mathscript lab - part i

Telemark University College

Department of Electrical Engineering, Information Technology and Cybernetics

Faculty of Technology, Postboks 203, Kjlnes ring 56, N-3901 Porsgrunn, Norway. Tel: +47 35 57 50 00 Fax: +47 35 57 54 01

So You Think You Can

MathScript HANS-PETTER HALVORSEN, 2011.09.07

Part I: Introduction to MathScript

ii

Preface

Purpose with this Lab

In this lab you will learn how to use a tool like MathScript (which has a similar syntax as MATLAB) to

solve control and simulation problems.

For additional information and resources:

http://home.hit.no/~hansha/?lab=mathscript

MathScript

MathScript is a high-level, text- based programming language. MathScript includes more than 800

built-in functions and the syntax is similar to MATLAB. You may also create custom-made m-file like

you do in MATLAB.

MathScript is an add-on module to LabVIEW but you dont need to know LabVIEW programming in

order to use MathScript.

http://home.hit.no/~hansha/?lab=mathscript

iii

What is LabVIEW?

LabVIEW (short for Laboratory Virtual Instrumentation Engineering Workbench) is a platform and

development environment for a visual programming language from National Instruments. The

graphical language is named G.

What is MATLAB?

MATLAB is a tool for technical computing, computation and visualization in an integrated

environment. MATLAB is an abbreviation for MATrix LABoratory, so it is well suited for matrix

manipulation and problem solving related to Linear Algebra.

MATLAB offers lots of additional Toolboxes for different areas such as Control Design, Image

Processing, Digital Signal Processing, etc.

What is MathScript?



you do in MATLAB.


order to use MathScript. If you want to integrate MathScript functions (built-in or custom-made

m-files) as part of a LabVIEW application and combine graphical and textual programming, you can

work with the MathScript Node.

In addition to the MathScript built-in functions, different add-on modules and toolkits installs

additional functions. The LabVIEW Control Design and Simulation Module and LabVIEW Digital

Filter Design Toolkit install lots of additional functions.

You can more information about MathScript here: http://www.ni.com/labview/mathscript.htm

How do you start using MathScript?

You need to install LabVIEW and the LabVIEW MathScript RT Module. When necessary software is

installed, start MathScript by open LabVIEW:

http://home.hit.no/~hansha/documents/software/LabVIEW.htmhttp://home.hit.no/~hansha/documents/software/MATLAB.htmhttp://home.hit.no/~hansha/documents/software/LabVIEW%20MathScript%20RT%20Module.htmhttp://home.hit.no/~hansha/documents/software/LabVIEW%20Control%20Design%20and%20Simulation%20Module.htmhttp://www.ni.com/labview/mathscript.htmhttp://home.hit.no/~hansha/documents/software/LabVIEW.htmhttp://home.hit.no/~hansha/documents/software/LabVIEW%20MathScript%20RT%20Module.htm

iv

In the Getting Started window, select Tools -> MathScript Window...:

v

Table of Contents

Preface ......................................................................................................................................................ii

Purpose with this Lab ...........................................................................................................................ii

MathScript............................................................................................................................................ii

Table of Contents ..................................................................................................................................... v

1 Introduction to MathScript ............................................................................................................. 6

1.1 Additional Resources ............................................................................................................... 6

1.2 Getting Started ........................................................................................................................ 6

1.3 Basic Operations ...................................................................................................................... 9

1.4 Matrices and Vectors............................................................................................................. 12

1.5 Solving Linear Equations ........................................................................................................ 15

2 Plotting.......................................................................................................................................... 17

2.1 Sub-plots ................................................................................................................................ 21

3 Functions ...................................................................................................................................... 23

3.1 Calling functions In MathScript ............................................................................................. 23

3.2 User-Defined Functions in MathScript .................................................................................. 23

3.3 Scripts .................................................................................................................................... 25

4 Flow Control.................................................................................................................................. 27

5 Additional Tasks ............................................................................................................................ 29

Appendix A MathScript Functions ...................................................................................................... 33

Basic Functions .................................................................................................................................. 33

Basic Plotting Functions .................................................................................................................... 33

Linear Algebra ................................................................................................................................... 34

6

1 Introduction to

MathScript

1.1 Additional Resources



you do in MATLAB.


order to use MathScript.

You need to install the LabVIEW MathScript RT Module in addition to LabVIEW before you start

this Lab Work.

Background information for this lab is described in detailed in the LabVIEW MathScript

Tutorial. The Tutorial consists of pdf documents, videos, example code, additional resources and

links.

You should also read MathScript Basics by Richard C Dorf and Robert H. Bishop -

Modern Control Systems.

The document Control Theory with MathScript Examples gives an overview of the basic

cybernetics theory needed for this Lab Work with lots of MathScript Examples.

Link to these documents are located here: http://home.hit.no/~hansha/?lab=mathscript

1.2 Getting Started

How do you start using MathScript? You need to install LabVIEW and the LabVIEW MathScript RT

Module. When necessary software is installed, start MathScript by open LabVIEW:

http://home.hit.no/~hansha/documents/software/LabVIEW%20MathScript%20RT%20Module.htmhttp://home.hit.no/~hansha/?tutorial=mathscripthttp://home.hit.no/~hansha/documents/labview/training/LabVIEW%20MathScript/Background/MathScript%20Basics.pdfhttp://home.hit.no/~hansha/documents/control/Documents/Lecture%20Notes/Control%20Theory%20with%20MathScript%20Examples.pdfhttp://home.hit.no/~hansha/?lab=mathscript

7 Introduction to MathScript

MathScript - Part I: Introduction to MathScript

In the Getting Started window, select Tools -> MathScript Window...:

The LabVIEW MathScript Window is an interactive interface in which you can enter .m file script

commands and see immediate results, variables and commands history. The window includes a

command-line interface where you can enter commands one-by-one for quick calculations, script

debugging or learning. Alternatively, you can enter and execute groups of commands through a script

editor window.

As you work, a variable display updates to show the graphical / textual results and a history window

tracks your commands. The history view facilitates algorithm development by allowing you to use the

clipboard to reuse your previously executed commands.



You can use the LabVIEW MathScript Window to enter commands one at time. You also can enter

batch scripts in a simple text editor window, loaded from a text file, or imported from a separate text

editor. The LabVIEW MathScript Window provides immediate feedback in a variety of forms, such

as graphs and text.



1.3 Basic Operations

Below we see MathScript and the Command window:

You use the Command window to enter commands to MathScript. The results are shown in the

Output Window. If we want to enter a collection of different commands that belong together, we

use the Script window.

Variables:

Variables are defined with the assignment operator, =. MathScript is dynamically typed, meaning

that variables can be assigned without declaring their type, and that their type can change. Values

can come from constants, from computation involving values of other variables, or from the output

of a function.

Note! MathScript is case sensitive! The variables x and X are not the same.

Example:

Type the following in the Command window:

>>y=16;

>>z=3;

>>y+z

Note! When you use a semicolon, no output will be displayed. Try the code above with and without

semicolon.

Other basic operations are:



>>16-3

>>16/3

>>16*3

Try them.

[End of Example]

Example:

Given the following simple function:

( )

We want to find ( )

In order to do that, we type the following in the Command Window:

First we need to define x:

x=2

Then we can define the function:

y = 3*x + 2

Then MathScript respond with the following answer:

y =

8

[End of Example]

Task 1: Math function

Given the following function:

( )

Find ( ) and ( )

[End of Task]

Task 2: Math function

Create the following mathematical expression in MathScript:



( )

is a function of and , i.e. ( ). Use MathScript to find ( ).

Tip! Use the following built-in functionality:

x^2

sqrt(x)

( ) log(x)

exp(x)

[End of Task]



1.4 Matrices and Vectors

Matrices and vectors (Linear Algebra) are the basic elements in MathScript and also the basic

elements in control design theory. So it is important you know how to handle vectors and matrices in

MathScript.

The Tutorial LabVIEW MathScript describes the basic syntax of Linear Algebra in

MathScript.

Vectors:

Given the following vector:

[ ]

This can be implemented in MathScript like this:

x =[1, 2, 3]

The colon notation is very useful for creating vectors:

Example:

x=1:5

This gives:

x =

1 2 3 4 5

x=1:0.5:3

This gives:

http://home.hit.no/~hansha/documents/labview/training/LabVIEW%20MathScript/LabVIEW%20MathScript.pdf



x =

1 1.5 2 2.5 3

[End of Example]

Matrices:

A general matrix may be written like this:

[

]

Basically, a matrix is a 2 dimensional array with rows and columns.

Example:

Given the following matrix:

*

+

MathScript Code:

A=[1, 2; 3, 4]

Note! , (or just an empty space) is used to separate elements in a row, while ; is used to define a

new row.

[End of Example]

How to get a subset of a matrix:

Example:

We want to find the value in the second row and the second column of matrix A:

A(2,2)

This gives:

ans =

4

We want to find the second row of matrix A:

A(2,:)

This gives:

ans =



3 4

We want to find the second column of matrix A:

A(:,2)

This gives:

ans = 2

4

[End of Example]

Task 3: Vectors and Matrices

Type the following vector in MathScript in the Command window:

[ ]

Type the following matrix in MathScript in the Command window:

*

+

Type the following matrix in MathScript in the Command window:

[

]

Use Use MathScript to find the value in the second row and the third column of matrix C.

Use MathScript to find the second row of matrix C.

Use MathScript to find the third column of matrix C.

[End of Task]

Task 4: Linear Algebra

(You may want to skip this task if you havent learned about Linear Algebra yet)

Given the matrix A, B and C:

*

+ *

+ *

+

Solve the following basic matrix operations using MathScript:



( ) ( )

( ) ( )

( )

( )

where eig = Eigenvalues, diag = Diagonal, det = Determinant

[End of Task]

Task 5: Matrices


Use MathScript to prove the following:

( ) ( )

( )

( )

( ) ( ) ( )

( ) ( )

where is the unit matrix

[End of Task]

1.5 Solving Linear Equations

Task 6: Equations


Given the equations:

Set the equations on the following form:



Find A and b.

Solve the equations, i.e., find , using MathScript.

[End of Task]

17

2 Plotting

MathScript has lots functions used for plotting:

Function Description Example

plot Generates a plot. plot(y) plots the columns of y against the indexes of the columns.

>X = [0:0.01:1];

>Y = X.*X;

>plot(X, Y)

figure Create a new figure window >>figure >>figure(1)

subplot Create subplots in a Figure. subplot(m,n,p) or subplot(mnp), breaks the Figure window into an m-by-n matrix of small axes, selects the p-th axes for the current plot. The axes are counted along the top row of the Figure window, then the second row, etc.

>>subplot(2,2,1)

grid Creates grid lines in a plot. grid on adds major grid lines to the current plot. grid off removes major and minor grid lines from the current plot.

>>grid

>>grid on

>>grid off

axis Control axis scaling and appearance. axis(*xmin xmax ymin ymax+) sets the limits for the x- and y-axis of the current axes.

>>axis([xmin xmax ymin ymax])

>>axis off

>>axis on title Add title to current plot

title('string')

>>title('this is a title')

xlabel Add xlabel to current plot xlabel('string')

>> xlabel('time')

ylabel Add ylabel to current plot ylabel('string')

>> ylabel('temperature')

legend Creates a legend in the corner (or at a specified position) of the plot

>> legend('temperature')

hold Freezes the current plot, so that additional plots can be overlaid >>hold on >>hold off

Below we see some examples of how to use the different plot functions:

18 Plotting


For line colors and line-styles we have the following properties we can use for the plot function:

Line Styles:

Marker specifiers:

19 Plotting


Colors:

We will use this in some tasks below.

20 Plotting


Task 7: Plotting

In the Command window in MathScript window input the time from t=0 seconds to t=10 seconds in

increments of 0.1 seconds as follows:

>>t=[0:0.1:10];

Then, compute the output y as follows:

>>y=cos(t);

Use the Plot command:

>>plot(t,y)

[End of Task]

Task 8: Plotting

Plot ( ) and ( ) for in the same plot.

Tip! You can use the hold on command in addition to the plot command or you can use the plot

command like this:

Plot(x1,y1, x2,y2, )

Make sure to add labels and a legend, and use different line styles and colors for the plots.

Note! We cannot use Greek letters in MathScript, but use, e.g., theta, x or another name for your

variable.

[End of Task]

Task 9: Plot of dynamic system

Given the following differential equation:

where

,where is the time constant

The solution for the differential equation is:

( )

Set and the initial condition ( )

21 Plotting


Plot the solution ( ) in the time interval

Add Grid, and proper Title and Axis Labels to the plot.

[End of Task]

2.1 Sub-plots

The subplot command enables you to display multiple plots in the same window or print them on the

same piece of paper. Typing subplot(m,n,p) partitions the figure window into an m-by-n matrix of

small subplots and selects the pth subplot for the current plot. The plots are numbered along the first

row of the figure window, then the second row, and so on.

The syntax is as follows:

subplot(m,n,p)

Example:

x=1:5;

subplot(2,1,1)

y=3*x+2;

plot(x,y)

subplot(2,1,2)

z=-2*x+2;

plot(x,z)

This gives the following plot:

22 Plotting


[End of Example]

Task 10: Sub-plots

Plot sin(x) and cos(x) in 2 different subplots.

Add Titles and Labels.

[End of Task]

23

3 Functions

3.1 Calling functions In MathScript

MathScript includes more than 800 built-in functions that you can use, e.g., in a previous task you

used the plot function.

Task 11: Statistics

Given the vector:

>>x=[1 2 5 6 8 9 3]

Find the mean value of the vector x.

Find the minimum value of the vector x.

Find the maximum value of the vector x.

[End of Task]

3.2 User-Defined Functions in MathScript

So far we have used the Command Window in MathScript, but now we will start using the Script

Window in MathScript.

MathScript includes more than 800 built-in functions that you can use but sometimes you need to

create your own functions.

To define your own function in MathScript, use the following syntax:

function outputs = function_name(inputs)

% documentation

Example:

We will create a simple function that finds the sum of two numbers:

24 Functions


function total = add(x,y)

% this function add 2 numbers

total = x+y;

Note! You need to save this as a file with extension .m with the same name as the function, i.e.,

add.m In this example are x and y inputs to the function, while total is an output from the function.

You may use the function like this:

%Example 1:

add(2,3)

%Example 2:

a=4;

b=6;

add(a,b);

%Example 3:

answer = add(a,b)

[End of Example]

Here is the procedure for creating a user-defined function in MathScript:

Task 12: User-defined function

Create a function calc_average that finds the average of two numbers.

25 Functions


Test the function afterwards as follows:

>>x=2;

>>y=4;

>>z=calc_average(x,y)

[End of Task]

3.3 Scripts

A script is a sequence of MathScript commands that you want to perform to accomplish a task. When

you have created the script you may save it as a m-file for later use.

Task 13: Script

Create a new Script where you use the function you created in the previous task. Call it several times

with different inputs.

[End of Task]

Task 14: Conversion

It is quite easy to convert from radians to degrees or from degrees to radians. We have that:

[ ] [ ]

This gives:

26 Functions


[ ] [ ] (

)

[ ] [ ] (

)

Create two functions that convert from radians to degrees (r2d(x)) and from degrees to radians

(d2r(x)) respectively.

Test the functions to make sure that they work as expected, e.g.:

>> r2d(2*pi)

ans =

360

>> d2r(180)

ans =

3.1416

[End of Task]

27

4 Flow Control

You may use different loops in MathScript

For loop

While loop

If you want to control the flow in your program, you may want to use one of the following:

If-else statement

Switch and case statement

Task 15: For Loop

Extend your calc_average function from a previous task so it can calculate the average of a vector

with random elements. Use a For loop to iterate through the values in the vector and find sum in

each iteration, e.g.:

mysum = mysum + x(i);

Test the function in the Command window

[End of Task]

Task 16: If-else

Depending on a variable a, different functions should be executed and plotted.

( )

( )

( )

Create a script that plot based on the value of the input .

[End of Task]

Task 17: Fibonacci Numbers

28 Flow Control


In mathematics, Fibonacci numbers are the numbers in the following sequence:

0, 1, 1, 2 , 3, 5, 8, 13, 21, 34, 55, 89, 144,

By definition, the first two Fibonacci numbers are 0 and 1, and each subsequent number is the sum

of the previous two. Some sources omit the initial 0, instead beginning the sequence with two 1s.

In mathematical terms, the sequence of Fibonacci numbers is defined by the recurrence relation:

with seed values:

Write a function in MathScript that calculates the N first Fibonacci numbers, e.g.,

>> N=10;

>> fibonacci(N)

ans =

0

1

1

2

3

5

8

13

21

34

Use a For loop to solve the problem.

Fibonacci numbers are used in the analysis of financial markets, in strategies such as Fibonacci

retracement, and are used in computer algorithms such as the Fibonacci search technique and the

Fibonacci heap data structure. They also appear in biological settings, such as branching in trees,

arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling

fern and the arrangement of a pine cone.

[End of Task]

29

5 Additional Tasks

Here are some additional tasks:

Task 18: If-else

Given the second order algebraic equation:

The solution (roots) is as follows:

{

where - there is no solution, - any complex number is a solution

Create a function that finds the solution for based on different input values for a, b and c, e.g.,

function x = solveeq(a,b,c)

Use if-else statements to solve the problems

Test the function from the Command window to make sure it works as expected, e.g.,

>> a=0, b=2, c=1

>> solveeq(a,b,c)

Compare the result with the results you get from the built-in roots function.

[End of Task]

Task 19: Basic Math function

Create a function that calculates the following mathematical expression:

( )

30 Additional Tasks


We can test the function like this:

Testing the function gives (in this example the name of our function is calcexpression):

>> x=2; y=2;

>> calcexpression(x,y)

ans =

16.8284

[End of Task]

Task 20: User-defined function

Create a function that uses Pythagoras to calculate the hypotenuse of a right-angled triangle, e.g.:

function h = pyt(a,b)

% ..

h =

Pythagoras theorem is as follows:

Note! The function should handle that a and b could be vectors.

[End of Task]

Task 21: Cylinder surface area

Create a function that finds the surface area of a cylinder based on the height (h) and the radius (r) of

the cylinder.

31 Additional Tasks


[End of Task]

Task 22: Create advanced expressions in MathScript

Create the following expression in MathScript:

( ) ( ) ( )

( )( )

Given

Find ( )

(The answer should be ( ) )

Tip! You should split the expressions into different parts, such as:

poly =

num =

den =.

f =

This makes the expression simpler to read and understand, and you minimize the risk of making an

error while typing the expression in MathScript.

[End of Task]

Task 23: Solving Equations

Find the solution(s) for the given equations:

32 Additional Tasks


[End of Task]

Task 24: Script

Given the famous equation from Albert Einstein:

The sun radiates of energy per day.

Calculate how much of the mass on the sun is used to create this energy per day.

How many years will it take to convert all the mass of the sun completely? Do we need to worry if

the sun will be used up in our generation or the next?

The mass of the sun is

[End of Task]

33

Appendix A MathScript

Functions

Basic Functions

Here are some descriptions for the most used basic MathScript functions.


help MathScript displays the help information available >>help

help

Display help about a specific function >>help plot

who, whos who lists in alphabetical order all variables in the currently active workspace.

>>who

>>whos

clear Clear variables and functions from memory. >>clear >>clear x

size Size of arrays, matrices >>x=[1 2 ; 3 4]; >>size(A)

length Length of a vector >>x=[1:1:10]; >>length(x)

format Set output format

disp Display text or array >>A=[1 2;3 4]; >>disp(A)

plot This function is used to create a plot >>x=[1:1:10]; >>plot(x)

>>y=sin(x);

>>plot(x,y)

clc Clear the Command window >>cls

rand Creates a random number, vector or matrix >>rand >>rand(2,1)

max Find the largest number in a vector >>x=[1:1:10] >>max(x)

min Find the smallest number in a vector >>x=[1:1:10] >>min(x)

mean Average or mean value >>x=[1:1:10] >>mean(x)

std Standard deviation >>x=[1:1:10] >>std(x)

Basic Plotting Functions


plot Generates a plot. plot(y) plots the columns of y against the indexes of the columns.

>X = [0:0.01:1];

>Y = X.*X;

>plot(X, Y)

figure Create a new figure window >>figure >>figure(1)

subplot Create subplots in a Figure. subplot(m,n,p) or subplot(mnp), breaks the Figure window into an m-by-n matrix of small axes, selects the p-th axes for the current plot. The axes are counted along the top row of the Figure window, then the second row,

>>subplot(2,2,1)

34 Appendix A MathScript Functions


etc.

grid Creates grid lines in a plot. grid on adds major grid lines to the current plot. grid off removes major and minor grid lines from the current plot.

>>grid

>>grid on

>>grid off

axis Control axis scaling and appearance. axis(*xmin xmax ymin ymax+) sets the limits for the x- and y-axis of the current axes.

>>axis([xmin xmax ymin ymax])

>>axis off

>>axis on title Add title to current plot

title('string')

>>title('this is a title')

xlabel Add xlabel to current plot xlabel('string')

>> xlabel('time')

ylabel Add ylabel to current plot ylabel('string')

>> ylabel('temperature')

legend Creates a legend in the corner (or at a specified position) of the plot

>> legend('temperature')

hold Freezes the current plot, so that additional plots can be overlaid >>hold on >>hold off

For more information about the plots function, type help plots.

Linear Algebra

Here are some useful functions for Linear Algebra in MATLAB:


rank Find the rank of a matrix. Provides an estimate of the number of linearly independent rows or columns of a matrix A.

>>A=[1 2; 3 4]

>>rank(A)

det Find the determinant of a square matrix >>A=[1 2; 3 4] >>det(A)

inv Find the inverse of a square matrix >>A=[1 2; 3 4] >>inv(A)

eig Find the eigenvalues of a square matrix >>A=[1 2; 3 4] >>eig(A)

ones Creates an array or matrix with only ones >>ones(2) >>ones(2,1)

eye Creates an identity matrix >>eye(2)

diag Find the diagonal elements in a matrix >>A=[1 2; 3 4] >>diag(A)

Type help matfun (Matrix functions - numerical linear algebra) in the Command Window for more

information, or type help elmat (Elementary matrices and matrix manipulation).

You may also type help for help about a specific function.


Faculty of Technology

Kjlnes Ring 56

N-3914 Porsgrunn, Norway

www.hit.no

Hans-Petter Halvorsen, M.Sc.


Department of Electrical Engineering, Information Technology and Cybernetics

Phone: +47 3557 5158

E-mail: [email protected]

Blog: http://home.hit.no/~hansha/

Room: B-237a

http://www.hit.no/mailto:[email protected]://home.hit.no/~hansha/

mathscript lab - part i

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