In the previous section, the derivation of the time response of the second-order system was based on the transfer-function derivation. Therefore, an implicit assumption was made that the initial conditions were zero. This section illustrates how the state-variable signal-flow graph method of the underdamped case yields a more versatile and complete solution in the time domain including the response to the initial condition terms.
The state-variable signal-flow graph for this system can be derived from Eq. (4.3) as follows:
Dividing through by s2, we obtain
we may rewrite Eq. (4.34) as
From Eq. (4.36) ...