applied mathematics in small and large firms - mdh/menu/general/column-content... · applied...

Applied Mathematics

in Small and Large Firms

George [email protected]

Västerås, Sweden, Nov 18, 2015

Mälardalens Högskola (MDH)

School for Education, Culture and Communication (UKK)

Master Program in Engineering Mathematics

Goals Mathematical Fields (selected)

1. Coproduction of scientific knowledge

Academia-Industry

Applied mathematics

Applied matrix analysis

2. Current and future technologies Mathematics of the Internet

Quantum computers and information

3. Established theories Wavelets

Applied mathematic structures

4. Flexibility, Independence, Specialization Project in mathematics

Degree project in mathematics

Speaker Series of the Master Program in Applied

Mathematics

Earlier talk held by distinguished speaker:

Christian Sohl, SAAB Electronic Defense

Applied Mathematics: Electromagnetics in Industry

Contents

• Pure and applied mathematics

• What are the math problems firms are trying to solve and why do they do it?

• Mathematics in different applications

• Unsolved problems

• Mathematics Curricula

• How to contact firms?

• Conclusions

How large is the field of mathematics?

Mathematics classification

• Current version: MSC 2010

• http://www.ams.org/mathscinet/msc/msc2010.html

• Example:

– 46-XX: Functional Analysis

– 46Bxx: Normed linear spaces and Banach spaces

– 46B22: Radon-Nikodym, Krein-Milman and related

properties

Total in 2015: 6675 fields

Pure research in Mathematics

• Ferma’s last theorem (1637)

• Problem in number theory: 11D41

• Ribet’s theorem: was a theorem between two conjectures

• Corollary of the modularity theorem of Taniyama-Shimura-Weil conjecture

• Andrew John Wiles’ proof (1995)

• Elliptic curves: 14H52

• Result: number theory 11G16

nnncba =+

Pure Mathematics

Pure and Applied Mathematics

Applied mathematics

• We return to the example:

– 46-XX: Functional Analysis

– 46Bxx: Normed linear

spaces and Banach spaces

– 46B22: Radon-Nikodym,

Krein-Milman and related

properties

Radon-Nikodym Theorem

• A result in Measure Theory

• Any absolutely continuous measure λ wrt some measure μ (μ could be a

Lebesgue measure of Haar measure) is given by the integral of some L1

function f

• The function f is called Radon-Nikodym derivative

• All Hilbert spaces have the Radon-Nikodym property

∫=E

fdE µλ )(

µ

λ

d

df =

Dual Hahn-Banach theorem

Using abstract space theories

Controlling rolling of ships

• A ship has 6 degrees of freedom, rolling is one of these

• QTAGG has sensors for the movement of the ship and for engine and propeller

properties

• One problem in control is to have a model of the process to be controlled

• After having a model, a model-predictive control (MPC) algorithm can be used to

reduce rolling

Wärtsilä, RT-flex 96C 14 cyl engine

Connection for the signals used for the model

State-space models

tt

ttt

Cxy

BuAxx

=

+= −1

Xu(t) y(t)

npmnnn

pnmLCBAR××× ℜ×ℜ×ℜ=∈=

,,),,(

A,B,C are matrices of appropriate dimensions

R = is an abstract space with many dimensions

What is the distance between a R1 and R2 ?

Speed control

Propeller control

Waves

Winds

Ship movement 6DOF

State-space models

• Let P be a (n x n) matrix in GL(n)

• GL = usual notation for “general linear groups”

• There is a property that if R=(A,B,C) then any

• P ◦ (A,B,C)=(P-1AP, B-1P, CP) is equivalent to R

• This means we cannot simply compare R1 with R2 but we need to slide one along the equivalent classes e.g. using the dual Hahn- Banach theorem

• However GL is not compact. One method is to seek maximally compact subgroup

• Then a max distance to a tangent space is found with the Jacobi Eigenvalue algorithm

• This is the distance between models R1 and R2

Connection math – applications

1. Radom-Nikodym property not fulfilled, find an appropriate compact subspace

2. Find that there is an invariant group

3. Find an abstract distance

4. Apply the optimization (Hahn-Banach, Jacobi)

5. We got the distance

• Without these steps, one cannot define a correct distance as the distance could be arbitrarily large or small depending on how the model was obtained.

Flatness Control in Cold Rolling Mills - ABB

• Measurement roll• Control system

Stressometer® Systems– for Flatness Measurement & Control

Flatness Control

Simultaneously control a large number of actuators of which several have similar

flatness effect

SVD = Singular Value Decomposition

'VUG ⋅Σ⋅=

Adaptive – Predictive Control in Orthogonal Space

Target Flatness

Orthogonal

Space

-Decoupling –

SVD(G)=USV’

Map to reduced control space

Decoupled Predictive ControllersMap back to real space

Adaptation

Observer

- Gains -

Range

Slew Rate

Step limits

Limits

Mill ActuatorsStrip

Stressometer

N x PPI regulators

PPI

Predictor

Lambda

AWO

Model

Library

SVD Space Mapping

Application for cold rolling of metals

The historical perspective

• Least squares: Gauss 1801 (predict orbit of Ceres) => 1960

• Hermite orthogonal polynomials 1849 => 1970

• Chebyshev orthogonal polynomials 1854 => 1980

• Gram-Schmidt ortogonalization: 1907 => 1980

• Hahn-Banach: 1930 => 1985

• Givens transformation: 1952 => 1985

• Housholder transformation: 1964 => 1985

• Singular value decomposition Golub: 1965 => 2007

• Matrix Perturbation, Steward: 1990 => 2008

Contents







• Conclusions

Academic vs. Industry Carrier

• Academic: high-level specialization around a certain field,

journals and conferences. Unlimited time to solve problems of

abstract nature. Focused publications.

• Industry: research through wide fields required by the

development of a product. Short time to solve a very specific

problem. Eclectic publications.

Typical Industrial Organization

Pre-study MSc / PhD Post-study Patent Product

Product Projects Dept. Customer ProjectsProduct Projects Customer ProjectsSystem Development Sales

Technical MathFormal SW dev.

and test methodsProcess models

Demand

Prediction

Statistics

Financials

Large and Small Firms

• A key need for all firms is prediction

– Predict the market

– Predict quality of materials and services

– Predict production capacity

– Predict that your bridge will hold in strong winds

– Predict the quality of the customer’s customer

– Predict exchange rate

– Predict the share prices

• Prediction means statistics, models, inferences, distances, correlations, etc. : Mathematics

Large and Small firms today

• Firms are experiencing nowadays structural changes

• Large firms are large since they have a stable market, suppliers, production and services

• The prediction requirements are less demanding

• Small firms have less secure conditions, their prediction requirements are hard and key for their future

• Therefore today large firms are solving smaller mathematical problems and small firms are solving large and difficult mathematical problems

• Historically this was not always so and may change in the future

Historical differences

• Joseph Schumpeter,

economist (1883-1950)

• Business cycles

• Creative

destruction

• Monopoly gains

AT&T

Satellite communications, fax, sound motion

picture, negative feedback, long-distance TV,

wave nature of matter, stereo recording, radio

astronomy, digital computer, HF radar, transistor,

information theory (Claude Shannon), solar cell,

laser, big-bang echo, Unix, Internet, fiber optic

communication, C++, HDTV, quantum

computing

RAND Corp.

• Kenneth Arrow

• Richard Bellman (optimization)

• George Dantzig (simplex algorithm)

• John von Neumann

• Edmund Phelps

• Thomas Schelling

• ….

• 30 Nobel prizes

Today

• Large firms are focused on their current markets, typically no R&D for new markets

• Information about large investments in R&D would tumble stock prices of a firm

• Small firms are innovative but have less financing and less marketing strength

• Academia can provide an environment where new bold ideas can move into applications

• Creating links between firms and academia is a good way to open this deadlock and generate new value

Contents







• Conclusions

Error-free software (EFS) for industrial

products

• Critical applications in steel, manufacturing, marine, chemical, pharmaceutical, health-related industries: software errors have consequences

• EFS: an important goal when I started at ABB Automation

• Internet, clouds, iPads: makes the problem more complex

• Still very important today, still not solved

Siemens (http://w1.siemens.com)

Formal methods

• Many SW testing

methods are based

on best practices

• Industry would need

methods that have

formal proofs (i.e.

have an MSC number)

SW Formal Methods

• Statistically, SW code used in industrial control for traditional control (like PID) is less than 5%. Discrete states are 95% of the code

• Traditional control has good stability proof theories (34Dxx): Asymptotic properties, Lyapunov stability, perturbations, Popov stability, attractors, etc.

• There are few stability theories for discrete state spaces

• I focused on theories of stability for discrete states

State modeling

Current state

Control action

Next expected state as predicted

effect of the control action

State materialized instead of the expected

one when disturbances occur

Disturbance

Formal representation: state vectors X and transition matrices A.

Colored Petri net theory

nn AXX =+1

State modeling

• Problem: to classify types and possible unexpected transitions

• Some some transitions will be still unknown

• Challenge: distinguish the case of unknown disturbances from the case of a wrong model

• Model built using ontological assumptions

• Violations of ontological assumptions: instability

• Found a formal method by which a controller can determine itself if it has a wrong model

• Results in a PhD thesis in AI at Linköping University

Fuzzy states03B52, 03B72, 04A72, 46S40, 54A40, 90C70, 93C42, 94D05

• Improvement of the

prediction strength by

using fuzzy states

instead of discrete

states

• I published papers

about best formal

control architectures

and fault detection and

isolation

Fuzzy States and Fuzzy State Processors

• Lofti Zadeh (father of Fuzzy Mathematics, prof. at UCLA), put me in contact at a conference in Sydney with prof. Janos Grantner from Western Michigan University who made similar research

• I started as adj. prof at WMU

Complex systems research

Award DAAD19-01-1-

0431 from NSF DURIP

(US Defense Research

Instrumentation

Program). ABB together

with Western Michigan

University

Distributed software for

complex control systems

I proposed the creation of the IEEE

Journal on Industrial Informatics

Contents







• Conclusions

Electronics

• Electronics for critical applications.

Reconfigurable FPGA using states modeling

failure modes and reconfigurations

• Requires a layer of “Electronic Monitoring and

Recovery” (EMR)

• General mathematical description of an EMR

• A discrete state / probabilistic state method

Industrial Production and Quality

Supervision

• KPI’s (Key Performance Indicators): are actually Lebesque / Haar distances

• Find a method to compute the distance between heterogeneous state spaces

• Link to statistical distances (Kullback-Leibler div.)

• Use Big Data Architectures and methods to find distances / measures and related predictions by automatic means

• Automatic Markov chain generation from data (Big Data Analytics)

Process modeling

• Applications with different mathematical Invariants

• Example: a stationary process means its probability distribution does not change

• Symplectic geometry (Weil): connected to Hamiltonian formulation of the classical mechanics.

• Movement of a ship => Kinetic + Potential Energy => differentiable manifold => Hamilton => State phase

• A general Hamiltonian / Symplectic method

More Industrial Problems

• Invariants via Inverse methods

• Invariants via ICA (independent component analysis): better measures of independence

• Signal processing: describe mathematically the difference between filtering, denoising and detrending. Example: article on EMD (empirical mode decomposition) finds a smooth function subtracted from the processed signal

• Fusion methods mapping statistical inferences (like Bayes) to state spaces of dynamical systems

Computational Fluid Dynamics

Iowa University, USA

Automatic mapping between CFD models and state models

Contents







• Conclusions

Christian Sohl’s experience

Christian Sohl, Electromagnetics Modeling George Fodor, Control Modeling

Curriculum - evaluation

Contents







• Conclusions

Organization

Pre-study MSc / PhD Post-study Patent Product

Product Projects Dept. Customer ProjectsProduct Projects Customer ProjectsSystem Development Sales

Technical MathFormal dev. and

test methodsProcess models

Demand

Prediction

Statistics

Financials

Three Steps Tutorial for how to contact firms

Required work: about 1 month

• Step 1: Find a firm that has the right profile

• Step 2: Make a pre-study of your own

– Find out what are the firm’s products and services

– Read the firm’s patents, conference and journal

articles

– Build a hypothesis about what are the firm’s

challenges

– Write a study about the solution to the challenge

you have identified

Cont. how to contact firms

– Have a detailed presentation of the problem and

solution on max 6 pages and a one page poster

– Train the presentation and arguments with a friend

– It is good if you have the presentation on some

homepage that can be easily accessed

• Step 3: Call the HR department (mail or phone)

– Ask to be put in contact with a manager responsible

with the problem you identified since you have a

results

– Make the presentation to the manager

Conclusions

Mathematical abstractions

1. Measurement theory, distances, orthogonality

2. Space projections (mappings) and special functions

3. Invariants

4. Representation of change in space, time and in related projected spaces. Change predictions.

5. Solution tools for Inverse Problems

6. Duality theorems

7. Symbolic mappings and statistical / Bayesian inferences

Summary of today’s presentation

• In what way is pure mathematics related to new applications?

• What kind of mathematical problems is a firm trying to solve?

• Is there a difference between the mathematics of the small

and of the large firm? Between firms and academia?

• Mathematics applications in different fields and some

problems

• How can a student / researcher get in contact with firms using

mathematics?

Thank You!

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