1. Slide# 1 212 Ketter Hall, North Campus, Buffalo, NY 14260 www.civil.buffalo.edu Fax: 716 645 3733 Tel: 716 645 2114 x 2400 Control of Structural Vibrations Lecture #7_1 Active Control - Algorithms Instructor: Andrei M. Reinhorn P.Eng. D.Sc. Professor of Structural Engineering

2. Slide# 2

3. Slide# 3 Issues in Active Control  Control Logic - Algorithms  Control Considerations - Stability, etc.  Control Implementations - Force Generation  Physical Implementations  Full Scale Implementations

4. Slide# 4 Control Algorithms  Method 1: Optimal Control  Method 2: Poles Assignment  Method 3: Instantaneous optimum  Method 4: Independent Modal Space Control  Method 5: Bounded State Control  Method 6: H 2 and H  Control  Method 7: Sliding Mode Control  Method 8: Fuzzy Logic Control

5. Slide# 5 Control Considerations  Stability - Liapunov Considerations  Controlability  Observability  Spill-Over

6. Slide# 6 Control Implementations  Force Generation  Time Delay  Robustness

7. Slide# 7 Block Diagram of Control

8. Slide# 8 Control Algorithms  Method 1: Optimal Control  Method 2: Poles Assignment  Method 3: Instantaneous optimum  Method 4: Independent Modal Space Control  Method 5: Bounded State Control  Method 6: H 2 and H  Control  Method 7: Sliding Mode Control  Method 8: Fuzzy Logic Control

9. Slide# 9 Structure Equations Structure’s Equation Control Force Effective Equation with Control

10. Slide# 10 Structure Equations Structure’s Equation Denote Variables as:

11. Slide# 11 State Space Equation 0

12. Slide# 12 Quadratic Performance Index

13. Slide# 13 Optimized Solution Solutions of the above leads to basic equations to determine control forces u(t):

14. Slide# 14 Closed Loop (Feed Back Loop)

15. Slide# 15 Closed Loop Control Ricatti Equation

16. Slide# 16 Closed Loop Control

17. Slide# 17 Closed Loop Control For time independent P :

18. Slide# 18 Output Control Computed output Cz(t)

19. Slide# 19 Open-Closed Loop (Feed Back-Feed Forward)

20. Slide# 20 Open-Closed Loop Control Open Loop Control Closed Loop Control Excitation

21. Slide# 21 Open Loop (Feed Forward)

22. Slide# 22 Open Loop Control  The solution needs the information of loading history and its derivative. Can’t work for earthquake type loading

23. Slide# 23 Active Tendon System

24. Slide# 24 Example of Control

25. Slide# 25 Example’s Parameters

26. Slide# 26 Example’s Equations

27. Slide# 27 Optimization Parameters

28. Slide# 28 SDOF System on Shaking Table at University at Buffalo (SUNY)

29. Slide# 29 Active Tendon System

30. Slide# 30 Frequency Response Function

31. Slide# 31

32. Slide# 32 Effect of Weighting Matrices

33. Slide# 33 Control Algorithms  Method 1: Optimal Control  Method 2: Poles Assignment  Method 3: Instantaneous optimum  Method 4: Independent Modal Space Control  Method 5: Bounded State Control  Method 6: H 2 and H  Control  Method 7: Sliding Mode Control  Method 8: Fuzzy Logic Control

34. Slide# 34 Method 2: Poles Assignment Desired eigenvalue for matrix: A+BG

35. Slide# 35 Determining Gain Matrix Determinant equation:

36. Slide# 36 Determining Gain Matrix

37. Slide# 37 Example of Poles Assignment

38. Slide# 38 Example of Poles Assignment

39. Slide# 39 Example of Poles Assignment

40. Slide# 40 Example of Poles Assignment

41. Slide# 41 Example of Poles Assignment

42. Slide# 42 Example of Poles Assignment Displacements at the two floors:

43. Slide# 43 Control Algorithms  Method 1: Optimal Control  Method 2: Poles Assignment  Method 3: Instantaneous optimum  Method 4: Independent Modal Space Control  Method 5: Bounded State Control  Method 6: H 2 and H  Control  Method 7: Sliding Mode Control  Method 8: Fuzzy Logic Control

44. Slide# 44 Method 3: Instantaneous Optimum

45. Slide# 45 Method 3: Instantaneous Optimum Modal formulation:

46. Slide# 46 Method 3: Instantaneous Optimum

47. Slide# 47 Method 3: Instantaneous Optimum

48. Slide# 48 Instantaneous Closed Loop

49. Slide# 49 Example of Instantaneous Control Data: m i =345.6 ton k i =3,404x10 5 kN/m  1 =  2 =2%  I =5.79, 17.18, 27.98, 37.82, 46.38, 53.36, 58.53 rad/s m d =29.63 ton (=2%W 1 ) c d = 25 ton /m/sec (=7.3%) k d = 957.2 kN/m

50. Slide# 50 Example of Instantaneous Control

51. Slide# 51 Example of Instantaneous Control

52. Slide# 52 Example of Instantaneous Control

53. Slide# 53 Control Algorithms  Method 1: Optimal Control  Method 2: Poles Assignment  Method 3: Instantaneous optimum  Method 4: Independent Modal Space Control  Method 5: Bounded State Control  Method 6: H 2 and H  Control  Method 7: Sliding Mode Control  Method 8: Fuzzy Logic Control

54. Slide# 54 Method 4: Independent Modal Space Control (IMSC)

55. Slide# 55 Method 4: Independent Modal Space Control (IMSC)

56. Slide# 56 Control Algorithms  Method 1: Optimal Control  Method 2: Poles Assignment  Method 3: Instantaneous optimum  Method 4: Independent Modal Space Control  Method 5: Bounded State Control  Method 6: H 2 and H  Control  Method 7: Sliding Mode Control  Method 8: Fuzzy Logic Control

57. Slide# 57 Method 5: Bounded State Control

58. Slide# 58 Method 5: Bounded State Control

59. Slide# 59 Method 5: Bounded State Control

60. Slide# 60 Method 5: Bounded State Control Prucz, Soong and Reinhorn, 1983

61. Slide# 61 Method 5: Bounded State Control

62. Slide# 62 Method 5: Bounded State Control