
A simulation diagram like the one shown in Figure 2.25 is easily constructed. State variables x
1
and
x
2
are chosen as the output y and first derivative dy=dt, respectively.
The second-order system is critically damped or overdamped if a
2
1
4a
0
0. In this case, it is
equivalent to two cascaded first-order systems as shown in Figure 2.26.
The parameters K, a, and b are related to a
0
, a
1
, and b
0
according to
K ¼ b
0
, a ¼
a
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a
2
1
4a
0
p
2
, b ¼
a
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a
2
1
4a
0
p
2
(2:105)
State variable x
1
is again the system output y; however, the second state variable x
2
is no longer the
output derivative dy=dt.
For an nth-order linear system model with ...