Solved–Lab Assignment 4 –Solution

Description

• Objective

The main goal of this assignment is to design and implement P ,P I,P D, and P ID controllers for cart’s position system.

• Your responsibility

Your responsibility is to answer all questions which have been asked throughout this assignment and submit all your answers in addition to Matlab codes and Simulink results.

• Questions

• 1. Derive the closed loop transfer function of a proportional position controller for the cart system(analytically).Identify and !_n from the closed loop transfer function and compute them for K_p = [5; 10; 20; 50](Since you have derived a similar result in assignment 2, part 9 you can just plug in new coe cients)[10 marks]

• 2. Implement (experimentally) a proportional position controller in your Simulink model and plot the step response for K_p = [5; 10; 20; 50]. Explain as K_p increases, how do the rise time, overshoot, steady state error and damping change?[15 marks]

• 3. Tuning value of K_p, nd an appropriate gain for proportional controller, i.e. change the value of K_p to get a step response with relatively low rise time while over shoot is relatively small.

Give the value of K_p and plot the step response. Report corresponding values for the rise time, maximum overshoot, settling time and steady state error from experimental results. [15 marks]

4. Compare the experimental step response with theoretical step response. Explain some of the reasons for the di erence between the two step responses. [10 marks]

4. Estimate the frequency response of the closed loop system (position control) using ! = [1; 2; 5; 10; 20; 50]^rad_s for Kp = 10 and Kp = 50 and plot the Bode diagram. [15 marks]

4. Plot both the Bode diagram you found in previous lab experimentally for the open-loop system(position as the output) and Bode diagram in question 5, on one gure. Explain the e ect of a proportional feedback controller on frequency response. [5 marks]

4. Add a di erentiating block and design a PD controller in the form of

C_{P D}(s) = K_p + K_d:s

1. 1. Using a xed K_p (K_p you found in question 3), increase the value of K_d gradually and explain the e ect of adding a di erentiating operator on step response of the system. Report the best K_d and plot step response of the system with P D controller along with best P controller(Question 3) on one gure. [10 marks]

1. 2. Try to tune K_d and K_p, simultaneously. Report best combination of K_d and K_p you have found and plot the step response of the new P D controller. [10 marks]

8. Remove the di erentiating block and replace it with an integrator to design a PI controller of the form

C_{P I} (s) = K_p + ^K_sⁱ

1. 1. Using a xed K_p (K_p you found in question 3), increase the value of K_i gradually and explain the e ect of adding an integrator operator on step response of the system. Report the best K_i and plot step response of the system with P I controller along with best P controller(Question 3) on one gure. [10 marks]

1. 2. Try to tune K_i and K_p, simultaneously. Report best combination of K_i and K_p you have found and plot the step response of the new P I controller. [10 marks]

9. Now add a di erentiating block, to design a PID controller of the form

C_{P ID}(s) = K_p + ^K_sⁱ + K_d:s

1. Using values of K_d and K_p (in question 7), increase the value of K_i gradually and explain the e ect of adding an integrator operator on step response of the system. Report the best K_i and plot step response of P ID controller along with the best P D controller(Question 7) on one gure. [10 marks]

2. Using values of K_i and K_p (in question 8), increase the value of K_d gradually and explain the e ect of adding a di erentiating operator on step response of the system. Report the best K_d and plot step response of P ID controller along with the best P I controller(Question 8) on one gure. [10 marks]

3. Try to tune all the three parameters K_p; K_i; K_d simultaneously to nd a suitable step response.(report the parameters and plot the step response) [20 marks]

4. By keeping two of the parameters xed and increasing the third one, complete following chart. Write the e ect of each of these controller gains on di erent parameters of the step response.[15 marks]

Table 1: PID controller

Gain Rise Time Overshoot Settling time Steady state error

K_p

K_d

^Ki

3