### 7.5. EXAMPLE FOR THE DESIGN OF A SECOND-ORDER CONTROL SYSTEM

In this section we consider the design and resulting performance of the second-order system by means of cascade and minor-loop rate-feedback techniques. This problem is useful in unifying concepts which were introduced in Chapters 4, 5, and 6, together with the design techniques illustrated in this chapter.

Let us consider the second-order system illustrated in Figure 7.18*a*. We will assume that the original forward-loop transfer function *G*_{0}(*s*) is given by

The closed-loop transfer function, *C*(*s*)/*R*(*s*), is given by

or

**Figure 7.18** Design of a second-order system.

Comparing Eqs. (4.3) and (7.44), we observe that the undamped natural frequency *ω*_{n} and damping ratio *ζ* of the system are given by

and

*ζ* = 0.417.

If the system is subjected to a unit step input, the transient response will have the form shown in Figure 4.4 (interpolate between *ζ* = 0.4 and *ζ* = 0.6). The maximum percent overshoot can be obtained from Eq. (4.33) and is ...