212 Ketter Hall, North Campus, Buffalo, NY 14260 civil.buffalo

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

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

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: H2 and H Control • Method 7: Sliding Mode Control • Method 8: Fuzzy Logic Control

5. Control Considerations • Stability - Liapunov Considerations • Controlability • Observability • Spill-Over

6. Control Implementations • Force Generation • Time Delay • Robustness

7. Block Diagram of Control

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: H2 and H Control • Method 7: Sliding Mode Control • Method 8: Fuzzy Logic Control

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

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

11. State Space Equation 0

12. Quadratic Performance Index

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

14. Closed Loop (Feed Back Loop)

15. Closed Loop Control • Ricatti Equation

16. Closed Loop Control

17. Closed Loop Control For time independent P :

18. Output Control Computed output Cz(t)

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

20. Open-Closed Loop Control • Closed Loop Control Excitation Open Loop Control

21. Open Loop (Feed Forward)

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

23. Active Tendon System

24. Example of Control

25. Example’s Parameters

26. Example’s Equations

27. Optimization Parameters

28. SDOF System on Shaking Tableat University at Buffalo (SUNY)

29. Active Tendon System

30. Frequency Response Function

32. Effect of Weighting Matrices

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: H2 and H Control • Method 7: Sliding Mode Control • Method 8: Fuzzy Logic Control

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

35. Determining Gain Matrix Determinant equation:

36. Determining Gain Matrix

37. Example of Poles Assignment

38. Example of Poles Assignment

39. Example of Poles Assignment

40. Example of Poles Assignment

41. Example of Poles Assignment

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

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: H2 and H Control • Method 7: Sliding Mode Control • Method 8: Fuzzy Logic Control

44. Method 3: Instantaneous Optimum

45. Method 3: Instantaneous Optimum Modal formulation:

46. Method 3: Instantaneous Optimum

47. Method 3: Instantaneous Optimum

48. Instantaneous Closed Loop

49. Example of Instantaneous Control Data: mi=345.6 ton ki =3,404x105 kN/m 1= 2=2% I=5.79, 17.18, 27.98, 37.82, 46.38, 53.36, 58.53 rad/s md=29.63 ton (=2%W1) cd = 25 ton /m/sec (=7.3%) kd = 957.2 kN/m

50. Example of Instantaneous Control