##please solve all this PID controller and Found A$)$: a$,$b$,$ B$),$C$)$ read the picture and take your time to solve it's Control system course and please type all the calculation and explain it and type the method you slove by and thanks $.$

Project Problem statement

Consider a plant having nominal model

$G_0(s)=\frac{B_0((s))}{A_0((s))}=\frac{s+3}{((s)+4)(s+6)}$

Assume a one degree of freedom control loop with controller, $C(s),$ of the form

$C(s)=\frac{P(s)}{L(s)}=\frac{P_{1}s+P_0}{L_{1}s+L_0}$

In our design we want to guaranty the stability just at the start with proper pole$-$zero placement. A parametric solution will enable us to make some ajustment on the speed of response and accuracy. To that end, we have decided to use a controller with several variables $(P_1,P_0,L_1$ and $L_0)$ that needs to be found in terms of parameter $"$ a $",$ also representing a pole of the characteristic equation. In this project, a $4$th$-$order characteristic equation