Consider the system of Problem 8.10-10.

(a) Plot the root locus for this system, and find the value of \(K>0\) for which the system is stable.

(b) Find the time constant for the system with \(K=1\).

(c) Design a phase-lead controller such that the characteristic-equation root in part (b) moves to yield a time constant of \(0.5 \mathrm{~s}\) for \(K=1\).

(d) By computer, verify the design in part (c) yields the correct characteristic-equation zero.