The unity feedback system of Figure P7.1 has a transfer function

and is to follow a ramp input, r(t) = tu(t), so that the steady-state output position differs from the input position by 0.01 of the input velocity (e.g., e(∞) = 1/K_v = 0.01). The natural frequency of the closed-loop system will be ω_n = 5 rad/s.

Find the following:

a. The system type

b. The values of K and α

c. The closed-loop system’s damping ratio, ζ

d. If K is reduced to 4 and α=0.4, find the corresponding new values of e(∞), ω_n, and ζ.

C(s) G(s) = K E(s) s(s+ a)