STEP RESPONSE OF FIRST-ORDER SYSTEM
A first-order system with input x(t) and output y(t) can be described by the ODE
y& + ay = bx + cx&
or by the transfer function
H (s) =
cs + b
s+a
which has a pole at spole = -a, a DC gain H(0) = b/a, and a high-frequency gain H(∞) = c. For a
step input of magnitude m, i.e., x(t) = m·u(t), one can easily determine the following information
about the step response y(t):
y (0 + ) = H (∞) ⋅ m = cm
y ss = H (0) ⋅ m =
τ=
1
s pole
=
mb
a
1
a
Thus, the step response of the first-order system in general takes the following form for t ≥ 0:
−t
y (t ) = ( y ss − y (0 + ) ) ⋅ ⎛⎜1 − e τ ⎞⎟ + y (0 + )
⎝
⎠
However, it is common to have first-order systems with c = 0, such that y(0+) = 0. In this special
(but common) case, the form of the step response (for t ≥ 0) simplifies to:
−t
y (t ) = y ss ⋅ ⎛⎜1 − e τ ⎞⎟
⎝
⎠
In either case, the shape of the response can be determined by noting that the distance from
equilibrium [i.e., |yss – y(t)|] at integer multiples of the time-constant τ is reduced to 37% (i.e.,
approximately one-third) of the corresponding distance at the immediately preceding multiple of
τ. The general-case first-order step response can be plotted as follows:
2%
5%
yss
14%
37%
100%
95%
86%
98%
63%
y(0+)
0
1
2
3
4
t /τ
Note that the above graph was drawn assuming that yss > y(0+); if the converse is true, then the
shape of the curve will be inverted.
ME 3103, Villanova University
© 2014 J.O.M. Karlsson