Analysis. Idea/Purpose. Matematisk Modellering FK (FRT095) Welcome to Mathematical Modelling FK (FRT095) Written report. Project

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1 Matematisk Moellering FK (FRT095) Course homepage: Welcome to Mathematical Moelling FK (FRT095) Aners Rantzer Department of Automatic Control, LTH 4.5 högskolepoäng ; betyg U/G 4 h lectures 00 h project Project Project supervision from Mathematics, Mathematical Statistics, Automatic Control. Project plan. An A4-paper prepare after consulting the supervisor. Sen to course responsible by January 3. Use with subject line FRT095. Written report Oral presentation (share among all group members) Opposition (all team members together) Written opposition report 4 persons per project (possibly less) Mathematical moelling Why an How? Why moelling? Natural sciences: Moels for analysis (unerstaning) Engineering sciences: Moels for synthesis (esign) Specification: Moel of a goo technical solution White boxes: Physical moeling Moel erive from funamental physical laws Black boxes: Statistical methos an machine learning Moel erive from measurement ata System Ientification / Time Series Analysis Gray boxes: Combination of the two See the website instructions: Cover sheet Summary Written report Table of Contents Main Text Presentation of problem: What is the purpose of the moel? Summary of use literature Theory/Metho Implementation 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 All Three Moelling Phases Must be Describe. Problem structure Formulate purpose, requirements for accuracy 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? Matematical moel an requirement specification Engineering Ethics Iea/Purpose Analysis Relevant for the Pi-program? Experiment Synthesis Ethical linear algebra? Ethical mathematical moelling? Implementation Thanks to Maria Henningsson Pi-02 for suggesting the next few slies.

2 Our calculations show that... Knowlege Gives You Power an Responsibility What is behin the numbers? What assumptions are mae? What limitations are there? Your expert role will give you an avantage What assumptions are mae? What limitations are there? "Essentially, all moels are wrong, but some are useful." - George E. P. Box. Example: The CitiCorp Builing Example 2: The Parental Leave Insurance What percentage of your income o you get? Example 3: Mortage Securities Mathematical moelling Why an How? Why moelling? Natural sciences: Moels for analysis (unerstaning) Engineering sciences: Moels for synthesis (esign) Specification: Moel of a goo technical solution White boxes: Physical moeling Moel erive from funamental physical laws Black boxes: Statistical methos an machine learning Moel erive from measurement ata System Ientification / Time Series Analysis Gray boxes: Combination of the two White boxes Principles an analogies: Hyraulics Example. A hyraulic system: Physical moeling (white boxes) Analogies between ifferent fiels Dimensione an imensionless variables Q Q2 Subsystems an ifferential-algebraic equations p a p p 2 pb Q 3 Q 4 Q 5 Incompressible flui. Pressures: p a, p, p 2, an p 3. Volume flows: Q, Q 2, Q 3, Q 4, an Q 5. 2

3 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. Intensity variables pressure voltage temperature force For both of them aition rules hol. Analogies (cont ) Analogies (cont ) Intensity variations C t (intensity)=flow C "capacitance": hyraulic: A/(ρ ) 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 = φ(intensity) intensity = ϕ(flow) Often linear relationship in the electrical case - nonlinearly in the other (may be approximate by linear for small changes of variables) More phenomena Energy flows Intensity variations L "inuctance" hyraucs: ρl/a electrics: inuctans heat: mechanics: mass balance equations! L t (flow)=intensity 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.) 3

4 White boxes Dimension analysis Physical variables have imensions. E.g., Physical moeling (white boxes) Analogies between ifferent fiels Dimensione an imensionless variables Subsystems an ifferential-algebraic equations where [ensity]= M L 3 [force]= M L T 2= M LT 2 M=[mass], T=[time], L=[length] Physical connections must be imensionally correct. Example: Bernoulli s law Dimensionless quantities an scaling In Bernoulli s law v= 2 h you have [v/ h]= LT (LT 2 L) 0.5 = v/ h is an example of imensionless quantity. 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 Note that the ratio (velocity) 2 /(length) is almost constant Which physical phenomenon can be thought to be the cause? 2 min problem White boxes 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, ) Physical moeling (white boxes) Analogies between ifferent fiels Dimensione an imensionless variables Subsystems an ifferential-algebraic equations Block moels Examples of more general connection: Boxes linke by ientifying the output of one with input of another. Series connection of two state moels gives a new state moel. u y = u 2 y 2 S S 2 Block moels (à la Simulink) often requires a preetermine causality which can bee problematic. We want to be able to moel in general without first etermining what is input an what is output. State moels for two separate components: Connection: φ = ω J ω = τ + τ 2 φ = φ 2 τ 2 = τ 3 φ 2 = ω 2 J 2 ω 2 = τ 3 + τ 4 The resulting moel is not exactly a state moel. 4

5 Linear ifferential-algebraic equations (DAE) Nonlinear ifferential-algebraic equations (DAE) Eż= Fz+ Gu If E were non-singular, one coul write ż= E Fz+ E Gu which is a vali state moel. If 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 φ φ 0 0 τ ω J 2 φ 2 = ω τ φ τ ω ω τ 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 Example: Penulum A penulum with length L an position coorinates(x, y) moves accoring to the equations ẋ=u ẏ= v u=λ x v= λ y L 2 = x 2 + y 2 Differentiating the fifth equation gives 0= ẋx+ ẏy= ux+vy Differentiating a secon time gives an a thir time 0= ux+uẋ+ vy+ vẏ = λ x 2 + u 2 +(λ y )y+ v 2 = λ L 2 y+ u 2 + v 2 0= λ L 2 3 v Finally, we have erivative expressions for all variables! Black Boxes Mathematical moelling Why an How? Why moelling? Natural sciences: Moels for analysis (unerstaning) Engineering sciences: Moels for synthesis (esign) Specification: Moel of a goo technical solution White boxes: Physical moeling Moel erive from funamental physical laws Black boxes: Statistical methos an machine learning Moel erive from measurement ata System Ientification / Time Series Analysis Gray boxes: Combination of the two Singular Value Decomposition (SVD) Statistical moeling from ata (static black boxes) Dynamic experiments (ynamic black boxes) Step response Frequency response Correlation analysis Gray boxes Preiction error methos Differential-Algebraic Equations revisite A matrix M can always be factorize [ ] Σ 0 M= U V 0 0 with Σ iagonal an invertible an U, V unitary: Σ= σ... σ n U U=I V V= I Diagonal elements of Σ are calle singular values of M an correspon to the square roots of the eigenvalues of M M. Computation of SVD is very numerically stable. M [ ] Example of SVD = U = [ ] 2 }{{} U [ ] Σ [ What oes it mean if a singular value is zero? What oes it mean if it is near zero? V ] [ ] 2 }{{} V Goo chilren can have many names Collect all the ata into a large matrix. Then compute the SVD: y () y (2)... y (N) y 2 () y 2 (2)... y 2 (N) σ 0. = U... V 0 σ y p () y p (2)... y p (N) p }{{} Σ Singular values σ i in ecreasing orer on the iagonal of Σ. The first columns of U give the irection of the main ata area. Principal Component Analysis: By replacing the small singular values σ i with zeros focuses on the essential. The name factor analysis is sometimes use as a synonymous, since large singular values σ i highlight important factors. 5

6 Principal Component Analysis (PCA) Data from a bi-imensional Gaussian istribution centere in (, 3): Example: Image processing What oes this picture represent? M = Principal component(0.878, 0.478) has stanar eviation 3. Next component has stanar eviation. [Källa: Wikipeia] Example: Image processing with SVD Example: Image processing with SVD >> [U,S,V]=sv(M) U = S = roun(u*s*v ) = roun(u*s2*v ) = Example: Image processing Example: Correlations genes-proteines The original image has 897-by-598 pixels. Tacking re, green an blue vertically gives a 269-by-598 matrix. Truncating all but 2 singular values gives the left picture. 20 gives the right. Black Boxes Cancer research: microarrays (glass) with human genes are expose to healthy cells, then to sick ones. Make a SVD of the ata to fin out which genes are important! Tomorrow Statistical moeling from ata (static black boxes) Dynamic experiments (ynamic black boxes) Step response Frequency response Correlation analysis Gray boxes Preiction error methos Differential-Algebraic Equations revisite More on black an grey moels Registere stuents will get a project Karl Johan Åström will lecture on bicycle moelling! 6