.4

0.2

0.2

0.51

2

2R

R

u0

LR

R

x =

=

eNL

oo

R = R + R1 2

u

PP1022.Nr.133ETH

IMRT

Measurement

Science

and

Technology


ExampleElectrical Loading of a Thermocouple

B10

19

ϑ1

ϑ2

uRQ RLuΔϑ Vˆ ˆ

Δϑ = ϑ ϑ [K] bei ϑ > ϑ

1

1 2

2

Messverstärker V

Thermoelement TE

uuRVRL

Q

Δu

Δϑu

iB10

20L

B

ˆΔϑ

gu ,uˆ ˆV Δϑ

PP1022.Nr.134ETH

IMRT

Measurement

Science

and

Technology


Cause: Voltage Divider with badly adapted parameters

ideal would be:•

RQ

= 0 and / or•

RL

= ∞then the loading path would be tight

Electrical Loading of a Thermocouple

ΔϑB

1315

ˆ

gu,Δϑ

_

+

euΔϑ

gu,Δϑ

uΔϑ

uΔϑ

Σ

Σ

RQ

+_

uΔϑˆ

[V]

[V]

[V]

[°C]

iL

g

RL

1

[A]

gu ,u

_

+

eΣ

Δϑ

[°C]

B uΔϑΔu = e [V]

V [V]

[V]

[V]

u ,uV

u V

u V

u V

nominalthermocouple TE


thermocouple TE(source Q)

amplifier V(load L)

non-ideal sensor process S

nominalamplifier V0 0

virtual nominal sensor process S 0

Example

PP1022.Nr.135ETH

IMRT

Measurement

Science

and

Technology


Modelling, step 5: Disturbances, back-loading and non-ideal transfer response

B11

28

–

+

e (t)y

Σ

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

v (t) z (t)M M

MM

u(t)

v(t)

y (t)0

–

+

e (t)y

Σ0

=

process P


disturbedmeasurement

quantities

erroneous,estimating

measurementresults


error quantitiesdue to non ideal

measurement

nominal domainprocess domain

measurementquantities of interest

error quantitiesdue to loading

nominal process P

y (t)dist

0

Two types of measurement errors:

1.

error quantities due toback-loading

2.

error quantities due tonon-ideal measurement


PP1022.Nr.136ETH

IMRT

Measurement

Science

and

Technology


Statements


•

Deviations within the estimated measurement results arise compared to the measurement quantities at the non-

instrumented process.

•

Loading deviations are not observed in the measurement results, since the measurand changed by loading.

•

Whether loading errors bother somebody or are acceptable has to be judged by the measurement personal (quality management).

PP1022.Nr.137ETH

IMRT

Measurement

Science

and

Technology

Detailed model in state space representation (time domain)Modelling, step 6: Disturbances, back-loading and non-ideal transfer response

non-ideal, dynamic, disturbed and loading system


loadingquantities

input quantities

outputquantities

Σ

Σ

Σ

A

B C

D

E

F

G

J

H

u(t) y(t)

v(t) z(t)

B12

30

x(t)x(t). x(0)

(t) (t) (t) (t)

(t) (t) (t) (t)

(t) (t) (t) (t)

= + +

= + +

= + +

x A x Bu E vy C x Du F vz Gx Hu Jv

(t) (t) (t)

(t) (t) (t)

(t) (t) (t)

= =

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

x A B E x x

y C D F u P u

z G H J v v

or


PP1022.Nr.138ETH

IMRT

Measurement

Science

and

Technology

SynthesisMeasurement process M and process P

Modelling, step 7: Disturbances, back-loading and non-ideal transfer response

B12

92

z (t)P

u (t)P

v (t)P

y (t)P

z (t)M

u (t)M

v (t)M

y (t)M

PP

PM

PP

PM

S {P , P }P M

u (t)1

u (t)

u (t)

u (t)

1

2

2

y (t)

y (t)

y (t)

y (t)

1

2

1

2

process P



observed process OPdescribed by

Redheffer-Star-Product Matrix S

"P" = parameter matrix

Special series connectionin state space representation:

Special multiplication of the twoparameter matrices PM

and PPwithin the so-called



PP1022.Nr.139ETH

IMRT

Measurement

Science

and

Technology


Model: Special series connection in the state space representation

Special Multiplication of twoparameter matrices PM

und PPin the so called


Reconstruction of Process and Measurement Process:

Inversion of the Redheffer-Star-Product Matrix S

Matlab-Function"starp"

PP1022.Nr.140ETH

IMRT

Measurement

Science

and

Technology

•

Version 2 of error analysis

delivers information about erroneous measurement results and

about error quantities ey

(t) as well.

• The error model of signal and system theory enables a qualitative and quantitative error analysis.

•

It delivers the non-ideal transfer response of the measurement process.

•

A sufficient knowledge of the non-ideal transfer response of the measurement process is an indispensable prerequisite for any uncertainty analysis.

•

The error model of signal and system theory has to be applied to

any small sub-structure of the measurement chain, if contributions to errors and uncertainties are expected.


ˆ(t)y

Supplement

→ Module "Nonideal Measurement Process"


PP1022.Nr.141ETH

IMRT

Measurement

Science

and

Technology


Properties of the Measurement Error

PP1022.Nr.142ETH

IMRT

Measurement

Science

and

Technology


Properties of the Measurement Error

Systematic and Random Error

Supplement

→ Module "Properties of the Measurement Errors"

In signal theory we distinguish between deterministic and random

signals.

As a consequence of deterministic and random causes

we get for a nonideal measurement process

systematic errors (bias errors)

and / or

random errors.

Regardless of the causes we state:

e (t) e (t) e (t) [{y}]y y ysyst rand= +


PP1022.Nr.143ETH

IMRT

Measurement

Science

and

Technology


Time Constant and Time Variable Error

Supplement


time-constant error ey

and

time-variable error ey

(t)


PP1022.Nr.144ETH

IMRT

Measurement

Science

and

Technology


Direct and Alternating Component of the Error

Supplement


One can subdivide the systematic error eysyst

(t) and the random error eyrand

(t) again, because every signal y(t) consists of a

constant direct component yDC

and an

alternating component yAC

= Δy.

The direct component yDC

corresponds to the arithmetic mean value μy

.

Therefore an error may consist of three components:

e e e ey y y yDC syst rande e [{y}]e y yy syst rand

= + Δ + Δ =

= μ + Δ + Δ

Definition: Components of the Error


PP1022.Nr.145ETH

IMRT

Measurement

Science

and

Technology


Mean Error

Supplement


We are seldom interested in the details of errors changing with time.We would be pleased by summarising mean information.These are defined according to signal and system theory.

We apply usual concepts of stochastics to the error ey

, and getarithmetic mean value, variance and mean square value:

Definition: Mean Error

Definition: Variance of the Error

Definition: Mean Square Error

{e } [{y}]e yyμ =M

2 2 2{ e } {( e e ) }e y y yy syst rand2 2 2[{y }]e esyst rand

σ = Δ = Δ + Δ =

= σ + σΔ Δ

M M

2 2 2{e } {( e e ) }e y e y yy y syst rand2 2 2 2[{y }]e e ey syst rand

ψ = = μ + Δ + Δ =

= μ + σ + σΔ Δ

M M

Supplement

→ Module "Averaging of One Variable"



PP1022.Nr.146ETH

IMRT

Measurement

Science

and

Technology


Mean Error

Supplement


According to the "Pythagoras of Signal Theory"

there is a relation between these three characteristic error values:

Supplement

→ Module "Averaging of One Variable"

2 2 2 2[{y }]e e ey y yψ = μ + σ

Root Mean Square Error

2 2 2 [{y}]e e e ey y syst systψ = μ + σ + σΔ Δ

In order to be able to compare the quadratic errors with the measurement quantity y we take the root and get the root mean square error (rms-error).



PP1022.Nr.147ETH

IMRT

Measurement

Science

and

Technology


Ideal Measurement Process –Nominal Measurement Process

PP1022.Nr.148ETH

IMRT

Measurement

Science

and

Technology


model M of non idealmeasurement process

models of correctmeasurement results

estimating,incorrect and uncertain

measurement results

idealnon ideal

measurementerrorsnominal model M

of measurements process0

process P


B11

65

y(t)

y(t)ˆ

Σ+

_e (t)y

u(t)

y (t) = y(t)0ˆ

(P )P

(P )M

I

Ideal Measurement Process = Nominal Model M0

Definition: Fundamental Equation of Metrologyˆ(t) (t)= ⋅y I y

Definition: Measurement Errorˆ(t) (t) (t) =− = yy y e o

Supplement

→ Module "Ideal Measurement Process"


PP1022.Nr.149ETH

IMRT

Measurement

Science

and

Technology


Ideal Measurement Process = Nominal Model M0

has the following properties

• linear

• non dynamic

• stable

• unit transfer value matrix I

(just as many result

quantities as measurement quantities: (PM) = (PP))

• no feedback paths to the precedent process(load transfer function equals zero)

• insensitive (robust) against disturbing quantities(disturbance transfer function equals zero)

Supplement

→ Module "Ideal Measurement Process"


PP1022.Nr.150ETH

IMRT

Measurement

Science

and

Technology


Nonideal Measurement Process

PP1022.Nr.151ETH

IMRT

Measurement

Science

and

Technology


Four transfer responses of nonideal measurement

•

«measurement transfer response»on the measurement path uM

(t) → yM

(t)

•

«disturbance transfer response»on the disturbance path vM

(t) → yM

(t)

•

«measurement / load transfer response»on the measurement / load path uM

(t) → zM

(t)

•

«disturbance / load transfer response»on the disturbance / load path vM

(t) → zM

(t)

y(t) = u (t) y (t) = y(t)M M

B12

06

v (t) z (t)M M

measuring process M

Supplement



PP1022.Nr.152ETH

IMRT

Measurement

Science

and

Technology


Causes of nonideal measurement1. nonideal transfer response behaviour

(structure, parameter)2.

disturbance quantities and disturbance paths

3. load quantities and load paths

B13

60

z (t)M

u (t)M

v (t)M

y (t)MΣ

Σ


measurementpath

disturbancepath

disturbance /load path

measurement /load path

loadingquantities


= disturbedmeasurementquantities y (t)

estimatingquantities y(t)ˆ

dist

Supplement



PP1022.Nr.153ETH

IMRT

Measurement

Science

and

Technology


Extended Fundamental Law of MetrologyTransfer Function Matrixof Measuring Process M is ideal, ifa. Transfer Function Matrixof Measurement Path

equals unit matrix I:

andb. Transfer Function Matrixof Disturbance Path

equals zero matrix 0:

andc. Transfer Function Matrixof Loading Path

equals

zero matrix 0:

B11

64

z (t)M

u (t)M

v (t)M

y (t)M

= 0!

= I!

= 0!

= 0!

Σ

Σ

idealmeasurement process M

measurementpath

disturbancepath

disturbance /load path

measurement /load path

loadingquantities


disturbedmeasurementquantities y (t)


results y(t)ˆdist

M M M M(t) (t) und (t) (t)= =0 0z u z v

M Mˆ(t) (t) (t)= =y y v0

M Mˆ(t) (t) (t) (t)= = =I Iy y u y M

M M

M(t(tt

)( )

))

(t⎡ ⎤ ⎡ ⎤⎡ ⎤

=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦v

y I 00z 0

u

Ideal Measurement Process M0

Supplement



PP1022.Nr.154ETH

IMRT

Measurement

Science

and

Technology


Causes of nonideal measurement

Secondary causes•

disturbing quantities vM

(t) on the disturbance path are unequal zero or unequal the nominal values

• loading quantities zM

(t) on the load path are unequal zero or unequal the nominal values

Main causenonideal transfer response of the measurement process M

Supplement



PP1022.Nr.155ETH

IMRT

Measurement

Science

and

Technology


Causes of nonideal transfer response behaviour

• nonlinear transfer function of the measurement path (usual situation)

• dynamic behaviour of the measurement path (differential equation instead of linear algebraic equation)

• time-dependent parameters of the transfer function of the measurement path (drift, random results)

• unstable transfer function of the measurement path (rather rare)

Supplement



PP1022.Nr.156ETH

IMRT

Measurement

Science

and

Technology


• model structure-

inadequate equation type

- inadequate order

- unconsidered information paths

- unconsidered disturbing quantities

• model parameter

• computational aspects-

numerical, rounding errors

- inversion difficulties-

no one-to-one functions

-

no quadratic matrices-

inverse dynamics

Supplement


Causes of for nonideal transfer response behaviour

Model Errors in Sensor Process S and Reconstruction Process R


PP1022.Nr.157ETH

IMRT

Measurement

Science

and

Technology


Means for the minimisation of measurement errors

Due to secondary causes•

insulating, shielding, measurement, correcting, filtering, controlling

Due to main cause•

design, planning, correcting, reconstructing, filtering

Supplement



PP1022.Nr.158ETH

IMRT

Measurement

Science

and

Technology

Calibration Process for Error Identification


PP1022.Nr.159ETH

IMRT

Measurement

Science

and

Technology

After Modelling:Calibration (Identification)

of processes and measurement processes

• Aim–

Determination of parameters of signals and systems concerned

•

Tasks–

measurement path calibration

–

disturbance path calibration–

back-loading path calibration

•

Documentation–

models of signals (transfer response functions)

–

models of systems (transfer response functions)


PP1022.Nr.160ETH

IMRT

Measurement

Science

and

Technology

after Modelling:Calibration (Identification)

of processes and measurement processes

B11

30

u(t)

–

+

e (t)yΣ

y(t)

y(t)

u (t) y (t)

–

+ e (t)y

Σ

refref

u (t) y (t)M M

process P

non idealmeasurement

process M


measurementresults


measurementerrors

measurement mode

calibration mode

generating processof reference quantities

< unknown

< known

< known

< known< known< known

< unknown

measurmenterrors

calibration process C


PP1022.Nr.161ETH

IMRT

Measurement

Science

and

Technology

Results of calibration (identification)Example

At least, steady-state response function of processes and measurement processes

B04

41

y

y

y0

SP

ˆ

ˆ

pdy yˆ

nominal steady-statetransfer response line (ideal)

estimatedsteady-state

transfer respons line(non-ideal)

conditionaldistribution density functionof random deviations


PP1022.Nr.162ETH

IMRT

Measurement

Science

and

Technology

Two components of measurement results:

systematicerror contributions make a measurement result

"incorrect"

randomerror contributions make a measurement result

"uncertain"


PP1022.Nr.163ETH

IMRT

Measurement

Science

and

Technology

Example: Quantising process with sine input signal

0 1 2 3 4 5 6 7 8

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time t [s]

sign

als

u [V

]

quantexample1.m; B1295.eps

original signal u(t)error signal e

q(t)

quantised signal uq(t)

Question: Deterministic or random error?


PP1022.Nr.164ETH

IMRT

Measurement

Science

and

Technology

B05

38

idealising, unreal situations

systematicand

randomerrors

Untreatedresults

of acquisitions and measurementscan be

correctonly

incorrectonly

uncertain

incorrectand

uncertain

noerrors

onlysystematic

errors

onlyrandomerrors

Therfore they contain

real situation


PP1022.Nr.165ETH

IMRT

Measurement

Science

and

Technology

Qualification Process for Uncertainty Declaration


PP1022.Nr.166ETH

IMRT

Measurement

Science

and

Technology

Measurement uncertainties u

are determined in a separate

qualification process Q

and are attached to the


determined in the


y


PP1022.Nr.167ETH

IMRT

Measurement

Science

and

Technology

y (t)ˆ

y(t)

B10

67

–

+

e (t)yΣ

ˆ

u (t)yˆ

y(t)ˆ

e (t)systˆ

non-idealmeasurement

process M


measurement errors(unknown) erroneous,

estimated measurementresults (raw data)

qualification process Q

estimatedtotal

measurementuncertainties

partial measurementuncertaintiesdesiredstatistic certainties

model of non idealmeasurement process

number ofmeasurements

apriori knowledge aboutunknown errors

correctionprocess CORR

estimates of systematicmeasurement errorsby calibration

model of ideal (nomional)measurement process

correctedmeasurement

results

corr

uncertaintyevaluation

process UE


PP1022.Nr.168ETH

IMRT

Measurement

Science

and

Technology

y = μ ; u [{y}] P = … %yˆ μ,σ

B09

55

ˆ

chosen statistic security for the measurement result

unit of the measurement quantity

calculated uncertainty of the measurement result

estimating mean measurement result, corrected by the estimated systematic measurement errors

symbol of measurement result

for


Measurement Result

PP1022.Nr.169ETH

IMRT

Measurement

Science

and

Technology


Relations?

PP1022.Nr.170ETH

IMRT

Measurement

Science

and

Technology


Relations?

Common roots -

common tools

Σ

B05

92

u(t)y(t)

x(t)

y (t)

x (t)

+

–

e (t)y

x (t)

L

obs obs

obs

obs

u(t)u(t)

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

x(t)

process P

observer controller OC

process model PMclosed loop observer CLO

ideal measurement MU


ideal measurement MY


not measurable

Σ

B05

91

x(t)u(t)

x(t)

w(t)e (t)x

+

–

y(t)

K co

controller CO

idealmeasurement

idealactuating



controlled process COP

observing system

PP1022.Nr.171ETH

IMRT

Measurement

Science

and

Technology


Relations?

Common roots -

common toolsDuality

Transpositioncontrolled process observing system

Σ

– K

A

Bx(t)x(t)

x(0).

B05

86

+

+

(M)

(M) (N) (N)Σ

– L

A

x (t)x (t)

x (0).

B05

85

Cy (t)+

+

obsobs

obs

obs

(N)(N)

(N) (P)

(P)

A BK

x(t)x(t)

x(0)

.B

0588ACO

(N) (N)

(N)

A LC

x (t)x (t)

x (0)

.

B05

87Aobs

obs obs

obs

(N) (N)

(N)

PP1022.Nr.172ETH

IMRT

Measurement

Science

and

Technology


Relations?

Supplement→ Module "Two Fundamental Control Structures"

y (t) = y(t)Py (t) = u (t)Cy (t) = u (t)C Pnom

B13

70

op {...}_1 op{...}

control process C process P


y (t) = y(t)Ry (t) = u (t)Sy(t) = u (t)S R

B13

71

op{...}1_

op {...}ˆ

sensor process S reconstruction process Rmeasurement process M

B12

27

process P


controlprocess C

measurementprocess M

actuationprocess A


original systemfollows the inverse system

control is alwaysModel-Based Control

inverse systemfollows the original system

Generic Loop of Information and Physical Power

measurement is alwaysModel-Based Measurement


PP1022.Nr.173ETH

IMRT

Measurement

Science

and

Technology

Conclusion

• Considerations and conclusions hold regardless of areas of expertise and of instrumental realisations

• Limitation to well-known terms of propaedeutic domains increases transparency and acceptance

• Theoretic tools are derived from mathematics, stochastics, statistics, signal theory, system theory

• Without model-based measurements, no statements about measurement quality are allowable

• Properties of measurement processes as well as properties of measurement results have to be disclosed in quality statements

PP1022.Nr.174ETH

IMRT

Measurement

Science

and

Technology

Conclusion

All results, shown here, are universally applicable,not only in the measurement or control field.

Analysis, Scepticism and Validationare mandatory

anywhere and anytime.

Basic Structures in Metrology -...

Documents

Transcript of Basic Structures in Metrology -...