For the loop shaping design show the desired closed loop

Question

Take Home Project

For the plant TF;

G_p(s) = 10(s+10)/s³+24.8s²+105s+180

Design a feedback control a) using Root Locus method, b) using PID control and c) using the loop shaping design method taught in class. The control designs should try to achieve the following specs, if possible, and will be graded accordantly for how good this is done. Make conclusions for individual designs and for comparisons, but be brief.

Design Specs:

The steady state error to a step input should be zero. The bandwidth, (pc, of the design should not exceed 50 rad/sec, and as such the initial response rate should not be greater than 1/tc or 50 units/s. The phase margin should be greater than 50 degrees, and the gain margin greater than 10 dB, with an overshoot of less than 20%. The mid-frequency disturbance rejection (for ω < (1/10)ω_c, should be _~ 20 dB or greater. The settling time should be less than 0.5 sec. The design should strive to closely meet the 50 rad and 50 units/s specs without exceeding them.

Show the following:

Show all the work, including all Matlab commands and commands used for the plots. Show the step responses of the closed loop designs and the corresponding zoom-in figures for the initial rise times with you calculations of units/sec of the initial response. On the bode plots place a cursor point by the crossover frequency of the gain plot to show cross over frequency and gain at that point, place a cursor on the cross-over frequency on the phase plot to show the phase for PM, and a cursor on the gain at about 5 rad (one decade below spec cross-over) to show mid-frequency gain for disturbance rejection.

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. Show the final root locus design in Matlab with the final closed loop pole location selected on the root locus grid, which shows the open loop proportional gain, the complex poles, and the frequency. Write down the final controller TF of this design.

For the PID design, show the best design you can come up with to meet requirements and achieve the fastest possible rise time within the spec, and write down the controller TF

For the Loop shaping design show the desired closed loop gain design, and the plot of the actual loop gain design to match the desired one in order to meet the specs. Write down the controller TF of this design.