intro tom at lab

9/21/2015

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Biosystems Analysis and Modeling – Department of Biological & Ecological Engineering

Introduction to MATLAB

Biosystems Analysis and Modeling


Overview

• Introduction

• Plots

• Conditional statements

• Numerical method

• Initial value problems (ODE’s) – Single ODE

– 2 simultaneous ODE’s (predator-prey equations)

– 3 simultaneous ODE’s (Bioreactor problem)

• Simulink Basics

• Simulink Models


Getting MATLAB

• Two choices

1) For engr students – access MATLAB through CITRIX

Environments – requires no local software. Access via the My-

CoE website:

(http://engr.oregonstate.edu/computing/personal/149

2) Install MATLAB locally using OSU’s site license. Instructions

are at: http://engr.oregonstate.edu/computing/mathworks. You

will have to provide your ONID email as described in the

instructions. You will be able to use MATLAB on your local

machine.


Introduction

• MATLAB stands for MATrix LABoratory

• High performance language for technical computing

• MATLAB is both computer programming language and software environment

• The MATLAB environment is command oriented

• There are a number of add-on software modules, called toolbox , available for MATLAB, that are used to for specialized computations or specific tasks


Command

Window

Command

History

Workspace


MATLAB Variables

• Variable names in MATLAB are case sensitive

• Variables are not declared before they are used

• Variable names can contain up to 63 characters

(MATLAB 6.5 and newer)

• Variables names cannot start with number. They must

start with a letter and can be followed by digits, letters or

underscore

• ans Default variable name for results

• Comments can be added to MATLAB using (%) symbol

• Built-in variables: e.g. pi Value of Π

a=3

A=5

http://engr.oregonstate.edu/computing/personal/149

http://engr.oregonstate.edu/computing/mathworks

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Basic Commands

• Define variables

• Calculating mode

(different arithmetic

operations)

• Rounding off

– Round

– Floor

– Ceil

– Fix

a=3

b=5

c=a+b

d=pi*b

d=pi/d

a=1.6689

round(a)

ceil(a)

floor(a)

fix(a)


Matrices

>> A=[2 3 4; 5 6 7]

A =

2 3 4

5 6 7

A(i,j) gives the element of ith row

and jth column

A (2,3)=7

A(1,2) + A(2,3) = 9

>> eye(3)

ans =

1 0 0

0 1 0

0 0 1

>> ones(3,2)

ans =

1 1

1 1

1 1

>> zeros(2,3)

ans =

0 0 0

0 0 0

>> zeros(2)

ans =

0 0

0 0

>> rand(2,3)

ans =

0.6557 0.8491 0.6787

0.0357 0.9340 0.7577

The following functions populate matrices


Matrix/Vector Operations

>> A*B

ans =

5 23 13

21 71 61

33 77 137

>> A.*B

ans =

3 8 21

5 30 63

0 27 0

>> A=[1 2 3; 5 6 7; 8 9 0]

A =

1 2 3

5 6 7

8 9 0

A’ =

1 5 8

2 6 9

3 7 0

>>B=[3 4 7; 1 5 9; 0 3 -4]

B =

3 4 7

1 5 9

0 3 -4

A+B =

4 6 10

6 11 16

8 12 -4

>> A/B

ans =

0.2885 0.1346 0.0577

1.8269 -0.4808 0.3654

3.2692 -1.8077 1.6538

>> A./B

ans =

0.3333 0.5000 0.4286

5.0000 1.2000 0.7778

Inf 3.0000 0

Populate a matrix

Transpose a matrix

Transpose (.'), power (.^), complex conjugate transpose ('), matrix power (^)

Matrix Addition

Matrix Multiplication Matrix Division

Element Multiplication Element Division


Subsetting - Colon Operator (:)

• J:K is the same as [J, J+1, ..., K]

• J:D:K is the same as [J, J+D, ...,

J+m*D] where m = fix((K-J)/D)

• A(:) is all the elements of A,

regarded as a single column

• A(:,J) is the Jth column of A

• A(J:K) is [A(J),A(J+1),...,A(K)]

• A(:,J:K) is [A(:,J),A(:,J+1),...,A(:,K)]

A =

1 2

3

4 5

6

7 8

9

0 -1

-2

B =

4 5 6

7 8 9

Creating new matrix from existing matrix

A = [1,2,3; 4,5,6; 7,8,9; 0,-1,-2]

B = A(2:3,:) = [4 5 6; 7 8 9]

C = A(3:4,1:2) = [7 8; 0 -1]

C =

7

8

0

-1


Some Useful Built-in Functions

• mean(A)

• max(A), min (A)

• sum(A)

• sqrt(A)

• median(A)

• eig(A)

• rank(A)

• rot90(A), flip(A)

• std(A)

• det(A)

• inv(A)

• tril(A)

• triu(A)


Array Creation

• linspace (Linearly spaced points)

– linspace(x1,x2) will generate 100 points between x1 and x2

– linspace(x1,x2,n) will generate n points between x1 and x2

>>linspace(0,10,6)

ans=

0 2 4 6 8 10

• logspace (logarithmically spaced points)

– logspace(x1,x2) will generate 100 points between 10x1 and 10x2

– logspace(x1,x2,n) will generate n points between 10x1 and 10x2

>>logspace(-1,2,4)

ans=

0.1000 1.0000 10.0000 100.0000

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Plots plot() (for plotting 2-D data) and plot3() (for plotting 3-

D data)

Example 1:

x = 0:0.1:12; % defines x for value from 0

to 12 by 0.1

a = 3; b =2; c = -.2; % parameter values

y = a + b*x + c*x.*x; % values for y. Note

x.*x because we want scalar multiplication

plot(x,y)

t= linspace(0,50,200);

x=exp(-0.1.*t).*sin(t);

y=exp(-0.1*t).*cos(t);

z=t;

subplot(2,1,1)

plot(x,y)

xlabel('x'), ylabel('y')

subplot(2,1,2)

plot3(x,y,z) xlabel('x'), ylabel('y'),zlabel('z')

title('Example','FontSize',12)

Example 2: 𝑥 = 𝑒−0.1𝑡 ∗ sin(𝑡)

𝑦 = 𝑒−0.1𝑡 ∗ cos(𝑡) 𝑦 = 𝑎 + 𝑏𝑥 + 𝑐𝑥2

0 2 4 6 8 10 12-2

-1

0

1

2

3

4

5

6

7

8

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

x

y

-1 -0.50 0.5

1

-1-0.5

00.5

10

50

x

Example

y

z

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

x

y

-1 -0.50 0.5

1

-1-0.5

00.5

10

50

x

Example

y

z


Mesh, surf and contours

-100

10

-10

0

10-1

0

1

-100

10

-10

0

10-1

0

1

-5 0 5

-5

0

5

-50

5

-50

5

-1

0

1

x=-7.5:0.5:7.5; % generate vector x

y=x; % generate vector y

[X,Y]=meshgrid(x,y); % generate evenly

spaced data points in x-y plane (grid

points)

r=sqrt(X.^2+Y.^2)+eps; % eps is a small no.

(built in) to avoid division by zero

z=sin(r)./r;

subplot(2,2,1); mesh(X,Y,z); % mesh plot of

z

subplot(2,2,2);

surf(X,Y,z); % surface plot of z

subplot(2,2,3);

contour(X,Y,z,20); % generate 20 no. of 2-D

contour lines for z

subplot(2,2,4);

contour3(X,Y,z,20); % Same contour plot in

3-D


Conditional Controls and Loops

for loop

i = 0;

for t=0:0.02:50

i = i+1;

y(i) = cos(t)

end

while loop

While condition

statements

end

Early termination

EXAMPLE:

x=24;

while(1)

x=x-5

if x<0, break

end

end

if

else

elseif

a = 4;

b = 4;

if (a<b)

j = -1;

elseif (a>b)

j = 2;

else

j = 3

end


Using Function Files

• Solve a simple equation with 3

variables (free fall velocity)

𝑣 =𝑔 ∗ 𝑚

𝑐𝑑∗ tanh

𝑔 ∗ 𝑚

𝑐𝑑∗ 𝑡

function v=freefall(t,m,cd)

g=9.81; %acceleration

v=sqrt(g*m/cd)*tanh(sqrt(g*m/cd)*t);

Save the file as freefall.m

Call the function in the MATLAB command window

v=freefall(10,70,.25);

fprintf('The velocity is %0.2f m/s \n',v)


Example - Numerical Differentiation

DIFFERENTIATION

• Forward difference approximation

𝑓′ 𝑥𝑖 =𝑓 𝑥𝑖+1 − 𝑓 𝑥𝑖

ℎ+ 𝑂(ℎ)

• Backward difference approximation

𝑓′ 𝑥𝑖 =𝑓 𝑥𝑖 − 𝑓 𝑥𝑖−1

ℎ+ 𝑂(ℎ)

• Centered difference approximation

𝑓′ 𝑥𝑖 =𝑓 𝑥𝑖+1 − 𝑓 𝑥𝑖−1

2ℎ+ 𝑂(ℎ2)

h h


Example – Numerical Differentiation (continued)

𝑓 𝑥 = 𝑒−2𝑥 − 𝑥

Find f’(x) using centered approximation at x=2

clear all

f=@(x)exp(-2*x)-x; % Describe the f(x)

x=2; % What point do you want to find the derivative

h=[0.5:-0.05:0.1]; % make a vector [0.5 0.45 … 0.1 ]

n=length(h);

for i=1:1:n

df(i)=(f(x+h(i))-f(x-h(i)))/(2*h(i));% Centered

finite-difference approximation

end

fprintf('h df\n');

for i=1:1:n

fprintf('%0.2f %0.4f\n',h(i),df(i))

end

Analytical solution

f’(2) = -1.0366

Given:

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Some Useful Matlab Links

General Overview of Matlab:

• http://web.mit.edu/acmath/matlab/unified/fall07/UnifiedMatlabF07/UnifiedIntroMatlabF07.pdf

• http://www.mathworks.com/help/index.html - Matlab web site

• http://www.mathworks.com/products/matlab/examples.html - many programming examples

• http://www.mathworks.com/help/index.html - Step-by-Step guides to Matlab and Simulink

• http://www.math.mtu.edu/~msgocken/intro/intro.html - many programming examples

• http://people.duke.edu/~hpgavin/matlab.html - links to additional Matlab tutorials

Regression:

• MIT: http://web.mit.edu/acmath/matlab/course16/16.62x/16.62x_Matlab.pdf - slide deck that

includes general overview of MatLab + data analysis functions + regression functions

• MathWorks: http://www.mathworks.com/help/matlab/data_analysis/linear-regression.html - covers

linear regression only

Videos:

• http://www.mathworks.com/products/matlab/videos.html - From the makers of Matlab

• https://www.youtube.com/playlist?list=PL7CAABC40B2825C8B – YouTube Matlab playlist

http://web.mit.edu/acmath/matlab/unified/fall07/UnifiedMatlabF07/UnifiedIntroMatlabF07.pdf



http://www.mathworks.com/help/index.html


http://www.mathworks.com/products/matlab/examples.html

http://www.mathworks.com/products/matlab/examples.html



http://www.math.mtu.edu/~msgocken/intro/intro.html

http://www.math.mtu.edu/~msgocken/intro/intro.html

http://people.duke.edu/~hpgavin/matlab.html

http://people.duke.edu/~hpgavin/matlab.html

http://web.mit.edu/acmath/matlab/course16/16.62x/16.62x_Matlab.pdf

http://www.mathworks.com/help/matlab/data_analysis/linear-regression.html



http://www.mathworks.com/products/matlab/videos.html

http://www.mathworks.com/products/matlab/videos.html

https://www.youtube.com/playlist?list=PL7CAABC40B2825C8B

https://www.youtube.com/playlist?list=PL7CAABC40B2825C8B

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