Section 7–1 / Introduction 399Although the frequency response of a control system presents a qualitative picture of the
transient response, the correlation between frequency and transient responses is indirect, ex-
cept for the case of second-order systems. In designing a closed-loop system, we adjust the
frequency-response characteristic of the open-loop transfer function by using several de-
sign criteria in order to obtain acceptable transient-response characteristics for the system.
Obtaining Steady-State Outputs to Sinusoidal Inputs. We shall show that the
steady-state output of a transfer function system can be obtained directly from the si-
nusoidal transfer function—that is, the transfer function in which sis replaced by jv,
wherevis frequency.
Consider the stable, linear, time-invariant system shown in Figure 7–1. The input and out-
put of the system, whose transfer function is G(s), are denoted by x(t)andy(t), respectively.
If the input x(t)is a sinusoidal signal, the steady-state output will also be a sinusoidal sig-
nal of the same frequency, but with possibly different magnitude and phase angle.
Let us assume that the input signal to the system is given by
[In this book “ ”is always measured in rad/sec. When the frequency is measured in
cycle/sec, we use notation “f”. That is, .]
Suppose that the transfer function G(s)of the system can be written as a ratio of two
polynomials in s; that is,
The Laplace-transformed output Y(s)of the system is then
(7–1)
whereX(s)is the Laplace transform of the input x(t).
It will be shown that, after waiting until steady-state conditions are reached, the fre-
quency response can be calculated by replacing sin the transfer function by jv. It will
also be shown that the steady-state response can be given by
whereMis the amplitude ratio of the output and input sinusoids and fis the phase
shift between the input sinusoid and the output sinusoid. In the frequency-response test,
the input frequency vis varied until the entire frequency range of interest is covered.
The steady-state response of a stable, linear, time-invariant system to a sinusoidal
input does not depend on the initial conditions. (Thus, we can assume the zero initial
condition.) If Y(s)has only distinct poles, then the partial fraction expansion of Equa-
tion (7–1) when x(t) = X yields
= (7–2)
a
s+jv
+
a–
s-jv
+
b 1
s+s 1
+
b 2
s+s 2
+p+
bn
s+sn
Y(s)=G(s)X(s)=G(s)
vX
s^2 +v^2
sinvt
G(jv)=Mejf=M/f
Y(s)=G(s)X(s)=
p(s)
q(s)
X(s)
G(s)=
p(s)
q(s)
=
p(s)
As+s 1 BAs+s 2 BpAs+snB
v= 2 pf
v
x(t)=Xsinvt
G(s)
X(s)x(t)
Y(s)Figure 7–1 y(t)
Stable, linear, time-
invariant system.