fractional order lqr for optimal control of civil structures abdollah shafieezadeh*, keri ryan*,...
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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
Email: [email protected]
2007 ASME DETC 3RD 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
)(tfbxaxxm
f(t)
f(t)FI FD
FS
)(tfkxxcxa
kxFS
xcFD
xmFI
Example of A Fractional Order System
Mathematical Definition
• Definitions of fractional derivatives and integrals
• Rienmann-Liouville• Grunvald-Letnikov• Caputo• Miller-Ross
Caputo (1967)
1
0
1 0n
k
kk xssXstxdt
dL
Modified Oustaloup’s approximation algorithm for Sα by Xue et al.
hb
10
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 H2 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 β1 and β2 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 Strategiesloop 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) Klqr 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
ResultsEl Centro Earthquake
0 10 20 30 40-2
-1
0
1
2
1st S
tory
Max D
rift
0 10 20 30 40-1
-0.5
0
0.5
1
1st S
tory
Max A
cc
0 10 20 30 40-2
-1
0
1
2
Time (sec)
2n
d S
tory
Max D
rift
0 10 20 30 40-2
-1
0
1
2
Time (sec)
2n
d S
tory
Max A
cc
WO Control
LQRCase (2a)
ResultsNorthridge Earthquake
0 10 20 30-6
-4
-2
0
2
4
1st S
tory
Max D
rift
0 10 20 30-2
-1
0
1
2
3
1st S
tory
Max A
cc
0 10 20 30-4
-2
0
2
4
Time (sec)
2n
d S
tory
Max D
rift
0 10 20 30-4
-2
0
2
4
Time (sec)
2n
d S
tory
Max A
cc
WO Control
LQRCase (2a)
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 H2 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 TestingHybrid 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.