Closed loop requirements and analysis of the plant

Question 1.1. Using the formula from your notes, calculate the minimum damping ratio of the dominant poles for an overshoot of o_s ≤ 10%.

Question 1.2. Using the formula from your notes, calculate the minimum damped frequency, ωd, of the dominant poles for a peak time t_p ≤ 0.5 sec.

Question 1.3. Sketch the area in the s-plane where the dominant poles must lie in order to meet both the specifications.

Question 1.4. Record the values of kg , T₁ and T₂.

Question 1.5. Record the transfer function and comment on it. Record and comment on the plant poles.

Include the print-out of the s-plane plot and explain it.

Question 1.6. Predict the characteristics of the open loop step response. Explain why.

Question 1.7. Include the print-out of this response. Explain its characteristics.

Proportional controller design by root-locus

Question 2.1. Include the print-out of the root locus. Explain its characteristics.

Question 2.2. From the root locus plot, determine whether both design specifications are simultaneously possible using proportional feedback alone. Show the area where the dominant poles must lie in order to meet both the specifications on the root locus plot. Record the value of kp at the minimum required value of ωd. Record the poles for this value of kp. Determine ζ and the predicted overshoot.

Question 2.3. Record and comment on the closed loop transfer function. Record the closed loop damping
ratio, natural frequency and poles and compare these to the values predicted from the root locus. Include the print-out of the step response. Record os and tp. Comment on the response. Does your design meet the specifications? Compare the overshoot and peak time to the values predicted from the root locus.

Question 2.4. Use the mouse on the root locus screen plot to determine the limiting value of kp for system stability. Calculate the value by hand using the Routh-Hurwitz criterion. Show your workings.

Frequency domain analysis

Question 3.1. Include a print-out of the Bode plot. Explain its main characteristics. Confirm that the gain margin is equal to the limiting value of gain for closed loop stability obtained in the previous section.

Question 3.2. Include a print-out of the Bode plot of the open-loop compensated system. Show the gain and phase margins on the plot. Comment on the plot.

Question 3.3. State the Nyquist criterion for this system.

Question 3.4. Include a print-out of the Nyquist plot (including the gain and phase margins) of the open- loop compensated system. Explain its main characteristics.

Question 3.5. Include a print-out of the Nichols chart of the open-loop compensated system kptt(s). Show the gain and phase margins on the plot. Show on the plot the maximum closed loop gain, Mr, resonant phase lag, φr , and maximum phase lag. Record the frequencies that these occur. Also show the closed loop bandwidth, ωb.

Question 3.6. Include a print-out of the closed loop frequency response. Record Mr, φr , ωr , ωb and maximum phase lag and its frequency. Compare these values to those predicted by the Nichols chart.

Question 3.7. Include a print-out of the closed loop system sensitivity frequency magnitude response.

Comment on the response and its significance. Record the maximum gain of the sensitivity function, MS . Compare this value to that predicted by the Nyquist plot.

PID/P+D controller design

Question 4.1. What will be the system type number a if PID controller is used on this plant? For zero steady state error (to unity step references) for this plant, is an integrator required in the controller? If there is a steady disturbance at the plant input, is integral action required in the controller for zero steady state error? Show why.

Question 4.2. Explain why integral action requires anti-windup. What are the other disadvantages of including integral action in the controller?

Question 4.3. Include the print-out of the step response of your final design. From the plot, measure os and tp. Does your design meet the specifications? Record the final values of kp and kd. Record the transfer function K(s), the system open loop transfer function L(s) and the closed loop transfer function T (s).

Question 4.4. Include the print-out of the Bode plot. Draw its asymptotes and explain its characteristics.

Question 4.5. Record the gain margin, phase margin, and associated frequencies. Compare these to the values obtained in the previous section. Include the print-out of the Bode plot and show the gain margin and phase margin on the Bode plot.

Question 4.6. Include the print-out of the s-plane plot. Record and comment on the closed loop poles and zeros.

Pole placement using state feedback

Question 5.1. Record the state space representation produced by MATLAB.

Question 5.2. Record the transformed state space representation and the transformation/observability matrix S. From G(s), calculate the state space representation by hand, and compare the results to those given by MATLAB.

Question 5.3. Record and explain how you calculated the desired pole positions.

Question 5.4. Record the values of K. Calculate the state feedback controller gain matrix, K, by hand
using the method in your notes and verify the MATLAB result.

Question 5.5. Include the print-out the closed loop step response. From the plot, measure os and tp . Does your design meet the specifications? If not, explain why.

Question 5.6. Briefly give the main disadvantages of state feedback.

Question 5.7. Suggest how the reference tracking of the closed loop system may be improved.