from verbal models to mathematical models – a didactical concept not just in metrology
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etrology. e a s u r e m e n t S c i e n c e a n d T e c h n o l o g y. From Verbal Models to Mathematical Models – A Didactical Concept not just in Metrology Karl H. Ruhm Institute of Machine Tools and Manufacturing (IWF), ETH Zurich, Switzerland [email protected] - PowerPoint PPT PresentationTRANSCRIPT
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
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
2 2 2
x2 2 2
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y2 2 2
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u u u u p u u uu v w F
t x y z x x y z
v v v v p v v vu v w F
t x y z y x y z
w w w w p w w wu v w
t x y z z x y z
zF
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!
7ETH
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
Some Structured Assumptions for Metrology
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
Some Structured Assumptions for Metrology
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
Some Structured Assumptions for Metrology
17ETH
IMathematical Models
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y2 2 2
2 2 2
2 2 2
u u u u p u u uu v w F
t x y z x x y z
v v v v p v v vu v w F
t x y z y x y z
w w w w p w w wu v w
t x y z z x y z
zF
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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
x2 2 2
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y2 2 2
2 2 2
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u u u u p u u uu v w F
t x y z x x y z
v v v v p v v vu v w F
t x y z y x y z
w w w w p w w wu v w
t x y z z x y z
zF
Navier-Stokes Equations Kármán-Vortex Street(courtesy Cesareo de La Rosa Siqueira)
20ETH
Models in a Hierarchy
21ETH
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
24ETH
Useful Models of Processes
25ETH
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
26ETH
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.
29ETH
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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Mathematical Model of a Dynamic Sensor Process
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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Mathematical Model of a Dynamic Sensor Process
ExampleResistance Thermometer (RTD)
The graphical result of the model design
35ETH
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.
36ETH
Properties of Process Models
We assignStructures and Parametersof mathematical equations
toProperties
of theProcess Under Modelling
(PUMO)
37ETH
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
38ETH
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
Temporal and Spatial Behaviour of Process Models
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.
40ETH
Temporal and Spatial Behaviour of Process Models
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)
41ETH
Temporal and Spatial Behaviour of Process Models
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
43ETH
IIModels and Metrology
44ETH
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
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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
Mathematical Modelof the
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
Mathematical Modelof the
Reconstruction Process R
As soon asOpR{…} = OpS
–1{…},
we get as desiredOpS
–1{…}·OpS{…} = I.
Supplement → Module "Reconstruction Process – A Survey"
58ETH
IIIModels and Structures
59ETH
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.
60ETH
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
62ETH
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
63ETH
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
Randomness and Probabilistic Concepts
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.
66ETH
Randomness and Probabilistic Concepts
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
Randomness and Probabilistic Concepts
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.
68ETH
Randomness and Probabilistic Concepts
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
Randomness and Probabilistic Concepts
Random quantities intended to be measured.Which ones? You have to choose!
70ETH
Randomness and Probabilistic Concepts
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"
73ETH
Mathematical Models of Averaging ProcessesExample of Application
Correlation Technique in Flow Measurement
74ETH
Mathematical Models of Averaging ProcessesCorrelation Technique in Flow Measurement
75ETH
Mathematical Models of Averaging Processes
Example of ApplicationCorrelation Technique in Flow Measurement
76ETH
Randomness and Probabilistic Concepts
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.
77ETH
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
Nonideal Versus Ideal
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
Nonideal Versus Ideal
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
Nonideal Versus Ideal
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?