2.      For the system below, the root locus is shown at the right.

Determine the following.

(10 pts) In this question, assume that speed of response considerations dictate that the closed loop poles must lie to the left of s = -2 (whether the poles are real or complex).  Determine the pole locations that produce the largest damping ratiom z, for the closed loop complex poles, and determine the gain that will produce that damping ratio.

• z = _____
• K_p = ______

        To get the maximum damping ratio, draw a line tangent to the root locus from the origin - tangent at the point with the smallest angle (to get the largest damping).  That's shown in the figure below.

We extended that line out to -10 on the real axis.  That makes computing the angle easy.  The angle off the horizontal is ,448 radians, and cos(.448) is .90, so that's the damping ratio.  Note that everything is copasetic because the closed loop pole on the root locus is at s = - 3.7 +/- j1.7 or thereabouts, clearly to the left of s = -2.  We just need to compute the gain at that point.  The root locus gain is the product of pole distances divided by the product of the zero distances.  We get that to be about 31.13.  (Well, that's a few too many places, but . . .)
• z = 0.9
• K_p = 31


(10 pts) Estimate the location of the third pole for the gain in the first part of this question.  Show that pole on the plot above, and label it.

To compute the third pole, compute the gain at an arbitrary point (an arbitrary closed loop pole) on the real axis segment between - 2 and -10.  Adjust the closed loop pole until the gain is reasonable close.

• Closed loop pole ~ -4.26
• Actually, we'll give perfect credit for anything around 4.