
an approach similar to the method used for first-order systems with an input derivative term
present is employed. An artificial variable z(t) is introduced, and the output y( t) is expressed as a
linear combination of z(t) and its two derivatives. The result is
d
2
dt
2
z(t) þ a
1
d
dt
z(t) þ a
0
z(t) ¼ u(t)(2:71)
y(t) ¼ b
0
z(t) þ b
1
d
dt
z(t) þ b
2
d
2
dt
2
z(t)(2:72)
The simulation diagram of the second-order system in Equation 2.70 is shown in Figure 2.14. Note
the use of the dot notation, short for differentiation with respect to time. It is clear that a direct
connection from the input u(t) to the output y(t) exists only when b
2
, the coefficient of the input
second