lab assignment

University of the District of Columbia

Control Systems Lab

Experiment #3

Laplace transform

L = laplace(F) is the Laplace transform of the sym F with default independent variable t. The default return is a function of s. If F = F(s), then laplace returns a function of z: L = L(z).

By definition, L(s) = int(F(t)*exp(-s*t),t,0,inf).

L = laplace(F,z) makes L a function of z instead of the default s:

laplace(F,z) <=> L(z) = int(F(t)*exp(-z*t),t,0,inf).

L = laplace(F,w,u) makes L a function of u instead of the default s (integration with respect to w).

laplace(F,w,u) <=> L(u) = int(F(w)*exp(-u*w),w,0,inf).

Examples:

syms a t y

f = exp(-a*t);

h1=laplace(f)

pretty(h)

1

-----

a + s

laplace(f, y)

syms t s

h2=laplace(dirac(t - 3), t, s)

syms a s t w x F(t)

laplace(t^5) returns 120/s^6

laplace(exp(a*s)) returns -1/(a-z)

laplace(sin(w*x),t) returns w/(t^2+w^2)

laplace(cos(x*w),w,t) returns t/(t^2+x^2)

laplace(x^(3/2),t) returns (3*pi^(1/2))/(4*t^(5/2))

laplace(diff(F(t))) returns s*laplace(F(t),t,s) - F(0)

Inverse Laplace transform in MATLAB

F = ilaplace(L) is the inverse Laplace transform of the sym L with default independent variable s. The default return is a function of t. If L = L(t), then ilaplace returns a function of x:

F = F(x).

By definition, F(t) = int(L(s)*exp(s*t),s,c-i*inf,c+i*inf)

where c is a real number selected so that all singularities

of L(s) are to the left of the line s = c, i = sqrt(-1), and

the integration is taken with respect to s.

F = ilaplace(L,y) makes F a function of y instead of the default t:

ilaplace(L,y) <=> F(y) = int(L(y)*exp(s*y),s,c-i*inf,c+i*inf).

F = ilaplace(L,y,x) makes F a function of x instead of the default t:

ilaplace(L,y,x) <=> F(y) = int(L(y)*exp(x*y),y,c-i*inf,c+i*inf),

integration is taken with respect to y.

Examples:

syms s

f1 = 1/s^2;

h3=ilaplace(f1)

t

syms s

f2=1/(s*(s+1)^2)

h4=ilaplace(f2)

1 - t*exp(-t) - exp(-t)

syms s

f3=(3*s+2)/(s^2+2*s+10)

h5=ilaplace(f3)

pretty(h5)

/ sin(3 t) \

exp(-t) | cos(3 t) - -------- | 3

\ 9 /

syms s t w x y f(x)

ilaplace(1/(s-1)) returns exp(t)

ilaplace(1/(t^2+1)) returns sin(x)

ilaplace(t^(-5/2),x) returns (4*x^(3/2))/(3*pi^(1/2))

ilaplace(y/(y^2 + w^2),y,x) returns cos(w*x)

ilaplace(laplace(f(x),x,s),s,x) returns f(x)

ASSIGNMENT

Find the Laplace transform corresponding to problems of 3.3, 3.4 as well as the inverse Laplace corresponding to problems 3.7 and 3.8 of the text book. charles L. phillips fourth edition