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Solve an IVP using the unilateral Laplace transform. The IVP problem from the recitation

on Friday, June 2, 2023, will be solved using a Laplace transform approach. Here is the problem

restatement:

y + 4y = 4u

y(0) = 1.3

u(t)

=

[1

0

uk (t)

Note that u is periodic for t ≥ 0, in other words, u(t +T)

T

=

= 1 second. Answer the following:

1. First, compute the unilateral Laplace transform of a single rectangular pulse, denoted uo,

1 t € [0, 0.5]

t> 0.5

so it can be argued that

t = [0, 0.5]

t≤ (0.5, 1)

=

(one period), t≥0

uo(t)

Argue that ROCuo is the entire complex plane.

2. Now compute the unilateral Laplace transform of a delayed rectangular pulse (delayed by k

seconds, where k is a positive integer),

=

0 t = [0, k)

te [k, k +0.5], k = 1,2,3,...

0 t>k+0.5

In fact, show ûk = ûoe-ks. In other words, the Laplace transform of a time-shifted function

has a simple relation with the transform of the unshifted function.

3. The input to the system u can be expressed as a superposition of these shifted pulses,

u(t) for all t ≥ 0. The period is

∞

u(t) = Σ uk(t), t> 0

k=0

1

∞

û(s) = Σûk(s)

(1)

k=0

What is the ROC associated with u, though? Hint: it is no longer the entire complex plane. 4. If s € ROCu, show that the geometric series formula can be applied to (1). What is the sum

in this case?

5. Apply the unilateral Laplace transform to the ODE to write a complete expression for the

unilateral Laplace transform of the IVP solution that is valid for t € [0, k]. In other words,

compute ŷ.

6. The inverse Laplace transform is

Since s = o + jw, where o is in the ROC of ŷ, then ds

=

y(t) = lim

1

y(t) = lim

R→∞ j2π o-jR

=

= eat wo

2π

=

Wo

2π

= eat wo

kwo, k

This integral can be approximated by a Riemann sum by discretizing w as w =

. . .‚ —2, —1, 0, 1, 2,… The frequency “step" is wo, i.e. dw wo. The Riemann sum shares

many similarities with the Fourier series synthesis formula,

y(t) ~ Σ ŷ(o + jkwo)e(o+jkwo)t

k=-∞

y(t) ~

R→∞ 2π

po jR

[ +1² (8)est ds.

R

#Lio

R

2π

∞

Σ ŷ (o + jkwo) ej kwot

k=-∞

Wo

2 (163-

2π

jdw and integral can be written as

+ jw)e(o+jw)t

20000

ŷ(o) + 2 Σ Re ŷ(o

k=1

(20

ŷ(0) +2ΣRe [ŷ(0 + jkwo)ej kwot]

k=1

=

Use Matlab to numerically approximate y on the interval t = [0,3] using wo

(confirm this is in the ROC of y), and the following limits on k,

2

t dw.

=

=

1

1)

+jkwo) ej kwot

Use a time grid so that the time step is ts 200000

This will ensure at least four time

steps fall within one period of the highest frequency sinusoid in the sum.

Graph the numerical approximations of u and y for t = [0,3] seconds. Note that Gibbs

phenomenon is present in u.

=

0.1 rad/s, o = 1