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: abdshafiee@cc.usu.edu

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.