mattalltech.com

%{
Author:     Matthew B. Pare
Date:       February 21, 2009
Course:     EE 3303 - Linear System Analysis
Assignment: Worksheet 3?, Due February 24, 2009
 
Copyright 2009 Matthew B. Pare
 
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation.
 
This program is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
for more details.
 
You should have received a copy of the GNU General Public License
along with this program as the file LICENSE.txt; if not, please see
http://www.gnu.org/licenses/old-licenses/gpl-2.0.txt.
%}
 
%---------------- House Keeping ----------------%
clear, clc;                       % Clear Variables and Screen
syms a s t w x;                        % Setup 
 
display('%-------------- Problem 5 --------------%');
% Find the Laplace transform of each of the following functions:
display('%------ Part b ------%');
display('% LaTeX Problem: f(t) = 3t^4e^{-2t}u(t)');
 
% Is the heaviside (step) function uncessary here? Is it implied?
f5b = 3*t^4 * exp(-2*t) * heaviside(t);
 
L5b = laplace(f5b);
 
display(L5b);
 
%LaTeX Solution: 72\, \left( s+2 \right) ^{-5}
display(latex(L5b));        % KICK ASS!!!
 
display('%------ Part c ------%');
display('% LaTex Problem: 2tu(t) - 4 ''\frac{d}{dt}\delta(t)');
f5c = 2*t-4*diff(dirac(t));
 
L5c = laplace(f5c);
 
display(L5c);
 
%LaTeX Solution: 
display(latex(L5c));        % KICK ASS!!!
 
display('%------ Part d ------%');
display('% LaTex Problem: 2e^{-(t-1)}u(t)');
f5d = 2*exp(-(t-1));
 
L5d = laplace(f5d);
 
display('LaTex Solution: ');
 
%LaTeX Solution: 
display(latex(L5d));        % KICK ASS!!!
 
display('%-------------- Problem 7 --------------%');
display('%------ Part a ------%');
f7a = (2*t+4);
 
L7a = laplace(f7a);
 
display('LaTex Solution: '), display(latex(L7a));
 
 
display('%------ Part b ------%');
L7b = laplace(4+3*exp(-2*t));
 
display('LaTex Solution: '), display(latex(L7b));
 
display('%------ Part c ------%')
L7c = laplace(6*sin(3*t) + 8 * cos(3*t));
display('LaTex Solution: '), display(latex(L7c));
 
display('%------ Part d ------%');
L7d = laplace(exp(-2*t)*cosh(4*t));
display('LaTeX Solution: '), display(latex(L7d));
 
 
display('%-------------- Problem 8 --------------%');
display('%------ Part a ------%');
L8a = laplace(2*t * heaviside(t-4));
 
display('LaTex Solution: '), display(latex(L8a));
 
display('%------ Part b ------%')
L8b = laplace(5*cos(t)*dirac(t-2));
 
display('LaTex Solution: '), display(latex(L8b));
 
display('%------ Part c ------%')
L8c = laplace(exp(-t)*heaviside(t-a));
 
display('LaTex Solution: '), display(latex(L8c));
 
display('%------ Part d ------%');
L8d = laplace(sin(2*t)*heaviside(t-a));
 
display('LaTex Solution: '), display(latex(L8d));
 
%-------------- Problem 9 --------------%
display('%------ Part a ------%');
L9a = laplace((t-4)*heaviside(t-2));
display('LaTeX Problem: '), display(latex(L9a)), display('LaTex Solution: '), display(latex(L9a));
display('%------ Part b ------%');
L9b = laplace(2*exp(-4*t)*heaviside(t-1));
display('LaTex Solution: '), display(latex(L9b));
 
display('%------ Part c ------%');
L9c = laplace(5*cos(2*t-1));
display('LaTex Solution: '), display(latex(L9c));
 
 
 
display('%------ Part d ------%');
L9d = laplace(6*(heaviside(t-2) - heaviside(t-4)));
display('LaTeX Problem: '), display(latex(L9d)), display('LaTex Solution: '), display(latex(L9d));
 
 
display('%------------- Problem  18 -------------%')
display('%------ Part a ------%');
L18a = laplace(heaviside(t) + heaviside(t-1) + heaviside(t-2) - 3*heaviside(t-3));
display('LaTex Solution: '), display(latex(L18a));
 
display('%------ Part b ------%');
L18b = laplace(2*t*(heaviside(t) - heaviside(t-1)) + heaviside(t-1) - 2*t*(heaviside(t-3) - heaviside(t-4)));
 
display('LaTex Solution: '), display(latex(L18b));
 
%---------------- Notes ----------------%
% Readabout matlabFunction(f), poly(A), poly2sym(C), setVar, simplify,
% solve, subexpr, subs, svd, 
% http://www.mathworks.com/access/helpdesk/help/toolbox/control/getstart/f1-1016555.html

%{
Author:     Matthew B. Pare
Date:       February 26, 2009
Course:     EE 3303 - Linear System Analysis
Assignment: EE 3303 MatLab Overview for Laplace
 
Copyright 2009 Matthew B. Pare
 
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation.
 
This program is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
for more details.
 
You should have received a copy of the GNU General Public License
along with this program as the file LICENSE.txt; if not, please see
http://www.gnu.org/licenses/old-licenses/gpl-2.0.txt.
%}
 
%---------------- House Keeping ----------------%
%clear, clc;                       % Clear Variables and Screen
 
%---------------- Example ----------------%
%clear, clc;
 
 
% Decending order 
num1 = [1 1];
num2 = [1 3];
 
num = 5 * conv(num1, num2);
 
% Decending order for power of s!!!
den1 = [1 0];
den2 = [1 2];
den3 = [1 8 25];
 
% Convolution only works for steps of 2 so extra step
den12 = conv(den1, den2);
 
den = conv(den12, den3);
 
% Transform, this is not essential but you get a nice output...
H = tf(num, den);
 
display(H);
 
% Numerator, Denominator
% R will represent the zeros in the numerator
% P will represent the poles in the denominator
% K will represent the remainder (Numerator greater order than denominator)
[r, p, k] = residue(num, den);
 
% The ouput will share indexes for each term, so you can reconstruct these
% by matching the numerator and denominator for each term...
display(r), display(p), display(k);
 
 
% 2 * abs(k) * exp^(-a*t)*cos(w*t*o)); "Formula in the book"
% K is the magnitude of the numerator, w is the mag of denom, a is 
 
% Go backwards
[num, den] = residue(r,p,k);
 
display(num), display(den);
 
% We can take advantage of this by
ralt = r(1:2);  % Alternate numerator
palt = r(1:2);  % Alternate denumenator, just first two terms
 
 
% Output K 
[numalt, denalt] = residue(ralt, palt, k);
 
% This outputs as H(s) = \frac{-0.49s + 1.45}{s^2 + 8s + 25}
display(numalt), display(denalt);
 
syms a s t w x;                        % Setup 
 
h = (5*s^2 + 20*s + 15)/(s^4 + 10*s^3 + 41*s^2+50*s);
 
hoft = ilaplace(h);
 
display(hoft);
 
 
%---------- Num greater than denominator example ----------%
%clear, clc;                 % Remove this stuff to see what we just did...
 
% S^3 / (S+1)(s+2)
syms s;
 
num = [1 0 0 0];
den1 = [1 1];
den2 = [1 2];
 
den = conv(den1, den2);
 
display(tf(num, den));
 
% SO OUR OUTPUT IS H(s) = \frac{8}{s+2} - \frac{1}{s+1} + s - 3
% And now we can take the inverse transform, where 
% s is the first der of del, e^-t, e^(-2t)
%[r, p, k] = residue(num, den);
 
% If you want to take an actual transform you have to use the symbolic
% toolbox. Symbolic tool box is matlabs very odd way of establishing a
% variable... I defined these up at the top, maybe I should move them?
H2 = s^3/(s^2 + 3*s + 2);
 
display(ilaplace(H2));
display(latex(ilaplace(H2)));