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Second order system step response
x s (t)

dx s (t)
dt

t
FIGURE 2.23: Step response of a second-order system and the derivative of its response.

The step response of a second-order system with a zero on the left half plane has the
form

1
x step (t) + ̇ x step (t)

and response with a zero on the right half s-plane (make the (s∕a + 1) term (s∕a − 1) in
the above equation and the result that follows (Figure 2.24) has the form
1
x (t) − ̇ x (t)
step step

The Effect of an Additional Pole The system becomes third order when an
additional pole exists in addition to the two complex conjugate poles.

( )
⎛ ⎞
1 1
⎜ ⎟
s ⎜ s 2 2
b + 1 ⎝ 2 n + n s + 1 ⎟
⎠
If the b = is 3 to 5 times to the left of , the effect of an additional pole on the
n n
step response is negligable. As b gets close to , the effect is to increase the rise time
n
and hence slow the response of the system (Figure 2.24).

(b) x (t)
s

(c)

t
FIGURE 2.24: Step response of a second-order system and the effect of a zero: (a) without
zero, (b) zero on the left hand plane, (c) zero on the right hand plane.