modelling and simulation with matlab: a case study in … · modelling and simulation with matlab:...

School of Electrical, Electronic and

Communications Engineering,

University College Dublin, Ireland

Dr Paul Curran

[email protected]

MATLAB Virtual Conference, March 26, 2014 1

a case study in problem-based learning

Modelling and Simulation with

MATLAB:

Layout of presentation

• Basics and background

• Initial module design decisions

• Introductory material: lectures and

laboratories

• Student work from the class of 2013

• Conclusion


The basics

• 167 students registered to module in 2013

• Mostly year three (junior year)

• 5 credits

• One semester

• 2 hours in lecture theatre and 4 hours in

laboratory per week

• The module is a first course on numerical

methods


... there is evidence of an ongoing decline in the

candidates’ willingness to struggle for success.


Leaving Certificate Examination 2005. Mathematics. Higher Level.

Chief Examiner’s Report.

Background: Perseverance


“Have patience. All things are difficult before they

become easy”. Sa’dī (circa 1210 – 1292)

quotes taken from BrainyQuote.

Perseverance


“Have patience. All things are difficult before they

become easy”. Sa’dī (circa 1210 – 1292)

The perceived wisdom is that a failure to

persevere can be equated to a failure to learn.

Perseverance

Thinking inside the box


... a noticeable decline in the capacity of candidates

to engage with problems that are not of a routine and

well-rehearsed type.

Leaving Certificate Examination 2005. Mathematics. Higher Level. Chief

Examiner’s Report.

Thinking inside the box


... a noticeable decline in the capacity of candidates

to engage with problems that are not of a routine and

well-rehearsed type.

Leaving Certificate Examination 2005. Mathematics. Higher Level. Chief

Examiner’s Report.

There is a strong tendency among the

student body to compartmentalise their

learning.

“Experience is the teacher of all things”. Gaius Julius Caesar (100 – 44 BC)

learn to do by doing


Practice:

“Experience is the teacher of all things”. Gaius Julius Caesar (100 – 44 BC)



The students will best learn how to apply numerical

methods to the solution of engineering problems by

actually applying numerical methods to the solution

of engineering problems.

Practice:



The student body is varied. The engineering

disciplines of Biosystems, Civil, Electrical,

Electronic, Mechanical, Materials and

Structural are all represented.

Practice:



The students will best learn how to apply numerical

methods to the solution of engineering problems by

actually applying numerical methods to the solution

of engineering problems of some interest to them.

Practice:


“A man who carries a cat by the tail learns

something he can learn in no other way”.

Samuel Clemens “Mark Twain” (1835 – 1910)

The fundamental dichotomy


“A man who carries a cat by the tail learns

something he can learn in no other way”.

Samuel Clemens “Mark Twain” (1835 – 1910)

We learn from doing it wrong, not from doing

it right.



“Learning too soon our limitations, we never learn

our powers”. Mignon McLaughlin(1913 – 1983)



We gain confidence from doing it right, not

from doing it wrong.

“Learning too soon our limitations, we never learn

our powers”. Mignon McLaughlin(1913 – 1983)



We learn from doing it wrong, not from doing it

right.

We gain confidence from doing it right, not from

doing it wrong.



We learn from doing it wrong, not from doing it

right.

We gain confidence from doing it right, not from

doing it wrong.

Practice but receive rewards

Why Matlab ?


Computation

Visualization

Programming

Why Matlab ?


Computation

Visualization

Programming

Computation: simplicity of initial use instils confidence.

Programming: significantly extends problem solving

power and efficiency.

Visualisation: renders results very accessible.

Why Matlab ?


MATLAB permits an expert to do

amazing things.

Why Matlab ?


MATLAB also permits a non-expert to do

amazing things.

MATLAB permits an expert to do

amazing things.

An Engineering Motto


“In theory, there is no difference between theory and

practice. But in practice, there is”.

Lawrence “Yogi” Berra (1925 – )



Problems should be as real as possible.

“In theory, there is no difference between theory and

practice. But in practice, there is”.

Lawrence “Yogi” Berra (1925 – )



Embrace the term “modelling” in the title of the

module. The students would have to

mathematically model as their first step in solving

many of the problems.

Modelling


The students found this to be the most

challenging part of the module. It

challenges compartmentalisation and

demands assimilation of previous learning.

It is neither rote nor routine.

Lecture notes will adopt a large

number of standardised styles for

the presentation of different types

of material.

Lectures:


>> a = 1; % initialise a

>> b = 2; % initialise b

>> fa = (a^2) – 2 % determine value of f(a)

>> fb = (b^2) – 2 % determine value of f(b)

If f(a) and f(b) have opposite sign:

>> c = a – (fa*(b – a))/(fb – fa) % update c using old a and b

>> fc = (c^2) – 2 % determine value of f(c)

If f(c) = 0 stop. If f(c) and f(a) have same sign:

>> a = c; % update a

else:

>> b = c; % update b

22 xxfRegula Falsi

Style for command

window MATLAB code

Lecture notes will adopt a large

number of standardised styles for

the presentation of different types

of material.

Lectures:


>> a = 1; % initialise a

>> b = 2; % initialise b

>> fa = (a^2) – 2 % determine value of f(a)

>> fb = (b^2) – 2 % determine value of f(b)

If f(a) and f(b) have opposite sign:

>> c = a – (fa*(b – a))/(fb – fa) % update c using old a and b

>> fc = (c^2) – 2 % determine value of f(c)

If f(c) = 0 stop. If f(c) and f(a) have same sign:

>> a = c; % update a

else:

>> b = c; % update b

a0 = 1.0000 , b0 = 2

a1 = 1.3333 , b1 = 2

a2 = 1.4000 , b2 = 2

a3 = 1.4118 , b3 = 2

a4 = 1.4138 , b4 = 2

a5 = 1.4141 , b5 = 2

a6 = 1.4142 , b6 = 2

22 xxfRegula Falsi

Style for command

window MATLAB code

Style for presentation

of numerical results

Introductory Laboratories: basics


Test Problem:

Using both the matrix inverse command, inv, and the

left matrix divide command, \ , solve this system of

equations.

3423

02

12

zyx

zyx

yx

Introductory Laboratories: basics


Test Problem:

“Wow! That was

easy”.

Using both the matrix inverse command, inv, and the

left matrix divide command, \ , solve this system of

equations.

3423

02

12

zyx

zyx

yx

Introductory Laboratories:

MATLAB files


Test Problem: Write a MATLAB file to

calculate and output N+1 successive states of the

logistic map having parameter μ and initial condition

x0 . ...

nnn xxx 11


MATLAB files


A typical output

might be as shown.

0 20 40 60 80 100 1200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

n

x

Logistic map for = 3.5 , x0 = 0.1


MATLAB files


A typical output

might be as shown.

0 20 40 60 80 100 1200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

n

x



MATLAB files


A typical output

might be as shown.

Explore!

0 20 40 60 80 100 1200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

n

x


Introductory Laboratories: MATLAB files


function y = my_logistic_map(mu,x0,N)

% y = my_logistic_map(mu,x0,N): iterate map x(n+1) = mu x(n)(1-x(n)) for parameter mu lying between 0

% and 4 and for initial condition x(0) = x0 lying between 0 and 1. Under these conditions the logistic

% map maps the interval [0,1] into itself. Iterate N times and produce output as vector y of N+1

% elements being the initial value x0 and N subsequent iterations in order of iteration.

% Version 1: 2014 Paul Curran

% Revision: not applicable

if nargin ~= 3 error('Not enough input arguments: my_logistic_map requires 3 inut arguments mu, x0 and

N.'); end

if nargout ~= 1 error('Not enough output arguments: my_logistic_map requires 1 output argument y being

a vector of N+1 element'); end

if mu < 0 || mu > 4 error('Invalid range: input argument mu must lie between 0 and 4'); end

if x0 < 0 || x0 > 1 error('Invalid range: input argument x0 must lie between 0 and 1'); end

if N ~= fix(N) || N <= 0 error('Invalid range or type: input argument N must be a positive integer');

end

y = zeros(N+1,1); % Set aside memory to store required N+1 values of the first N+1 successive states

y(1) = x0; % Store initial state as first element of y

x = x0; % initialise state variable x

for count = 1:N

x = mu*x*(1-x); % employ logistic map to determine next state, i.e. iterate

y(count+1) = x; % Store current state in appropriate location in vector y

end

end











N.'); end






end




for count = 1:N



end

end

Descriptive function name











N.'); end






end




for count = 1:N



end

end

Extensive comments

Version information

Help file












N.'); end






end




for count = 1:N



end

end

Extensive comments

No bad code, such as assignment

of memory inside loop

Error checking with fully

descriptive error messages

Version information

Help file


Test Problem: Using ode45 determine an

approximate solution to the ordinary differential

equation ...

Introductory Laboratories: ode45


012

2

2

ytd

ydy

td

yd



ode45 will permit the students to test the

performance of their own first principles

ordinary differential equation solvers. Indeed

numerous MATLAB commands will permit

testing in general and this is another reason

for adopting the package.



A typical output

might be as shown.

Students can

investigate dynamic

plots.

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

y

dy/d

t

Van der Pol equation for = 0.1

Explore!



For example in

exploring the

Lorenz equation

and sensitive

dependence on

initial conditions

one student

received bonus

grades for this:



1

11

6

8 910

121314

15

16

7

1718 19

20

-5

0

5

10

15

20

-15

-10

-5

0

5

10

-5

0

5

10

15

20

Code given to draw

shaft. Problem is to

draw crossbar.

patch



1

11

6

8 910

121314

15

16

7

1718 19

20

-5

0

5

10

15

20

-15

-10

-5

0

5

10

-5

0

5

10

15

20

Code given to draw

shaft. Problem is to

draw crossbar.

Reuse, don’t

rewrite!

patch

One minor project was concerned with a

10 bar linkage from 1991 Ford Mustang

convertible roof mechanism. The

mechanism was animated on the website

of J. Michael McCarthy (University of

California at Irvine) although the link

has since ceased to work.

Minor Project: Static Equations


gggggggggggggggggggg

0

1

2

3

4

5

6

7

8

9 1011

40o

20o

20o

40o

12

20o

45o



The crank drives the mechanism.

The problem is treated as pseudo

static, with all angles determined

for a given crank angle.

gggggggggggggggggggg

0

1

2

3

4

5

6

7

8

9 1011

40o

20o

20o

40o

12

20o

45o



The first rule of perseverance:

ignore what you cannot do and find

something that you can do. There

are no big problems, only large

collections of small problems.

0

1

2

3

4

5

6

7

8

9 1011

40o

20o

20o

40o

12

20o

45o

The student must derive a number

of closure equations.



0

1

2

3

4

5

6

7

8

9 1011

40o

20o

20o

40o

12

20o

45o

For example we may obtain two

expressions for the vector from

node 1 to node 4.

Employing Cartesian coordinates we

obtain the closure equation:



0

1

2

3

4

5

6

7

8

9 1011

40o

20o

20o

40o

12

20o

45o

sin5.21

cos5.2

sin2sin2

cos2cos2

4,33,1

4,33,1

This equation can be solved by

employing the Newton-Raphson

algorithm.

Reuse, don’t rewrite!

As a solution I had in mind something like this:





In general the

solution which the

students actually

obtained looked

more like this,

without the

animation having

been completed.

Problem 2.5: Space-shuttle

... model mission STS-30, the 29th Space Shuttle

mission, ... produce an animation of the flight of the

vehicle from launch to the point where the external

tank is jettisoned, ...

Team Minor Project:

Dynamic Equations


Team Minor Project:

Dynamic Equations


Sensible models for numerous effects

such as reduction with altitude of air

density and force of gravity, changing

drag, decrease in mass, changing throttle

setting, staging etc. had to be chosen.

Team Minor Project:

Dynamic Equations

54

MATLAB Virtual

Conference, March 26, 2014

Donal Bourke,

Sean Cleary and

Deborah Hayden.

Snapshots from

report of Sean

Cleary.

Team Minor Project:

Dynamic Equations

55

Nose cones

modelled using

ogives. Indicative

of quality of

reporting required

and received.

MATLAB Virtual


Team Minor Project: Dynamic

Equations

56

MATLAB Virtual


Team Minor Project: Dynamic

Equations

57 MATLAB Virtual Conference, March 26, 2014

https://www.dropbox.com/sh/pe8y5axenp2u5he/fvWCFR4jxT

Problem 2.4: 2-body problem

Set up the equations of motion for the planet earth

taking into account the gravitational effects of the sun

and of the planet Jupiter. ... Visualise by creating an

animation, ...

Team Minor Project:

Dynamic Equations


Anton Katayev, Peadar Keegan and Carl Kelly.

Carl solves a number of similar problems on his website:

Team Minor Project:

Dynamic Equations


http://carl-kelly.site44.com/projects/cele/cele.html#top

Team Minor Project:

Dynamic Equations


Problem 2.2: Vehicle Suspension

... describe the performance of the half vehicle ...

Choose your own model for the road surface but be

sure to include a seam, i.e. a sharp step, some

undulation and a stochastic component.

Team Minor Project:

Dynamic Equations


Team Minor Project:

Dynamic Equations

MATLAB Virtual Conference,

March 26, 2014

Report of Ciaran Devenney:

Start with

mathematical model

of vehicle.

Team Minor Project:

Dynamic Equations


Report of Ciaran Devenney:

Start with

mathematical model

of vehicle.

Team Minor Project:

Dynamic Equations


Next model

road.

Ciaran Devenney, Dominic Dolan and Ronan Keane.

Team Minor Project:

Dynamic Equations


../../../Text/Courses/NumericalMethods/Car.mp4

Team Minor Project:

Dynamic Equations


This project shows what the non-expert can

achieve with MATLAB, producing an

animation which is physically meaningful and

totally accessible. On seeing it my two year old

son announced “I love that”.

Conclusion


Thanks to Dr Barry Brophy, UCD

for video production.

modelling and simulation with matlab: a case study in … · modelling and simulation with matlab:...

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