412 C H A P T E R 6: Application to Control and Communications
we need to multiply polynomials to getY(s)before we perform partial fraction expansion to gety(t).
(a) Find out about the MATLAB functionconvand how it relates to the multiplication of polynomials.
LetP(s)= 1 +s+s^2 andQ(s)= 2 + 3 s+s^2 +s^3. Obtain analytically the productZ(s)=P(s)Q(s)and
then useconvto compute the coefficients ofZ(s).
(b) Suppose thatX(s)= 1 /s^2 , and we haveN(s)=s+1,D(s)=(s+ 1 )((s+ 4 )^2 + 9 ). Useconvto find
the numerator and the denominator polynomials ofY(s)=N 1 (s)/D 1 (s). Use MATLAB to findy(t), and
to plot it.
(c) Create a function that takes as input the values of the coefficients of the numerators and denominators
ofX(s)and of the transfer functionH(s)of the system and provides the response of the system. Show
your function, and demonstrate its use with theX(s)andH(s)given above.
6.11. Feedback error—MATLAB
Control systems attempt to follow the reference signal at the input, but in many cases they cannot follow
particular types of inputs. Let the system we are trying to control have a transfer functionG(s), and the
feedback transfer function beH(s). IfX(s)is the Laplace transform of the reference input signal, andY(s)
the Laplace transform of the output, then the close-loop transfer function is
Y(s)
X(s)
=
G(s)
1 +G(s)H(s)
The Laplace transform of the error signal isE(s)=X(s)−Y(s)H(s),G(s)=1
s(s+ 1 )(s+ 2 )
and H(s)= 1(a) Find an expression forE(s)in terms ofX(s),G(s), andH(s).
(b) Letx(t)=u(t)and the Laplace transform of the corresponding error beE 1 (s). Use the final value
property of the Laplace transform to obtain the steady-state errore 1 ss.
(c) Letx(t)=tu(t)(i.e., a ramp signal) andE 2 (s)be the Laplace transform of the corresponding error
signal. Use the final value property of the Laplace transform to obtain the steady-state errore 2 ss. Is
this error value larger than the one above? Which of the two inputsu(t)andr(t)is easier to follow?
(d) Use MATLAB to find the partial fraction expansions ofE 1 (s)andE 2 (s)and use them to finde 1 (t)and
e 2 (t)and then plot them.
6.12. Wireless transmission—MATLAB
Consider the transmission of a sinusoidx(t)=cos( 2 πf 0 t)through a channel affected by multipath and
Doppler. Let there be two paths, and assume the sinusoid is being sent from a moving transmitter so that
a Doppler frequency shift occurs. Let the received signal ber(t)=α 0 cos( 2 π(f 0 −ν)(t−L 0 /c))+α 1 cos( 2 π(f 0 −ν)(t−L 1 /c))where 0 ≤αi≤ 1 are attenuations,Liare the distances from the transmitter to the receiver that the signal
travels in theith pathi=1, 2,c= 3 × 108 m/sec, and the frequency shiftνis caused by the Doppler
effect.
(a) Letf 0 =2 KHz,ν=50 Hz,α 0 =1,α 1 =0.9, andL 0 =10,000 meters. What would beL 1 if the two
sinusoids have a phase difference ofπ/ 2?
(b) Is the received signalr(t), with the parameters given above butL 1 =10,000, periodic? If so, what
would be its period and how much does it differ from the period of the original sinusoid? Ifx(t)is
the input andr(t)the output of the transmission channel, considered a system, is it linear and time
invariant? Explain.
(c) Sample the signalsx(t)andr(t)using a sampling frequencyFs=10 KHz. Plot the sampled sentx(nTs)
and receivedr(nTs)signals forn= 0 to 2000.