Question

Consider the signal x(t) that has a Laplace transform of the form: s2-3s +1 X (s) where a, a , β, and γ are real constants E Fs +G Write a MATLAB FUNCTION called "values" that will take the values of αι, α2,β, and y and compute the constants A, B, C, D, E, F, and G. The Matlab function will also take these computed values and plot the function x(t) (meaning you will have to plot the obtained values in the inverse Laplace transform and plot the resulting function. You do not have to show the inverse Laplace transform but only compute the coefficients and time constants of different exponents and sinusoids and use these to plot the resulting x(t). Remember that you studied several methods in class to break a complicated rational function of S into simpler rational functions of S. You can use any of these methods or a combination of them here. Also note that Matlab is good in solving a set of equations so you are not limited to only one method.

Answer 1

Function file : (save this file with "values.m" name)

function [A,B,C,D,E,F,G] = values(a1,a2,b,y)
syms s;
A = (s^2-3*s+1)/((s+a2)*(s+b)^3*(s^2+y^2));
A = subs(A,s,a1);A = double(A);
B = (s^2-3*s+1)/((s+a1)*(s+b)^3*(s^2+y^2));
B = subs(B,s,a2);B = double(B);
E = (s^2-3*s+1)/((s+a1)*(s+a2)*(s^2+y^2));
E = subs(E,s,b);E = double(E);
D = ((2*s-3)-E*((s+a2)*(s^2+y^2)+(s+a1)*(s^2+y^2)+2*s*(s+a1)*(s+a2)))/((s+a1)*(s+a2)*(s^2+y^2));
D = subs(D,s,b);D = double(D);
C = (2-D*((s+a2)*(s^2+y^2)+(s+a1)*(s^2+y^2)+2*s*(s+a1)*(s+a2))-E*(2*(s^2+y^2)+4*s*(s+a1)+4*s*(s+a2)+2*(s+a1)*(s+a2)))/(2*(s+a1)*(s+a2)*(s^2+y^2));
C = subs(C,s,b);C = double(C);
G = (1-A*a2*b^3*y^2-B*a1*b^3*y^2-C*a1*a2*b^2*y^2-D*a1*a2*b*y^2-E*a1*a2*y^2)/(a1*a2*b^3);
F = (-3-A*(b^3*y^2+3*a2*b^2*y^2)-B*(b^3*y^2+3*a1*b^2*y^2)-C*(a2*b^3*y^2+a1*b^3*y^2+2*a1*a2*b*y^2)-D*(a2*b^3*y^2+a1*b^3*y^2+a1*a2*y^2)-E*((a1+a2)*y^2)-G*((a1+a2)*b^3+3*a1*a2*b^2))/(a1*a2*b^3);
end

Program file : (save this file with any name you want) (Enter the value of a1,a2,b and y)

clear all
clc

a1 = 1; % enter any value of this constants
a2 = 10;
b = 3;
y = 4;

[A,B,C,D,E,F,G] = values(a1,a2,b,y);
clear s;
syms s t;
x_s = A/(s+a1)+B/(s+a2)+C/(s+b)+D/(s+b)^2+E/(s+b)^3+(F*s+G)/(s^2+y^2);
x_t = ilaplace(x_s,s,t)
time = 0:0.1:10;time = time';
x_t = subs(x_t,t,time);
x_t = double(x_t);
plot(time,x_t)