Final answer:
To determine the Root Locus of the control system G(s)H(s) = k ((s+20)/(s³+15s²+50s)), we plot the roots of the characteristic equation in the complex plane and analyze their movement as parameter k varies to determine system stability.
Step-by-step explanation:
The student asked about finding the Root Locus of the given control system G(s)H(s) and assessing the system's stability. The transfer function provided is k ((s+20)/(s³+15s²+50s)). The general approach to finding the root locus involves plotting the possible locations of the system poles in the complex plane as the parameter k varies from zero to infinity and analyzing how the roots of the characteristic equation, which determine system stability, move in the s-plane.
The characteristic equation is derived from the denominator of the closed-loop transfer function. For the system to be stable, all roots of the characteristic equation must be in the left half of the s-plane (i.e., have negative real parts). Based on the provided function, we can write the characteristic equation as s³+15s²+50s+k(s+20) = 0. Root locus methods or software tools can then be used to plot the locus and examine stability as k changes.