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Research ArticleVibration Control of Wind Turbine Blade Based on
Data Fittingand Pole Placement with Minimum-Order Observer
Tingrui Liu and Lin Chang
College of Mechanical & Electronic Engineering, Shandong
University of Science & Technology, Qingdao 266590, China
Correspondence should be addressed to Tingrui Liu;
[email protected]
Received 26 September 2017; Accepted 1 April 2018; Published 24
May 2018
Academic Editor: Marc Thomas
Copyright © 2018 Tingrui Liu and Lin Chang. This is an open
access article distributed under the Creative Commons
AttributionLicense, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is
properlycited.
Vibration control of wind turbine blade withminimum control cost
is investigated. Realization ofminimum control cost is based onpole
placement with minimum-order observer (PPMO). The blade analysis
employs a novel compromise method between the 2Dairfoil analysis
method and the 3D coupled blade body analysis method based on data
fitting. It not only ensures certain accuracy,but also greatly
improves the speed of calculation. The Wilson method, developed on
the basis of the blade momentum theory, isadopted to optimize the
structural parameters of the blade, with all parameters fitted as
general model Sin6 (Sum of Sine) fittingcurves. Also the
aerodynamic coefficients based on data obtained by Xfoil software
are fitted. Pole placement technology based onminimum-order
observer is applied to control unstable vibrations of vertical
bending and lateral bending with minimum controlcost characterized
by the energy consumption of the controller.The pole placement
technology is a novel pole assignment techniquebased on self-poles
derived from constant stable eigenvalues, which can effectively
avoid the mismatch problems caused by poleselection.The superiority
of PPMO can be apparently demonstrated by comparison of linear
quadratic regulator (LQR). Analyticalproof of the control accuracy
and feasibility analysis of the physical realization of the PPMO
algorithm are also investigated byexperimental platform of
hardware-in-the-loop simulation.
1. Introduction
The flutter analyses of wind turbine blades usually includetwo
cases: typical blade section analysis and the three-dimensional
coupled blade body analysis. The former iswidely used in quick
study of flap/lag vibrations or bend-ing/twist vibrations of single
cross-section with a wide rangeof structural and external
parameters [1, 2]. An aeroservoe-lastic model was presented to
study the microtab control ofthe blade section undergoing moderate
stall flutter and deepstall flutter separately [1]. Effects of
aeroservoelastic pitchcontrol for flutter suppression on wind
turbine blade sectionsubjected to combined flap/lag motions were
studied, whichwas dedicated to solving destructive flapwise and
edgewiseinstability of stall-induced flutter [2]. Although this
methodis simple and convenient, it cannot fully reflect the
vibrationbehavior of the whole blade body.The latter method
involvesthe study of dynamic analysis of coupled displacements
andaeroelastic characteristics of the whole blade body [3–6],which
can more accurately characterize the dynamics of
the blade. Li et al. [3] presented the analysis of
dynamiccharacteristics of horizontal axis wind turbine blade,
wherethe mode coupling among axial extension, flap
vibration,lead-lag vibration, and torsion was emphasized. Lee et
al.[4] investigated the performance and aeroelastic
character-istics of wind turbine blades based on flexible
multibodydynamics, a new aerodynamic model, and the
fluid-structureinteraction approach. Members of our project group
[5]also investigated the stall-induced aeroelastic stability ofwind
turbine blade with bending-bending-twist coupling.Andersen et al.
[6] investigated the fatigue load reductionapplying trailing-edge
flaps on wind turbine blade that isanother structural form of the
blade, rather than the generalblade structure. Load control of wind
turbine was analyzed,with requirements for load monitoring sensors,
sensingtechnologies, and applications discussed in [7]. However,all
this literature involves complicated blade structures
andaeroelastic behavior, in which their solutions involve
bothcomplex decoupling problems and decomposition of aerody-namic
forces along the blade spanwise direction.Whether it is
HindawiShock and VibrationVolume 2018, Article ID 5737359, 12
pageshttps://doi.org/10.1155/2018/5737359
http://orcid.org/0000-0002-2680-7289https://doi.org/10.1155/2018/5737359
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2 Shock and Vibration
based on numerical analysis or by finite element simulation,the
solution process must be a time-consuming process[5]. Needless to
say, actual vibration control on the basisof the actual controller
hardware is impossible due to thecomplexity of these algorithms or
complicacy of the controlsystem hardware itself.
Structural failures observed in recent years of large
windturbine blades have been mostly attributed to severe
vibra-tions in flapwise and edgewise directions [8]. Some
earlierstudies involved the flap/lead-lag stability analysis based
oneigenvalue problem [9] and stability control based on fuzzyPID
algorithm and LQR controller [10], with the actualanalyzed
structure being single blade section.Members of ourgroup
investigated a variety of methods of intelligent controlfor
flap/lead-lag flutter, also based on single blade section [2].As
for analysis of the whole blade body as shown in [3], thenumerical
difference methods adopted to solve the systemintegrodifferential
equations of the flap vibration and lead-lag vibration are so
complex that it is impossible to realize therelevant control
algorithms in the actual hardware.
Since the cross-sections of different sizes experiencedifferent
loads, the combined effects of different loads havefar surpassed
those of the same loads of the typical sectionsof the same sizes
described in related references [1, 2, 9, 10].Therefore, the
effectiveness of design parameters of cross-sections of different
sizes needs to be confirmed by aeroelasticstability analysis for
safety reasons. At the same time, thecontrol cost of flap/lead-lag
vibrations and the hardwareimplementation of the control algorithm
are not mentionedin above related researches. The “control cost”
here mainlyrefers to the amplitude of the controller, the energy
consumedby the controller, or the power required to drive the
controller.
In present work, a novel compromise method (NCM)between the 2D
typical section analysis method and the3D coupled blade body
analysis method is investigated tostudy vertical and lateral
vibrations of blade tip. Then theanalysis accuracy might be between
the accuracies of the twostructural model analysis methods due to
the approximatecompromise behavior.The NCMnot only preserves the
basiccharacteristics of typical section model, but also extends
theintegral behavior of blade body along the spanwise directionbut
avoids the decoupling problem of partial differentialequations.
More importantly, the vibration control of blademodel based on NCM
is more efficient and easier to beimplemented in controller
hardware. In addition, realizationof minimum control cost is based
on pole placement withminimum-order observer (PPMO), which can be
depictedas pole placement technology based on
minimum-orderobserver.The PPMO is applied to control unstable
vibrationsof vertical/lateral directions based on self-poles
derived fromconstant stable eigenvalue analysis, which is exactly
theadvantage of the method NCM. At the same time, theminimum-order
observer is applied in PPMO to estimate theunmeasurable
variables.
It should be stated that the emphasis of present studyis on the
application of the control method, but not on theaerodynamic
nonlinearities associated with dynamic stall inwind turbines [1].
However, the control algorithm proposedhere can also provide a
method to solve the problem afterlinearization of the stall
nonlinear system.
Z
y
Windspeed U
X
Chordwise
Attack angle
Lateralbending
Verticalbending
U
Pretwist angle
Blade sectionlocated at a
distance r from
Rotating plane
E0
Er1
Er2
The hub E0
ΩrV0
Figure 1: The blade structure and coordinate system.
2. Theoretical Model
2.1. Equations ofMotions of Blade Section. Consider the
bladestructure and coordinate system in Figure 1, the length of
theblade is 𝑅 = 30m in 𝑥 direction and it has the rotating speedΩ,
normal to the plane of rotation. The origin of the rotatingaxis
system is located at the rigid root. The
characteristiccross-sectional dimension, chord length is 𝑐(𝑟) at a
distance 𝑟from the hub 𝐸0. It is assumed that the blade section
locatedat the distance 𝑟 is connected by 𝐸𝑟1 in lead-lag direction
𝑦and by𝐸𝑟2 in flapwise direction 𝑧, where both𝐸𝑟1 and𝐸𝑟2 arerigidly
connected to the hub 𝐸0. The wind velocity is denotedby 𝑈. The
attack angle is denoted by 𝛼, with pretwist angleby 𝜃. It is
assumed that the rotating plane will not changedue to yaw, and as
the design of the pretwist angle increasesthe torsional stiffness,
the elastic torsional displacements ofdifferent blade sections can
be approximately neglected. Therotating speed Ω can be determined
by 𝑈, according tosubsequent structural parameters in Section
2.2.
Consider the single blade section in Figure 1; the equa-tions of
motions in 𝑧 and 𝑦 directions are derived usingLagrange’s equations
as depicted in [2]. Meanwhile consider-ing dynamic stiffening
effect in [9], the equations of vertical(𝑧) and lateral (𝑦) motions
of blade section are expressed asfollows:
�̈� cos (𝜃) − ̈𝑦 sin (𝜃) + 2Ω𝜉𝑧𝜔𝑧 (�̇� cos (𝜃) − ̇𝑦 sin (𝜃))+
Ω2𝜔2𝑧 (𝑧 cos (𝜃) − 𝑦 sin (𝜃))= 12𝜌𝑎𝜌−1𝑏 𝑐𝑉20 (𝐶𝐿 cos𝜓 + 𝐶𝐷
sin𝜓)
(1a)
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Shock and Vibration 3
a
Fitted curveData
0.2 0.4 0.6 0.8 10r/R
−0.5
0
0.5
1
1.5
a b
0
0.2
0.4
0.6
0.8
0.2 0.4 0.6 0.8 10
(a) 𝑎 and 𝑎𝑏
0.2 0.4 0.6 0.8 10r/R
0
0.5
1
1.5
(r
ad)
0
20
40
60
b(k
g/m
)
0.2 0.4 0.6 0.8 10
Fitted curveData
(b) 𝜌𝑏 and 𝜃
Figure 2: The optimized data of the key parameters and
corresponding fitting curves.
̈𝑦 cos (𝜃) + �̈� sin (𝜃) + 2Ω𝜉𝑦𝜔𝑦 ( ̇𝑦 cos (𝜃) + �̈� sin (𝜃))+
Ω2𝜔2𝑦 (𝑦 cos (𝜃) + 𝑧 sin (𝜃))= 12𝜌𝑎𝜌−1𝑏 𝑐𝑉20 (𝐶𝐿 sin𝜓 − 𝐶𝐷 cos𝜓)
,
(1b)
where the incoming velocity is𝑉0, with incoming angle being𝜓 = 𝛼
+ 𝜃; the mass per length of blade section is describedby 𝜌𝑏; the
natural angular frequencies of motions in 𝑦 and𝑧 directions are
described by 𝜔𝑦 and 𝜔𝑧, respectively; thedamping ratios in 𝑦 and 𝑧
directions are denoted by 𝜉𝑦 and𝜉𝑧, respectively; the Lift
coefficient and Drag coefficient aredescribed as 𝐶𝐿 and 𝐶𝐷,
respectively.2.2. Optimal Design of Airfoil Parameters. As the
actualsectional position is different, the structural parameters
ofairfoil and blade body are also changed. In order to takefull
account of the stability performance, the blade structureis
optimized by Wilson method [11, 12], and the aeroelasticmodel and
the aerodynamic coefficients are fitted basedon the blade element
momentum (BEM) theory [13, 14].The design uses 21 cross-sections,
with airfoil type beingNACA63-215.
TheWilsonmethod is based on BEM theory.Themethodaims to maximize
the utilization coefficient of wind energy.When designing the
blade, it can take into account theinfluence of blade tip loss and
airfoil lift-drag ratio on theoptimum aerodynamic performance of
the blade, so thedesigned blade has high wind energy conversion
efficiency.Hence the incoming velocity is 𝑉0 and the incoming angle
𝜓can be expressed as follows:
𝑉0 = √[(1 − 𝑎)𝑈]2 + [(1 + 𝑎𝑏)Ω𝑟]2 (2a)tan𝜓 = tan (𝛼 + 𝜃) = (1 −
𝑎)𝑈(1 + 𝑎𝑏)Ω𝑟 (2b)
where 𝑎 and 𝑎𝑏 are the initial axial induction factor andangular
induction factor, respectively. According to BEMtheory, they can be
deduced as follows:
𝑎 = 14 sin2𝜓/𝜎 (𝐶𝐿 cos𝜓 + 𝐶𝐷 sin𝜓) + 1 (3a)𝑎𝑏 = 14 sin𝜓 cos𝜓/𝜎
(𝐶𝐿 sin𝜓 − 𝐶𝐷 cos𝜓) − 1 (3b)
where 𝜎 is the blade solidity.The acquisition of 𝑎 and 𝑎𝑏 can be
iteratively repeated by
the Wilson method as follows [11]:
(a) Assume a reasonable set of initial values of 𝑎 and 𝑎𝑏.(b)
The incoming angle 𝜓 and attack angle 𝛼 are calcu-
lated by (2a) and (2b).(c) 𝐶𝐿 and 𝐶𝐷 are obtained from the
corresponding
airfoil aerodynamic data library according to 𝛼.(d) A new set of
parameters of 𝑎 and 𝑎𝑏 is obtained by
using (3a) and (3b).(e) The steps are repeated continuously
until the two
adjacent calculated values of 𝑎 and 𝑎𝑏 converge.The final
calculated values of 𝑎 and 𝑎𝑏, as well as their
curves fitted as expressions of generalmodel Sin6 (Sumof
six-order Sine) [15], are shown in Figure 2(a). Herein, 𝑟 stands
forrotation radius of single cross-section.
According to the chord length formula [12], each chordlength
𝑐(𝑟) at distance 𝑟, can be calculated as
𝑐 (𝑟) = 8𝜋𝑎𝑟 (1 − 𝑎) sin2𝜓𝐵0𝐶𝐿 (1 − 𝑎)2 cos𝜓, (4)where 𝐵0 = 2 is
the number of blades.The curve 𝑐(𝑟) can alsobe fitted as general
model Sin6 expression (see Table 1).
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4 Shock and Vibration
Table 1: Coefficients and errors of the fitted parameters.
Items Parameters 𝑓(𝑥)𝑎 𝑎𝑏 𝑐 𝑅𝑓 𝜌𝑏 𝜃 𝐶𝐿 𝐶𝐷𝑎1 0.5392 0.3246 5.561
5.33 21.66 1.401 0.8691 3.309𝑏1 3.047 3.545 3.943 2.92 4.656 2.976
2.073 0.351𝑐1 −0.3229 0.9545 −0.3818 1.233 0.2604 0.9264 0.04122
1.488𝑎2 0.2553 0.7905 141.9 0.1832 25.49 0.921 0.307 0.715𝑏2 6.71
9.677 8.576 18.32 8.564 4.827 4.013 2.037𝑐2 0.692 0.5523 −0.3956
1.246 −0.2689 2.951 −0.05805 −1.58𝑎3 0.077 0.6918 139.1 8.023 12.5
0.1057 0.1994 2.658𝑏3 12.02 11.39 8.63 28.74 12.53 11.32 5.892
0.4024𝑐3 0.8957 2.719 2.71 −3.652 0.7943 2.427 −0.05113 −1.663𝑎4
0.03021 0.1294 0.332 0.3119 4.097 0.03815 0.1706 −0.01812𝑏4 19.04
18.53 18.44 12.93 21.25 17.08 0.7129 2.987𝑐4 0.2307 1.815 −0.4086
2.527 −1.891 2.236 0.8469 −1.528𝑎5 0.01978 0.08002 0.2582 4.864
2.297 0.009333 0.1377 0.01948𝑏5 25.12 24.63 24.52 3.297 25.08 26.92
7.613 3.949𝑐5 0.07678 1.495 −1.307 4.787 −1.4 5.012 −0.1192 1.586𝑎6
0.01366 0.0478 0.1401 8.004 0.3702 −0.009578 0.08546 0.000341𝑏6
31.36 30.07 30.59 28.82 36.38 34.03 9.298 4.816𝑐6 −0.1892 1.455
−1.185 −0.566 −5.521 0.4797 −0.1933 −1.575SSE 0.002921 0.01901
0.06146 0.09117 0.1135 0.000607 0.07078 0.004993R-square 0.9755
0.9913 0.9896 0.9958 0.9997 0.9998 0.9956 0.9992Adjusted R 0.8368
0.9423 0.9304 0.9721 0.9982 0.9986 0.9926 0.9979RMSE 0.0312 0.07961
0.04526 0.01743 0.06151 0.01422 0.05321 0.02131
At the same time, the linear density 𝜌𝑏 of the blade can
beapproximately estimated and fitted in Figure 2(b), accordingto
the structure parameter 𝑐(𝑟) and an expression of a densityfactor𝑅𝑓
= 𝜌𝑎𝑐2/𝜌𝑏 in terms of the air density 𝜌𝑎 in [15]. Also,the density
factor𝑅𝑓 has the corresponding fitting expressionand fitting
coefficients (see Table 1). In addition, consideringthe influence
of the theories of blade cascades [16] on theattack angle, the
optimal angle of attack 𝛼 can be obtainedaccording to the Reynolds
number.Then the pretwist angle 𝜃is also obtained in Figure 2(b)
according to (2a) and (2b).
The fitting curves of the 4 parameters in Figure 2 donot mean
that they are more accurate than the optimizeddata itself. However,
the equation of the fitting curve can bedirectly substituted into
the aeroelastic system for integraloperation. As mentioned above,
the analysis and calculationprocess can be simplified as much as
possible. Equation (5)is the expression of the fitted six-order
Sin6 model of therelated parameters, with the corresponding
coefficients andthe errors in the fitting process displayed in
Table 1.
𝑓 (𝑥) = 6∑𝑖=1
𝑎𝑖 sin (𝑏𝑖𝑥 + 𝑐𝑖) , (5)where 𝑎𝑖, 𝑏𝑖, and 𝑐𝑖 are the
corresponding structural items; 𝑥 =𝑟/𝑅 for 𝑎, 𝑎𝑏, 𝑐, 𝜌𝑏, and 𝜃; 𝑥 =
𝛼 for 𝐶𝐿 and 𝐶𝐷.
After using graphical methods to evaluate the goodnessof
fitting, some items of the goodness-of-fit statistics for
parametric models should be examined. The items includethe sum
of squares due to error (SSE), R-square, Adjusted R-square, and
root mean squared error (RMSE), which can befound in Table 1.
It can be seen in Table 1 that the SSE values closer tozeros
indicate that the models have smaller random errorcomponents and
that the fitting will be more useful forprediction; the R-square
values closer to 1 indicate that agreater proportion of variance is
accounted for by the model,which means that the fitting explains
greater proportion ofthe total variation in the data about the
average. Also, theadjusted R statistics closer to 1 indicate a
better fitting, with aRMSE value closer to 0 indicating a fitting
that is more usefulfor prediction as an estimate of the standard
deviation of therandom component in the data.
2.3. The Fitted Aerodynamic Coefficients.
Aerodynamicscharacteristics of airfoil are the basis of aerodynamic
perfor-mance design of blades. Meanwhile the acquisition of
airfoilaerodynamic data is often the most difficult part of
bladedesign. The accuracy and completeness of the aerodynamicdata
obtained are of great significance to the analysis ofthe dynamic
behavior of the blade. The blade of the windturbine usually
operates at the angle of attack range of −180∘∼180∘ with a wide
range of Reynolds number. Hence theaerodynamic coefficients of
airfoils in corresponding rangesare also required. Considering the
actual situation of the
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Shock and Vibration 5
Data DataCL CD
−1
−0.5
0
0.5
1
1.5
Aero
dyna
mic
coeffi
cien
ts
0.50 1 1.5−1 −0.5−1.5 (rad)
Figure 3: The curves of the aerodynamic lift coefficient 𝐶𝐿 and
theaerodynamic drag coefficient 𝐶𝐷 against the angles of attack
𝛼.
present design, it is preliminarily estimated that the angle
ofattack will change between −90∘ ∼90∘. The acquisition stepsof
aerodynamic coefficients are as follows [11]:
(a) Firstly, the lift and drag coefficients within a range
ofsmall attack angles (−15∘ ∼15∘) are obtained by usingXfoil
software [11].
(b) Using the worksheet “TableExtrap” in AirfoilPrepv2p2 program
that prepares the corresponding airfoildata for AeroDyn software
[17], the range of smallattack angles is extended to range of −90∘
∼90∘.
(c) The validity of the fitting processes of
structuralparameters is demonstrated by the error data dis-played
in Table 1; the aerodynamic coefficients canalso be fitted by the
acquired data within the rangeof −90∘ ∼90∘ [15].
Figure 3 shows the curve of the aerodynamic lift coeffi-cient 𝐶𝐿
(data 𝑦1) versus the angle of attack 𝛼 (data 𝑥1), aswell as the
aerodynamic drag coefficient 𝐶𝐷 (data 𝑦2) againstthe angle of
attack 𝛼 (data 𝑥2), according to the acquired datafrom above steps
(a∼c). The structural items of coefficients(𝐶𝐿 and 𝐶𝐷) and the
corresponding errors in the fittingprocess are also displayed in
Table 1.
It should be emphasized that, in view of the actualsituation of
the design, only the aerodynamic coefficientswere fitted to the
angles of attack within the range of −90∘∼90∘. If the large angles
of attack and deep stall-inducedsituation are put into
consideration, this design method canalso be applied to the angles
of attack within the range of−180∘ ∼180∘.2.4. Governing System and
Solution. The 3D coupled bladebody analysis method fully considers
all the cross-sections
along the spanwise length, and the cross-section displace-ments
have complex coupling behaviors. Its solution needsstrip
theoretical decomposition, aerodynamic decomposi-tion, and
decoupling using Galerkin method [15]. The NCMmethod used here only
extends the integral behavior of the3D coupled blade body analysis
method on the basis of the2D typical section analysis method.
Inserting (5) into (1a)and (1b) and considering the integral
behavior along theblade spanwise corresponding to 𝑟 from 0 to R
result in theequations governing the motions of the aeroelastic
system asfollows:
𝑀 ̈𝑞 + 𝐶 ̇𝑞 + 𝐾𝑞 = 𝑄, (6)where 𝑞 is the variable vector.
Furthermore, in order to carryon subsequent analysis, (6) needs to
be transformed into thestate space expression as follows:
�̇� = 𝐴𝑌 + 𝐵𝑌0 = 𝐶𝑌, (7)
where 𝑌 = [𝑧 𝑦 d𝑧/d𝑡 d𝑦/d𝑡]T is the new variable vector.Some
cases intended to highlight the effects of the
damping ratios and wind velocities on the vibration andstability
of different sectional radius 𝑟will be presented. Basicstructure
parameters are as follows: 𝜔𝑧 = 2 rad/s, 𝜔𝑦 =3 rad/s, 𝜉𝑧 = 𝜉, 𝜉𝑦 =
2𝜉. Figure 4 illustrates the real parts ofboth flap eigenvalues of
vertical direction and lag eigenvaluesof lateral direction at
different positions 𝑟 (from 1/6R to Rat intervals of 1/6R), under
conditions of wind velocities 𝑈from 5m/s to 50m/s at intervals of
5m/s and damping ratios𝜉 from 0 to 0.1 at intervals of 0.01. All
the real parts ofeigenvalues in both directions at every position 𝑟
are less thanzeros, which means absolute stability. In fact, the
eigenvalueshere involve only homogeneous equation structure in
theleft-hand side terms in (7), which only concerns the massmatrix
coefficients, dampingmatrix coefficients, and stiffnessmatrix
coefficients in (1a) and (1b). If there is no aerodynamicforce,
most fundamentally, this implies constant stability ofthe
structure. Therefore, in view of this characteristic, nomatter how
the change of wind speed and damping ratio,eigenvalues of
homogeneous equation in (7) are also boundto be stable.
Furthermore, in view of this characteristic, anovel pole assignment
technology based on self-poles derivedfrom these stable eigenvalues
of homogeneous equation partin (7) is proposed, which is named as
PPMO technology inSection 3.
As is known to all, the first step in the pole-placementdesign
approach is to choose the locations of the desiredclosed-loop
poles. The most frequently used approaches areto choose such poles
based on experience and the quadraticoptimal control approach.
Otherwise, the system responsesbecome very fast, with very large
amplitudes; or the polescannot balance the acceptable responses and
the amount ofcontrol energy required. And the pole assignment
technologybased on self-poles derived from these stable eigenvalues
canjust overcome these shortcomings and save the trouble infinding
the right poles.
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6 Shock and Vibration
00.05
0.1
10 20 3040 50
U
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
Flap
eige
nval
ues
00.05
0.1
10 20 3040 50
U
−10
−8
−6
−4
−2
0
2
Lag
eige
nval
ues
Figure 4:The real parts of flap eigenvalues of vertical
direction and lag eigenvalues of lateral direction at different
positions 𝑟 from 1/6R to Rat intervals of 1/6R.
3. Application of Pole Placement withMinimum-Order Observer
(PPMO)
3.1. Pole Placement Based on Self-Poles. Different from
spec-ifying only dominant closed-loop poles in the
conventionaldesign approach [18], the present self-pole placement
speci-fies all closed-loop poles including poles of two
displacementvariables and two velocity variables. The assignment
ofdisplacement poles can be efficiently performed owing
toaccuratelymeasured displacements. However the assignmentof
velocity poles requires the inclusion of a state observerin the
system. There is also a requirement for the closed-loop poles to be
placed at arbitrarily chosen locations, whichimplies that the
system should be completely state control-lable. As depicted above,
constant stability of the structure in(7) without action of
aerodynamic forces decides the desiredlocations where the regulator
poles can be placed, which isexactly what is said of the self-pole
placement. The regulatorpoles are the eigenvalues of𝐴−𝐵𝐾, where𝐾 is
determined bythe control signal 𝑢 = −𝐾𝑌. The method of self-pole
place-ment avoids the difficulty of selecting the desired poles
andeffectively improves the performance of the pole
assignment,which shows the rapidity of the configuration process
and theeffectiveness of the configuration results. Present study
showsthat the selection of the desired closed-loop poles or
thedesired characteristic equation based on self-pole placementhas
not only the response speed superiority, but also thesuperiority in
dealing with the sensitivity to disturbance andmeasurement
noise.
According to Ackermann’s formula [19], the state-feedback gain
matrix 𝐾 can be determined as follows:
𝐾 = [0 0 0 1] [𝐵 𝐴𝐵 𝐴2𝐵 𝐴3𝐵]−1 𝜙 (𝐴) , (8)where 𝜙 satisfies the
desired characteristic equation:
|𝑠𝐼 − 𝐴 + 𝐵𝐾| = 𝑠4 + 𝛼1𝑠3 + 𝛼2s2 + 𝛼3s + 𝛼4 = 0. (9)3.2.
Minimum-Order Observer. During the pole-placementprocess, it is
assumed that all state variables are available
1
Minimum–order observer
A
B
Transformation
u−K
+
+1
sC
Y0
Y1 = [xaXb]
Figure 5: The feedback system with a minimum-order observer.
for feedback. In present study, velocity variables are
unmea-surable for direct feedback and need to be observed.
Noticethat, in order to deal with velocity variables, one needsthe
derivative of displacement variables, which presents adifficulty
due to the reason that differentiation amplifiesnoise. Hence it is
important to estimate the velocity variablesaccurately. Because
only two velocity variables need to beestimated, so the observer
called minimum-order observer.Figure 5 shows the feedback system
with a minimum-orderobserver.
The PPMO method exactly consists of the self-poleplacement
technology and the structure of minimum-orderobserver. Firstly,
both the system and state variables 𝑌 of (7)are split into two
parts, based on displacement variables 𝑥𝑎and velocity variables 𝑥𝑏,
as follows:
[�̇�𝑎�̇�𝑏] = [𝐴𝑎𝑎 𝐴𝑎𝑏𝐴𝑏𝑎 𝐴𝑏𝑏][
𝑥𝑎𝑥𝑏] + [
𝐵𝑎𝐵𝑏] 𝑢
𝑌0 = [1 0] [𝑥𝑎𝑥𝑏] ,(10)
where 𝑥𝑏 are the state variables of the minimum-orderobserver
system, then the feedback state variables 𝑥𝑏 of thetransformation
system in Figure 5 can be written as follows[19]:̇̃𝑥𝑏 = (𝐴𝑏𝑏 −
𝐾𝑒𝐴𝑎𝑏) 𝑥𝑏 + 𝐴𝑏𝑎𝑥𝑎 + 𝐵𝑏𝑢 + 𝐾𝑒𝐴𝑎𝑏𝑥𝑏, (11)
-
Shock and Vibration 7
with the error (𝑒) equation for the minimum-order
observerdescribed as
̇𝑒 = (𝐴𝑏𝑏 − 𝐾𝑒𝐴𝑎𝑏) 𝑒 = (𝐴𝑏𝑏 − 𝐾𝑒𝐴𝑎𝑏) (𝑥𝑏 − 𝑥𝑏) , (12)where the
state observer gain matrix 𝐾𝑒 is an 3 × 1 matrixand can also be
deduced according to Ackermann’s formulaas follows:
𝐾𝑒 = 𝜙 (𝐴𝑏𝑏) [[[[
𝐴𝑎𝑏𝐴𝑎𝑏𝐴𝑏𝑏𝐴𝑎𝑏𝐴2𝑏𝑏
]]]]
−1
[[[
001]]], (13)
where 𝜙 satisfies the characteristic equation |𝑠𝐼−𝐴𝑏𝑏| and canbe
deduced as follows:
𝜙 (𝐴𝑏𝑏) = 𝐴3𝑏𝑏 + 𝛼1𝐴2𝑏𝑏 + 𝛼2𝐴𝑏𝑏 + 𝛼3𝐼, (14)where 𝛼𝑖 is the
linear coefficient of corresponding character-istic equation.
The closed-loop poles of the PPMO system in Figure 5comprise the
closed-loop poles due to pole placement fromthe eigenvalues of
matrix (𝐴−𝐵𝐾) and the closed-loop polesdue to the minimum-order
observer from the eigenvalues ofmatrix (𝐴𝑏𝑏 −𝐾𝑒𝐴𝑎𝑏). Hence the
pole-placement design andthe design of the minimum-order observer
are independentof each other. The characteristic equation of the
PPMOsystem can be derived as |𝑠𝐼 − 𝐴 + 𝐵𝐾||𝑠𝐼 − 𝐴𝑏𝑏 + 𝐾𝑒𝐴𝑎𝑏|.
Inother words, one step is to calculate the feedback gain matrix𝐾
with a feedback law of 𝑢 = −𝐾𝑌1, which has closed-looppoles at the
specified values of all the self-poles; the otherstep is to compute
a state-feedback matrix 𝐾𝑒 such that theeigenvalues of (𝐴𝑏𝑏 −
𝐾𝑒𝐴𝑎𝑏) are those specified in the lasthalf of the self-poles
(corresponding to the two velocity statevariables).
It should be emphasized that a necessary and sufficientcondition
for arbitrary pole placement is that the systembe completely state
controllable. In view of the constantstability of the homogeneous
equation structure in (7), thesystem here is not only completely
controllable which canbe determined by the rank of the
controllability matrix withrank [𝐵 𝐴𝐵 𝐴2𝐵 𝐴3𝐵] ≡ 4, but also
completely observablewhich can be determined by the rank of the
observablematrixwith rank [𝐶 𝐶𝐴 𝐶𝐴2 𝐶𝐴3]𝑇 ≡ 4.4. Analysis and
Discussions
4.1. Divergent Cases and PPMO Control. Figures 6(a)-6(b)show the
uncontrolled displacements (a) and those controlledby PPMO (b)
under the condition of wind speed of 𝑈 =20m/s. The uncontrolled
displacements exhibit unstablestates, with vibration amplitudes
more than the blade lengthof 𝐿 = 30m in a very small range of time,
which is in fact anstate of extremely divergent instability, from
the point of viewof numerical simulation. The controlled cases with
PPMOcontrol in Figure 6(b) present the states of convergent
stabilitywith very small vibration amplitudes, which
demonstratesideal control effects. Two characteristics of PPMO
controlperformance need to be emphasized, one is small
vibration
amplitude as mentioned above, and the other is very
smallsteady-state value which implies slight offset of
vibrationdisplacements of blade tip.
The linear quadratic regulator (LQR) is often used toanalyze
vibration control of rotating structure. In [20], thehost structure
with sensor-actuator pairs embedded alongthe spanwise direction was
considered. The total output wasfed to a controller and then two
types of LQR methodswere used to investigate rotating effects.
Figure 6(c) illustratescomparisons of control effects between PPMO
method andLQR algorithm. As can be seen, the LQR control results
haverelatively large offsets, which also means greater amount
ofcontrol consumption which will be discussed later.
In order to verify that the PPMO control method can becommonly
used at that given external wind velocities. Theother cases of
different wind speeds are investigated. Figure 7shows comparisons
of control effects between PPMO andLQR under conditions of wind
speeds of 𝑈 = 50m/s (a)and 𝑈 = 5m/s (b), respectively. From the
magnitude of thevibration, or from the steady-state value, almost
the sameconclusion can be obtained, which proves the robustness
ofPPMO control algorithm.
In addition, for the linear control strategy, the controlcost
(control consumption of energy) often affects the imple-mentation
of the control measures seriously. Therefore, themagnitude of the
controller itself needs further exploration.Figure 8 shows
amplitudes of PPMOcontroller and LQR con-troller versus different
state variables, respectively. Herein, “𝑖= 1, 2” refers to the
first two displacement variables; “𝑖 = 3, 4”refers to the latter
two velocity variables. It can be seen thatthe amplitude of PPMO
controller is much smaller than theamplitude of LQR controller
under various wind speeds.
From the point of view of the “control cost” of
controllerhardware mentioned above, another advantage of
PPMOcontrol over LQR control is that the implementation ofLQR
control requires full state-feedback, which will notonly increase
the number of the hardware sensors used formeasurement, what is
more, the measurement process itselfof the state variables of
velocities is a complex process.
In addition, the implementation of PPMO is an automaticprocess,
including the configuration of self-poles which isalso automatic.
Performance of LQR control depends on thechoice of weighting Q/R
matrices which have no analyticsolution. So the choice of Q/R
matrices is artificial, which isanother drawback of LQR
control.
4.2. Feasibility Analysis of Hardware Implementation of Con-trol
Algorithm. In view of the large wind power system beingmostly
controlled byPLC, some intelligent control algorithmscannot be
implemented in PLC hardware because of theircomplexity. At present,
most of wind turbine control systemalgorithms employ fuzzy control,
which can be implementedin PLC system. However, the fuzzy algorithm
itself is tedious.For the other neural network algorithm, although
the numer-ical simulation of control effect is more ideal, because
of thecomplex integration problem, it is difficult to be
implementedin the PLC hardware [21]. Hence the feasibility analysis
of thehardware implementation of PPMO algorithm needs
furtherdiscussion.
-
8 Shock and Vibration
z(R, t)/(m)y(R, t)/(m)
−250
−200
−150
−100
−50
0
50
100
150
Unc
ontro
lled
disp
lace
men
ts
2 4 6 8 100t (s)
(a) The uncontrolled displacements
2 4 6 8 100t (s)
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
Con
trolle
d di
spla
cem
ents
z(R, t)/(m)y(R, t)/(m)
(b) The displacements controlled by PPMOmethod
2 4 6 8 100t (s)
−2.5
−2
−1.5
−1
−0.5
0
0.5
z(R
, t) (
m)
PPMOLQR
−0.5
0
0.5
1
1.5
2
2.5
3
3.5y
(R, t
) (m
)
PPMOLQR
2 4 6 8 100t (s)
(c) Comparisons of control effects between PPMO and LQR
Figure 6: The uncontrolled displacements (a) and those
controlled by PPMO (b) under the condition of wind speed of 𝑈 =
20m/s, withcomparisons of control effects between PPMO and LQR
demonstrated in (c).
In present study, a hardware-in-the-loop simulation plat-form is
built by OPC technology [21] between PLC controllersystem and
MATLAB/SIMUILINK simulation environmentto test the feasibility of
hardware implementation. Figure 9shows the experimental system,
including the main interfaceof the human-machine interface (HMI),
and the PPMO con-trol interface, respectively. The experimental
system consistsof PLC controller system andMATLAB-OPC computer.
PLCcontroller system runs the entire PPMO algorithm and sendsthe
controller output to MATLAB-OPC computer to drivethe aeroelastic
system and accepts signal that is the outputof aeroelastic system
from MATLAB-OPC computer. Theaeroelastic system model is run in
MATLAB/SIMULINK
environment. The signals of the control process can bedisplayed
by HMI which is connected to PLC (CPU 224).
Figure 10 shows the touchscreen interface in HMI anddisplays the
experimental signals of the vertical and lateraldisplacements
without control (a) and those under PPMOcontrol (b) based on the
condition of wind speed of 𝑈 =20m/s. Compared with the uncontrolled
cases in Figure 6(a),Figure 10(a) shows almost the same change
trends of verticaland lateral displacements without control. Not
only can thetrend of change be shown in HMI, but the specific
valuesof change can also be shown, such as the lateral
displace-ment in Figure 10(a). Also, Figure 10(b) demonstrates
thecontrolled cases as depicted in Figure 6(b). Similar trends
and
-
Shock and Vibration 9
PPMOLQR
−0.5
0
0.5
1
1.5
2
2.5
3z(R
, t) (
m)
2 4 6 8 100t (s)
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
y(R
, t) (
m)
2 4 6 8 100t (s)
PPMOLQR
(a) Comparison under𝑈 = 50m/s
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
z(R
, t) (
m)
2 4 6 8 100t (s)
PPMOLQR
2 4 6 8 100t (s)
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4y
(R, t
) (m
)
PPMOLQR
(b) Comparison under𝑈 = 5m/s
Figure 7: Comparisons of control effects between PPMO and LQR
under conditions of wind speeds of 𝑈 = 50m/s (a) and 𝑈 = 5m/s
(b),respectively.
magnitude values can be displayed, with similar
conclusionsobtained.
It should be noted that the data matching and
parametertransmission planning of OPC technology are not
discussedhere, which is another specific and complex problem.What
ismore, the speed of PPMO control in experimental platformis
faster, compared with the hardware implementation ofconventional
fuzzy control by OPC technology in [21], whichmight be due to the
simplified matrix operations in PPMOalgorithm.
4.3. Performance Improvement of Theoretical PPMO
Control.Consider the controlled cases by PPMO in Figure 6(b),
the
vibration frequencies of both vertical and lateral
displace-ments are relatively large. Especially for the vertical
displace-ment 𝑧 in Figure 6(c), compared with LQR control,
verticalvibration of PPMO exhibits greater frequency
fluctuations,which is actually a real-time drawback. To improve the
per-formance of PPMO control, an improved scheme to
integratePPMOmethod and LQRmethod together is
highlightedwithresponse effects demonstrated. The first step of the
improvedscheme is to determine the expected poles by the
quadraticoptimal control method [19], then to compare the
expectedpoles with the self-poles of the system, and then to
choosethe best poles between them. The second step is to
executivePPMO control programme on the basis of the best poles.
-
10 Shock and Vibration
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1A
mpl
itude
s of P
PMO
cont
rolle
r
1 2 3 4 50State viariables (i = 1, 2, 3, 4)
U = 5Ｇ/Ｍ
U = 10Ｇ/Ｍ
U = 15Ｇ/Ｍ
U = 20Ｇ/Ｍ
U = 50Ｇ/Ｍ
−200
−150
−100
−50
0
50
100
Am
plitu
des o
f LQ
R co
ntro
ller
1 2 3 4 50State viariables (i = 1, 2, 3, 4)
U = 5Ｇ/Ｍ
U = 10Ｇ/Ｍ
U = 15Ｇ/Ｍ
U = 20Ｇ/Ｍ
U = 50Ｇ/Ｍ
Figure 8: Amplitudes of PPMO controller and LQR controller
versus different state variables.
Figure 9: The experimental system.
Still taking the cases of Figure 6 under wind speed of𝑈 = 20m/s
for example, Figure 11 demonstrates the verticaland lateral
responses controlled by the improved scheme.Theimproved scheme, not
only for vertical displacement but alsofor lateral displacement,
not only relative to PPMO control,but also relative to LQR control,
not only in the frequencyfluctuations, but also in the steady-state
values, even inthe vibration amplitudes, has achieved greatly
improvedperformance.
It should be stated that this performance improvement isonly a
theoretical improvement, and it only fully combinesthe advantages
of PPMO control and LQR control and doesnot consider the
shortcomings of LQR control. Therefore,this scheme has some
unavoidable drawbacks in controlcost, automation, and hardware
implementation of controlalgorithms as mentioned above. Its
implementation can onlybe expected in hardware performance
improvement andcontrol cost reduction in the future.
5. Conclusions
In this study, vibration control for vertical bending andlateral
bending based on pole placement with minimum-order observer is
investigated by numerical simulation andhardware-in-the-loop
simulation platform. Some concludingremarks can be drawn from the
results:
(1) Structural modeling is investigated based on a NCMmethod
between the 2D typical section analysismethod and the 3D coupled
blade body analysismethod, which implies a compromise result that
isbound to occur, but it can greatly reduce the diffi-culty of
processing. Aeroelastic equation is realizedby fitting of
structural parameters and aerodynamiccoefficients based on
six-order Sin6 model, whichcan greatly simplify the integral
behavior along thespanwise length and, at the same time, reduce the
sys-tem variables (completely eliminate the
aerodynamicvariables).
(2) PPMO method combines the advantages of poleplacement
andminimum-order observation, which ismuch better than LQR control
either from vibrationamplitude or from control cost. Pole
assignment isbased on self-poles, which is themost essential
featureof PPMO control. Otherwise, the control perfor-mance of the
proposedmethod is greatly reduced dueto the difficulty of finding
poles and the matchingproblem of configuration poles.
(3) Feasibility analysis of hardware implementation ofcontrol
algorithm is verified by hardware-in-the-loopsimulation platform.
The platform consists of PLCcontroller system connected with HMI,
MATLAB
-
Shock and Vibration 11
(a) The vertical and lateral displacements without control
(b) The vertical and lateral displacements under PPMO
control
Figure 10: The vertical and lateral displacements without
control (a) and those under PPMO control (b).
2 4 6 8 100t (s)
−8
−6
−4
−2
0
2
4
6
8
10
12
Resp
onse
s bas
ed o
n th
e im
prov
ed sc
hem
e
z(R, t)/(m)y(R, t)/(m)
×10−3
Figure 11: Responses based on the improved scheme.
simulation environment, and the OPC link betweenthe two
structures. In the experiment, the runningspeed of the platform and
the accuracy of the dis-placement display further demonstrate the
advan-tages of PPMO control.
(4) In view of the fact that both simulation and exper-iment are
the analysis and discussion of the con-trol theory itself, some
future work needs to be
further envisaged. Practical engineering applicationof PPMO
control can be realized via external pitchmotion which can be fed
back to the aeroelasticsystem.The pitchmotion can be driven by a
hydraulicsystem. Hydraulic systems have a rich variety ofcomponents
and are easy to be controlled and shouldbe easier to realize pole
placement.
Conflicts of Interest
The authors declare that there are no conflicts of
interestregarding the publication of this article.
Acknowledgments
This work is supported by the National Natural ScienceFoundation
of China (Grant no. 51675315) and the GraduateInnovation Project of
SDUST (Grant no. SDKDYC180333).
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