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CLOSED LOOP CONTROL 71
x(t)

k
m f(t)
c

f(t) x(t)
Input Mass-spring-damper Output

FIGURE 2.21: Mass-spring system dynamics and its position control.

Further, we will consider the step response when there is an additional pole and zero
( )
s
+ 1 2
a n
( )
s + 1 s + s + 2
2
b n n
The utility of this is that given transient response specifications for a control system
(PO%, t ) we can determine where the dominant poles should be in order to meet the
s
specifications. Even though our system may not be second order, second-order system
pole-zero locations can provide a good starting point in design, especially if the higher
order system can be made to have a dominant second-order dynamics.
Consider the step response of a second-order system (Figure 2.21);
m̈ x + c ̇ x + kx = f

let f(t) = kr(t)
c k k
̈ x + ̇ x + x = r
m m m
c k 2
Let = 2 , = and take the Laplace transform of the differential equation with
m n m n
zero initial conditions,
x(s) 2 n
=
2
r(s) s + 2 s + 2
n n
1
If r(t) is a step input, r(s) = , the response x(s) is given by
s
2 n 1
x(s) = ⋅
2
s + 2 s + 2 s
n n
Using the partial fraction expansions and taking the inverse Laplace transform, the response
can be found as
( )
√ √
2
2
x(t) = 1 − e − n t cos 1 − t + √ sin 1 − t for 0 ≤ < 1 range
n
n
1 − 2