Final answer:
The question involves sketching the root locus for different transfer functions to determine system stability. It follows standard rules such as branch determination and asymptote analysis. Hand sketches and MATLAB simulations are typically used in conjunction to illustrate the root locus.
Step-by-step explanation:
The question asks to sketch the root locus for different transfer functions L(s) with respect to a variable gain K. Root locus is a graphical method used in control systems to determine the stability of a system as the gain K varies, and it follows specific rules. The various L(s) provided represent different system dynamics, and the sketching of the root locus involves following six root locus rules: determining the number and location of root locus branches, real-axis segments, asymptotes, their angles and intersection with the real axis, departure and arrival angles, and breakaway and break-in points.
Part (a) presents a system with a characteristic equation of a second-order polynomial, whereas parts (b), (c), and (d) have more complex transfer functions with higher order dynamics and additional poles at the origin. To provide a correct sketch of the root locus for these systems, one must analyze the poles and zeros of L(s), considering the effects of each on the overall system stability.
Typically, a software tool like MATLAB is used to accurately plot the root locus, but hand sketches based on precise calculations can also provide insight into system behavior. Note that examples provided in your question are not directly applicable to solving the current problem, as they touch on a variety of topics unrelated to the root locus technique.