
A simulation diagram for an LTI system offers a conveni ent way of defining the states and
revealing whether a direct path (no integrators) exists from the input to the output. Figure 4.11
shows a simulation diagram for the third-order system
d
3
dt
3
y(t) þ a
2
d
2
dt
2
y(t) þ a
1
d
dt
y(t) þa
0
y(t) ¼ b
3
d
3
dt
3
u(t) þb
2
d
2
dt
2
u(t) þ b
1
d
dt
u(t) þ b
0
u(t)(4:173)
in what is known as observer canonical form (Ogata 1998). This form clearly shows the direct path
from the input u to the output y when b
3
6¼ 0. For the case when b
3
¼0, the state x
3
is equal to the
output y and a direct path exists from u to
_
x
3
. For a unit step input, the following is true if b
3
¼0:
y(0
þ
) ¼ y(0
),
_