The first-order homogeneous solution is of the form of an exponential function y
h
(t) = e
−λt
where λ = 1/τ. The total response y(t) is the sum of two components
y(t) = y
h
(t) + y
p
(t)
= Ce
−t/τ
+ y
p
(t) (14)
where C is a constant to be found from the initial condition y(0) = 0, and y
p
(t) is a particular
solution for the given forcing function f (t). In the following sections we examine the form of y(t)
for the ramp, step, and impulse singularity forcing functions.
1.2.1 The Characteristic Unit Step Response
The unit step u
s
(t) is commonly used to characterize a system’s response to sudden changes in its
input. It is discontinuous at time t = 0:
f(t) = u
s
(t) =
(
0 t < 0,
1 t ≥ 0.
The characteristic step response y
s
(t) is found by determining a particular solution for the step
input using the method of undetermined coefficients. From Table 8.2, with a constant input for
t > 0, the form of the particular solution is y
p
(t) = K, and substitution into Eq. (13) gives K = 1.
The complete solution y
s
(t) is
y
s
(t) = Ce
−t/τ
+ 1. (15)
The characteristic response is defined when the system is initially at rest, requiring that at t = 0,
y
s
(0) = 0. Substitution into Eq. (14) gives 0 = C + 1, so that the resulting constant C = −1. The
unit step response of a system defined by Eq. (13) is:
y
s
(t) = 1 − e
−t/τ
. (16)
Equation (16) shows that, like the homogeneous response, the time dependence of the step response
depends only on τ and may expressed in terms of a normalized time scale t/τ. The unit step char-
acteristic response is shown in Fig. 6, and the values at normalized time increments are summarized
in the fourth column of Table 1. The response asymptotically approaches a steady-state value
y
ss
= lim
t→∞
y
s
(t) = 1. (17)
It is common to divide the step response into two regions,
(a) a transient region in which the system is still responding dynamically, and
(b) a steady-state region, in which the system is assumed to have reached its final value y
ss
.
There is no clear division between these regions but the time t = 4τ, when the response is within 2%
of its final value, is often chosen as the boundary between the transient and steady-state responses.
The initial slope of the response may be found by differentiating Eq. (16) to yield:
dy
dt
¯
¯
¯
¯
t=0
=
1
τ
. (18)
The step response of a first-order system may be easily sketched with knowledge of (1) the system
time constant τ, (2) the steady-state value y
ss
, (3) the initial slope ˙y(0), and (4) the fraction of the
final response achieved at times equal to multiples of τ.
7