Signals and Systems - Electrical Engineering

232 C H A P T E R 3: The Laplace Transform

3.22. Poles, stability, and steady-state response

The steady-state solution of stable systems is due to simple poles in thejaxis of thes-plane coming from

the input. Suppose the transfer function of the system is

H(s)=

Y(s)

X(s)

=

1

(s+ 1 )^2 + 4

(a) Find the poles and zeros ofH(s)and plot them in thes-plane. Find then the corresponding impulse

responseh(t). Determine if the impulse response of this system is absolutely integrable so that the

system is BIBO stable.

(b) Let the inputx(t)=u(t). Findy(t)and from it determine the steady-state solution.

(c) Let the inputx(t)=tu(t). Findy(t)and from it determine the steady-state response. What is the

difference between this case and the previous one?

(d) To explain the behavior in the case above consider the following: Is the inputx(t)=tu(t)bounded?

That is, is there some finite valueMsuch that|x(t)|<Mfor all times? So what would you expect the

output to be knowing that the system is stable?

3.23. Responses from an analog averager

The input–output equation for an analog averager is given by

y(t)=

1

T

∫t

t−T

x(τ)dτ

wherex(t)is the input andy(t)is the output. This equation corresponds to the convolution integral.

(a) Change the above equation so that you can determine from it the impulse responseh(t).

(b) Graphically determine the outputy(t)corresponding to a pulse inputx(t)=u(t)−u(t− 2 )using the

convolution integral (letT= 1 ) relating the input and the output. Carefully plot the input and the

output. (The output can also be obtained intuitively from a good understanding of the averager.)

(c) Using the impulse responseh(t)found above, use now the Laplace transform to find the output

corresponding tox(t)=u(t)−u(t− 2 ). Let againT= 1 in the averager.

3.24. Transients for second-order systems—MATLAB

The type of transient you get in a second-order system depends on the location of the poles of the system.

The transfer function of the second-order system is

H(s)=

Y(s)

X(s)

=

1

s^2 +b 1 s+b 0

and let the input bex(t)=u(t).

(a) Let the coefficients of the denominator ofH(s)beb 1 = 5 andb 0 = 6. Find the responsey(t). Use

MATLAB to verify the response and to plot it.

(b) Suppose then that the denominator coefficients ofH(s)are changed tob 1 = 2 andb 0 = 6. Find the

responsey(t). Use MATLAB to verify the response and to plot it.

(c) Explain your results above by relating your responses to the location of the poles ofH(s).

3.25. Effect of zeros on the sinusoidal steady state

To see the effect of the zeros on the complete response of a system, suppose you have a system with a

transfer function

H(s)=

Y(s)

X(s)

=

s^2 + 4

s((s+ 1 )^2 + 1 )