(Solved): Given a unity feedback system with: G(s)=(s+9)(s+6)(s+10)K(s+3);andH(s)=1 a) Evaluate the pe ...

Given a unity feedback system with: $G(s)=\frac{K(s+3)}{(s+9)(s+6)(s+10)};\text{ and }H(s)=1$ a) Evaluate the performance of the uncompensated system in MATLAB. b) For a $20\%$ overshoot, find $\%\mathrm{MP}$ and the peak time $T_p$ and the dominant closed loop poles, ie, must find $-\xi {\omega }_n\pm j{\omega }_d$. c) Next, design theoretically a PD controller to reduce the peak time to two-thirds of that of the uncompensated system. First you must find the new dominant closed loop poles $-\xi {\omega }_n\pm j{\omega }_d$. In doing so, you must find the angle of the zero ${\theta }_z$ and the location of the zero of the compensator $z_c$. Write the PD compensator as $G_{PD}(s)=K\left(\mathrm{s}+z_c\right)$. d) Analyze the complete root locus of the PD-compensated system $G_{PO}(s)G(s)$ Draw the new root locus and plot the step response also Determine loop gain $K$ for the PD-compensated system either manually or from the root locus plot at $20\%$ overshoot. Hint: you can find $\mathrm{K}$ theoretically from the magnitude condition. e) Finally choose an ideal integral compensator (or PI controller) of the form $G_{PI}(s)=\frac{s+0.1}{s}$ to reduce the steady-state error to zero for a step input. Hence the overall system with the PID controller will be as: $G(s)=\frac{K(s+0.1)(s+3)\left(s+z_c\right)}{s(s+9)(s+6)(s+10)}$ f) Find the loop gain $K$ for the PID-compensated system and analyze the system to be sure that all requirements have been met, you need to find: a) Step response of the PID-compensated system with the found value of $K$ in S). b) Compare step response for uncompensated system and PID-compensated system. c) Compare the peak time for uncompensated system and PID-compensated system. d) Compare the steady-state error for uncompensated system and PID-compensated system. 1. Discuss and conclude the results obtained.