The plant transfer function for a particular process is

$G_p(s)=\frac{8-s}{s^2+2s+3}$

We wish to investigate the use of proportional control action with this plant.

a$.$ Obtain the root locus and determine the range of values of the proportional gain $K_P$ for which the system is stable.

b$.$ Determine the value of $K_P$ required to give a time constant of $=\frac{2}{3}.$

c$.$ Plot the unit$-$step response of the plant for $K_P=1.$ A process containing a negative sign in the numerator of its transfer

function is called a "reverse reaction" process. What is the eflect of the negative sign in the numerator of $G_p(s)?$