ADAPTIVE FEEDFORWARD CONTROL DESIGN FOR GUST …

ADAPTIVE FEEDFORWARD CONTROL DESIGN FOR GUSTLOADS ALLEVIATION AND LCO SUPPRESSION

Wang, Y.∗ , Da Ronch, A.∗∗ , Ghandchi–Therani, M.∗∗ , Li, F.∗∗China Academy of Aerospace Aerodynamics, Beijing, 100074, China∗∗University of Southampton, Southampton, S017 1BJ, United Kingdom

Keywords: gust loads alleviation, LCO suppression, adaptive feedforward control, aeroelastic model.

Abstract

An adaptive feedforward controller is designedfor gust loads alleviation and limit cycle oscil-lations suppression. Two sets of basis functions,based on the finite impulse response and modi-fied finite impulse response approaches, are in-vestigated to design the controller for gust loadsalleviation. Limit cycle oscillations suppressionis shown by using the modified finite impulse re-sponse controller. Worst case gust search is per-formed by using a nonlinear technique of modelreduction to speed up the costs of calculations.Both the “one–minus–cosine” and Von Kármáncontinuous turbulence gusts of different intensi-ties were generated to examine the performanceof controllers. The responses of these two typesof gust can be reduced effectively by finite im-pulse response controller in the whole process,while the modified finite impulse response con-troller is found to increase the loads during theinitial transient response. The above two typesof gust induced limit cycle oscillations were usedto test the modified finite impulse response con-troller. Results show that it can suppress limitcycle oscillations to some extent.

1 Introduction

To reduce the impact of air transport on the envi-ronment and improve aircraft global efficiency, alightweight solution with high aspect ratio wingsis usually needed [1]. For such aircraft, the fre-quencies of rigid body motions and aeroelasticvibrations have the tendency to get closer to each

other, which increases the nonlinear aeroelasticcoupling between flight mechanics and structuraldynamics. Thus, atmospheric turbulence gustssignificantly excite structural vibrations. Thesevibrations cause dynamic structural loads and in-fluence the rigid body motions of the aircraft,which decreases handling qualities and passengercomfort. In practical conditions, gusts can inducelimit cycle oscillations (LCOs), which are non-diverging, self-sustained, fixed-frequency struc-tural vibration phenomena, and can cause fatigueissues and reduce the aircraft operational life.

Aeroelastic active control has been investi-gated for several decades, especially for fluttersuppression and gust loads alleviation [2, 3, 4, 5].Feedback control strategies, e.g. linear quadraticregulator (LQR), optimal control algorithm, H∞

robust control synthesis, are relatively mature andhave been used extensively in aeroservoelasticityfor gust loads alleviation. When prior knowledgeof disturbance is available, a feedforward controlstrategy is generally preferred over feedback con-trol for disturbance rejection. One of the advan-tages of feedforward control is that there is notime delay between the disturbance and controlcompensation, which means that corrective ac-tion can be taken before the output has deviatedfrom the set point. It is possible to design a feed-forward controller to alleviate gust load responseby measuring the vertical gust speed ahead of air-craft using light detection and ranging (LIDAR)beam airborne wind sensor [6]. Reference [7] de-veloped a linear adaptive feedforward controllerusing orthonormal basis function for gust loadsalleviation on linear aeroelastic model and good

1

WANG, Y., DA RONCH, A., GHANDCHI–THERANI, M., LI, F.

performances were found.The aim of this paper is to design an appropri-

ate feedforward controller to control the responseof an aerofoil system, including gust loads allevi-ation and LCOs suppression. The feedforwardcontroller can be realized by two different basisfunctions, finite impulse response (FIR) modeland modified finite impulse response (MFIR)model. Section 2 introduces the two degrees offreedom (DoF) aerofoil aeroelastic system usedin this paper. The adaptive feedforward con-trol design is formulated in Sec. 3. The turbu-lence gust models are reviewed in Sec. 4. Anefficient worst–case–gust search for the “one–minus–cosine” family is presented in Sec. 5 us-ing a nonlinear reduced order model [5]. Resultsfor cases with strong nonlinear effects are shownin Sec. 6. Finally, conclusions are given.

2 Aeroelastic model

The aeroelastic model used in this paper, shownin Fig. 1, is a typical aerofoil section with twodegrees of freedom that define the motion abouta reference elastic axis (e.a.). The plunge de-flection is denoted by h, positive downward, andα is the angle of attack about the elastic axis,positive with nose up. The semi–chord length isb. The nondimensional distances from the mid-chord to the elastic axis and from the elastic axisto the centre of gravity are ah and xα, respec-tively. The aerofoil is equipped with a mass-less trailing–edge flap with hinge at a distancecb from the midchord. The flap deflection, δ, isdefined relative to the aerofoil chord. The mo-tion is restrained by two springs, Kξ and Kα, andthe model is assumed to have a horizontal equi-librium position at h = α = δ = 0. The systemcontains structural damping in both degrees offreedom, Cξ and Cα. The equations in dimen-sion form with a polynomial nonlinearity for therestoring forces are

mh+Sαα+Cξh

+Kξ(h+βξ3h3 +βξ5h5) =−L (1)

Sαh+ Iαα+Cαα

+Kα(α+βα3α3 +βα5α

5) = M (2)

where m, Sα, and Iα are the aerofoil sectionalmass, the first and second moment of inertia ofaerofoil about elastic axis. The lift, L, is definedpositive upward according to the usual sign con-vention in aerodynamics. The moment aroundthe elastic axis is M. The plunge displacement, h,is positive downward, as it is conventionally donein aeroelasticity. The unsteady aerodynamics ismodeled with strip theory and the incompressibletwo-dimensional classical theory of Theodorsen.The model 1 is formulated in first order and con-tains 12 states. More details can be found in Ref.[8].

b bcb

bh

x b

Undeformed position

h

e.a. c.g.

K

K

Fig. 1 : Schematic of an aerofoil section withtrailing–edge flap; the wind velocity is to theright and horizontal; e.a. and c.g. denote, respec-tively, the elastic axis and centre of gravity

3 Feedforward control

A feedforward control system consists two chan-nels, see Fig. 2. One channel is the disturbancetransfer path and the other one is the control path.Denote wg the atmospheric gust disturbance, andby wg the gust measured by the on–board LIDARsensor. In this paper, it is assumed that the mea-sured gust, wg, is identical to the true atmosphericgust, wg. The transfer function of the physicalplant (in this case, the aerofoil section) betweenthe gust disturbance and the structural response isdenoted by H; G indicates the transfer function ofthe physical plant between the control effector (inthis case, the trailing edge flap) and the structuralresponse, and G indicates the approximate trans-fer function of −G. The feedforward controller

1The code can be obtained from Da Ronch, A., [email protected]

2

mailto:[email protected]

mailto:[email protected]

Adaptive Feedforward Control Design for Gust Loads Alleviation and LCO Suppression

H

Lidar

Beam

wg(t) x(t)

Gcŵg(t)

Ĝ

Gu(t)

y(t)

+

+e(t)

Adaptive

Filteringua(t)

ŵg(t)

coefficients

Fig. 2 : Block diagram of the adaptive feedforward control applied to a physical plant with transferfunction H

is represented by Gc, and u(t) indicates the con-troller output, e.g. the rotation of the trailing edgeflap.

The object of this study is to design the feed-forward controller Gc to control the gust responseof the nonlinear aerofoil model. Theoretically,the ideal feedforward controller Gci is

Gci =−HG−1 (3)

The method used to evaluate the controller model(transfer function) is based on a prediction errorminimization (PEM) approach.

First, PEM is used to identify G, an approxi-mation model of the control path G

G≈−G (4)

which is done by performing an experiment, e.g.using an external signal u(t) to inject into the flapas the excitation signal, and the response signaly(t) as the output signal. Then, using G to filterthe measured gust signal wg(t) and then get anoutput response ua(t), which is

ua(t) = Gwg(t) (5)

Here it is assumed that wg(t) is equal to wg(t). Inthe disturbance path, the relationship of responsex(t) and gust wg(t) is

x(t) = Hwg(t) (6)

Substituting (5) into (6) yields to

x(t)≈−HG−1ua(t) (7)

Finally, the information about the physical sys-tem for feedforward control can be identified byusing ua(t) as input and x(t) as output. Adap-tive strategy is used to ensure the robustness ofthe feedforward controller. The coefficients ofthe controller are computed by an adaptive fil-tering algorithm. In practice, x(t) is replaced bye(t), which is the error between the disturbanceresponse and the control path response measuredby a sensor.

The controller is considered as a discrete lin-ear time invariant system

u(t) = Gc(q)wg(t) (8)

where wg(t) is the input signal and u(t) is thecorresponding output signal. Gc(q) is transferoperator represented the controller, where q isthe forward shift operator, qwg(t) = wg(t + 1),and q−1 is the delay (backward shift) operator,q−1wg(t) = wg(t−1). The corresponding transferfunction Gc(z), z ∈C (the complex plane), whichis formulated as

Gc(z) =n

∑k=1

LkBk(z) (9)

where Bk(z) are basis functions, Lk are the cor-responding coefficients, and n is the model or-der. In this paper two kinds of basis functionsbased on FIR model and MFIR model were usedto approximate the ideal feedforward controller.These two methods are discussed in the follow-ing section.

3


3.1 Finite Impulse Response model

A FIR model of Eq. (9) corresponds to the choice

Bk(z) = z−k,k = 1,2, · · · ,n (10)

where n is the order of model chosen by the bal-ance of the approximation accuracy and compu-tational cost. The FIR model structure is shownin Fig. 3. So the FIR model means using a num-ber of transfer functions with zero poles to ap-proximate the controller.

++

u(t)

ŵg(t) z-1

++

z-1

...L1

z-1...

...

L2 Ln

Fig. 3 : Structure of FIR model

3.2 Modified Finite Impulse Response model

If the real system dynamics have a slow pole,then the model order n will be very large to pro-vide an accurate approximation to the true dy-namics. One strategy to overcome this is to in-stead choose

Bk(z) =n

∏k=1

1z−ξi

k = 1,2, · · · ,n; i = 1,2, · · · ,k (11)

where {ξi}i=1,2,...,n is a set of chosen poles [9],which means injecting a priori knowledge intobasis functions. The MFIR model structure isshown in Fig 4. For the treatment of multiplecomplex poles, see Ref. [9] for guidelines.

+

+u(t)

ŵg(t)

+

+

...

...

L1 L2 Ln...

1

1

z 2

1

z

1

nz

Fig. 4 : Structure of MFIR model

3.3 Exponentially weighted recursive least-square algorithm

The coefficients of the basis functions are calcu-lated by exponentially weighted recursive least-square algorithm. Donate a cost function by

ε(N) =N

∑i=1

λN−i|e(i)|2

0 < λ≤ 1,N = 1,2, · · · (12)

where N is the total number of time steps, λ is theforgetting factor, and e(i) is the error between thedesired response e(i) and the FIR or MFIR modeloutput z(i) at time i

e(i) = e(i)− z(i) = e(i)−L(N)ΦT (i) (13)

the vector ΦT (i) = [ua1(i),ua2(i), · · · ,uan(i)]T is

the output of every basis function of the FIR orMFIR model, and L(i) = [L1(i),L2(i), · · · ,Ln(i)]is the corresponding coefficient vector, or calledtap weight vector. The adaptive algorithm in-cludes the following steps:

• Initialize

L(0) = 0,

P(0) = δ−1I, where δ is a small positiveconstant (e.g. 1.0).

• Iterate for each instant of time, N =1,2, · · · , compute

4


π(N) = P(N−1)Φ(N) (14)

k(N) =π(N)

λ+ΦT (N)π(N)(15)

ε(N) = d(N)−LT (N−1)Φ(N) (16)L(N) = L(N−1)+ k(N)ε(N) (17)

P(N) = λ−1P(N−1)

−λ−1k(N)ΦT (N)P(N−1) (18)

where P(N) is the inverse correlation matrix,k(N) is a gain vector, and π(N) is a middle vari-able which is used to increase the computationaccuracy. The forgetting factor λ should be care-fully chosen. By default, the value 1.0 is used.More details about this algorithm can be found inRef. [10].

4 Turbulence gust model

For aircraft certification, gust loads are criticalfor structural sizing. Two types of atmosphericgust model were used in this paper. The discretemodel based on the “one–minus–cosine” shape isformulated as

wg(t) =w0

2cos(

2π

Hg

Ub(t− t0)

)t0 ≤ t ≤ t0 +

HgbU

(19)

where w0 is the gust intensity normalized by thefreestream speed U , Hg is gust wave length nor-malized by aerofoil semi–chord. The physicaltime is t. The Von Kármán spectrum is usedfor the continuous model. According to MIL-F-8785C, the vertical spectrum function is

φw(ω) =σ2

wLw

πU·

1+ 83

(1.339Lw

ω

U

)2[1+(1.339Lw

ω

U

)2]11/6 (20)

where φw(ω) is the energy spectral density, σwis the turbulence intensity, ω is the circular fre-quency, and Lw represents the turbulence scalelength. The turbulence is generated by passing aband–limited white noise through an appropriate

forming filter with transfer function

Hw(s) =σw

√1π· Lw

U (1+A(s))

1+B(s)

A(s) = 2.7478Lw

Us+0.3398

(Lw

U

)2

s2

B(s) = 2.9958Lw

Us+1.9754

(Lw

U

)2

s2

+0.1539(

Lw

U

)3

s3 (21)

The turbulence intensity is

σw = 0.1w20 (22)

where w20 is the wind speed at 20 feet (6 m). Typ-ical value for light turbulence, the wind speed at20 feet is 15 knots; for moderate turbulence, thewind speed is 30 knots; and for severe turbulence,the wind speed is 45 knots.

5 Worst–case–gust search

Regarding to an aircraft system, when the fre-quency of gust is close to a frequency of onemode, the response would become larger, whichis the worst case gust corresponding to thismode. In this paper, the worst–case–gust searchfor “one–minus–cosine” family was performed.To reduce computational cost, reduced–ordermodel (ROM) combined with kriging interpola-tion technique was used to perform the worst–case–gust search. The detailed derivation ofROM can be found in Ref. [1]. Assuming thepitch and plunge frequencies of the aerofoil sys-tem are fα and fξ, respectively, the wave lengthsof the worst case gust corresponding to these twomodes are

Hgα=

Ufαb

,Hgξ=

Ufξb

(23)

which can be represented in nondimensionalform

Hgα=

2πUωαb

= 2πU∗,Hgξ=

2πUωξb

=2πU∗

ω(24)

where ωα is the pitch mode circular frequencyand ωξ is the plunge mode circular frequency, ω

5


is the ratio between ωξ and ωα. U∗ is reducedvelocity, equal to U/bωα. Equation (24) showsthat the worst case gust wave length for pitch andplunge are influenced by freesteam velocity andcorresponding circular frequency.

6 Results

In this work, results are presented for a non-linear aeroelastic model of a pitch–plunge aero-foil. The model is representative of a wind tun-nel test rig installed at University of Liverpool.The aeroelastic parameters are summarized in Ta-ble 1 (see, e.g. Ref. [1] for their definitions). Theair mass ratio is represented by µ, ζα and ζξ arethe viscous dampings of plunge and pitch, andrα is nondimensional radius of gyration aboutthe elastic axis. The values of βξ3 and βξ5

forthe linear system are 0. The wind tunnel aeroe-lastic model has the following parameters whichwere measured at the University of Liverpool:wing semichord b = 0.175 m, pitch circular fre-quency ωα = 28.061 rad/s, plunge circular fre-quency ωξ = 16.629 rad/s.

Table 1: Aeroelastic parameters

Parameter Valueω 0.593µ 69.000ah -0.333xα 0.090rα 0.400ζα 0.015ζξ 0.015βξ3 1741.881βξ5

638721.901

6.1 Validation results

The predictions of the coupled aeroelastic modelare compared to data measured from the aeroe-lastic model in the low–speed, open–loop windtunnel of the University of Liverpool. Figure 5shows the numerical and the wind tunnel experi-mental results of damped natural frequency anddamping ratios for varying freestream speeds.The predictions for the pitch mode agree well

with the measured data. Some difference are no-ticed for the plunge mode. More comparisons be-tween aeroelastic model and experiment can befound in [8].

air speed [m/s]

ω d [

Hz]

0 5 10 152

3

4

5ModelWT Data

plunge

pitch

(a) Damped frequency

air speed [m/s]

ζ

0 5 10 150

0.05

0.1

0.15

0.2

0.25ModelWT Data

pitch

plunge

(b) Damping ratio

Fig. 5 : Eigenvalues tracing for varyingfreestream speed from simulation and wind tun-nel measurements of the aerofoil test rig

The nonlinear aeroelastic model exhibits in-teresting dynamics below the flutter speed of15.28 m/s, and this is shown in Fig. 6. Whereasat the lower speed of 8 m/s the aerofoil responseis well damped , a LCO arises at 13 m/s.

6.2 Worst–case–gust search

Identification of the worst case gust allows iden-tifying for critical gust loads used to verify theeffectiveness of the control approaches. The ap-proach to model reduction [1, 5, 8, 11] was used

6


time [s]

α [d

eg]

0 2 4 6-6

-3

0

3

68 m/s13 m/s

(a) Pitch

time [s]

h [

mm

]

0 2 4 6

-1

0

1

28 m/s13 m/s

(b) Plunge

Fig. 6 : Open loop/ uncontrolled responses for aninitial angle of attack of 6 deg at speeds of 8 and13 m/s

to generate a reduced model, which allows retain-ing the nonlinear dynamics exhibited by the orig-inal coupled aeroelastic model, is smaller in sizeand computationally more efficient. For modelprojection, 3 modes were used (see Ref. [1]).This model was used in combination with theKriging approach to predict the worst–case–gustprofile.

6.2.1 Light “one–minus–cosine” gust

An intensity of 0.005 “one–minus–cosine” gustwas investigated first, and results are shownin Fig. 7. The solid curves in the top plotare nonlinear ROM (NROM) results, while thesolid squares are computed by nonlinear full–order model (NFOM). The figures show that the

Hg [ ]

α [d

eg]

5 10 15 20 25-0.04

-0.02

0

0.02

0.04

0.06NROMNFOM

(a) Pitch

Hg [ ]

h [

mm

]

5 10 15 20 25-0.3

-0.2

-0.1

0

0.1

0.2NROMNFOM

(b) Plunge

Fig. 7 : Worst–case–gust search for the"one–minus–cosine" gust of intensity 0.005 atfreestream speed of 8 m/s; the vertical line de-notes the gust wavelength calculated from Eq. 24

NROM and NFOM results agree well. The worstcase gust wavelengths for pitch and plunge atintensity of 0.005 are 10 and 16 semi–chords.When freestream speed is 8 m/s, U∗ is 1.629and ω is 0.593, Hgα

and Hgξcalculated by Eq.

24 are 10.235 and 17.260 semi–chords, whichare shown in Fig. 7 as dashed lines. The worstcase gust corresponding to pitch mode has a goodagreement between search result and the valuecalculated by Eq. 24, while there is a small dif-ference for the plunge mode.

6.2.2 Strong “one–minus–cosine” gust

For a strong gust intensity of 0.1, the struc-tural nonlinearities influence the prediction of

7


the worst–case–gust wavelength compared to thelinear case, which are shown in Fig. 8. The

Hg [ ]

α [d

eg]

5 10 15 20 25-0.8

-0.4

0

0.4

0.8

1.2LROMNROM

(a) Pitch

Hg [ ]

h [

mm

]

5 10 15 20 25-6

-4

-2

0

2

4LROMNROM

(b) Plunge

Fig. 8 : Worst–case–gust search for the "one–minus–cosine" gust of intensity 0.1 at freestreamspeed of 8 m/s; the vertical dashed and dotted linedenote the worst case gust wavelength of linearand nonlinear system, respectively

solid curves are the linear system results by us-ing linear reduced–order model (LROM), whilethe dashed dot curves are corresponding to thenonlinear system using NROM. The worst casegust wavelengths of the linear system for pitchand plunge at intensity of 0.1 are the same as thecase of light intensity of 0.005. In terms of thenonlinear system, the worst case gust wavelengthof pitch at this intensity is still 10 semi–chords,while the worst case gust of plunge is 12, whichis different from the linear system and the valuecomputed by Eq. (24). It is because of the non-

linear hardening effect when encountering stronggust. This effect reduces the gap between pitchand plunge vibration frequencies. Nonlineari-ties also deteriorate the worst case gust for pitchmode which increase its amplitude.

6.3 Gust loads alleviation

Gust loads alleviation to both discrete “one–minus–cosine” and continuous Von Kármán at-mospheric turbulence will be performed by us-ing FIR and MFIR controllers. Corresponding toeach type of gust, light and strong intensities willbe investigated. The freestream speed used in thissection is low speed of 8 m/s, a higher speed of13 m/s will be considered in Sec. 6.4.

6.3.1 Light Atmospheric Turbulence

The nonlinear open loop and control responses tomultiple cycles of “one–minus–cosine” gust forpitch and plunge at intensity of 0.005 and wave-length of 10 semi–chords (worst case for pitchmode) by using FIR and MFIR controllers areshown in Fig. 9. Both FIR and MFIR controllersexhibit the ability to alleviate gust loads responsefor pitch. Nearly half of the pitch amplitude isreduced by the FIR and MFIR controllers fromaround 0.15 deg to 0.08 deg at steady stage. Air-craft will benefit from this gust loads alleviationeffects, e.g. fatigue resistance and improving pas-senger comfort. The FIR control response takesa long time to converge to a constant amplitude,while the MFIR controller shows a fast conver-gence in 5 seconds. The reason is that MFIRcontroller considers the poles of the system in thebasis functions, see Sec. 3.2.

The controllers designed in the paper aim toalleviate the pitch response. The plunge controlresponse is here verified, see Fig. 9. The plungeresponse amplitudes of the two controllers areclose to the open loop at steady stage, while themean values increase some. There are some peakvalues for pitch and plunge by using the MFIRcontroller at the initial period, which are biggerthan the open loop responses , while the controlresponses by using FIR controller are stable andchanging gradually in the whole control process.

8


time [s]

α [d

eg]

0 5 10 15-0.2

-0.1

0

0.1

0.2Open loopFIRMFIR

Gust Profile

(a) Pitch

time [s]

h [

mm

]

0 5 10 15-0.4

-0.2

0

0.2Open loopFIRMFIR

(b) Plunge

Fig. 9 : Open loop and control responses to amulti–cycle “one–minus–cosine” gust with in-tensity of 0.005 and wavelength of 10 semi–chords at freestream speed of 8 m/s

time [s]

δ [d

eg]

0 1 2 3 4 5

-1

-0.5

0

0.5

1FIRMFIR

Fig. 10 : Flap rotation to a multi–cycle “one–minus–cosine” gust with intensity of 0.005 andwavelength of 10 semi–chords at freestreamspeed of 8 m/s

The initial five seconds output of the FIR andMFIR controllers, which are the rotation input ofthe flap in this case, are shown in Fig. 10.

The flap rotation of the MFIR controller isvery large at the beginning, while the values ofthe FIR controller are close to zero. At steadystage, the flap rotation of the MFIR controller be-come smaller but it is still bigger than the FIRcontroller. All the flap rotation and correspond-ing velocity of the two controllers are smallerthan the practical experimental limits: -7 deg≤ δ≤ 7 deg, δ≤ 15 Hz.

time [s]

α [d

eg]

0 5 10 15 20-0.04

-0.02

0

0.02

0.04Open loopFIR

Gust Profile

(a) Pitch

time [s]

h [

mm

]

0 5 10 15 20-0.3

-0.15

0

0.15

0.3Open loopFIR

(b) Plunge

Fig. 11 : Open loop and control responses toa Von Kármán gust (h = 10,000 m, intensity =10−2, ‘light’) with peak amplitude intensity of0.005 at freestream speed of 8 m/s

The Von Kármán turbulence model was usedto generate the continuous gust. Light turbu-lence gust was computed at 10,000 meters height

9


and freestream speed of 8 m/s with light inten-sity of 10−2. The peak amplitude of gust in-tensity is 0.005, which is the same as the light“one–minus–cosine” gust. The open loop andcontrol responses to this gust using FIR con-troller are shown in Fig. 11. Like in “one–minus–cosine” gust control, the MFIR controller ex-hibits even worse control characteristics at the be-ginning stage which is not shown here. The FIRcontroller has effect control on light Von Kármáncontinuous gust for pitch, while small differenceis found for plunge between open loop and con-trol responses. The FIR controller reduces theamplitude and mean value of the pitch response,which will decrease the gust impact on aircraftfatigue and extend the operation life. The flap ro-tation of the FIR controller shown in Fig. 12 andcorresponding velocity are within the experimen-tal setting limits.

time [s]

δ [d

eg]

0 5 10 15 20-0.1

-0.05

0

0.05

0.1FIR

Fig. 12 : Flap rotation to a Von Kármán gust (h= 10,000 m, intensity = 10−2, ‘light’) with peakamplitude intensity of 0.005 at freestream speedof 8 m/s

6.3.2 Strong Atmospheric Turbulence

A stronger gust intensity is here investigated toverify the control approach in cases where thestructure nonlinearity is strongly affects the sys-tem responses.

The open loop and control responses to “one–minus–cosine” gust with intensity of 0.1 andwavelength of 10 semi–chords (worst case forpitch mode) are shown in Fig. 13. Comparedwith light intensity of “one–minus–cosine” gust,

the FIR controller shows better performancefor strong “one–minus–cosine” gust, which con-verged to stable state quickly. The beginningstage of MFIR control is still not good. Bothcontrollers show effective gust loads alleviationfunctions for pitch. The amplitude of pitch wasreduced from around 3.9 deg to 2.1 deg. Themean values of plunge response by using the twocontrollers was increased from around -1.5 mmto -1.8 mm. The flap rotation is shown in Fig. 14.The initial flap rotation of the MFIR controllereven gets to the setting limits.

time [s]

α [d

eg]

0 2 4 6 8 10

-2

0

2

4Open loopFIRMFIR

Gust Profile

(a) Pitch

time [s]

h [

mm

]

0 2 4 6 8 10-4.5

-3

-1.5

0

1.5

3Open loopFIRMFIR

(b) Plunge

Fig. 13 : Open loop and control responses toa multi–cycle “one–minus–cosine” gust with in-tensity of 0.1 and wavelength of 10 semi–chordsat freestream speed of 8 m/s

10


time [s]

δ [d

eg]

0 2 4 6 8 10-8

-4

0

4

8FIRMFIR

Fig. 14 : Flap rotation to a multi–cycle “one–minus–cosine” gust with intensity of 0.1 andwavelength of 10 semi–chords at freestreamspeed of 8 m/s

time [s]

α [d

eg]

0 5 10 15 20

-0.8

-0.4

0

0.4

0.8Open loopFIR

Gust Profile

(a) Pitch

time [s]

h [

mm

]

0 5 10 15 20-4

-2

0

2

4Open loopFIR

(b) Plunge

Fig. 15 : Open loop and control responses toa Von Kármán gust (h = 10,000 m, intensity= 10−5, ‘severe’) with peak intensity of 0.1 atfreestream speed of 8 m/s

Figure 15 shows the open loop and control re-sponses to Von Kármán turbulence gust by usingFIR controller. The gust was generated by VonKármán turbulence model at altitude of 10,000meters and freestream speed of 8 m/s with severeintensity of 10−5. The peak amplitude of gust in-tensity is 0.1. The control responses of pitch andplunge show similar characteristics to the resultsof light Von Kármán turbulence gust. The flap ro-tation of FIR controller is shown in Fig. 16. Allthe values are located in the setting range.

time [s]

δ [d

eg]

0 5 10 15 20-2

-1

0

1

2FIR

Fig. 16 : Flap rotation to a Von Kármán gust (h =10,000 m, intensity = 10−5, ‘severe’) with peakintensity of 0.1 at freestream speed of 8 m/s

6.4 Limit Cycle Oscillations suppression

It was observed that the nonlinear system exhibitsLCOs below the flutter speed above a speed of12.45 m/s. In this section, the suppression ofa gust–induced LCO response is investigated ata freestream speed of 13 m/s (85% of the flut-ter speed). A comparison between prediction andmeasured LCOs is reported in Ref. [12].

6.4.1 “One–minus–cosine” gust induced LCO

Figure 17 shows the open loop and control re-sponses by using FIR and MFIR controllers interms of a one–cycle “one–minus–cosine” gustinduced LCO at 13 m/s. It is suggested that theMFIR controller has good effect to suppress LCOfor both degrees of freedom, which reduces thepitch amplitude from 6 deg to 4 deg and plunge

11


time [s]

α [d

eg]

0 2 4 6 8-5

-2.5

0

2.5

5Open loopFIRMFIR

Gust Profile

(a) Pitch

time [s]

h [

mm

]

0 2 4 6 8-10

-5

0

5

10Open loopFIRMFIR

(b) Plunge

Fig. 17 : Open loop and control responses toa one–cycle “one–minus–cosine” gust with peakintensity of 0.1 at freestream speed of 13 m/s

time [s]

δ [d

eg]

0 2 4 6 8-2

-1

0

1

2

3FIRMFIR

Fig. 18 : Flap rotation to a one–cycle “one–minus–cosine” gust induced LCO suppression atfreestream speed of 13 m/s

amplitude from 14 mm to 9 mm. The FIR con-troller doesn’t have reduction effect for the pitchand plunge responses. The reason can be foundfrom the output signal of controller shown inFig. 18, which is the input of flap. The outputof the FIR controller converges to zero quickly,while the output of the MFIR controller atten-uates slowly, which means that there is still ef-fective input for the flap to suppress LCO afterswitching off the gust input.

6.4.2 Von Kármán turbulence induced LCO

time [s]

α [d

eg]

0 2 4 6 8-5

-2.5

0

2.5

5Open loopMFIR

Gust Profile

(a) Pitch

time [s]

h [

mm

]

0 2 4 6 8-8

-4

0

4

8

open loopMFIR

(b) Plunge

Fig. 19 : Open loop and control responses to aVon Kármán turbulence gust induced LCO sup-pression at freestream speed of 13 m/s

A Von Kármán turbulence induced LCO andcontrol response by using MFIR controller areshown in Fig. 19. The Von Kármán turbulence

12


last for 3 seconds from the beginning. The ampli-tude of pitch and plunge responses were reducedfrom 6 deg to 5 deg and from 14 mm to 11 mm.Here the control results by using FIR controlleris omitted, which doesn’t have any LCO suppres-sion effect. The flap rotation input of the MFIRcontroller shown in Fig. 20 meet the limit re-quirements.

time [s]

δ [d

eg]

0 2 4 6 8-2

-1

0

1

2MFIR

Fig. 20 : Flap rotation and velocity to a Von Kár-mán turbulence gust induced LCO suppression atfreestream speed of 13 m/s

7 Conclusions

This paper investigated adaptive feedforwardcontrol for gust loads alleviation and limit cycleoscillations (LCOs) suppression. Two kinds ofbasis functions, finite impulse response (FIR) andmodified finite impulse response (MFIR), wereused to design the controller. The robustness wasincreased by using an adaptive control strategy.The test case is a nonlinear aeroelastic aerofoilsection, with pitch and plunge as degrees of free-dom. The following considerations can be drawnfrom the present work:

• The FIR controller shows good control ef-fect for multi–cycle “one–minus–cosine”and Von Kármán turbulence gust loads al-leviation, while the MFIR controller tendsto increase the amplitude during the initialtransient of the response.

• The MFIR controller was found adequateto reduce the amplitude of the “one–

minus–cosine” and Von Kármán turbu-lence gust–induced limit cycle oscillationsfor both pitch and plunge degrees of free-dom.

8 Acknowledgments

The first author is grateful to China ScholarshipCouncil (CSC) for the financial support received.The authors also acknowledge the ongoing re-search collaboration with the University of Liv-erpool and for their continuous feedback.

References

[1] Da Ronch, A., Badcock, K. J., Wang, Y., Wynn,A., and Palacios, R. N., “Nonlinear Model Re-duction for Flexible Aircraft Control Design,”AIAA Atmospheric Flight Mechanics Confer-ence, AIAA Paper 2012–4404, Minneapolis,MN, 13–16 August 2012, doi: 10.2514/6.2012-4404.

[2] Karpel, M., “Design for Active Flutter Sup-pression and Gust Alleviation Using State–Space Aeroelastic Modeling,” Journal of Air-craft, Vol. 19, No. 3, 1982, pp. 221–227, doi:10.2514/3.57379.

[3] Block, J. J. and Strganac, T. W., “Applied Ac-tive Control for a Nonlinear Aeroelastic Struc-ture,” Journal of Guidance, Control, and Dy-namics, Vol. 21, No. 6, 1998, pp. 838–845, doi:10.2514/2.4346.

[4] Vipperman, J. S., Clark, R. L., Conner, M.,and Dowell, E. H., “Experimental Active Con-trol of a Typical Section Using a Trailing–EdgeFlap,” Journal of Aircraft, Vol. 35, No. 2, 1998,pp. 224–229, doi: 10.2514/2.2312.

[5] Da Ronch, A., Tantaroudas, N. D., andBadcock, K. J., “Reduction of NonlinearModels for Control Applications,” 54thAIAA/ASME/ASCE/AHS/ASC Structures, Struc-tural Dynamics, and Materials Conference,AIAA Paper 2013–1491, Boston, MA, 08–11April 2013, doi: 10.2514/6.2013-1491.

[6] Schmitt, D., Rehm, W., Pistner, T., Diehl, H.,Navé, P., Jenaro-Rabadan, G., Mirand, P., andReymond, M., “Flight Test of the AWIATORAirborne LIDAR Turbulence Sensor,” 14th Co-herent Laser Radar Conference, Hunstvile, AL,June 2007.

13


[7] Zeng, J., Moulin, B., De Callafon, R., andBrenner, M., “Adaptive Feedforward Controlfor Gust Load Alleviation,” Journal of Guid-ance, Control, and Dynamics, Vol. 33, No. 3,2010, pp. 862–872, doi: 10.2514/1.46091.

[8] Da Ronch, A., Tantaroudas, N., and Jiffri,S.and Mottershead, J., “A Nonlinear Con-troller for Flutter Suppression: from Sim-ulation to Wind Tunnel Testing,” 55thAIAA/ASMe/ASCE/AHS/SC Structures, Struc-tural Dynamics, and Materials Conference,AIAA SciTech 2014, National Harbor, MD,11–15 January 2014, doi: 10.2514/6.2014-0345.

[9] Ninness, F. and Gustafsson, F., “A UnifyingConstruction of Orthonormal Bases for SystemIdentification,” IEEE TRANSACTIONS ON AU-TOMATIC CONTROL, Vol. 42, No. 4, 1997,pp. 515–521, doi: 10.1109/9.566661.

[10] Haykin, S., Adaptive Filter Theory, chap. 13,Prentice Hall Information and System Scienceseries, 3rd ed., 2001, pp. 566–569.

[11] Papatheou, E., Tantaroudas, N. D., Da Ronch,A., Cooper, J. E., and Mottershead, J. E.,“Active control for flutter suppression: anexperimental investigation,” International Fo-rum on Aeroelasticity and Structural Dynamics(IFASD), Bristol, U.K., 24–27 June 2013.

[12] Fichera, S., Jiffri, S., Wei, X., Da Ronch, A.,Tantaroudas, N. D., and Mottershead, J. E.,“Experimental and numerical study of nonlin-ear dynamic behaviour of an aerofoil,” ISMA2014 Conference, Leuven, BE, 15–19 Septem-ber 2014.

9 Contact Author Email Address

Wang, Y.: [email protected] Ronch, A.: [email protected]–Tehrani, M.: [email protected], F.: [email protected]

10 Copyright Statement

The authors confirm that they, and/or their company ororganization, hold copyright on all of the original ma-terial included in this paper. The authors also confirmthat they have obtained permission, from the copy-right holder of any third party material included in this

paper, to publish it as part of their paper. The authorsconfirm that they give permission, or have obtainedpermission from the copyright holder of this paper, forthe publication and distribution of this paper as part ofthe ICAS 2014 proceedings or as individual off-printsfrom the proceedings.

14

Documents

ADAPTIVE FEEDFORWARD CONTROL DESIGN FOR GUST …