Optimal Vibration Control of Guyed Masts

B.Błachowski, W.Gutkowski

Institute of Fundamental Technological Research

Polish Academy of Sciences

Warsaw, Poland

Motivations

Ice and windpressure are themost commonreasons of mastfailures.

Mechanical model

Wind forcesVelocity

measurement

Control forces

Performance output

Feedback control

Small displacements

Guyed mast dynamics

Finite element model

)()()()()( ttfttt uf uBBKqqDqM

Discrete time state-space model

nt)(Measureme

output) ce(Performan

)(

)(

2

1

211

kkk

kk

kc

kc

kkkkkt

tf

vxCy

xCz

η

ηxuBBAxx

Modal truncation

)()()()( tttt ccrrcc ηφηφηφq

Wind fluctuations

Davenport filter

)()( fSjH

tcoefficien drag density,air where

)1(4800)(

spectrum and forces Wind

2/322

2

2

1

1

0

sin

cos

0

sin

cos

da

f

da

da

da

da

f

Cρ

S

CA

CA

CA

CA

B

Controllability

modes ofnumber where

2rank 212

22

22

N

NN BABAABB

Classical criterion of controllability

Modal controllability index

NjjTuu

Tjj ,,2,1 φBBφ

Modal transformation

scoordinate modal)( where

)()()(2)( 2

t

tttt

c

uT

ccc

η

uBφηΩηΞΩη

Mode 1 Mode 4

Mode 7 Mode 23

Modal analysis shown displacements are out of scale

Control system scheme

Hydraulic actuator

Digital controller

Velocity measurement

Cable

2P

1P

Force

Control strategy

Quadratic Programming

State estimator

Controller

Wind

Control input Measured output

Model updating

Performance output

State estimator concept

2B

1B

Delay 2C

kv

ku

2B Delay

A

2C

ky

A

eK

kf

kx̂

1kx kx

1ˆ kx kx̂

Guyed mast

Guyed mast model

Estimator

Optimal estimator

klTlk

klTlk

T

E

EE

Rvv

QffMxx

)(

errors tmeasuremen and

)()(

esdisturbanc condition, Initial

000

TTkk

kT

kT

kkk

Tke

kkekkk

m

mk

111

21

222

12

0221

)(

,,1,0

givenˆ)ˆ(ˆˆ

state of estimate likelihood Maximum

QBBAAPM

MCRCMCCMMP

RCAPK

xxCyKuBxAx

x

Optimal control

Performance index

output eperformanc where

1

0

k

kTk

n

kk

Tknn

Tn

z

ΨuuΦzzzΦzJ

Quadratic programming with constraints

maxmin

2

UUU

VUWUU

T0JJ

1

1

0

nu

u

u

U

VUWUUUbLU

2minarg

TOPT

Numerical simulations

Conclusions

• A 3D FEM model of guyed mast under control forces wascreated

• External loading was included in the form of a stochasticwind model of Davenport spectrum

• Controllability of individual mode shapes of the mast wasdetermined

• A model based state estimator for the mast was constructed

• Numerical simulation of a control process was presented,displaying significant reduction in vibration amplitudes ofthe top of the mast

Acknowledgements

The research was carried out with the financial supportprovided by the State Committee for Scientific Research ofPoland (KBN) under grant No. 5T07A 001 23 and this isgratefully acknowledged.