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

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 ruhm@ethz.ch

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

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  1. 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 ruhm@ethz.ch Invited Plenary Lecture Joint International IMEKO TC1+TC7+TC13 Symposium 2011 Jena, Germany August 31st – September 2nd, 2011 28. 06. 2011 Version 02; 15.10.2011 www.mmm.ethz.ch/dok01/e0001000.pdf

  2. I Mathematical Models II Models and Metrology III Models and Structures IV Models and Randomness TU Karlsruhe, DE

  3. Points of View Are there different measurement concepts? • 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 No, there are only different procedures and tools.

  4. Points of View Indeed, there are quite particular interests of individual circles. Yes, but they do not concern essential aspects of Metrology.

  5. Points of View Are there global concepts in Metrology concerning common interests? Yes! • 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

  6. 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!

  7. Points of View Are there different approaches to measurement tasks? • 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 The combination does it!

  8. 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.

  9. Concentration on Few Terms process a defined fraction of the natural and man-made real world, always multivariable and dynamic quantity in the real world, time and space dependent model relates quantities of processes mathematical model relates quantities of processes by equations property and behaviour describe processes and quantities by parameters and solutions of the model equations Supplement → Slides "Process and System" Supplement → Conference Paper "Process and System – A Dual Definition"

  10. Concentration on Few Terms error a quantity appearing as a difference (deviation, discrepancy) between two defined quantities, deterministic and / or random uncertainty a parameter in Statistics, describing a property of a random quantity Supplement → Terminology List "Error and Uncertainty"

  11. Main Tools Mathematical Models describe processes by logical expression and mathematical functions This field is covered likewise by Signal and System Theory and Stochastics and Statistics A useful graphical visualisation is the Signal Effect Diagram (block diagram, flow chart, event map) Supplement → Slides "Process and System" Supplement → Conference Paper "Process and System – A Dual Definition"

  12. graphical model structures are important, they reflect logical and mathematical structures in an impressively descriptive way

  13. Points of View Some Structured Assumptions for Metrology • Process under Measurement PUM • with • Process P without Measurement Process • Measurement Process M without Process

  14. Points of View Some Structured Assumptions for Metrology • Process under Measurement PUM • Process P without Measurement Process • Measurement Process M without Process

  15. Points of View Some Structured Assumptions for Metrology • Quantity • with some hierarchically ordered sub-terms • concerning 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

  16. Points of View Some Structured Assumptions for Metrology • 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

  17. I Mathematical Models

  18. Two Worlds • 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")

  19. How Do Models Come In? Models in the real world (NASA) Models in the virtual, abstract world (intellectual products of the human mind) Kármán-Vortex Street Navier-Stokes Equations (courtesy Cesareo de La Rosa Siqueira)

  20. Models in a Hierarchy

  21. Types of Models qualitative ideas about something mind models modelling thoughts verbal statements opinion and prejudice ideas and visions etc. quantitative drawings, pictures, notes, articles, novels, instructions, theories, logical, mathematical and probabilistic equations, business plans, programs, flowcharts, acoustical and optical verbal documentation etc.

  22. What do Quantitative Models Describe? • 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 and 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.

  23. Ways to Mathematical Models 1. Analytical modelling by first principles mathematical 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 Note

  24. Useful Models of Processes

  25. Three Questions around a Process Model Supplement→ Module "Process and System"

  26. Describing Processes by Models 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?

  27. We start a model verbally with a-priory knowledge The description will be more or less appropriate elaborate detailed accurate qualitative It is a model already and it is useful since we can discuss it and it can be the base of first decisions

  28. We start a process model choosing quantitiesreal quantitiesandderived quantitieslikeefficiency, flexibility, utility, stability, robustness, observability, controllability, capacity,etc.

  29. Example Model of a pump as process P.

  30. Example We select and identify the group (vector) of quantities of interest. We decide which are independent (input) quantities and which are dependent (output) quantities. Here, the model of process P is identical with the set of two mathematical equations (operations), relating three quantities of interest, that's all! Processes are described by relations between quantities!

  31. Models of Dynamic Processesdescribe the processes by different types ofdifferential equationsandintegral equations,introducingvelocitiesand accelerations of quantitiesas additionalquantities

  32. Mathematical Model of a Dynamic Sensor Process Example Resistance Thermometer (RTD) simplifying assumptions: R(t) J (t) F (WIKA Pt100)

  33. Mathematical Model of a Dynamic Sensor Process Question Is the abstract mathematical model able to represent the real world? Answer Yesand No Yes, if only relations between distinguished quantities are considered No, if the overall existence and behaviour is meant.

  34. Mathematical Model of a Dynamic Sensor Process Example Resistance Thermometer (RTD) The graphical result of the model design

  35. Properties of Process ModelsMathematical and probabilistic equationsare characterised byStructures and ParametersStructuresare determined by assumptions and hypothesesParametersare determined by parameter identification (calibration)ThusStructures and Parametersare always hypotheses and estimates,prone tomodel errors and model uncertainties.

  36. Properties of Process ModelsWe assignStructures and Parametersof mathematical equationstoPropertiesof theProcess Under Modelling(PUMO)

  37. ExampleGeneral model of a dynamic process of second order (ODE):Applied model for an oscillating process (equation of motion):

  38. Temporal and Spatial Behaviour of Process ModelsA process will respond to changing input quantities.The way it responds is calledbehaviour.The behaviour dependson the structure,on the parametersof the process modeland on theinput quantities(excitation, impact, stimulation)

  39. Temporal and Spatial Behaviour of Process ModelsStandardised excitation functionsat the inputduring measurement and calibrationfor comparison purposes of process behaviour:impulse functionstep functionramp functionharmonic functionrandom function(e.g. Monte Carlo Simulation)etc.

  40. Temporal and Spatial Behaviour of Process ModelsWe get the behaviourby experiment(measurement)orby analysis (solution of a set of equations)consisting of thehomogenous solution(eigen-behaviour)and theparticular solution,combined in thegeneral solution(overall-behaviour)

  41. Temporal and Spatial Behaviour of Process ModelsGraphical Representation

  42. Process Models give descriptions they give no explanations (interpretation) explanations are searched by human beings or by programs using so-called artificial intelligence and expert knowledge

  43. II Models and Metrology

  44. The surroundings of metrological procedures Examples of processes • a hospital patient • a motor vehicle • a machine tool • a global positioning system (gps) • an education system

  45. First Steps to Models Ideas and A-Priory Knowledge Example Measurement of Humidity "Flight of Imagination" Leonardo da Vinci 1452 - 1519 Hard Model "The noblest pleasure is the joy of understanding"

  46. Definition of quantities intended to be measured Example Physical quantities of potential interest within a model of a process concerning a humid gas in a container.

  47. Model-Based Measurement Reconstruction of Non-Acquirable Process Quantities Example Measurement of the Terrestrial Circumference by Eratosthenes (276 – 194 BC) Supplement→ Example "Measurement of the Terrestrial Circumference"

  48. Model of Measurement Procedure 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! Supplement→ Module "Ideal Measurement Process – Nominal Behaviour"

  49. For now we assume hypothetically an Ideal Measurement Process MN with a Nominal Behaviour: In the model domain the result quantities have 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 a mathematical model by Supplement→ Module "Ideal Measurement Process – Nominal Behaviour"

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

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