Feedback Control Systems

1. Feedback Control Systems Dr. Basil Hamed Electrical & Computer Engineering Islamic University of Gaza

2. STABILITY OF LINEAR FEEDBACK

3. PROBLEM DEFINITION For people paralyzed from the neck down, the ability to drive themselves around in motorized wheelchairs is highly desirable.   A proposed system uses velocity sensors mounted in the headgear at 900 intervals, so that forward, left, right, or reverse directions can be commanded.   Output of the headgear sensor is proportional to the magnitude of the head movements.   The block diagram for this system is shown in figure 1.   Here, typical values for the time constants are 1 = 0.5 s, 3 = 1 s, and 4 = 1/4 s.

4. Using MATLAB do the following 1) Determine the limiting gain K = K1K2K3 for a stable system. 2) When the gain K is set equal to 1/3 of the limiting value, determine if the settling time to within 2% of the final value of the system is less than 4 s. 3) Determine the value of gain that results in a system with a settling time of 4 s.  Also, obtain the value of the roots of the characteristic equation when the settling time is equal to 4 s.

5. Block Diagram

6. Part 1)

7. Routh-Hurwitz Table Routh-Hurwitz Table:s3          1          14         0 s2          7        8+8K       0 s1          A          0          0 s0        8+8K       0          0 A = -(8+8K-98) = 98-8-8K              7               7 For stability, A > 0, therefore 0 < 98 - 8 - 8K Therefore range for stability is given by: 0 < K < 11.25

8. Part 2) the closed-loop transfer function was found to be:                   T(s) =                 8K                                             s3 + 7s2 + 14s + 8 + 8K. In calculating the settling time, we assume the validity of a second order approximation, allowing the use of the dominant pole pair to find settling time as:                   Ts =    4                             ζwnwhere ζwn = σd = -1 * real part of dominant poles. .

9. Part 2) K = 3.7333 Open-loop system Transfer function: 3.733 --------------------------------------------------- 0.125 s^3 + 0.875 s^2 + 1.75 s + 1 Closed-loop system Transfer function: 3.733 ---------------------------------------------------------- 0.125 s^3 + 0.875 s^2 + 1.75 s + 4.733 P =          -5.70955929441205         -0.645220352793972 +     2.4931592351599i         -0.645220352793972  -     2.4931592351599i settling_time =   6.19943246160627

10. Part 2)

11. Part 3) For the system to be practical, a settling time of 4 seconds is required. Making use of the second order equation, settling time Ts = 4 / ζwn where ζwn = - real part of the dominant closed-loop pole pair, the settling time is calculated for each K value from 0.1 to the limiting gain or until one yielding a result of 4 seconds is found. The step response, transfer functions and the roots of the characteristic equation are displayed for this value of K.

12. Part 3) For K = 1.5 settling time is approximated as 4 seconds. Open-loop transfer function with K = 1.5 Transfer function:                1.5 ---------------------------------------------------- 0.125 s^3 + 0.875 s^2 + 1.75 s + 1 Closed-loop transfer function with K = 1.5 Transfer function:                1.5 ------------------------------------------------------ 0.125 s^3 + 0.875 s^2 + 1.75 s + 2.5 Roots of the characteristic equation for K = 1.5 P =               -4.99999999999999               -1 +      1.73205080756888i               -1  -      1.73205080756888i Settling time for K = 1.5 settling_time =   3.9999

13. Part 3)

14. Animation

15. Problem 2

16. PROBLEM DEFINITION The goal of vertical takeoff and landing (VTOL) aircraft is to achieve operation from relatively small airport and yet operate as normal aircraft in level flight. An aircraft taking off in a form similar to a missile (on end) is inherently unstable. A control system using adjustable jets can control the vehicle

17. Block Diagram

18. Use MATLAB a) Find and plot closed loop poles in s-plane and discuss their location for K=100. b) Determine the range of gain K for which the system is stable, marginally stable and unstable. c) Determine and plot the roots of the characteristic equation for gain K obtained in part "b", which makes the system to be marginally stable and for selected gain K that makes the system unstable including poles locations from part "a" giving full comment. d) Plot step responses of the system for K=100, selected system gain, which makes the system to be unstable and the obtained gain K in part "b" which makes the system to be marginally stable, giving comments on the obtained results.

19. Part a) MATLAB results for K=100:   P1= -3.1269+7.9403i    ;    zeros: z1= -2   P2= -3.1269-7.9403i   P3= -2.7463

20. Part a)

21. Part b) • K<12.857143 - unstable • K=12.875143 - marginally stable • K>12.875143 - stable

22. Part C) The roots for selected K=2 (which make the system unstable) are: P1=-9.8531 P2=0.4266+0.4733i P3=0.4266-0.4733i

23. Part C)

24. Part C) The roots for K=12.857143 (from part b) are: P1=-9 P2=0+1.6903i P3=0-1.6903i zeros: z1=-2

25. Part C)

26. Part D) Output step responses for K=100, K=12.8571 & K=2 Output step response of the aircraft control system

27. Output step response for K=100

28. Output step response for K=12.85714

29. Output step response for K=2