Problems 411

(b) If we letG(s)=sin the previous feedback system, determine the overall transfer functionY(s)/X(s)

whereX(s)andY(s)are the Laplace transforms of the inputx(t)and the outputy(t)of this new feed-

back system. Sketch the magnitude response of the overall system and determine the type of filter

it is.

6.7. Feedback implementation of an all-pass system

Suppose you would like to obtain a feedback implementation of an all-pass filter

T(s)=

s^2

√

2 s+ 1

s^2

√

2 s+ 1

(a) Determine if theT(s)is the transfer function corresponding to an all-pass filter by means of the poles

and zeros ofT(s).

(b) Determine the feedforward transfer functionG(s)and the feedback transfer function H(s)of a

negative-feedback system that hasT(s)as its overall transfer function.

(c) Would it be possible to implementT(s) using a positive-feedback system? If so, indicate its

feedforward transfer functionG(s)and the feedback transfer functionH(s).

6.8. Filter stabilization

The transfer function of a designed filter is

G(s)=

1

(s+ 1 )(s− 1 )

which is unstable given that one of its poles is in the right-hands-plane.

(a) Consider stabilizingG(s)by means of negative feedback with a gainK> 0 in the feedback. Determine

the range of values ofKthat would make the stabilization possible.

(b) Use the cascading of an all-pass filterHa(s)with the givenG(s)to stabilize it. GiveHa(s). Would it be

possible for the resulting filter to have the same magnitude response asG(s)?

6.9. Error and feedforward transfer function

Suppose the feedforward transfer function of a negative-feedback system isG(s)=N(s)/D(s), and the

feedback transfer function is unity.

(a) Given that the Laplace transform of the error is

E(s)=X(s)[1−H(s)]
whereH(s)=G(s)/( 1 +G(s))is the overall transfer function of the feedback system, find an expres-
sion for the error in terms ofX(s),N(s), andD(s). Use this equation to determine the conditions under
which the steady-state error is zero forx(t)=u(t).
(b) If the input isx(t)=u(t), the denominatorD(s)=(s+ 1 )(s+ 2 ), and the numeratorN(s)= 1 , find an
expression forE(s)and from it determine the initial valuee( 0 )and the final valuelimt→∞e(t)of the
error.
6.10. Product of polynomials ins—MATLAB
Given a transfer function
Y(s)
X(s)

N(s)
D(s)
whereY(s)andX(s)are the Laplace transforms of the outputy(t)and of the inputx(t)of an LTI system,
andN(s)andD(s)are polynomials ins, to find the output

Y(s)=X(s)

N(s)

D(s)