Fractional Order LQR for Optimal Control of Civil Structures Abdollah Shafieezadeh*, Keri Ryan*, YangQuan Chen+ *Civil and Environmental Engineering Dept. +Electrical and Computer Engineering Dept. Utah State University Speaker: Abdollah Shafieezadeh ASME DETC 3 RD FDTA, Sept. 4, 2007

Outlines  Goals of structural control  Introducing fractional calculus  Optimization process  Combined FOC-LQR strategies  Results  Future works  Conclusions

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.  Fractional order filters, offering more features are applied here.

Outlines  Goals of structural control  Introducing fractional calculus  Optimization process  Combined FOC-LQR strategies  Results  Future works  Conclusions

Goals of Structural Control  Functionality  Safety  Human comfort  Flexibility for design

Outlines  Goals of structural control  Introducing fractional calculus  Optimization process  Combined FOC-LQR strategies  Results  Future works  Conclusions

What is Fractional Calculus?

A mass-damper-spring system Conventional models Hook’s law Ideal viscoelastic materials Second Newton’s law New fractional models f(t) FIFI FDFD FSFS Example of A Fractional Order System

Mathematical Definition Definitions of fractional derivatives and integrals Rienmann-Liouville Grunvald-Letnikov Caputo Miller-Ross  Caputo (1967)

 Modified Oustaloup’s approximation algorithm for S α by Xue et al. where Using Oustaloup’s approximation Numerical Solution

Outlines  Goals of structural control  Introducing fractional calculus  Optimization process  Combined FOC-LQR strategies  Results  Future works  Conclusions

Optimization Process  Analytical optimization Given a set of gains for output and input control force, LQR approach gives the best controller.  Numerical optimization The output is sensitive to chosen gains H 2 method leads to an optimal controller in the sense of 2-norm if the input disturbance is white noise.

Numerical Optimization Process  Performance Index  RMS response for frequent moderate events like wind  MAX response for extreme events like earthquake  Selection of β 1 and β 2 are based on the control objectives  64 artificially generated earthquakes are used in optimization part.

Outlines  Goals of structural control  Introducing fractional calculus  Optimization process  Combined FOC-LQR strategies  Results  Future works  Conclusions

Combined FOC-LQR Strategies  Case (1)  Case (2a)  Case (2b)  Case (3)

Combined FOC-LQR Strategies loop diagram Case (1), (2a), and (2b) Case (3)

Civil Structure Model  Governing Equation  State Space Model  Natural periods of the building are 0.3 and 0.14 seconds  Damping is 2% in each mode

Outlines  Goals of structural control  Introducing fractional calculus  Optimization process  Combined FOC-LQR strategies  Results  Future works  Conclusions

Results  Case (1) K lqr is constant and a search is done to find α  In other cases, Matlab Optimization Toolbox is used to find optimal gains and fractional orders

Response of Controllers to Artificial Ground Motions

Results El Centro Earthquake

Results Northridge Earthquake

Response of Controllers to Real Ground Motions

Results  The structural performance for El Centro earthquake is much better than for Kobe and Northridge earthquakes  Filter model: The Kanai-Tajimi filter used in optimization gives similar trend to real ground motions in frequency domain but not in time domain  Saturation limit: Larger ground motions require larger control force. Kobe and Northridge have PGA of 2.5 times larger than El Centro

Outlines  Goals of structural control  Introducing fractional calculus  Optimization process  Combined FOC-LQR strategies  Results  Future works  Conclusions

Future Works  A more realistic structure is considered. The building model is nonlinear which can form plastic hinges at the column ends. MR dampers which are more applicable replaced ideal actuators.  General H 2 robust control approach is used as the primary controller The performance is enhanced by introducing some filters for input disturbance, output, and actuator.

FHT Facility at University of Colorado

 part of the structure which is numerically hard to model is constructed at lab and tested  Other parts of the structure is numerically modeled in computer  The interaction between superstructure and substructure are applied by actuators Hybrid Testing

Outlines  Goals of structural control  Introducing fractional calculus  Optimization process  Combined FOC-LQR strategies  Results  Future works  Conclusions

Conclusion  Several combinations of FOC and LQR were considered.  64 artificially generated earthquakes were used to optimize the controller gains.  Case (2a) gives the best performance. It reduces the performance index by 36% compared to LQR.  Controllers led to the same trend in performance for real earthquakes as the artificial ones.