$1.$ Design a Lead Compensator: Design a lead compensator using any of the methods discusses. The general form of a lead compensator is: Gc$($s$)=$ Kc $($s $+$ zc$)/($s $+$ pc$),$ where zc is the zero, pc is the pole $($pc $>$ zc$),$ and Kc is the gain. Note that both the zero and the pole are on the real axes by design. Your design must include the derivation of these parameters in detail supported by relevant drawings. $2.$ Verify the Phase Constraint Algebraically: Verify algebraically that the designed compensator ensures the desired poles satisfy the phase constraint. $3.$ Root$-$Locus Diagram for Compensated System: Plot the root$-$locus diagram of the compensated system and graphically demonstrate that the root$-$locus passes through the desired dominant poles. Note: Up to this point, the focus is to ensure the root$-$locus passes through the desired dominant poles derived from your unique design criteria. In the next steps, analyze the timedomain response of the compensated system and discuss your results based on the following criteria: $4.$ Step Response Analysis: Plot the step response of the compensated system. Using the plot, determine graphically the following values: $$ Rise Time $($tr$)$ Settling Time $($ts$)$ Peak Time $($tp$)$ Maximum Overshoot $($Mp$)$ Steady$-$State Error $($ess$)5.$ Discussion: Evaluate whether your compensator satisfies the given requirements and discuss the results.