Homework 9, due April 18, 2018

[goodwine]

Reading: last section of Ogata.

The problem is from Chapter 19 from here, page 11. You may or may not want to refer to it (it is optional and allowable to do so).

Consider a long beam pinned at one end with a mass on the other end. You can apply a torque on the pinned end, and the goal is to control the position of the mass with the torque. The transfer function from the torque to mass position is given by

X(s)/Tau(s) = 6.28/s^2

and your controller is lead controller give by

C(s) = Tau(s)/E(s) = 500(s+10)/(s+100).

Draw the Nyquist contour for this system to show that it is stable.

Clearly, our model for the transfer function from the torque to the position of the mass does not include any of the flexible modes of the beam. If we consider only the first mode, we can think of the beam as a spring, so it will have a second-order effect on the relationship between the torque and the mass. Specifically, assume that the real system is better described by

X(s)/Tau(s) = 6.28/s^2 + 12.56/(s^2 + 0.707 s + 28).

Of course, there are second, third and higher modes that even this model neglects. It is a question of engineering judgement whether to try to include higher modes.

Instead of adding in the second term (12.56/(s^2 + 0.707 s + 28)), let's just keep the original transfer function and treat the flexible modes as unmodeled dynamics.

Consider the set of plants, G(s) = G0(s) ( 1 + Delta) = 6.28/s^2 (1 + Delta) where abs(Delta) < 2 abs(w^2/(28 - w^2 + 0.707 i w)), i.e., let

W2(s) = 2 s^2/(s^2 + 0.707 s + 28).

Show that the infinity norm of T W2 is greater than one. Check whether the system with the specific second term added in is stable. From the robust stability condition, is it guaranteed to be stable or unstable?

Replace the controller with

C(s) = Tau(s)/E(s) = (5 x 10^(-4)) (s + 0.01)/(s + 0.1).

What can you say about guarantees for stability of the system with this controller?