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Matlab & Ss

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Lab 1: Introduction to MATLAB

Warm-up MATLAB is a high-level programming language that has been used extensively to solve complex engineering problems. The language itself bears some similarities with ANSI C and FORTRAN. MATLAB works with three types of windows on your computer screen. These are the Command window, the Figure window and the Editor window. The Figure window only pops up whenever you plot something. The Editor window is used for writing and editing MATLAB programs (called M-files) and can be invoked in Windows from the pull-down menu after selecting File | New | M-file. In UNIX, the Editor window pops up when you type in the command window: edit filename (‘filename’ is the name of the file you want to create). The command window is the main window in which you communicate with the MATLAB interpreter. The MATLAB interpreter displays a command >> indicating that it is ready to accept commands from you. • View the MATLAB introduction by typing >> intro at the MATLAB prompt. This short introduction will demonstrate some basic MATLAB commands. • Explore MATLAB’s help capability by trying the following: >> help >> help plot >> help ops >> help arith • Type demo and explore some of the demos of MATLAB commands. • You can use the command window as a calculator, or you can use it to call other MATLAB programs (M-files). Say you want to evaluate the expression [pic], where a=1.2, b=2.3, c=4.5 and d=4. Then in the command window, type: >> a = 1.2; >> b=2.3; >> c=4.5; >> d=4; >> a^3 + sqrt(b*d) - 4*c ans = -13.2388 Note the semicolon after each variable assignment. If you omit the semicolon, then MATLAB echoes back on the screen the variable value.
Arithmetic Operations There are four different arithmetic operators: + addition - subtraction * multiplication / division (for matrices it also means inversion) There are also three other operators that operate on an element by element basis: .* multiplication of two vectors, element by element ./ division of two vectors, element-wise .^ raise all the elements of a vector to a power. Suppose that we have the vectors x = [[pic],[pic], ..., [pic]] and y = [[pic],[pic], ...,[pic]]. Then x. * y = [[pic],[pic], ..., [pic]] x./y = [[pic],[pic], ..., [pic]] x.ˆp = [[pic],[pic], ...,[pic]] The arithmetic operators + and - can be used to add or subtract matrices, scalars or vectors. By vectors we mean one-dimensional arrays and by matrices we mean multi-dimensional arrays. This terminology of vectors and matrices comes from Linear Algebra. Example: >> X = [1,3,4] >> Y = [4,5,6] >> X + Y ans= 5 8 10 For the vectors X and Y the operator + adds the elements of the vectors, one by one, assuming that the two vectors have the same dimension. In the above example, both vectors had the dimension 1 × 3, i.e., one row with three columns. An error will occur if you try to add a 1 × 3 vector to a 3 × 1 vector. The same applies for matrices. To compute the dot product of two vectors (i.e. [pic]), you can use the multiplication operator * . For the above example, it is: >> X*Y' ans = 43 Note the single quote after Y. The single quote denotes the transpose of a matrix or a vector. To compute an element by element multiplication of two vectors (or two arrays), you can use the .* operator: >> X .* Y ans = 4 15 24 That is, X.*Y means [1×4, 3×5, 4×6] = [4 15 24]. The ‘.*’ operator is used very often (and is highly recommended) because it is executed much faster compared to the code that uses for loops.
Complex numbers MATLAB also supports complex numbers. The imaginary number is denoted with the symbol i or j, assuming that you did not use these symbols anywhere in your program (that is very important!). Try the following: >> z = 3 + 4i % note that you do not need the ‘*’ after 4 >> conj(z) % computes the conjugate of z >> angle(z) % computes the phase of z >> real(z) % computes the real part of z >> imag(z) % computes the imaginary part of z >> abs(z) % computes the magnitude of z You can also define the imaginary number with any other variables you like. Try the following: >> img = sqrt(-1) >> z = 3 + 4*img >> exp(pi*img)
Array indexing In MATLAB, all arrays (vectors) are indexed starting with 1, i.e., y(1) is the first element of the array y. Note that the arrays are indexed using parenthesis (.) and not square brackets [.] as in C/C++. To create an array having as elements the integers 1 through 6, just enter: >> x=[1,2,3,4,5,6] Alternatively, you can use the : notation, >> x=1:6 The notation above creates a vector starting from 1 to 6, in steps of 1. If you want to create a vector from 1 to 6 in steps of say 2, then type: >> x=1:2:6 Ans = 1 3 5 Try the following code: >> ii=2:4:17 >> jj=20:-2:0 >> ii=2:(1/10):4 Extracting or inserting numbers in a vector can be done very easily. To concatenate an array, you can use the [ ] operator, as shown in the example below: >> x=[1:3 4 6 100:110] To access a subset of the array, try the following: >> x(3:7) >> length(x) % gives the size of the array or vector >> x(2:2:length(x))
Allocating memory You can allocate memory for one-dimensional arrays (vectors) using the zeros command. The following command allocates memory for a 100-dimensional array: >> Y=zeros(100,1); >> Y(30) ans = 0 Similarly, you can allocate memory for two-dimensional arrays (matrices). The command >> Y=zeros(4,5) defines a 4 by 5 matrix. Similar to the zeros command, you can use the command ones to define a vector containing all ones, >> Y=ones(1,5) ans= 1 1 1 1 1
Special characters and functions Some common special characters used in MATLAB are given below:


Some special functions are given below: length(x) - gives the dimension of the array x find - Finds indices of nonzero elements. Examples: >> x=1:10; >> length(x) ans = 10 The function find returns the indices of the vector X that are non-zero. For example, I = find(X>100), finds all the indices of X when X is greater than 100. So for the above Example: >> find(x> 4) ans = 5 6 7 8 9 10
Control flow MATLAB has the following flow control constructs: • if statements • switch statements • for loops • while loops • break statements The if, for, switch and while statements need to terminate with an end statement. Examples: IF: x = -3; if x > 0 str = 'positive'; elseif x < 0 str = 'negative'; elseif x == 0 str = 'zero'; else str = 'error'; end

What is the value of ’str’ after execution of the above code?

WHILE: x = -10; while x < 0 x = x + 1; end What is the value of x after execution of the above loop? FOR loop: X=0; for i=1:10 X=X+i; end

The above code computes the sum of all numbers from 1 to 10. BREAK: The break statement lets you exit early from a for or a while loop: x = -10; while x < 0 x = x + 2; if x == -2 break; end end MATLAB supports the following relational and logical operators: Relational Operators Symbol Meaning = Greater than equal > Greater than == Equal ˜ = Not equal Logical Operators Symbol Meaning & AND | OR ˜ NOT
You can plot arrays using MATLAB’s function plot. The function plot (.) is used to generate line plots. The function stem(.) is used to generate “picket-fence” type of plots. Example: >> x = 1:20; >> plot(x) %see Figure 1 >> stem(x) % see Figure 2
Example of a plot generated using the plot command is shown in Figure 1, and example of a plot generated using the stem function is shown in Figure 2. More generally, plot(X,Y) plots vector Y versus vector X. Various line types, plot symbols and colors may be obtained using plot(X,Y,S) where S is a character string indicating the color of the line, and the type of line (e.g., dashed, solid, dotted, etc.). Examples for the string S include:

[pic] You can insert x-labels, y-labels and title to the plots, using the functions xlabel(.), ylabel(.) and title(.) respectively. To plot two or more graphs on the same figure, use the command subplot. For instance, to show the above two plots in the same figure, type: >> subplot(2,1,1), plot(x) >> subplot(2,1,2), stem(x) The (m,n,p) argument in the subplot command indicates that the figure will be split in m rows and n columns. The ‘p’ argument takes the values 1, 2. . . m× n. In the example above, m = 2, n = 1, and, p = 1 for the top figure and p = 2 for the bottom figure. [pic] To get more help on plotting, type: help plot or help subplot.
Programming in MATLAB (M-files) MATLAB programming is done using M-files, i.e., files that have the extension .m. These files are created using a text editor. To open the text editor, go to the File pull-down menu, choose New, then M-file. After you type in the program, save it, and then call it from the command window to execute it. Say for instance that you want to write a program to compute the average (mean) of a vector x. The program should take as input the vector x and return the average of the vector. Steps: 1. You need to create a new file, called “average.m”. If you are in Windows, open the text editor by going to the File pull-down menu, choose New, then M-file. If you are in UNIX, then type in the command window: edit average.m. Type the following in the empty file ________________________________ function y = average(x) L = length(x); sum = 0; for i = 1:L sum = sum + x(i); end y = sum/L; % the average of x ________________________________ Remarks: y — is the output of the function “average” x — is the input array to the function “average” average — is the name of the function. It’s best if it has the same name as the filename. MATLAB files always need to have the extension .m 2. From the Editor pull-down menu, go to File | Save, and enter: average.m for the filename. 3. Go to the Command window to execute the program by typing: >> x = 1:100; >> y = average(x) ans = 50.5000 Note that the function average takes as input the array x and returns one number, the average of the array. In general, you can pass more than one input argument and can have the function return multiple values. You can declare the function average, for instance, to return 3 variables while taking 4 variables as input with the following statement: function [y1, y2, y3] = average(x1,x2,x3,x4) In the command window it has to be invoked as: >> [y1, y2, y3] = average(x1,x2,x3,x4)
MATLAB sound If your PC has a sound card, then you can use the function soundsc to play back speech or audio files through the PC’s speakers. The usage of this function is given below: soundsc(Y,FS) sends the signal in vector Y (with sample frequency FS) out to the speaker on platforms that support sound. Stereo sounds are played, on platforms that support it, when Y is an N-by-2 matrix. Try the following code, and listen to a 400-Hz tone: >> t=0:1/8192:1; >> x=cos(2*pi*400*t); >> soundsc(x,8192); Now, try listening to noise: >> noise=randn(8192,1); % generate 8192 samples of noise >> soundsc(noise,8192); The function randn generates Gaussian noise.
Loading and saving data You can load or save data using the commands load and save. To save the variable x of the above code in the file data.mat, type: >> save data.mat x Note that MATLAB’s data files have the extension .mat. To retrieve the data that was saved in the vector x, type: >> load data.mat The vector x is loaded in memory. To see the contents of memory use the command whos: >> whos Name Size Bytes Class X 1x8193 65544 double array Grand total is 8193 elements using 65544 bytes The command whos gives a list of all the variables currently in memory, along with their dimension. In our case, x contained 8193 samples. To clear up memory after loading a file, you may type clear all when done. That is very important, because if you do not clear all the variables in memory, you may run into problems with other programs that you will write that use the same variables.

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...Exam 2 Part A (a.) We first created the “trim” file to determine the required control (elevator and thurst setting) to achieve specific equilibrium flight by minimizing the left hand side of equations (2.5-34) which are equations for the time derivatives of the velocity, angle of attack, pitch rate, and pitch angle. The “transp” function is used in the “trim” file to determine the rate of changes for these specified controls. Once trimmed flight is achieved, the non-dimensional characteristics of this flight are used to determine the dimensional derivatives needed to set up the state space model for the trimmed flight. Once the dimensional derivatives are computed, the state space angle of attack, pitch rate, velocity, and pitch can be set up. Several characteristics of both the short period and phugoid of this state space model can be seen in Table 1. Table 1: A table of system characteristics for both the short term period and phugoid Characteristic | Short Period | Phugoid | Eigenvalues | -0.6252 +/- 1.3188i | -0.0021 +/- 0.0780i | Time Period (s) | 1.5995 | 486.9 | Frequency (s^-1) | 1.3188 | 0.078 | Settling Time (s) | 6.398 | 1947.6 | Period of Oscillation (s) | 4.7645 | 80.5 | Natural Frequency (s^-1) | 1.4594 | 0.0781 | Damping Ratio | 0.4284 | 0.0263 | Part B We were tasked with plotting the Argand diagrams associated with the individual phugoid and short-period modes. An argand diagram is a scaled representation of the eigenvectors,......

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Design Machine

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