The preceding section has indicated several important relationships between open-loop and closed-loop transfer functions. This is very important in the design of control systems for the case where the closed-loop transfer function is specified and it is desired to determine the open-loop transfer function. A typical problem might specify the desired velocity constant; then use is made of Eq. (5.35) in Section 5.4 which gave the velocity constant in terms of the closed-loop poles and zeros. The problem is to determine the resulting linear-state-variable feedback system.
Let us illustrate the procedure by considering the following problem. It is desired that the closed-loop characteristics of a unity-feedback control system be given by the following parameters:
ωn = 50 rad/sec, Kv = 35/sec, ζ = 0.707
What form of closed-loop transfer function will satisfy these requirements? Let us first try a simple quadratic control system having a pair of complex-conjugate poles. From Eq. (5.37), such a system has a velocity constant given by
Therefore, a simple quadratic control system having a pair of complex-conjugate poles will satisfy these specifications. From Eq. (4.18),
For a damping ratio ...