A model of a flexible structure is given by where n is the natural frequency of the flexible mode, and is the corresponding damping

A model of a flexible structure is given by

where ωn is the natural frequency of the flexible mode, and £ is the corresponding damping ratio. In general, it is difficult to know the structural damping precisely, while the natural frequency can be predicted more accurately using well-established modeling techniques. Assume the nominal values of ωn - 2 rad/s, f - 0.005, and* = 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) less than 200 seconds and (2) an overshoot of less than 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. CP12.6 The industrial process.

2 (1 + kw,?)s + 2%ws + wn G(s) + 25w,s + w,?) s?(s?