Signals and Systems - Electrical Engineering

Problems 231

(b) Ify(^1 )( 0 )= 1 andy( 0 )= 1 are the initial conditions for the above differential equation, findY(s). If

the input to the system is doubled—that is, the input is 2 x(t)with Laplace transform 2 X(s)—isY(s)

doubled so that its inverse Laplace transformy(t)is doubled? Is the system linear?

(c) Use MATLAB to find the poles and zeros and the solutions of the differential equation when the

input isu(t)and 2 u(t)with the initial conditions given above. Compare the solutions and verify your

response in (b).

3.19. Differential equation, initial conditions, and impulse response—MATLAB
The following functionY(s)=L[y(t)]is obtained applying the Laplace transform to a differential equation
representing a system with nonzero initial conditions and inputx(t), with Laplace transformX(s):

Y(s)=

X(s)

s^2 + 2 s+ 3

+

s+ 1

s^2 + 2 s+ 3

(a) Find the differential equation iny(t)andx(t)representing the system.

(b) Find the initial conditionsy′( 0 )andy( 0 ).

(c) Use MATLAB to determine the impulse responseh(t)of this system. Find the poles of the transfer

functionH(s)and determine if the system is BIBO stable.

3.20. Different responses—MATLAB
LetY(s)=L[y(t)]be the Laplace transform of the solution of a second-order differential equation
representing a system with inputx(t)and some initial conditions,

Y(s)=

X(s)

s^2 + 2 s+ 1

+

s+ 1

s^2 + 2 s+ 1

(a) Find the zero-state response (response due to the input only with zero initial conditions) forx(t)=u(t).

(b) Find the zero-input response (response due to the initial conditions and zero input).

(c) Find the complete response whenx(t)=u(t).

(d) Find the transient and the steady-state response whenx(t)=u(t).

(e) Use MATLAB to verify the above responses.

3.21. Poles and stability
The transfer function of a BIBO-stable system has poles only on the open left-hands-plane (excluding the
jaxis).
(a) Let the transfer function of a system be

H 1 (s)=

Y(s)

X(s)

=

1

(s+ 1 )(s− 2 )

and letX(s)be the Laplace transform of signals that are bounded (i.e., the poles ofX(s)are on the

left-hands-plane). Findlimt→∞y(t). Determine if the system is BIBO stable. If not, determine what

makes the system unstable.

(b) Let the transfer function be

H 2 (s)=

Y(s)

X(s)

=

1

(s+ 1 )(s+ 2 )

andX(s)be as indicated above. Find

lim

t→∞

y(t)

Can you use this limit to determine if the system is BIBO stable? If not, what would you do to check

its stability?