Root locus is a graphical method used in control systems to analyze and design the placement of poles in a closed$-$loop system by varying the gain, $\backslash ($ K $\backslash ).$ It is particularly valuable in designing PID controllers, as it helps to visualize how the system poles move in response to changes in proportional, integral, and derivative terms. By examining the root locus plot, control engineers can select PID parameters to achieve desired system behavior, such as improved stability, reduced overshoot, and faster settling time.

Compared to other tuning methods like Ziegler$-$Nichols, root locus provides a deeper insight into the pole$-$zero placement and system dynamics. Ziegler$-$Nichols is often quicker and simpler for initial PID tuning, especially for straightforward applications, as it uses a set of predefined rules to set the controller parameters. However, Ziegler$-$Nichols lacks the precision and flexibility of root locus, particularly in complex or highly nonlinear systems, and can lead to higher overshoot. In contrast, root locus allows for a more tailored approach to PID tuning, offering the advantage of fine$-$tuning and customizing the response at the cost of requiring more time and expertise to interpret and adjust the plot effectively.

Based on your self$-$study of root locus and its application in designing PID control, you are required to design PID controller that fulfill requirement for the system mentioned as follows:

System Transfer Function:

$\backslash [$

G$($s$)=\backslash$frac$\{5\}\{$s$($s$+3)\}$

$\backslash ]$

Design Requirements:

$-$ Overshoot less than $\backslash (10\backslash \%\backslash )$

$-$ Settling time, $\backslash ($ T$\_\{$s$\}<mn>2</mn><mtext></mtext><mi>\backslash </mi><mi>)</mi>$ seconds

Discuss your solution in detail, including aspects such as controller background, tuning$/$configuration necessary to achieve the assigned task, and controller robustness and performance. All explanations, but not limited to$,$ such as derivation steps, result comparisons before improvement and after improvement, as well as other related details should be clearly described and easy to understand. Cite any references used in your writing as necessary. Your report should demonstrate a thorough understanding of solving the given task. Submit your assignment report in both Word and PDF formats.