Transformada de Laplace

Table of Contents

Ejemplo 4.1

clearvars
syms a t s
x(t,a) = exp(-a*t)*heaviside(t)
x(t, a) = 
X(s,a) = laplace(x(t,a))
X(s, a) = 
Falta especificar la región de converhencia que se encuentra con el procedimeinto análitico.
X_1(s,a)=int(x(t,a)*exp(-s*t),t,0,inf)
X_1(s, a) = 
assume(a,'real')
assumeAlso(real(s) + real(a) > 0)
X_1(s,a)=int(x(t,a)*exp(-s*t),t,0,inf)
X_1(s, a) = 
fplot(x(t,2))
syms rs is real
s = rs + j*is
s = 
X(rs,is,a)=X(s,a)
X(rs, is, a) = 
fmesh(abs(X(rs,is,2)),[-1.5 1.5 -2.5 -1.5])

Ejemplo 4.2

clearvars
syms t s
x_a(t) = dirac(t)
x_a(t) = 
X_a(s) = laplace(x_a(t))
X_a(s) = 
1
int(x_a(t)*exp(-s*t),t,0,inf)
ans = 
1
x_b(t) = heaviside(t)
x_b(t) = 
X_b(s) = laplace(x_b(t))
X_b(s) = 
int(x_b(t)*exp(-s*t),t,0,inf)
ans = 
assume(real(s)>0)
int(x_b(t)*exp(-s*t),t,0,inf)
ans = 
Realizar la gráfica del "espectro de magnitud"

Ejemplo 4.3

clearvars
syms t s
X_a(s) = (7*s-6)/(s^2-s-6)
X_a(s) = 
x_a(t) = ilaplace(X_a(s))
x_a(t) = 
fplot(x_a(t)*heaviside(t))
X_d(s) = (8*s+10)/((s+1)*(s+2)^3)
X_d(s) = 
x_d(t) = ilaplace(X_d(s))
x_d(t) = 
fplot(x_d(t)*heaviside(t) )

Ejemplo 4.4

clearvars
syms t s
X_b(s) = (2*s^2+7*s+4)/((s+1)*(s+2)^2)
X_b(s) = 
num = [2 7 4];
den = [conv([1 1],conv([1 2],[1,2]))];
[r,p,k] = residue(num,den)
r = 3×1
3.0000 2.0000 -1.0000
p = 3×1
-2.0000 -2.0000 -1.0000
k = []
x_b(t) = ilaplace(X_b(s))*heaviside(t)
x_b(t) = 

Ejemplo 4.5

Este ejercicio utiliza directamente la función laplace
syms a b t; x_a = sin(a*t)+cos(b*t);
X_a = laplace(x_a)
X_a = 
X_a = collect(X_a)
X_a = 
Implementa el inciso b)

Ejemplo 4.6

clearvars
syms t s
u(t) = heaviside(t)
u(t) = 
x(t)=(t-1)*[u(t-1)-u(t-2)]+[u(t-2)-u(t-4)]
x(t) = 
fplot(x(t), [-2,7])
X(s) = laplace(x(t))
X(s) = 
simplify(X(s))
ans = 

Ejemplo 4.7

syms t s
X(s) = (s+3+5*exp(-2*s))/((s+1)*(s+2))
X(s) = 
x(t) = ilaplace(X(s))
x(t) = 

Ejemplo 4.8

syms s t
syms b a real
x(t) = exp(-a*t)*cos(b*t)*heaviside(t)
x(t) = 
X(s) = laplace(x(t))
X(s) = 

Ejemplo 4.9

syms s t
u(t) = heaviside(t);
x(t) = t*(u(t)-u(t-2))+(-2*t+6)*(u(t-2)-u(t-3))
x(t) = 
fplot(x(t),[-1 5])
X(s) = collect(laplace(x(t)))
X(s) = 

Ejemplo 4.10

syms s t tau
syms a b real
x(t) = exp(a*t)*heaviside(t);
g(t) = exp(b*t)*heaviside(t);
C(s) = laplace(x(t))*laplace(g(t))
C(s) = 
c(t) = ilaplace(C(s))
c(t) = 
simplify(c(t))
ans = 
c1(t) = int(exp(a*tau)*exp(b*(t-tau)),tau, 0,t)
c1(t) = 

Ejemplo 4.11

syms s t
Y(s) = (10*(2*s+3))/(s*(s^2+2*s+5))
Y(s) = 
y_0=limit(s*Y(s),s,inf)
y_0 = 
0
y_i = limit(s*Y(s),s,0)
y_i = 
6
y(t)= ilaplace(Y(s))
y(t) = 
y(0)
ans = 
0
limit(y(t),t,inf)
ans = 
6
sympref('HeavisideAtOrigin', 1)
ans = 
1
fplot(y(t)*heaviside(t),[-1,10])

Ejemplo 4.12

clearvars
syms s t y(t) x(t) Y(s) X(s) yy
Li= diff(y(t),t,2)+ 5* diff(y(t),t) + 6*y(t)
Li = 
Ld = diff(x(t),t)+x(t)
Ld = 
ecu = Li == Ld
ecu = 
LI = laplace(Li)
LI = 
LI = subs(LI,[laplace(y(t), t, s), y(0),subs(diff(y(t), t), t, 0)],[Y(s),2,1])
LI = 
LD = laplace(Ld)
LD = 
LD = subs(LD,[laplace(x(t), t, s),x(0)],[X(s),0])
LD = 
ecu = LI==LD
ecu = 
ecu = subs(ecu,X(s),laplace(exp(-4*t)*heaviside(t)))
ecu = 
ecu=subs(ecu,Y(s),yy);
Y(s)=simplify(solve(ecu,yy))
Y(s) = 
Y(s)= partfrac(Y(s))
Y(s) = 
y(t) = ilaplace(Y(s))*heaviside(t)
y(t) = 
fplot(y(t),[-1,5])