DESIGN OF A RECEPTANCE-BASED ACTIVE AEROELASTIC CONTROLLER IN THE PRESENCE OF PARAMETRIC UNCERTAINTIES S. Fichera * , T.A.M. Guimarães ** , S. Jiffri *** , D.A. Rade **** , J.E. Mottershead * * University of Liverpool, School of Engineering, Liverpool L69 3GH, United Kingdom , ** UFU - Federal University of Uberlândia, School of Mechanical Engineering, Brazil , *** Swansea University, College of Engineering, Swansea SA2 8PP, United Kingdom , **** ITA - Technological Institute of Aeronautics, Division of Mechanical Engineering, Brazil Keywords: Active Aeroelastic Control, Receptance Method, Uncertainty propagation, Polynomial Chaos Expansion Abstract This paper presents a numerical investigation of the effects of parametric uncertainties propagated through Polynomial Chaos Expansion on the de- sign of a Receptance-based active controller for aeroelastic systems. The test-case is representa- tive of an experimental rig featuring a subsonic flexible wing with multiple control surfaces. The uncertainty is introduced in the Young’s modu- lus of the main spar. Such uncertainty is firstly propagated to assess the open loop behavior of the aeroelastic system in terms of flutter velocity and frequency responses. A Receptance-based controller is then designed deterministically with the goal of increasing the flutter boundary and its performance is tested against the uncertain aeroe- lastic system. Finally, the PDFs of the receptance control gains are evaluated and discussed. 1 Introduction The presence of uncertainties in aeroelastic sys- tems has been carefully reviewed by Beran et al. in [1], the authors highlight the importance of ad- dressing it both from a structural and an aerody- namic prospective, the latter being not subject of this works. The ability of modeling it as part of the aeroelastic design is essential when flut- ter mechanisms are closely spaced [7] or when in presence of nonlinearities [4]. Manan et al. [3] looked to the problem of uncertainty in the struc- tural parameters of and aeroelastic model via stochastic expansion methods in the frequency domain. This work, in the context of aeroelasticity, parametric uncertainties and active control, at- tempts to evaluate the Receptance Method [5] in the presence of aleatory uncertainty for the de- sign of an Active Aeroelastic Controller (AAC), to understand its limits and to propose a way forward. The propagation technique used for this study is Polynomial Chaos Expansion (PCE) [8]. A numerical model, based on an underly- ing Finite Element model coupled with unsteady aerodynamic, describing the MODular FLEXible aeroelastic wing (MODFLEX) apparatus [2], is adopted in the present investigation. The paper is divided as follows: after the above introduction, Section 2 presents the nu- merical MODFLEX model used for investigat- ing the problem, Section 3 discusses the Polyno- mial Chaos (PC) expansion used for addressing the propagation of the uncertainties, while Sec- tion 4 introduces the Repentance Method (RM) used for partially assigning the poles of the sys- tem. The results are discussed in Section 5, fol- lowed by the conclusions in Section 6. 1
Transcript
DESIGN OF A RECEPTANCE-BASED ACTIVE AEROELASTICCONTROLLER IN THE PRESENCE OF PARAMETRIC
UNCERTAINTIES
S. Fichera∗ , T.A.M. Guimarães∗∗ , S. Jiffri∗∗∗ , D.A. Rade∗∗∗∗ , J.E. Mottershead∗∗University of Liverpool, School of Engineering, Liverpool L69 3GH, United Kingdom , ∗∗UFU -
Federal University of Uberlândia, School of Mechanical Engineering, Brazil , ∗∗∗SwanseaUniversity, College of Engineering, Swansea SA2 8PP, United Kingdom , ∗∗∗∗ITA - Technological
Institute of Aeronautics, Division of Mechanical Engineering, Brazil
Keywords: Active Aeroelastic Control, Receptance Method, Uncertainty propagation, PolynomialChaos Expansion
Abstract
This paper presents a numerical investigation ofthe effects of parametric uncertainties propagatedthrough Polynomial Chaos Expansion on the de-sign of a Receptance-based active controller foraeroelastic systems. The test-case is representa-tive of an experimental rig featuring a subsonicflexible wing with multiple control surfaces. Theuncertainty is introduced in the Young’s modu-lus of the main spar. Such uncertainty is firstlypropagated to assess the open loop behavior ofthe aeroelastic system in terms of flutter velocityand frequency responses. A Receptance-basedcontroller is then designed deterministically withthe goal of increasing the flutter boundary and itsperformance is tested against the uncertain aeroe-lastic system. Finally, the PDFs of the receptancecontrol gains are evaluated and discussed.
1 Introduction
The presence of uncertainties in aeroelastic sys-tems has been carefully reviewed by Beran et al.in [1], the authors highlight the importance of ad-dressing it both from a structural and an aerody-namic prospective, the latter being not subject ofthis works. The ability of modeling it as partof the aeroelastic design is essential when flut-ter mechanisms are closely spaced [7] or when in
presence of nonlinearities [4]. Manan et al. [3]looked to the problem of uncertainty in the struc-tural parameters of and aeroelastic model viastochastic expansion methods in the frequencydomain.
This work, in the context of aeroelasticity,parametric uncertainties and active control, at-tempts to evaluate the Receptance Method [5] inthe presence of aleatory uncertainty for the de-sign of an Active Aeroelastic Controller (AAC),to understand its limits and to propose a wayforward. The propagation technique used forthis study is Polynomial Chaos Expansion (PCE)[8]. A numerical model, based on an underly-ing Finite Element model coupled with unsteadyaerodynamic, describing the MODular FLEXibleaeroelastic wing (MODFLEX) apparatus [2], isadopted in the present investigation.
The paper is divided as follows: after theabove introduction, Section 2 presents the nu-merical MODFLEX model used for investigat-ing the problem, Section 3 discusses the Polyno-mial Chaos (PC) expansion used for addressingthe propagation of the uncertainties, while Sec-tion 4 introduces the Repentance Method (RM)used for partially assigning the poles of the sys-tem. The results are discussed in Section 5, fol-lowed by the conclusions in Section 6.
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S. FICHERA , T.A.M. GUIMARÃES , S. JIFFRI , D.A. RADE , J.E. MOTTERSHEAD
2 Description of the numerical aeroelasticmodel
As anticipated in the introduction, the proposedprocedure is implemented on the twin numeri-cal model of the MODFLEX experimental rig.The MODFLEX apparatus has been developedin the recent years at the University of Liverpoolas a test-bench for experimental investigations ofactive control strategies in aeroelastic systems.The rig is representative of a flexible wing withmultiple control surfaces; the main structure iscomposed by a load-bearing aluminum alloy sparto which are single-point connected four aerody-namic sectors and a tip sector, as shown in Fig-ure 1. The main specifications of the MODFLEXaeroelastic flexible wing are summarized in Table1.
mm
mm
mm 1234
Fig. 1 : MODFLEX drawing.
Table 1: MODFLEX main specifications.
wing data dimensionwing span 1 m
wing sector 0.248 msector chord (c) 0.3 m
aerofoil NACA 0018mass axis position 0.5 x c
flexural axis position 0.5 x c
The experimental model has a numerical twinwhich features a Finite Element beam structurecoupled with potential aerodynamics (DoubletLattice - DLM). Numerical analyses and windtunnel tests showed that the aeroelastic wing ex-periences a typical bending-torsional flutter insta-bility at a wind speed of Vf = 13 m/s. A detaileddescription of the rig can be found in [2]. In thiswork, all the structural and aerodynamic infor-
mation for both assessing the uncertainty prop-agation and for designing the Receptance-basedcontroller are computed with Nastran/DMAP [6].The configuration used in this work features asingle input, the Trailing Edge Outer (TEO), andtwo displacement outputs, perpendicular to theplane of the wing, located at three quarters of thespan of the wing, ahead (OP1) and aft (OP2) themain spar. This has been done for assuring con-sistency with previous experimental works of thesame authors, see [2]. However, the procedurecan be immediately extended for the MIMO case.Using a generic formulation, the aeroelastic sys-tem can be described as follows
where MMMsss, CCCsss, and KKKsss are the mass, damping,and stiffness structural matrices, respectively; fff aaais the vector of generalized aerodynamic forces(GAFs), fff mmm represents the external force, q isthe dynamic pressure and qqq is the vector of thesystem states. The dimension of the generalizedmodel that indicates the number of modes re-tained is m; specifically, the first mode representsthe static deflection of the trailing edge outer andthe remaining nine the elastic modes of the sys-tem. The GAFs are computed in the reduced-frequency k domain by using the classical DLMtheory, which is
fff aaa = HHHam(k;M∞)qqq. (2)
Since the HHHam(k;M∞) is computed in a discretemanner, an interpolation is necessary for evaluat-ing the unsteady aerodynamics on the entire do-main of interest. Eq. 1 can be rewritten as followsby moving to the LHS the aerodynamic term
MMMsssq̈qq+(CCCsss−q(c
2V) · real(HHHam))q̇qq+
(KKKsss−q · img(HHHam))qqq = 0.(3)
By taking the inverse of the LHS of Eq. 3,and pre- and post-multiplying by the eigenvec-tors matrix after having opportunely selected thesingle-input and multiple-output nodes, the trans-fer functions in physical coordinates can be eval-uated. Figure 2 shows the FRF evolution ob-
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DESIGN OF A RECEPTANCE-BASED ACTIVE AEROELASTIC CONTROLLER IN THE PRESENCEOF PARAMETRIC UNCERTAINTIES
10-4
14 6
5
10-3
13
Velocity (m/s) Excitation - (Hz)
4
H(
) A
mp
litu
de
- (
m /
de
g)
12
10-2
3
11 2
10-1
Fig. 2 : MODFLEX FRF evolution with velocityfor deterministic case - OP1.
tained for the first output point, between an air-flow speed of 11m/s up to 14m/s, for the deter-ministic solution. It is evident that that the modescoalesce approximately at 12.5m/s.
3 Propagation of uncertainty though themodel
The MODFLEX numerical model is augmentedby introducing an uncertainty in the value ofthe main-spar elastic modulus. The influence ofparameter uncertainty on the aeroelastic stabil-ity is then investigated using the classical Poly-nomial Chaos Expansion (PCE) technique [8]which main polynomial is reported in Eq. 4 thatexpresses the response for one parameter in onedimension
u = β0 +β1ξ+β2(ξ2−1)+β3(ξ
3−3ξ)+
β4(ξ4−6ξ
2 +3)+ . . .(4)
where ξ is the independent standard Gaussianrandom variable, in this case the Young’s mod-ulus, and βi are the terms of the orthogonal poly-nomial. Specifically, Hermite polynomials havebeen used and the elastic modulus (assumed tobe Gaussian) is defined by a standard deviation,σ, equal to 3.3% of the nominal value. The Latin-hypercube sampling technique is used for select-ing a small sample of individuals (N = 20), forwhich the solution of the aeroelastic problem wascomputed numerically. The PC expansions werethen fitted by using the same base but changing
the surrogate outputs in relation to the differentstage of the investigation. Initially a surrogatemodel was created for evaluating the envelope ofthe V-g flutter diagram (as shown hereinafter inSubsection 3.1). Subsequently, the same sampleof individuals was used to create the PDFs of theReceptance-based controller gains.
3.1 V-g flutter diagram for the expandedmodel
Due to the aeroelastic behavior shown by MOD-FLEX, the first two modes are considered themost relevant for this study, whereas the third isretained for verification purposes. This led to 3 x2 polynomials (frequency and damping) expan-sion of the 2nd order to be computed. The V-g
0 5 10 15 20 25
Velocity [m/s]
0
2
4
6
8
Fre
q. [
Hz]
0 5 10 15 20 25
Velocity [m/s]
-50
0
50
Da
mp
. [%
]
11st bending
1st torsion
1st in-plane
Fig. 3 : V-g flutter diagram due to a parametricuncertainty in the Young’s modulus.
flutter diagram bounds shown in Figure 3 werethen computed by evaluating such polynomialsfor a 1000 sampling points. It is evident that theuncertainty on the elastic modulus has the effectof propagating an uncertainty on the flutter ve-locity (bounds 12.7 - 13.3 m/s), being the PFDsolution highlighted in Fig. 4; it is interesting tonotice here that the probability of flutter onset isslightly higher for lower velocities. Moreover,from a closer inspection of the diagram, it is evi-dent that the most significant effect is on the nat-ural frequency of the first bending mode that re-sults in a spread of the zero-crossing position ofthe damping of the torsional mode.
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S. FICHERA , T.A.M. GUIMARÃES , S. JIFFRI , D.A. RADE , J.E. MOTTERSHEAD
12.5 13 13.5
Flutter [m/s]
0
0.5
1
1.5
2
PD
F
Fig. 4 : PDF for flutter onset.
3.2 Uncertain Frequency Response Function(UFRF)
Following a similar procedure to that describedabove, and in a similar manner to the approachpresented by Manan et al. [3], where PCE wasused to fit the coefficients of the polynomial func-tion representing the system response, the enve-lope of the uncertain frequency response func-tions at incremental airflow velocities have beencomputed, as shown Fig. 5. A close inspectionenables one to conclude that the first vibrationmode, which represents the first bending mode,is more susceptible to the uncertainty in the elas-tic modulus. Moreover, as the airflow velocity in-creases, it becomes evident that coalescence oc-curs as the second mode migrates towards thefirst. In fact, the aeroelastic uncertainty propa-gation leads to a stochastic transfer function, andmainly the poles and zeros for the first mode(bending mode), as expected, are more suscep-tible. Figures 5.a, 5.b and 5.d depict the FRFsa two velocities below the flutter speed and oneabove, in all the cases the first peak is the mostaffected by the uncertainty, however, the overallshape of the FRFs is not significantly impactedby the variation of the elastic modulus. Of spe-cific interest is instead Fig. 5.c that shows theFRF envelope in proximity of the flutter veloc-ity. At this airspeed, even a small change in theelastic modulus can trigger the transition from astable to an unstable condition, with the disap-pearance of the first peak.
0 2 4 6 8 10
Excitation - (Hz)
10-4
10-2
100
102
FR
F -
Am
plit
ud
e (
m/d
eg
)
(a) Airspeed = 5 (m/s)
2 4 6 8 10
Excitation - (Hz)
10-3
10-2
10-1
100
FR
F -
Am
plit
ud
e (
m/d
eg
)
(b) Airspeed = 10 (m/s)
1 2 3 4 5 6 7 8 9 10
Excitation - (Hz)
10-3
10-2
10-1
100
FR
F -
Am
plit
ud
e (
m/d
eg
)
(c) Airspeed = 12 (m/s)
2 4 6 8 10
Excitation - (Hz)
10-2
100
FR
F -
Am
plit
ud
e (
m/d
eg
)
(d) Airspeed = 14 (m/s)
Fig. 5 : Stochastic FRF results for different val-ues of airspeed - OP1.
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DESIGN OF A RECEPTANCE-BASED ACTIVE AEROELASTIC CONTROLLER IN THE PRESENCEOF PARAMETRIC UNCERTAINTIES
4 Uncertainty propagation to theReceptance-based controller
The final step of this procedure sees the imple-mentation of a feedback controller aimed at in-creasing the flutter velocity of the aeroelastic sys-tem. This reflects numerically what was experi-mentally validated recently by the same authorsin [2]. The main contribution of this work is thatthe uncertainty is included in the model and itseffects are propagated to the controller design.The controller design method chosen for com-puting the feedback gains is the so-called Re-ceptance Method developed by Ram and Mot-tershead [5] and its main strength lies in not re-quiring any direct knowledge of the system be-sides the frequency response function at the lo-cations of the poles. The receptance method isapplicable to any frequency response function,but is readily understood in terms of receptances.Consider a generic dynamic system which recep-tance matrix is H(s) that has open loop eigen-pairs equal to {λk, vk} for k = 1,2, . . . ,2n. Viapartial-state feedback, proportional to displace-ments (gm) and velocities (fm), p eigenvalues µkfor k = 1,2, . . . , p can be assigned while the re-maining, µk = λk for k = p+1, p+2, . . . ,2n, areretained. It has been demonstrated in [5] that, bydefining
rµk, j = H(µk)b j j = 1,2, . . . ,m (5)
with b j input distribution vector and m number ofinputs, and
αµk, j = (µkf jT +g j
T )wk k = 1,2, . . . , pj = 1,2, . . . ,m
(6)
the part of the system related to the assignedpoles can be expressed as
wk = αµk,1rµk,1 +αµk,2rµk,2 + · · ·+αµk,mrµk,m
k = 1,2, . . . , p.(7)
The problem can be rewritten in matrix form inEq. 8, where Pk and Qk matrix are defined in 9,and the values of the gains are computed after a
judicious choice of αµk, j.
P1...
PpQp+1
...Q2n
f1...
fmg1...
gm
=
α1...
αp0...0
(8)
Pk =
µkwT
k 0 . . . 0 wTk 0 . . . 0
0 µkwTk . . . 0 0 wT
k . . . 0...
... . . . ......
... . . . ...0 0 . . . µkwT
k 0 0 . . . wTk
Qk =
λkvT
k 0 . . . 0 vTk 0 . . . 0
0 λkvTk . . . 0 0 vT
k . . . 0...
... . . . ......
... . . . ...0 0 . . . λkvT
k 0 0 . . . vTk
(9)
4.1 PCE of the controller gains
The Receptance Method briefly presented abovehas been implemented in the numerical proce-dure developed within this work for assigningfrequency and damping of the poles associatedwith the first bending and first torsion mode, withthe aim of increasing the flutter velocity of thesystem. As described in Section 2, the aeroelasticsystem features a single input (b j = b1) and twooutputs. The receptance matrix is computed to-gether with input distribution vector b1 and con-stitutes the rµk,1 matrix presented in Eq. 5. Suchmatrix represents the two FRFs between the twooutputs and the single input. The FRFs are com-puted at an airspeed of 10m/s that assures a safetymargin with respect to the deterministic fluttervelocity. The target of the aeroelastic controlleris to increase the damping of both poles by 10%and to reduce the frequency of the first (bend-ing) by 10% while increasing that of the secondmode (torsion) by the same amount, always withrespect to the deterministic case. The stochasticresults and the procedure are highlighted in Fig.6. Even if the controller has always the same ob-jective, the FRFs change due to the uncertaintyin the elastic modulus and consequently the gainsalso change. A PCE, based on the same sample ofindividuals discussed in Section 3, was used for
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S. FICHERA , T.A.M. GUIMARÃES , S. JIFFRI , D.A. RADE , J.E. MOTTERSHEAD
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02
Real Axis (sec-1
)
-6
-4
-2
0
2
4
6
Imag A
xis
(sec
-1)
O - O.L. Stochastic Poles and Zeros
X - C.L. Assigned Poles and Zeros
Fig. 6 : MODFLEX pole-zero map from stochas-tic results for the first two pairs of poles.
deriving the surrogate model of the Receptance-based controller design procedure. Once such amodel is constructed, it is possible to evaluate thefm and gm gains on the entire domain of interestand compute their PDFs, as shown in Fig. 7.
5 Discussion of the Results
The analyses presented highlight the potentialdetrimental impact of uncertainty in the elasticmodulus on the onset of flutter instability. Fig-ure 3 shows the range of airspeed in which theflutter can occur, while Fig. 5.c, by represent-ing the envelope of FRFs near the flutter veloc-ity, highlights the transition between a stable andan unstable behavior. Figure 7 shows the PDFfor the two pairs of gains due to the considereduncertainty. As expected, and in agreement withthe previous experimental activities, the f1 andf2 gains, that are in feedback to the velocity, aresmaller compared to g1 and g2 gains that are infeedback to the displacements. This ratio, ofcourse, depends by the choice of the placed poles,and it can be different if the controller objectiveis not to increase the flutter stability margin ofthe system. Figure 8 shows the auto- and cross-correlation of the gains distributions and high-lights the linear relationship between the f gainsas well as between the g gains, while is evidentthat the cross-relationships are non-linear. Onepossible use of this information is to choose, forthe uncertain system, the pairs of gains that havethe highest PDFs (by using in conjunction withFigs. 7 and 8) and this will lead to a system that,even if sub-optimal for the deterministic case,
will be less sensitive to the uncertainty.
6 Conclusions
The results presented above are the first steptowards a stochastic, Receptance-based, activeaeroelastic controller. The variation in the flut-ter velocity due to the elastic modulus uncer-tainly, highlights the importance of consideringit in the design of a controller. The procedure im-plemented in this work proved to be effective inpropagating such uncertainty to the frequency re-sponse functions first, and to the controller gainslater. PCE has been used at different levels forcreating surrogate models that, in end, resulted inPDFs useful for gaining information on the con-troller gains. The proposed use of the auto- andcross-correlation matrix of the gains (Fig. 8) asmean for making the system less sensitive to theuncertainty, will be subject of further investiga-tions.
References
[1] Philip Beran, Bret Stanford, and ChristopherSchrock. Uncertainty Quantification in Aeroe-lasticity. Annual Review of Fluid Mechanics,49(1):361–386, 1 2017.
[2] S Fichera, S Jiffri, and J E Mottershead. Designand wind tunnel test of a MODular aeroelasticFLEXible wing (MODFLEX). In Proceedings ofthe International Conference on Noise and Vibra-tion Engineering ISMA 2016, Leuven, Belgium,2016.
[3] A. Manan and J. E. Cooper. Prediction of un-certain frequency response function bounds usingpolynomial chaos expansion. Journal of Soundand Vibration, 329(16):3348–3358, 2010.
[4] Chris L. Pettit and Philip S. Beran. Effects ofParametric Uncertainty on Airfoil Limit CycleOscillation. Journal of Aircraft, 40(5):1004–1006, 9 2003.
[5] Y.M. M. Ram and J. E. Mottershead. Multiple-input active vibration control by partial poleplacement using the method of receptances.Mechanical Systems and Signal Processing,40(2):1–9, 2013.
[6] W.P. Rodden and E.H. Johnson. MSC/NASTRAN
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DESIGN OF A RECEPTANCE-BASED ACTIVE AEROELASTIC CONTROLLER IN THE PRESENCEOF PARAMETRIC UNCERTAINTIES
10 12 14 16 18
F1
0
0.5
1
PD
F
8 10 12 14 16
F2
0
0.5
1
1.5
PD
F40 50 60 70 80
G1
0
0.05
0.1
0.15
PD
F
40 50 60 70 80
G2
0
0.05
0.1
0.15
PD
F
Deterministic = 70.42Deterministic = 72.40
Deterministic = 13.51Deterministic = 14.97
Fig. 7 : Stochastic gains PDF.
Fig. 8 : Stochastic gains auto- and cross-correlation.
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S. FICHERA , T.A.M. GUIMARÃES , S. JIFFRI , D.A. RADE , J.E. MOTTERSHEAD
Aeroelastic Analysis: User’s Guide, Version 68.MacNeal-Schwendler Corporation, 1994.
[7] Carl Scarth, Jonathan E. Cooper, Paul M. Weaver,and Gustavo H C Silva. Uncertainty quantifi-cation of aeroelastic stability of composite platewings using lamination parameters. CompositeStructures, 116(1):84–93, 2014.
[8] N. Wiener. The Homogenous Chaos. AmericanJournal of Mathematics, 60(4):897–936, 1938.
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