For the feedback control system shown below with the process TF, Gp(s) = [10/(s+30)(s²+30s+709)], and for a step input command, design the controller Gc(s),

a) Using the Root Locus method

b) Using the PID controller

c) Using the loop shaping design method.

The design specifications are as follows: The bandwidth, ω_c, of the control system design should be limited to 100 rad/s or up to 100 units/s speed for the plant actuator is permitted, without exceeding the bandwidth limit. The steady state error to a step input should be zero. The PM should be greater or equal to 50 degrees, and the GM should be greater or equal to 10 dB, with an overshoot of less than 20%. The noise or disturbance rejection of the control system to disturbances at the mid-frequency range should be greater or equal to 10 times (i.e. 20 dB at mid-frequency or 17 dB at 0.1ω_c gain for the open loop TF). The settling time should be less or equal to 0.3 s.

The assignment will be scored on how well the designs meet the specs. That is meet or even exceed the specs. But if the specs cannot be met with a particular control design, do the best you can. Also, student will be scored on how well instructions are followed and the results are organized and displayed. For the graduate students, expectations will be higher than the undergraduates.

Show the following:

For each of the control designs show all the work, including all Matlab commands and commands used for the plots. Show the bode plots of the open loop TF (the plant and the controller). On the bode plots show the PM, GM, and Mid-frequency gain for disturbance rejection by placing appropriate cursor points on these plots. Show the step responses of the feedback control system designs and the corresponding zoom-in figures for the initial rise times, with your calculations for the speed of the control system. For each of the designs write down the final controller TF. Discuss briefly final conclusions for the 3 designs.

For the final Root Locus (RL) design, briefly explain the logic for choosing the pole/zero locations to shape the RL for the open loop TF. On the RL show the chosen "sgrid" and the selected design point for the closed loop pole location with a cursor point that displays the open loop TF gain K, the dominant poles, the damping, and the frequency.

For the Loop Shaping design, in addition, also show the desired closed loop gain TF, and the overlay plot of the actual loop gain design to match the desired one in order to meet the specs.