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Category: ECSE 403

Description

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1 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.

2 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.

3 Questions

1. Derive the closed loop transfer function of a proportional position controller for the cart

system(analytically).Identify ζ and ωn from the closed loop transfer function and compute

them for Kp = [5, 10, 20, 50](Since you have derived a similar result in assignment 2, part 9

you can just plug in new coefficients)[10 marks]

2. Implement (experimentally) a proportional position controller in your Simulink model and

plot the step response for Kp = [5, 10, 20, 50]. Explain as Kp increases, how do the rise time,

overshoot, steady state error and damping change?[15 marks]

3. Tuning value of Kp, find an appropriate gain for proportional controller, i.e. change the value

of Kp to get a step response with relatively low rise time while over shoot is relatively small.

1

Give the value of Kp 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 difference between the two step responses. [10 marks]

5. Estimate the frequency response of the closed loop system (position control) using ω =

[1, 2, 5, 10, 20, 50] rad

s

for Kp = 10 and Kp = 50 and plot the Bode diagram. [15 marks]

6. 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 figure. Explain the

effect of a proportional feedback controller on frequency response. [5 marks]

7. Add a differentiating block and design a PD controller in the form of

CP D(s) = Kp + Kd.s

(a) Using a fixed Kp (Kp you found in question 3), increase the value of Kd gradually and

explain the effect of adding a differentiating operator on step response of the system.

Report the best Kd and plot step response of the system with P D controller along with

best P controller(Question 3) on one figure. [10 marks]

(b) Try to tune Kd and Kp, simultaneously. Report best combination of Kd and Kp you

have found and plot the step response of the new P D controller. [10 marks]

8. Remove the differentiating block and replace it with an integrator to design a PI controller

of the form

CP I (s) = Kp +

Ki

s

(a) Using a fixed Kp (Kp you found in question 3), increase the value of Ki gradually

and explain the effect of adding an integrator operator on step response of the system.

Report the best Ki and plot step response of the system with P I controller along with

best P controller(Question 3) on one figure. [10 marks]

(b) Try to tune Ki and Kp, simultaneously. Report best combination of Ki and Kp you

have found and plot the step response of the new P I controller. [10 marks]

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

CP ID(s) = Kp +

Ki

s

+ Kd.s

2

(a) Using values of Kd and Kp (in question 7), increase the value of Ki gradually and

explain the effect of adding an integrator operator on step response of the system.

Report the best Ki and plot step response of P ID controller along with the best P D

controller(Question 7) on one figure. [10 marks]

(b) Using values of Ki and Kp (in question 8), increase the value of Kd gradually and

explain the effect of adding a differentiating operator on step response of the system.

Report the best Kd and plot step response of P ID controller along with the best P I

controller(Question 8) on one figure. [10 marks]

(c) Try to tune all the three parameters Kp, Ki

, Kd simultaneously to find a suitable step

response.(report the parameters and plot the step response) [20 marks]

(d) By keeping two of the parameters fixed and increasing the third one, complete following

chart. Write the effect of each of these controller gains on different parameters of the

step response.[15 marks]

Table 1: PID controller

Gain Rise Time Overshoot Settling time Steady state error

Kp

Kd

Ki

3

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