CP12.5 A model of a flexible structure is given by where n is the natural frequency of the flexible mode, and is the

CP12.5 A model of a flexible structure is given by

where ωn is the natural frequency of the flexible mode, and ζ is the corresponding damping ratio. In general, it is difficult to know the structural damp

ing precisely, while the natural frequency can be predicted more accurately using well-established modeling techniques. Assume the nominal values of

ω ζ n = = 2 rad s, 0.005, and k = 0.1.

a. Design a lead compensator to meet the following specifications: (1) a closed-loop system response to a unit step input with a settling time (with a 2%

criterion) Ts ≤ 200 s and (2) a percent overshoot of P O. . ≤ 50%.

b. With the controller from part (a), investigate the closed-loop system unit step response with

ζ = 0, 0.005, 0.1, and 1. Co-plot the various unit step responses and discuss the results.

c. From a control system point of view, is it preferable to have the actual flexible structure damping less than or greater than the design value? Explain.

G(s)= (1+ kw)s+2wns + w $ (s + 25wns + w)