Due Date: $5/12/2024$

Consider the following closed$-$loop control system:

Figure $1$: Block Diagram of a System with a Controller D$($S$)$

$1-$ Design a PID controller for the system in figure. $1$ such that $\backslash (\backslash$mathrm$\{$e$\}\_\{\backslash$mathrm$\{$Ss$\}\}=0\backslash$mathrm$\{$v$\},\backslash$mathrm$\{$O$\}.\backslash$mathrm$\{$S$\}.=25\backslash \%\backslash )$ and $\backslash (\backslash$mathrm$\{$T$\}\_\{\backslash$mathrm$\{$s$\}\}=2\backslash$mathrm$\{$sec$\}\backslash ),$ where $\backslash ($ G$($s$)=\backslash$frac$\{1\}\{$S$\backslash$left$($S$^\{3\}+20$ S$^\{2\}+124$ S$+240\backslash$right$)\}\backslash ).$

$2-$ Simulate the system of figure$1$ using the controller designed in step $1($MATLAB$).$

$3-$ Plot the output step response and the error versus time for the uncompensated and compensated system.

Report should include:$-$

$1-$ Theory of different types of controllers with the characteristics of each type and overall comparison between them with detailed theory of PID controller.

$(2$ Points$)$

$2-$ Detailed design steps of the PID compensator used in the project.

$(4$ Points$)$

$3-$ Response of the system before and after compensation. Plot the error versus time and include the MATLAB code and the block diagram.

$(3$ Points$)$

$4-$ Conclusion: comments on the results obtained.

$(2$ Points$)$

$5-$ Presentation and Oral discussion

$(4$ Points$)$