From Verbal Models to Mathematical Models – A Didactical Concept not just in Metrology

ETH

From Verbal Models to Mathematical Models –

A Didactical Concept not just in Metrology

Karl H. RuhmInstitute of Machine Tools and Manufacturing (IWF), ETH Zurich, Switzerland

[email protected]

Invited Plenary Lecture

Joint International IMEKO TC1+TC7+TC13 Symposium 2011

Jena, Germany

August 31st – September 2nd, 2011

28. 06. 2011Version 02; 15.10.2011

www.mmm.ethz.ch/dok01/e0001000.pdf

easurement

Science

and

Technology

etrology

2ETH

I Mathematical Models

II Models and Metrology

III Models and Structures

IV Models and Randomness

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TU Karlsruhe, DE

3ETH

Points of View

• model based measurement (soft sensors)

• knowledge based measurement

• intelligent measurement (smart sensors)

• learning measurement (neural sensors)

• fuzzy measurement

• cyber measurement

• robust measurement

• integrated and distributed measurement

Are there different measurement concepts?

No, there are only different procedures and tools.

4ETH

Points of View

Indeed, there are quite particular interests of individual circles.

Yes, but they do not concern essential aspects of Metrology.

5ETH

Points of View

• intentional definition and description of quantities

• quantities traceable back consistently to set standards

• calibration (identification) of instrumental processes

• stimulation of processes under measurement

• local acquisition of quantities, intended to be measured

• reconstruction in time and space

• verification of measurement results

Are there global concepts in Metrology concerning common interests?

Yes!

6ETH

Points of View

Procedures and tools may differ

but

integral constituents of these concepts are always

mathematical models,

at least in the background

NO EXCEPTIONS!

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Points of View

• Bottom Up Approach,starting from individual, very specific needs,remaining in a restricted perspective

• Top Down Approach,starting from the common Fundamental Axiom of Metrology,designing, judging and informing from a prospective position

Are there different approaches to measurement tasks?

The combination does it!

8ETH

Points of View

The following statements

will stick to the top down approach

and

will present examples in the bottom up approach,

they are supposed to apply to all fields of Metrology.

9ETH

Supplement → Slides "Process and System"Supplement → Conference Paper "Process and System – A Dual Definition"

Concentration on Few Terms

processa defined fraction of the

natural and man-made real world,always multivariable and dynamic

quantityin the real world, time and space dependent

modelrelates quantities of processes

mathematical modelrelates quantities of processes by equations

property and behaviourdescribe processes and quantities

by parameters and solutions of the model equations

10ETH Supplement → Terminology List "Error and Uncertainty"

Concentration on Few Terms

errora quantity appearing as a

difference (deviation, discrepancy)between two defined quantities,deterministic and / or random

uncertaintya parameter in Statistics,

describinga property of a random quantity

11ETH Supplement → Slides "Process and System"

Supplement → Conference Paper "Process and System – A Dual Definition"

Main Tools

Mathematical Modelsdescribe processes by

logical expression and mathematical functions

This field is covered likewise bySignal and System Theory

andStochastics and Statistics

A useful graphical visualisation is theSignal Effect Diagram

(block diagram, flow chart, event map)

12ETH

graphical model structures

are important,

they reflectlogical and mathematical structuresin an impressively descriptive way

13ETH

Points of View

• Process under Measurement PUM

with

• Process P without Measurement Process

• Measurement Process M without Process

Some Structured Assumptions for Metrology

14ETH

Points of View

• Process under Measurement PUM

• Process P without Measurement Process

• Measurement Process M without Process


15ETH

Points of View

Quantity

with some hierarchically ordered sub-termsconcerning measurement (measurand and resultant)

• quantity of no interest

• quantity of interest

• quantity intended to be measured

• quantity immeasurable

• quantity under measurement

• quantity actually measured

• quantity indirectly measured

• quantity resulting


16ETH

Points of View

Errors and Uncertainties

• are virtual quantities

• are models already

• are given by abstract mathematical definitions in theory

• are determined by calibrations and inference in practice


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IMathematical Models

2 2 2

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Substantial World of Reality• infinitely large• infinitely interconnected• infinitely dynamic• infinitely nonlinear

("Nature Loves to Hide")

Abstract World of Imagination• small• bounded and limited• defined• estimates more or less exactly the real world• manageable by today's tools

("Universe of Knowledge")

Two Worlds

19ETH

How Do Models Come In?

Models in the real world

(NASA)

Models in the virtual, abstract world(intellectual products of the human mind)

2 2 2

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Navier-Stokes Equations Kármán-Vortex Street(courtesy Cesareo de La Rosa Siqueira)

20ETH

Models in a Hierarchy

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Types of Models

qualitativeideas about something

mind modelsmodelling thoughtsverbal statements

opinion and prejudiceideas and visions

etc.

quantitativedrawings, pictures, notes, articles, novels, instructions,

theories, logical, mathematical and probabilistic equations,business plans, programs, flowcharts,

acoustical and optical verbal documentationetc.

22ETH

An Idealised Virtual World of Imagination

• reduced to limited and bounded extent• considering only essential relations• reduced to few orders• largely linearised • assuming deterministic relations to a large extent• allowing errors and uncertainties

What do Quantitative Models Describe?

The design of a model allows finite effort only.

Additionally, we need the «ideal» on the other hand,

↓

«the ideal» as a possibility with the probability zero.

and

23ETH

1. Analytical modelling by first principlesmathematical and probabilistic equations

2. Empirical modelling by experiment, by measurement(structure and parameter identification, calibration, regression)

at an original process(for example: measurement process, sensor process)

at a model process(for example: aircraft in wind tunnel)

Nearly all models have been designed both ways

Ways to Mathematical Models

Note

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Useful Models of Processes

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Three Questions around a Process Model

Supplement → Module "Process and System"

given: given: searched: scope of functions

1. input signals u system structure,system parameter p

→ output signals y transformation, control, convolution, forecast, transfer response, mappingsimulation, measurement

2. output signals y system structure,system parameter p

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

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

structure identification, parameter identification, correlation, calibration, test

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We describe quantities,some of which are intended to be measured

and

we describe relations between quantities.

Important,

we do not describe processes,we describe them only indirectly via quantities and their relations.

WHY SO?

Describing Processes by Models

27ETH

We start a model verbally with a-priory knowledge

The description will be more or less appropriateelaboratedetailedaccurate

qualitative

It isa model already

and it is usefulsince

we can discuss itand

it can be the base of first decisions

28ETH

We start a process model choosing quantities

real quantities

and

derived quantities

like

efficiency, flexibility, utility, stability, robustness, observability, controllability, capacity,

etc.

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Example

Model of a pump as process P.

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ExampleWe select and identify the group (vector) of quantities of interest.

We decide which areindependent (input) quantities

and which aredependent (output) quantities.

Here, the model of process Pis identical with the set of two mathematical equations (operations),

relating three quantities of interest, that's all!

Processes are described by relations between quantities!

31ETH

Models of Dynamic Processes

describe the processes by different types of

differential equations

and

integral equations,

introducing

velocities and accelerations of quantitiesas additionalquantities

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Mathematical Model of a Dynamic Sensor Process

ExampleResistance Thermometer (RTD)

simplifying assumptions:

(WIKAPt100)

R(t)

(t)F

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QuestionIs the abstract mathematical model

able to represent thereal world?

AnswerYes and No

Yes, if onlyrelations between distinguished quantities

are considered

No, if the overallexistence and behaviour is meant.

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ExampleResistance Thermometer (RTD)

The graphical result of the model design

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Properties of Process Models

Mathematical and probabilistic equationsare characterised by

Structures and Parameters

Structuresare determined by assumptions and hypotheses

Parametersare determined by parameter identification (calibration)

ThusStructures and Parameters

are always hypotheses and estimates,prone to

model errors and model uncertainties.

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Properties of Process Models

We assignStructures and Parametersof mathematical equations

toProperties

of theProcess Under Modelling

(PUMO)

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Example

General model of a dynamic process of second order (ODE):

Applied model for an oscillating process (equation of motion):

21 0 0

n

with

u [{u}] input quantity

y [{y}

y(t) a y(t) a y(t) b u(t

] output qu

) [{y}

antit

s ]

y

a ;b parameter

2c c 1h(t) h(t) h(t) a(t) v(t) h(t) f(t) ms

m m m m m

1

2

1 2

1

1

quantities

h m stroke

h ms velocity

h ms acceleration

f N force

parameter

m Nm s kg mass

c Nm s damping value

Nm stiffness value

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Temporal and Spatial Behaviour of Process Models

A process will respond to changing input quantities.

The way it responds is calledbehaviour.

The behaviour dependson the structure,

on the parametersof the process model

and on theinput quantities

(excitation, impact, stimulation)

39ETH


Standardised excitation functionsat the input

during measurement and calibrationfor comparison purposes of process behaviour:

impulse functionstep functionramp function

harmonic functionrandom function

(e.g. Monte Carlo Simulation)etc.

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We get the behaviourby experiment(measurement)

orby analysis

(solution of a set of equations)consisting of the

homogenous solution(eigen-behaviour)

and theparticular solution,combined in thegeneral solution

(overall-behaviour)

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Graphical Representation

42ETH

Process Models give

descriptions

they giveno explanations(interpretation)

explanationsare searched by human beings

or byprograms using so-called

artificial intelligenceand

expert knowledge

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IIModels and Metrology

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Examples of processes• a hospital patient• a motor vehicle• a machine tool• a global positioning system (gps)• an education system

The surroundings of metrological procedures

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First Steps to Models

Ideas and A-Priory Knowledge

Leonardo da Vinci1452 - 1519

"The noblest pleasure isthe joy of understanding"

Hard Model

ExampleMeasurement of Humidity

"Flight of Imagination"

46ETH

Definition of quantities intended to be measured

ExamplePhysical quantities of potential interest

within a model of a processconcerning a humid gas in a container.

1 1

g gasw waterwv water vapours saturated

x absolute humidityrelative humidity

p bar pressureC temperature

K absolute temperatureR Jkg K gas constantm kg

Quantities:

Indic

mas

es:

s

47ETH

Model-Based Measurement

Supplement → Example "Measurement of the Terrestrial Circumference"

Reconstruction of Non-Acquirable Process Quantities

ExampleMeasurement of the Terrestrial Circumference by Eratosthenes (276 – 194 BC)

21,2

1, 1[unit of circuc c lar length

360 50c ]

11,2 1,

22

,

36050 [unit of circular lc c ength]c

48ETH

Model of Measurement Procedure

Supplement→ Module "Ideal Measurement Process – Nominal Behaviour"

The most general Model of a Measurement Process M is simple.

All statements made up to now are valid here too.

Are there other aspects, are there new aspects?

No,with one exception:

The Fundamental Axiom of Metrology.

It is special for Metrology!

49ETH Supplement→ Module "Ideal Measurement Process – Nominal Behaviour"

For now we assume hypothetically an

Ideal Measurement Process MNwith a

Nominal Behaviour:

In the model domain theresult quantitieshave to equal the

unknown measured quantities y(t).

This is the nominal model of a measurement process.We call this concept

The Fundamental Axiom of Metrology

and we formalise it as amathematical model

by

ˆ(t)y

ˆ(t) (t)y I y

50ETH Supplement→ Module "Ideal Measurement Process – Nominal Behaviour"

The Fundamental Axiom of Metrology

•is a mathematical model

•is extremely simple

•is independent from instrumental realisations

•has far reaching consequences

↓

every design of a measurement process follows it

51ETH

The Nominal Measurement Process MNhas a most simple structure,

with transfer response values gn which equal one everywhere.

In a graphical representation of a process Pwith an ideal measurement process MN,

the measurement process MN is frequently omitted.

Control people like this version,we don't, because it does not reflect the real situation

with errors and uncertainties.

52ETH

We include the measurement process Min the famous measurement chain

(series connection)

This structure is too simple for different reasons.

ˆ(t) (t)y I y

53ETH

With instrumentation we have as a sub-process the sensor processes S,providing again physical quantities, but not

the desired result quantities ,given as numbers with the appropriate units.

So we had to add a second sub-process,connected in series with the sensor process S,

we call it reconstruction process R.

ˆ(t) (t)y I y

ˆ(t)y

54ETH

But, we live in a nonideal world.

ˆ(t) (t)y I y

What happens? Errors will arise!

55ETH

Mathematical Modelof the

measurement errors ey(t)

In fact, the error is definedas the difference between output and input y(t)

of the measurement process M.y(t)

y

!

!

!

!

or

o

y 1y

y 1 y

y y 0

e 0

r

or

ˆ(t( )t) (t) y y ye

Supplement → Module "Nonideal Measurement Process"

56ETH


Reconstruction Process R

By means of the Reconstruction Process Rthe

Fundamental Axiom of Metrologywill be fulfilled,

as soon as its transfer response function is realised asthe mathematically inverse model

of the given transfer response function of theSensor Process S.

Supplement → Module "Reconstruction Process – A Survey"

57ETH


Reconstruction Process R

As soon asOpR{…} = OpS

–1{…},

we get as desiredOpS

–1{…}·OpS{…} = I.

Supplement → Module "Reconstruction Process – A Survey"

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IIIModels and Structures

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In Metrologythe series connection of

Sensor Process S and Reconstruction Process Ris the most important one,

because this structure defines the basic task of measurement.

All other structures are derived from it.

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Important DetailSystematic structures for the inversion

of interconnected systems (models)

Supplement → Module "Inversion of Interconnected Systems"

61ETH

Inversion of an interconnected system (model)

ExampleReconstruction of a nonlinear (quadratic) sensor model,with pressure p as the input and current i as the output,

developed by analytical means.Main Result

Error Model and Error Compensation

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Inversion of an interconnected system (model)

ExampleReconstruction of a nonlinear (quadratic) sensor model,with pressure p as the input and current i as the output,

developed by empirical means (calibration table).Main Result

Error Model and Error Compensation

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IVModels and Randomness

TU Karlsruhe, D

64ETH

Randomness and Probabilistic Concepts

There are Questions

Are there differences in respect to deterministic concepts and procedures?

Why do deterministic procedures finally end up in uncertain results?

How are models of processes concerned?

What is Stochastics, what is Statistics?

Some Answers

65ETH


Are there differences in modellingin respect to deterministic concepts and procedures?

Not really

Instead of quantitieswe relate characteristic values and functions of the random quantities.

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Are there differences in modellingin respect to deterministic concepts and procedures?

Not really

With deterministic quantities we create deterministic relationsbetween these quantities,which describe processes

and

with random quantities we create deterministic (!) relationsbetween the characteristic values and functions of the random quantities,

which describe processes.

The tools are Signal and System Theory and Stochastics and Statistics.

67ETH


With deterministic quantities we create deterministic relationsbetween these quantities,which describe processes

and

with random quantities we create deterministic (!) relationsbetween the characteristic values and functions of the random quantities,

which describe processes.

These two concepts are extremely important when dealing withmeasurement errors and measurement uncertainties

inMetrology.

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Why dodeterministic procedures

finally end up inuncertain results?

The merging effects of several deterministic causesappear random to the observer,

who searches backwards for the causes of the observed effects.

They appear random by inspection but they are deterministic on principle!

69ETH


Random quantities intended to be measured.Which ones? You have to choose!

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QuestionDo random quantities of processes have something to do withrandom measurement errors and measurement uncertainties?

AnswerPer definition No!

71ETH

Mathematical Models of Averaging ProcessesExample

Averaging structure () for the determination of individual and jointcharacteristic values of two random quantities (mathematical models in Statistics)

Supplement → Module "Individual Averaging of Two and More Quantities"

72ETH

Mathematical Models of Averaging ProcessesExample

Matrix structure for the determination of joint and crosscharacteristic values of random quantities around a process

(mathematical models in Statistics)

Supplement → Module “Joint Averaging of Two and More Quantities"

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Mathematical Models of Averaging ProcessesExample of Application

Correlation Technique in Flow Measurement

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Mathematical Models of Averaging ProcessesCorrelation Technique in Flow Measurement

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Mathematical Models of Averaging Processes

Example of ApplicationCorrelation Technique in Flow Measurement

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All measurement concepts fordeterministic quantitiesremain unchanged for

random quantities.

(Note: Sensors do not sense whether acquired quantitiesare deterministic or random)

In order to get certain types of characteristic values and functionsof selected

random quantities,special statistical operators must be added.

On principle they deliver estimates only.The results are uncertain,

systematic and random errors appear.

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Conclusion

78ETH

Models are used everywhere in Metrology,often unconsciously

Models are found empirically and analyticallysimultaneously

Quantitative models are always logical, mathematical and probabilistic models

Foundations for quantitative models are besides othersMathematics

Signal and System TheoryStochastics and Statistics

Metrology needs theories urgently,but not its own theories!

Models Everywhere

79ETH

Discrepancies

between

the world of real processes

and

the world of virtual model imagination

lead to

Model Errors and Model Uncertainties.

Here too, onlyCalibration (Identification)

will do.

Nonideal Versus Ideal

80ETH


Types of deviations from the Ideal

Model errors concerning the model structure:

structure errors

• neglected influence quantities and state quantities

• unconsidered local and temporal influences

• inadequate structures of equations

• too low orders of system equations

• linearised nonlinear relations

Model errors concerning the model parameters:

parameter errors

• inaccurate and uncertain numerical values

81ETH


As usual, if we want to reduce

Model Errors and Model Uncertainties

arbitrarily,

we have to increase the

Effort

above average:

effort

lim errors,uncertainties 0

82ETH

Quality of mathematical models

• qualitative models deliver a lot already

• the simplest model serves better than no model

• quantitative models need not be absolutely exact

• however: better models → better information

• however: modelling errors might have consequences


83ETH

Modelling in Metrology

• improves the understanding of measurement procedures

• supports discussions with partners

• helps realise innovative ideas

• reduces the risk of faulty decisions

• develops analytical and holistic thinking

• is equally effective in all fields of Metrology

84ETH

Mathematical Modelling in Metrology – A Quotation

... how should a measurement task be approached methodically?

One possible answer could be:

"Take it as a statistical estimation problem and solve it optimally"

... and the solution?

"Minimising nonlinear functions with side conditions" –

… a classical task of applied and numerical Mathematics.

Jürg Weilenmann, Leica Geosystems

85ETH

• in the top down approachcommon statements for all fields of Metrology

• in the restricted number of systematic termsbetter overview without loss of generality

• in the general concept of the reconstruction process

solution of different types of tasks in one procedure

• in the error model within the measurement process

useful for the qualification of measurement results (GUM)

• in the incorporation of probabilistic conceptsbetter understanding of deterministic and random

influences and relations

Where do the didactical aspects show up?

Documents

From Verbal Models to Mathematical Models – A Didactical Concept not just in Metrology