Fractional order control of civil structures using MR Damper

1. Fractional order control of civil structures using MR Damper AbdollahShafieezadeh*, Keri Ryan*, YangQuan Chen+*Civil and Environmental Engineering Dept.+Electrical and Computer Engineering Dept.Utah State University Utah State University

2. Outline • Introduction • Fast Hybrid Testing • Robust Controllers • Filter Design • Fractional Order Filter • Results • Conclusions

3. Introduction • Optimal control theories have been studied intensely for Civil Engineering structures. • In most cases idealized models were used for both the structure and actuators. • The effect of filters has not been investigated intensely. • Fractional order filters, offering more features are applied here.

4. Dynamic Testing Techniques • Quasi-static loading test method (QST) • Shaking table test method (STT) • Effective force method (EFT) • Pseudo-dynamic test method (PDT) • Real time dynamic hybrid test method (RTDHT)

5. Pseudo-dynamic testing method • Applying slowly varying forces to a structural model • Motions and deformations observed in the test specimens are used to infer the inertial forces that the model would have been exposed to during the actual earthquake • Substructure techniques can be applied

6. Hybrid Testing • part of the structure (the physical model) whose hysteretic behavior is complex is constructed and tested • The rest part of the structure is numerically modeled in computer • The superstructure effects on substructure are applied by actuators

7. Closed Loop Diagram for the Test

8. FHT Facility at University of Colorado

9. FHT Facility at University of Colorado

10. Possible Controllers • The objective is to mitigate structural response in the presence of nonlinearity in structure. • Controllers should have robust stability and performance at different frequencies. • Robust control approach is one of the options. • H2 controllers • H∞ controllers • Controllers based on μ synthesis

11. Designing filters • Weights can incorporate our knowledge of the disturbance and performance objectives into the controller. • High loop gain is required in the low frequency range where the disturbance is most effective at that range. • Small loop gains at high frequency ranges are required for sensor noise error reduction and preventing saturation in actuators.

12. The shape of the filters Three filters are used Wd: disturbance shaping filter. Kanai-Tajimi filter is used to model earthquakes power spectrum. A second order transfer function is introduced to reshape the frequency content.

13. The shape of the filters We: weighting functions on desired system response and control force. Constant diagonal matrices are used here. Wact: actuator filter. This filter is used to have more flexibility on changing the loop shape to a desired configuration.

14. Optimization on filter parameters The following structure is considered for optimizing the gain weights • The performance of the system is sensitive to the selected parameters of filters. • An optimization process on control parameters were done.

15. Resultant System Transfer Functions Control Force Earthquake Drift Controller Structure Acceleration Noise • Following transfer functions are required to be checked • Transfer function from earthquake excitation to acceleration and drift • Transfer function from input noise to acceleration and drift • Transfer function from earthquake excitation to control force • Transfer function from input noise to control force

16. Resultant System Transfer Functions Control Force Earthquake Drift Controller Structure Acceleration Noise • Following transfer functions are required to be checked • Transfer function from earthquake excitation to acceleration and drift • Transfer function from input noise to acceleration and drift • Transfer function from earthquake excitation to control force • Transfer function from input noise to control force

17. Resultant System Transfer Functions Control Force Earthquake Drift Controller Structure Acceleration Noise • Following transfer functions are required to be checked • Transfer function from earthquake excitation to acceleration and drift • Transfer function from input noise to acceleration and drift • Transfer function from earthquake excitation to control force • Transfer function from input noise to control force

18. Resultant System Transfer Functions Control Force Earthquake Drift Controller Structure Acceleration Noise • Following transfer functions are required to be checked • Transfer function from earthquake excitation to acceleration and drift • Transfer function from input noise to acceleration and drift • Transfer function from earthquake excitation to control force • Transfer function from input noise to control force

19. Transfer Functions From Earthquake Excitation to Output Response

20. Transfer Functions From Input Noise to Output Response

21. Transfer Functions From Inputs to Control Force

22. f(t) FD f(t) FI FS Fractional Order Filters • What is fractional order calculus? • A mass-damper-spring system • Conventional models • Hook’s law • Ideal viscoelastic materials • Second Newton’s law • New fractional models

23. Fractional Order Filters • Definitions of fractional derivatives and integrals • Rienmann-Liouville • Grunvald-Letnikov • Caputo • Miller-Ross • Caputo (1967)

24. Modified Oustaloup’s approximation algorithm for Sα by Xue et al. Fractional Order Filters where Using Oustaloup’s approximation

25. Fractional Order Filters • Fractional filter is defined to finely tune the original H2 controller.

26. Fractional Order Filters • Shape of the fractional order filter for a=1.5, b=41.33, and c=45 • The controller configuration • The filter will slightly affect the response parameters that are more sensitive to controller parameters, like residual strain.

27. Results • Acceleration and drift response for Eq #23

28. Results • Acceleration and drift response for Eq #54

29. Results

30. Results • Comparing numerical and experimental results • Numerical and experimental results show similar trends • Maximum drift from numerical analysis are larger than maximum drift from FHT • FHT gives larger residual drift compared to numerical analysis

31. The H2 controller is designed for 28 dampers at each story level. But it still works for the case with 2 dampers. Introducing Wact improves the response considerably. Having fractional order filter as a pre filter to H2 controller reduces residual drifts for earthquakes of bin 2 by 3 percent. Conclusions