$8\mathrm{.}31.$ A process has an openloop transfer function $G_{M(s)}$ relating controlled and manipulated variables that has a steady$-$state gain of $3$ and two identical first$-$order lags in series with $5-$minute time constants.

$298$ PART TWO: Laplace$-$Domain Dynamics and Control

$($a$)$ Using a proportional controller, derive the closedloop characteristic equation. Sketch a root locus plot. Solve for the value of controller gain that gives a closed loop damping coefficient of $0\mathrm{.}3.$ What is the closedloop time constant using this controller gain?

$($b$)$ Using a proportional$-$integral controller with reset time set equal to $5$ minutes, repeat all the parts of $($a$)$