Signals and Systems - Electrical Engineering

Problems 235

(a) Carefully plot the half-wave rectified signaly(t)fromx(t).

(b) Lety 1 (t)be the period ofy(t)between 0 ≤t≤ 1. Show thaty 1 (t)can be written as

y 1 (t)=sin( 2 πt)u(t)+sin( 2 π(t−0.5))u(t−0.5)

or

y 1 (t)=sin( 2 πt)[u(t)−u(t−0.5)]

Use MATLAB to verify this. Find the Laplace transformX 1 (s)ofx 1 (t).

(c) Expressy(t)in terms ofy 1 (t)and find the Laplace transformY(s)ofy(t).

3.33. Polynomial multiplication—MATLAB
When the numerator or denominator is given in a factorized form, we need to multiply polynomials.
Although this can be done by hand, MATLAB provides the functionconvthat computes the coefficients
of the polynomial resulting from the product of two polynomials.
(a) Usehelpin MATLAB to find howconvcan be used, and then consider two polynomials

P(s)=s^2 +s+ 1 andQ(s)= 2 s^3 + 3 s^2 +s+ 1

Do the multiplication of these polynomials by hand to findZ(s)=P(s)Q(s)and useconvto verify your

results.

(b) The output of a system has a Laplace transform

Y(s)=

N(s)

D(s)

=

(s+ 2 )

s^2 (s+ 1 )((s+ 4 )^2 )+ 9 )

Useconvto find the denominator polynomial and then find the inverse Laplace transform using

ilaplace.

3.34. Feedback error—MATLAB
Consider a negative feedback system used to control a plantG(s)= 1 /(s(s+ 1 )(s+ 2 )). The outputy(t)of
the feedback system is connected via a sensor with transfer functionH(s)= 1 to a differentiator where
the reference signalx(t)is also connected. The output of the differentiator is the feedback errore(t)=
x(t)−v(t)wherev(t)is the output of the feedback sensor.
(a) Carefully draw the feedback system, and find an expression forE(s), the Laplace transform of the
feedback errore(t).
(b) Two possible reference test signals for the given plant arex(t)=u(t)andx(t)=r(t). Choose the one
that would give a zero steady-state feedback error.
(c) Use MATLAB to do the partial fraction expansions for the two error functionsE 1 (s), corresponding
to whenx(t)=u(t)andE 2 (s)whenx(t)=r(t). Use these partial fraction expansions to finde 1 (t)and
e 2 (t), and thus verify your results obtained before.