Modern Control System Theory and Design, 2nd Edition

4.4.  STATE-VARIABLE SIGNAL-FLOW GRAPH OF A SECOND-ORDER SYSTEM

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.

Figure 4.5   Percent maximum overshoot versus damping ratio for a second-order system.

Figure 4.6   Responses of a second-order control system for various root locations in the s-plane (conjugate roots are not shown).

The state-variable signal-flow graph for this system can be derived from Eq. (4.3) as follows:

Dividing through by s², we obtain

Defining

we may rewrite Eq. (4.34) as

From Eq. (4.36) ...