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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.