I Course introduction. I Models from physics (white boxes) Friday lecture. I Models from data (black boxes) I Mixed models (grey boxes)

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1 Outline Thursay lecture Welcome to Mathematical Moelling FK (FRTN45) Aners Rantzer Department of Automatic Control, LTH Course introuction Moels from physics (white boxes) Friay lecture Moels from ata (black boxes) Mixe moels (grey boxes) Matematisk Moellering FK (FRTN45) Project Project supervision from Mathematics, Mathematical Statistics, Automatic Control. Course homepage: Project plan. An A4-paper prepare after consulting the supervisor. Sen to course responsible by January 29. Use with subject line FRTN högskolepoäng ; betyg U/G 4 h lectures 00 h project Written report Oral presentation (share among all group members) Opposition (all team members together) Written opposition report 4 persons per project (possibly less) Written report All Three Moelling Phases Must be Describe See the website instructions: Cover sheet Summary Table of Contents Main Text Presentation of problem: What is the purpose of the moel? Summary of use literature Theory/Metho mplementation Results Evaluation/iscussion: Does the moel suit its purpose? Reference list Description of how the work is istribute within the group Presentation of the course theory part. Problem structure Formulate purpose, requirementsforaccuracy Break up into subsystems What is important? 2. Basic equations Write own the relevant physical laws Collect experimental ata Test hypotheses Valiate the moel against fresh ata 3. Moel with esire features is forme Put the moel on suitable form. (Computer simulation or peagogical insight? ) Document an illustrate the moel Evaluate the moel: Does it meet its purpose? Group : Vilken fågel sjunger? Group 2-3: Statistical Analysis of fmr ata Kan u skilja på en talgoxe och en gråsparv när u hör fågelsång utanför fönstret? Kvitteromat är en helt ny app som ientifierar fågelarter baserat på sången. Den är ock, enligt utvärering, känslig för störningar och ganska osäker i sitt beslut, å resultatet en presenterar är tre olika förslag på vilken art et är som sjunger. Detta projekts syfte är att ientifiera några av våra vanligaste fågelarter genom att analysera eras sång, (lättare och svårare inspelningar), och hitta lämpliga kriterier för säker klassificering. Eventuellt kan en jämförelse och utvärering göras mot kvitteromat. Verktyg för stationära stokastiska processer är använbara, tillsammans me information om signalens variation över ti. Data-material i mp3-format samt några minre program för Matlab kommer att tillhanahållas. Avisor: Maria Sansten, MatStat First meeting: Monay January 22 at 5.5 in avisors office The aim of the project is to analys fmr ata from a trial where the subject juge whether a pair of wors rhyme or not. The experiment con- siste of alternating 20-secon work an rest blocks an we want to ientify which parts of the brain that are active uring the task. To illustrate possible techniques an to make the project feasible you will stuy a single slice from the fmr (rather than the complete 3D voxel ata). The slice consists of a vieo sequence with 60 time points representing the experiments 320s runtime (at 2s temporal resolution). Active pixels can be etecte as those that exhibit a 0 sample (0 2=20s)frequency. Avisor: Johan Linström, MatStat First meeting: Monay January 22 at 5.5 in avisors office Final presentation: Thursay March at

2 Group 4-5: Teknologens väg till examen Group 6: Moellering av pappershelikoptrar Teknologens väg till examen är intressant att moellera för högskolorna, som kan använa moellen för att preiktera hur många teknologer som kommer att ta examen inom en viss ti. Vägen till examen kan ses som ett antal övergångar mellan iskreta tillstån, såsom registrering till respektive programtermin, stuieuppehåll, utbytesstuier eller inaktivitet, och avslutas me antingen examen eller stuieavbrott. Målet me etta projekt är att moellera stuievägen för LTH s teknologer utifrån terminsata från LADOK, och utföra simulering av en årskull stuenters väg genom utbilningssystemet. Avisor: Mattias Fält, Reglerteknik First meeting: Monay 22/ at 5.5 in avisors office Me hjälp av en sax kan man ganska lätt bygga en liten pappershelikopter som när man släpper en faller och roterar. Ganska snabbt blir rotationshastigheten jämn. Projektet går ut på att försöka moellera etta förlopp, att mäta rotationshastigeterna experimentellt, t ex genom att filma fallet och att försöka utveckla en moell som kan förklara rotationshastigheten. Avisor: Kalle Åström, Matematik LTH First meeting: Monay 22/ at 3.5 in avisors office Group 7: Structure an motion for soun Using several microphones it is possible to calculate the position of soun sources. f the microphone positions are known this is usually calle trilateration. f neither the soun sources nor the microphone positions are known, the problem is more challenging. The purpose of this project is to stuy an evelop mathematical moels for soun an use them in experiments with real ata for structure an motion for soun. There is a choice to focus more on the signal processing for the soun or to focus on the geometrical aspects of the positions of the microphones an the souns sources. Avisor: Kalle Åström, Matematik LTH First meeting: Monay 22/ at 3.5 in avisors office Group 8: Moelling traffic flow of future vehicles This project eals with the moelling of traffic flow of vehicles, all behaving the same, which is likely to be the case with future river-less vehicles. With simple assumptions, one can stuy relations between the uration of green/re traffic lights, the traffic flux an vehicle ensity on a rive through. How can the traffic flow be controlle an optimize? Traffic flow is a nonlinear phenomenon: the flux (number of vehicles per time unit) on a rive through can be obtaine with two ifferent total times spent in a vehicle. What is best for the passengers, the environment? This project will give some basic knowlege of nonlinear moelling an the formation of shock waves, valuable for any moeller. Many time- an spatial-epenent physical phenomena are moelle with the conservation law of mass an the resulting governing equation is often a hyperbolic partial ifferential equation. This continuum approach is use for flui flow but also for the flow of iscrete particles. Avisor: Stefan Diehl, Matematik LTH First meeting: Friay 9/ at 3.5 in avisors office Final presentation: Thursay March at Group 9: Stokastisk populationsynamik Ett mycket viktigt inslag i populationsmoeller är sk Markov Jump processes. ntuitivt, kan e beskrivas som processer är för et mesta häner ingenting på mycket korta tisintervall, men å et häner något, är effekten "ramatiskt" (exempelvis antalet friska i en population änras me +). Man kan beskriva såana processer me en sk Kolmogorov Forwar Equation (93) och en första numeriska algoritmen som implementerar ién gjores av Kenall (950), efter forskning av Feller från 940. Syftet är att applicera såana moeller på en lagom komplicera populationsynamisk process. Avisor: Mario Natiello, Matematik LTH First meeting: Monay 22/ at 3.5 in avisors office Final presentation: Thursay March at Mathematical moelling Why moelling? Natural sciences: Moels for analysis (unerstaning) Engineering sciences: Moels for synthesis (esign) Specification: Moel of a goo technical solution Physical moeling (white boxes, toay!) Moel erive from funamental physical laws Statistical methos (black boxes, Friay s lecture) Moel erive from measurement ata Singular Value Decomposition (SVD) Machine Learning System entification / Time Series Analysis Combination of the two (gray boxes) Moelling in three phases:. Problem structure Formulate purpose, requirementsforaccuracy Break up into subsystems What is important? 2. Basic equations Write own the relevant physical laws Collect experimental ata Test hypotheses Valiate the moel against fresh ata ea/purpose Experiment Matematical moel an requirement specification Analysis Synthesis 3. Moel with esire features is forme Put the moel on suitable form. (Computer simulation or peagogical insight? ) Document an illustrate the moel Evaluate the moel: Does it meet its purpose? mplementation 2

3 Engineering Ethics Our calculations show that... Relevant for the Pi-program? Ethical linear algebra? Ethical mathematical moelling? What is behin the numbers? What assumptions are mae? What limitations are there? "Essentially, all moels are wrong, but some are useful." - George E. P. Box. Thanks to Maria Henningsson Pi-02 for suggesting the next few slies. Knowlege Gives You Power an Responsibility Example: The CitiCorp Builing Your expert role will give you an avantage What assumptions are mae? What limitations are there? Example 2: The Parental Leave nsurance Example 3: Mortage Securities What percentage of your income o you get? My Own Research: Dynamic Buffer Networks Example: Ectogri Proucers, consumers an storages Examples: water, power, traffic, ata Discrete/continuous, stochastic/eterministic Multiple commoities, human interaction Problem: Scalable an aaptive methos for control. 3

4 Outline Principles an analogies: Hyraulics Thursay lecture Course introuction Moels from physics (white boxes) Analogies an imensions Moelica A moern moeling language Example. A hyraulic system: Q Q2 Friay lecture Moels from ata (black boxes) Mixe moels (grey boxes) p a p p 2 pb Q 3 Q 4 Q 5 ncompressible flui. Pressures: p a, p, p 2, an p 3. Volume flows: Q, Q 2, Q 3, Q 4, an Q 5. Principles an analogies: Electrics Principles an analogies: Heat Example 2. An electrical system: Example 3. A thermal system (heat transfer through a wall): R 3 R i 3 i 4 R 5 4 i 5 Värmekap. Värmekap. i i 2 v a v v 2 v b C C 2 T a q C q 4 C 2 3 q5 T T 2 T b Potentials v a, v b, v, an v 2 Currents i, i 2, i 3, i 4, an i 5 Two elements with thermal capacities C an C 2 separate by insulating layers. Heat flows: q 3, q 4 an q 4 Temperatures: T a, T b, T an T 2 Principles an analogies: Mechanics Analogies Exempel 4. A mechanical system: F F 2 k k 2 F a F b m m 2 m 3 v v 2 2 v 3 External forces: F a an F b Velocities: v, v 2 an v 3 Spring constants: k an k 2 Damping constants: an 2 Analogies: hyraulic - electric - thermal - mechanical Two types of variables: A. Flow Variables volume flow power flow heat flow spee B. ntensity variables pressure voltage temperature force For both of them aition rules hol. Analogies (cont ) Analogies (cont ) ntensity variations C (intensity) =flow t C "capacitance": hyraulic: A/(r ) electrical: kapacitans heat: thermal capacity mechanical: inverse spring constant Balance equations! (More complicate if the capacitance is not constant.) Losses Hyraulic: flow resistance Electrics: resistance Heat: thermal conuctivity Mechanics: friction flow = f(intensity) intensity = j(flow) Often linear relationship in the electrical case - nonlinearly in the other (may be approximate by linear for small changes of variables) 4

5 More phenomena Energy flows ntensity variations L "inuctance" hyraucs: rl/a electrics: inuctans heat: mechanics: mass balance equations! L (flow) =intensity t Can you make a general moeling theory base on flow an intensity variables? Note the following. pressure flow = power voltage ifference current = power force velocity = power torque angular velocity = power temperature heat flow = power temperature (more complicate if the inuctance is not constant.) Dimension analysis Example: Bernoulli s law Physical variables have imensions. E.g., where [ensity] =ML 3 [force] =M L T 2 = MLT 2 M =[mass], T =[time], L =[length] n Bernoulli s law v = 2 h you have [v/ p h] =LT (LT 2 L) 0.5 = v/ h is an example of imensionless quantity. Physical connections must be imensionally correct. Dimensionless quantities an scaling 2 min problem Some historical passanger ships: Kaiser Wilhelm the great, 898, 22 knots, 200 m Lusitania, 909, 25 knots, 240 m Rex, 933, 27 knots, 269 m Queen Mary, 938, 29 knots, 3 m Fin the relationship (except for a scaling by a imensionless constant) between a penulum perio time an its mass, its length an the acceleration of gravity, i.e., t = f (m, l, ) Note that the ratio (velocity) 2 /(length) is almost constant Which physical phenomenon can be thought to be the cause? Outline A Graphical Moelica Moel Thursay lecture Course introuction Moels from physics (white boxes) Analogies an imensions Moelica A moern moeling language Friay lecture Moels from ata (black boxes) Mixe moels (grey boxes) 5

6 A submoel can be opene Simple system in text format Documentation is important: nitialize at equilibrium! A preator-prey moel ( ẋ = x(a b y) ẏ = y( x g ) Re-using ol moels Mathematics of general connection: nheritance: State moels for two separate components: Mofication: Connection: f = w J w = t + t 2 f = f 2 t 2 = t 3 f 2 = w 2 J 2 w 2 = t 3 + t 4 The resulting moel is not exactly a state moel. Linear ifferential-algebraic equations (DAE) Nonlinear ifferential-algebraic equations (DAE) Eż = Fz+ Gu f E were non-singular, one coul write ż = E Fz+ E Gu which is a vali state moel. f E is singular, variables have to be eliminate to get a state equation. Using a DAE solver is often better, since elimination can estroy sparsity. Example: J 0 0 f f 0 0 t w J f 2 5 = w t f t w w t Differential-algebraic equations, DAE F(ż, z, u) =0, y = H(z, u) u: input, y: output, z: "internal variable" Special case: state moel ẋ = f (x, u), y = h(x, u) u: input, y: output, x: state 6