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Given a unity feedback system with: $G(s)=(s+9)(s+6)(s+10)K(s+3) ;andH(s)=1$ a) Evaluate the performance of the uncompensated system in MATLAB. b) For a $20%$ overshoot, find $%MP$ and the peak time $T_{p}$ and the dominant closed loop poles, ie, must find $−ξω_{n}±jω_{d}$. 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 $−ξω_{n}±jω_{d}$. In doing so, you must find the angle of the zero $θ_{z}$ and the location of the zero of the compensator $z_{c}$. Write the PD compensator as $G_{PD}(s)=K(s+z_{c})$. 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 $K$ theoretically from the magnitude condition. e) Finally choose an ideal integral compensator (or PI controller) of the form $G_{PI}(s)=ss+0.1 $ 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)=s(s+9)(s+6)(s+10)K(s+0.1)(s+3)(s+z_{c}) $ 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.

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