Transformada de Laplace

Table of Contents

Ejemplo 4.1 Ejemplo 4.2 Ejemplo 4.3 Ejemplo 4.4 Ejemplo 4.5 Ejemplo 4.6 Ejemplo 4.7 Ejemplo 4.8 Ejemplo 4.9 Ejemplo 4.10 Ejemplo 4.11 Ejemplo 4.12

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) =

int(x_a(t)*exp(-s*t),t,0,inf)

ans =

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) )

Terminar los otros ejemplos (si es posible)
Faltaría incluir pues pensamos que son señales causales

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 =

y_i = limit(s*Y(s),s,0)

y_i =

y(t)= ilaplace(Y(s))

y(t) =

y(0)

ans =

limit(y(t),t,inf)

ans =

sympref('HeavisideAtOrigin', 1)

ans =

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])