SYSTEMS CONTROL LABORATORY Introduction to MATLAB (Bring a 3.5 in floppy disk with you to save your work)
MATLAB is a high performance language for technical computing. It integrates computation, visualization and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation. It is particularly convenient for modeling, simulation, analysis, and design of dynamic control systems. The purpose of this lab project is to give you a quick introduction to MATLAB enabling you to use the software from the very beginning of the Control Systems course. To run MATLAB, doubleclick on the MATLAB icon. Once the program is loaded you will see the prompt " " in the command window. The command window is the main window in which we communicate with the MATLAB interpreter. Any thing typed after '%' sign is treated as a comment and ignored by MATLAB. A carriage return results in execution of a valid command in the command window. Part A: Examples of command line entries. In the command window, MATLAB commands are executed as soon as the return key is pressed. The following is a sequence of possible command line entries. Report Item #1 To get started with MATLAB recreate the following transactions and save them (in diary ) for the report. pwd % Gives current (present) working directory path(path, 'a:\') % Allows you to work from 'a' drive cd a: % Makes the 'a' drive the working drirectory Format compact % suppresses extra line-feeds diary my_file_name.txt % opens a text file named my_file_name.txt in 'a' drive % All subsequent transactions are written in this file. Remember to close the file at end of the session % Later, to close the file you will be typing diary off . % Explanation of the notation seen in the margin: YT: user typed entry; ML: Matlab's response. % Do not type YT and ML; that is not part of command line entry! % In the following, the MATLAB prompt " " is not shown % Example 1 - Scalars, vectors and matrices YT ML a=2.5 a= 2.5000 YT ML a=[1 a= 1 2 3 2 3] % " [ ] " signify a vector or a matrix % Assigns a scalar value to the variable "a".
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YT ML
a=[1, a= 1
2 , 3]
% both " , " and "blank space" act as delimiters; extra spaces are ignored
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3 5 6; 7 8 10] % a " ; " signifies end of a row of the matrix
YT ML
a=[1 2 3; 4 a= 1 4 7 2 5 8 3 6 10
YT ML
f= a(2,1) f= 4
% The element in Row 2 and Column 1 of matrix "a"
YT ML
g=a(2:3,2) % Will assign elements in rows 2 to 3 and column 2 of "a" to the variable "g" g= 5 8
YT ML
h=a(2,2:3) % Will assign elements in row 2 and columns 2 to 3 of "a" to the variable "h" h= 5 6 %The semicolon after ] suppresses printing on screen of the value of "b"
YT ML YT ML
b=[1 2;3 4];
d=inv(b) % Matlab will compute the inverse of matrix "b" and assign it to "d" and print it. d= -2.0000 1.0000 1.5000 -0.5000 % Example 2 - Plotting graphs and repetitive computations.
YT YT
t=[0 : 0.1 : 20]'; % " t " is a column vector with elements 0,.1,.2,...,20 - a total of 201 elements for i=1 : 1 : 201
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efficient.
x(i)=sin(t(i))*(t(i)^2); end y=sin(t) .* (t.^2); % alternate approach. Note that x = y = t 2 sin(t). Second method is more % The column vectors x and y are identical vectors of dimension 201. y is generated by using % array operators, an efficient way of performing repetitive computations. plot(t,x) % This command plots a x vs. t graph. With the graph window active, the plot can be printed % (just use the print command in the File menu). Next plot y vs. t for comparison. plot(t,y)
YT ML YT
Note: Use the key sequence ALT-TAB to switch among the MATLAB windows % Example 3 - Complex Arithmetic YT ML z=1+2i z= 1.0000 + 2.0000i YT ML r=abs(z) % absolute value or magnitude of z r= 2.2361
YT
% Assigns a complex value to z;
theta=angle(z) % angle of z in radians; multiply by (180/pi) for angle in degrees theta = 1.1071
ML
YT YT ML
fz=(z+1)/(z*(z^2-4*z+5)*(z-1)); % Evaluating complex expression: F(z) = fz fz = 0.1400 + 0.0200i % Example 4 - Factoring polynomials
(z +1) z(z - 4z + 5)(z -1)
2
YT YT YT
% F(s) = s 3 + 6s2 +11s + 6. We want to factor this polynomial. % In Matlab polynomials are defined as row vectors of the polynomial coefficients. pol1=[1 6 11 6]; rt1=roots(pol1); % computes the roots and stores the result in rt1 without printing on screen. rt1
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rt1 = -3.0000 -2.0000 -1.0000
YT ML
pol2=[1 4 rt2 = -2 -2
4]; rt2=roots(pol2) % You may have more than one command (statement) in a line!
% Note: no ";" at the end of rt2 statement. If there is no ";" Matlab prints the value of the variable. YT ML pol3=[1 1 1]; rt3=roots(pol3) rt3 = -0.5000 + 0.8660i -0.5000 - 0.8660i YT YT ML f1=poly(rt1); f1 f1 = 1.0000 6.0000 11.0000 6.0000 % given the roots, this will compute the polynomial coeffs.
% Example - 5 Partial Fraction Expansion. % Given a ratio of polynomials, G(s) = YT YT YT YT ML (3s + 7) , determine the corresponding PFE. (s + 6s2 +11s + 6)
3
% This is done by first defining the numerator and denominator polynomials. num1=[3 7]; den1=[1 6 11 6]; [r,p,k]=residue(num1,den1); % r is the vector of PFE coefficients; p the vector of roots of den1; % and k=0 if the degree of num1 < the degree of den1. r r= -1.0000 -1.0000 2.0000
YT
p
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p= -3.0000 -2.0000 -1.0000
YT ML
k k= []
% Example 6 - Plotting functions revisited. YT YT YT YT YT YT YT YT % Generate and plot the function: f (t) = 1 - 2e -0.24t cos(2t) t=[0 : 0.1: 20]; f=1-2*exp(-.24*t).*(cos(2*t)); % Array operator " .* " is needed to multiply two vectors element by element plot(t,f) xlabel(' The t axis ') ylabel(' The f axis ') title(' The plot of f versus t ') grid on diary off % The file my_file_name.txt is closed.
Print out the file "my_file_name.txt" and include it in the report. PS. You may also copy and paste the transactions in the Command Window into a text file and print it. Report Item #2 While the Figure window is active, use print command from the file menu to get a hard copy of The plot of f versus t ; include the plot in the report. PART B : MATLAB Programming: Files that contain MATLAB language code are called M-files. M-files can be functions that accept arguments and produce output, or they can be scripts that execute a series of MATLAB statements. We create M-files using a text editor, then use them as any other MATLAB function or command. Report Item #3 Rework example 6 above (with suggested modifications) using a script m-file named ex_6.m. Include printout of the m-file and the plot in the report. While in the command window, choose New and m-file from the file pull-down menu. An Edit/Debug window opens. Type the following commands in that window and save the file as ex_6.m. Then, in MATLAB command window type ex_6 and hit return.
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% Generate and plot the function: f (t) = 1 - 2e -0.24t cos(2t) , and its envelopes t=[0 : 0.1: 20]; f=1-2*exp(-.24*t).*(cos(2*t)); % Array operator " .* " is needed to multiply two vectors element by element plot(t,f) hold on % holds the graph for superimposed plots plot(t,(1+2*exp(-0.24*t)), '--') % gives a dashed line plot plot(t,(1+2*exp(-0.24*t)), '--') % gives a dashed line plot xlabel(' The t axis ') ylabel(' The f axis ') title(' The plot of f versus t by second method ') grid on hold off % removes the hold for future plots Report Item #4 Write a function m-file which prompts the user for a positive integer input, say n, and returns the sum of integers from 1 to n. Include the printout of the m-file in the report. % function m-file int_sum.m function y=int_sum(x) % first line of a function m-file must be the function statement n=input(' Type an integer and hit return: ') % prompting for input if ( (n-abs(round(n))) ~= 0 ) % check if input is not a posit...