design of control system

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)