Control Systems

AUK (Fall 2018) ELEG 421 Exam #2

Dr. Ashraf Zaher 1/12

Department of Electrical and Computer Engineering, College of Arts and Sciences, American University of Kuwait, P. O. Box 3323, Safat, 13034 Kuwait

ELEG 421 (Fall 2018) Exam #2

Title: Control Systems Score: / 100 /20 Student name: Student #: Scope: Chapter (5) Date: November 28, 2018

 Problems:

# Score Comments

1 /20

2 /20

3 /20

4 /20

5 /20

E.C. /20

 Student’s feedback:

Exam Rating: too easy [ ] easy [ ] fair [ ] somewhat difficult [ ] difficult [ ] too difficult [ ] Your expected mark: /100

 Notes:

1. Answer the first 5 questions, and attempt question 6 for extra credit 2. Time allowed is 120 minutes 3. There is a maximum of 10% extra credit! 4. Show all your work in details 5. Use matrix form to represent your state space equations 6. Use Matlab and/or Simulink, whenever possible 7. Attach any additional printouts to the same booklet 8. This is an open-book exam (textbooks and notes are allowed)

( Good Luck  )

AUK (Fall 2018) ELEG 421 Exam #2

Dr. Ashraf Zaher 2/12

Q1. The first order system, shown in Fig. (1), represents a process such that 2  T  4. i. For a unit impulse input, find the range of K such that the maximum value of

c(t) is limited to 5. Find the corresponding range of the control signal, u(t). ii. For T = 3 and K = 4, find:

 the settling time of c(t), for a unit step input,  the range of u(t), at steady state, when r(t) = 2 sin(5t), and  the value of e(t = 2), when r(t) is a unit ramp input.

[10+4+3+3 marks]

Figure (1)

U(s)

AUK (Fall 2018) ELEG 421 Exam #2

Dr. Ashraf Zaher 3/12

AUK (Fall 2018) ELEG 421 Exam #2

Dr. Ashraf Zaher 4/12

Q2. The following state space model describes the transfer function of G(s) between the input; R(s), and the output; C(s):

i. Find the range of K that results in a stable closed-loop system, and ii. Using the maximum value of K in (i), and after a long time, c(t) could be

approximated by: c(t)  A sin(2ft) + d, when r(t) is a unit step. Find A, f, d, and the dominant closed-loop poles.

[12+8 marks]

 

1 1

2 2

3 3

1

2

3

0 1 0 1

1 0 0 0

0 1 0 0

1 2 4

x x

x x u

x x

x

c x

x

                                      

        

  

Figure (2)

U(s) G(s)

AUK (Fall 2018) ELEG 421 Exam #2

Dr. Ashraf Zaher 5/12

AUK (Fall 2018) ELEG 421 Exam #2

Dr. Ashraf Zaher 6/12

Q3. For the block diagram, shown in Fig. (3), find: i. the range of K for stability, as a function in a,

ii. the output frequency, for Kmax of stability, if a = 8 and the input is a unit step, iii. the offset (ess) to a ramp input, when a = K = 2, and iv. the range of a for stability, if K is allowed to be – 0.2.

[6+4+4+6 marks]

Figure (3)

PlantController

AUK (Fall 2018) ELEG 421 Exam #2

Dr. Ashraf Zaher 7/12

AUK (Fall 2018) ELEG 421 Exam #2

Dr. Ashraf Zaher 8/12

Q4. For the system, shown in Fig. (4), J = 0.01, b = 0.2, and r(t) is a unit step input: i. Find the value of Kp that will result in an under damped response, with an

oscillation frequency d = 20 rad/s, in the absence of the external disturbance. Find the corresponding damping ratio, , and

ii. For Kp = 4, find the percentage change in the maximum overshoot, the settling time, and the final value of c(t), with and without the external disturbance, assuming d(t) = 0.1.

[8+12 marks]

Figure (4)

AUK (Fall 2018) ELEG 421 Exam #2

Dr. Ashraf Zaher 9/12

AUK (Fall 2018) ELEG 421 Exam #2

Dr. Ashraf Zaher 10/12

Q5. For the system, shown in Fig. (5), J = 0.1, B = 2, Kp = 10, and r(t) is a unit step input: i. Find the locations of the poles/zeros of the closed-loop system, when Kd = 0 and

Kd = 0.5, respectively. Determine the percentage change in the values of both  and n, in both cases, and

ii. For Kd = 3.2, find the dominant closed-loop pole, and use it to calculate the corresponding time constant, TCL, (using first-order approximation). Measure the settling time of the system and compare it to the calculated value (3TCL). Explain the discrepancy!

[12+8 marks]

Figure (5)

U(s)

AUK (Fall 2018) ELEG 421 Exam #2

Dr. Ashraf Zaher 11/12

AUK (Fall 2018) ELEG 421 Exam #2

Dr. Ashraf Zaher 12/12

Q6. For the system, shown in Fig. (6): i. Find suitable values for the PID controller gains such that:

 ess = 0  No overshoot  Settling time is less than 10 s, and

ii. If the control signal is limited to ± 1.0, simulate the system, using the above settings for the PID controller. Explain the result!

[12+8 marks]

Figure (6)