A Numerical Study on Active Control for Tiltrotor Whirl FlutterStability Augmentation

Joerg P. Mueller! Yves Gourinat Rogelio FerrerDynamics Engineer Professor Dynamics Engineer

Eurocopter SUPAERO EurocopterMarignane, France Toulouse, France Marignane, France

Tomasz Krysinski Benjamin KerdreuxDynamics Manager Dynamics Engineer

Eurocopter EurocopterMarignane, France Marignane, France

ABSTRACTThe use of active control to augment whirl flutter stability of tiltrotor aircraft is studied by means of a multibody simulation.The numerical model is based on a 1/5 scale semi-span aeroelastic wind tunnel model of a generic tiltrotor concept andpossesses a gimballed, stiff-in-plane rotor that is windmilling. A single-input single-output controller and two types ofmulti-input multi-output algorithms, Linear Quadratic Gaussian Control and Generalized Predictive Control, are studied.They are using measured wing deflections in order to calculate appropriate swashplate input. Results on the closed-loopbehavior of three wing and two gimbal natural modes are given. Robustness analyses with respect to major parameters likewing natural frequencies or structural damping are also briefly discussed. The rotor shear force is shown in the uncontrolledcondition and in presence of a controller in order to illustrate the whirl flutter mechanism. The single-input single-outputcontroller yielded substantial gain in stability and turned out to be most suitable for industrial application, whereas theLinear Quadratic Gaussian Regulator yielded even higher damping and still had good robustness characteristics.

NOTATION

f frequencyg controller gainh flight altitudem number of plant inputs

! Currently: Aeroelastics Engineer, AIRBUS, Bremen, GermanyRevised version of the paper “A Multibody Study on Single- and Multi-Variable Control Algorithms for Tiltrotor Whirl Flutter Stability Augmen-tation”, presented at the American Helicopter Society 4th Decennial Spe-cialist’s Conference on Aeromechanics, Fisherman’s Wharf, San Francisco,California, January 21-23, 2004.

p model orderq natural moder number of plant outputsu plant inputwr weight on command angles (GPC)x state vectorx estimator state vectory plant outputG plant transfer functionH controller transfer functionJ performance indexK controller gain matrix

1

!+ rotor progressive gimbal flap mode!" rotor regressive gimbal flap mode" damping, % critical#1c longitudinal cyclic swashplate input#1s lateral cyclic swashplate input#0 collective swashplate inputµ advance ratio, forward speed / blade tip speed$ parameter governing state estimator dynamics (LQG)% phase angle& wing degree of freedom (deflection angle)' angular velocity( rotor rotation speed

Subscripts

()b value related to wing beamwise bending()c value related to wing chordwise bending()d all pass()G plant related()H controller related()hp high pass()l p low pass()t value related to wing torsion()crit critical quantity (flutter point)

INTRODUCTION

Tiltrotor aircraft have been the subject of research and devel-opment since well before the 1950’s when the first flight ofthe Transcendental 1G took place. Considerable experiencein tiltrotor design was acquired with the research aircraftXV-3 and XV-15 that followed, but the first tiltrotor to enterservice in the near future will be the military V-22 Ospreyand the smaller BA-609 for the civil and parapublic market.The most recent European tiltrotor concept ERICA is pre-sented in Fig. 1 . Tiltrotor aircraft provides unique featuresas it combines the advantages of helicopters, notably ver-tical take off, landing and hover capabilities, with the highcruise speed and range of turboprop airplanes.

The tiltrotor concept is subject to an instability called whirlflutter, which occurs in high speed flight in the airplaneconfiguration, and which constitutes an important factor intiltrotor design as it can lead to very high dynamic loads. Itis therefore necessary to guarantee adequate stability mar-gins throughout the flight envelope.

Studies into aeroelastic stability augmentation focus basi-cally on the design of more efficient aircraft. A secondobjective is to extend stability boundaries with respect toflight speed. One major structural design parameter influ-encing whirl flutter stability is the wing (torsional-) stiffness.For this reason, today’s tiltrotor concepts have wing struc-tures with a high thickness to chord ratio, typically 21% to

23%, inducing high aerodynamic drag in cruise. Therefore,it would be advantageous to provide adequate stability byother means in order to reduce wing stiffness requirements.

Whirl flutter on tiltrotor aircraft was discovered duringfullscale wind tunnel testing of the XV-3 research aircraftin the early 1960’s. On an articulated rotor, the destabilizingeffect is generated by large in-plane forces that act on the py-lon. They become destabilizing in the high-inflow conditionand are the direct result of cyclic blade incidence variationsthat arise during nacelle pitch and yaw motion as the rotortip path plane is following the plane perpendicular to therotor mast with a time lag, Ref. 1 . Consequently, the phe-nomenon can occur in a combined manner on the pitch andyaw axes but also on one axis only. Furthermore, forwardand retrograde whirl is possible.

In the past, considerable work has been done on passive op-timization of wings and rotors. The influence of differentdesign parameters, like wing frequencies, wing sweep or ro-tor flap and lag dynamics, is presented in Ref. 2 . Formal op-timization techniques with the objective to find an optimalset of rotor design parameters are reported in Ref. 3 wherethe main parameters considered are pitch-flap and pitch-lagcouplings, blade flapping flexibility distribution and wingstiffness.

Active control has been studied with the objective of ex-panding the tiltrotor flight envelope and enhance maneuver-ability, versatility and ride qualities. In this context, the as-pects of gust response and maneuver loads with respect torotor loads, drive train loads and vibration behavior havebeen looked at in Refs. 4 and 5 . On the other hand, lesswork has been done on active control for aeroelastic stabilityaugmentation. The aim of the latter is to enable the aircraftto fly at higher speeds, to descend more steeply or to reducestructural stiffness and therefore weight. The advent of fly-by-wire control systems with powerful flight control com-puters has made feasible the implementation of controllersof higher complexity for stability augmentation. Recent ac-tuator technology is able to handle commands of requiredbandwidth, and the controller commands can therefore beapplied directly on the control system.

The potential of aeroelastic control to increase dynamic sta-bility margins has been studied in Refs. 6 to 12 . Thefirst three references deal with rather direct feedback to the

Fig. 1 . Artist’s illustration of the European ERICA ad-vanced tiltrotor concept.

2

swashplate. Either wing deformation (Ref. 6) or velocityand acceleration measurements (Refs. 7 , 8) are used forfeedback. As these control laws are basically single-inputsingle-output (SISO) or a combination of several SISO chan-nels, their effectiveness on multiple natural modes is lim-ited. In order to increase stability in a more comprehensivemanner, multi-input multi-output (MIMO) algorithms havealso been studied. In Ref. 9 , a Linear Quadratic Regula-tor with a state estimator is presented that is based on ana-lytical equations of the tiltrotor dynamics. In modal space,stable and unstable modes were separated and the controllerwas designed for the unstable modes only. A more recentMIMO control theory, the Generalized Predictive Control,was evaluated in Ref. 4 for maneuver load alleviation usinga multibody model and in Ref. 11 for flutter control usingwind tunnel tests.

As a complement to the cited publications, the present paperevaluates a SISO controller and two MIMO control algo-rithms by means of the same numerical modeling approach.Thus, a direct comparison of different control strategies ispossible. The SISO controller is based on a 180# out-of-phase feedback of wing motion to the swashplate whereasLinear Quadratic Gaussian Control and Generalized Predic-tive Control were chosen for MIMO studies. A general pur-pose multibody analysis tool is used as the central simula-tion engine.

Frequency and damping of three fundamental wing modesand two rotor flap modes are presented up to the closed-loopstability boundaries. A comparison of the robustness of thecontrol algorithms with respect to variations in wing nat-ural frequencies, structural damping, flight speed, rotationspeed, flight altitude, and signal noise are presented. Thework is done with the objective of developing the methodsand knowledge for application to subsequent wind tunneltests. Thus, system identification techniques have been ap-plied to describe the dynamics of the rotor system. Physicalillustration of controller action is given by comparing thecontroller generated rotor shear forces with the oscillationto be attenuated.

The paper is organized as follows: In the next section, thenumerical model and a brief validation are described. Thebasic equations underlying the control algorithms are de-scribed thereafter, together with a description of the iden-tification techniques that are used to determine the systemtransfer functions from swashplate input to wing deflection.The subsequent section summarizes the basic procedure ofthe numerical simulation and the analysis techniques that aregenerally applied in the time domain. The determination ofidentification and controller parameters is also presented indetail. A results section presents an evaluation of the threecontrol techniques, and the paper closes with the conclu-sions and gives recommendations about the most suitablealgorithms.

NUMERICAL MODEL

Model Configuration

For this initial investigation, a semi-span proprotor modelwith a cantilevered wing is chosen and the portside rotor ismodeled. This configuration contains the basic features oftiltrotor dynamics and has been the subject of much theoret-ical and experimental work. A windmilling rotor has beenchosen, as it was found in past research that this is a conser-vative approach. Due to this assumption, the dynamic modelis further simplified as the drive-train and the engine dynam-ics are less important. This is favorable for system identifi-cation and controller design and reduces the complexity ofthe numerical model, which means substantial advantages interms of computation speed and numerical stability.

The main degrees of freedom are three wing deflections andthe two gimbal flapping angles. The corresponding funda-mental wing natural modes beamwise bending, chordwisebending, and torsion are modeled as they contribute to theprincipal mechanism involved in whirl flutter. Cyclic flap-ping of the rotor is accounted for by the gimbal joint. Otherrotor flexibility is not included, even though rotor coning aswell as collective and cyclic lag dynamics do have some in-fluence on whirl flutter (Ref. 13) . These choices were madein the context of the development of control algorithmswhere it is important to keep the system order rather low.This allows focusing on the critical modes and therefore de-signing an efficient controller tailored to those modes.

The baseline for this work is a generic tiltrotor design in-spired by the new European ERICA concept (Ref. 14) . Itsdynamic characteristics have been translated to a 1/5 scaleaeroelastic model using Froude scaling as summarized inTable 1 . Here, the rotor blade inertias and static momentsare given with respect to the hub center. Contrarily to theERICA rotor, an equivalent three-bladed rotor is modeledin order to be as close as possible to an in-house wind tun-nel model. This equivalence has been achieved by keepinga similar Lock number and solidity for the three-bladed ro-tor as the four-bladed rotor model would have. The aero-dynamic coefficients are approximations taken from airfoildata. Flapping inertia of hub and control system as well ascontrol system mass are neglected. Furthermore, no springstiffness in the gimbal joint is taken into account. Finally,the nacelle containing engine and gear box is assumed to berigidly attached to the wing, the wing itself being modeledas one entity in terms of mass and inertia.

For modeling the principal aerodynamic forces, a simpleblade element model has been used. No inflow model hasbeen considered as the rotor induced flow field is less im-portant in the high-inflow condition. Furthermore, incom-pressible airfoil characteristics have been used that are justi-fied by the low velocity at the blade tip of well bellow Mach0.3 even in forward flight. Wing aerodynamics are not takeninto account since previous investigations showed that it has

3

Tabel 1 . Model data.

Parameter Symbol Value

Rotorradius R 729mmnumber of blades b 3solidity (thrust wtd.) ) 0.18rotation speed nr 950 rev/mindistance of rotor centerfrom conversion axis 331mm

mass mb 0.51 kgblade static moment ms 0.2 kgmblade flapping inertia I! 0.081 kgm2Lock number * 3.4twist (non-lin. distrib.) #t "35 #

lift coefficient (+ = 0) cl,0 0.34lift curve slope cl,+ 5.7 rad"1drag coefficient (+ = 0) cd,0 7.5 ·10"3quadratic drag coeff. cd,+ 0.8 rad"2

Pitch control systempitch-flap coupling ,3 "20 #

Hub and const. vel. jointmass mhub 1.07 kg

Winghalf wingspan 1480mmequivalent beam length 980mmmass mw 3.22 kgstructural damping "s 3%

Nacellemass mn 6.62 kg

a slightly stabilizing effect. The rotation axis is assumed tobe parallel to the forward speed vector of the aircraft.

A numerical simulation model based on this data was builtusing a general purpose multibody simulation tool with mul-tiple interfaces to industrial design tools and control soft-ware. The numerical model generation is partially based ona CAD virtual mockup of the wind tunnel model for geome-try and mass data, which highlights the benefits of integrat-ing the dynamic simulation within the virtual prototypingframework. After extraction of the main bodies, the articu-lations have to be added and rotor blade aerodynamics haveto be implemented in the model by explicit formulation offorce vectors. The industrial multibody code, being not spe-cially adapted to rotating systems and periodicity, providesno direct stability analysis and linearization mechanisms.Thus, the stability characteristics have been extracted fromtime domain response data after individual excitation of thenatural modes.

The major mechanical components are modeled with rigidbodies that are connected by different types of joints withdiscrete stiffness and damping. The main bodies are the ro-

x

yz

Fig. 2 . Multibody model, global view.

Tabel 2 . Natural frequencies of the fundamental wingmodes.

Characteristic Symbol Frequency , Hz

beamwise bending fb 5.1chordwise bending fc 7.2torsion ft 10.5

tor blades, the hub, the bodies related to the control system,the rotor mast, the nacelle, and the rigid wing beam. Themain articulations are the gimbal joint represented by anideal spherical joint, and the equivalent suspension mech-anism of the wing. The secondary joints are the rotor tiltactuator, which is fixed in the airplane configuration for thisstudy, the rotor drive articulation, a joint that prescribes thetrajectory of the vehicle in space and various articulationsbelonging to the control system and constant velocity joint.

The constant velocity joint is a V-22 like concept that isbased on three flexible drive links. It is designed to transferthe mast torque to the rotor hub that inclines due to cyclicpitch. Three drive links are connected on one side to theapices of a star fixed on the mast and on the other side to thehub. As the hub side prescribes a movement out of the planeperpendicular to the mast, the drive links are subjected toperiodic deformation. The periodic loads that are generatedin the drive links balance each other, resulting in a homoki-netic transmission of rotation from the mast plane to the hubplane.

The wing is approximated by a rigid beam whose length istwo thirds of the half wingspan. This choice results, for agiven wing tip deflection, in the deformation angle at thewing tip that a uniform beam would have. The wing stiff-ness is modeled by three angular spring-dampers at the beamarticulation point. They are oriented in the beamwise, chord-wise and torsion directions to represent the three fundamen-tal natural modes of the wing. These spring-dampers also

4

rotor blade

aerodynamic forces

pitch change bearing

hub

gimbal jointrotor mast pitch rod

rotating swashplate(synchronized with mast rotation)

non-rotating swashplate

cyclic pitch input

collective pitch input

rotor drive

nacellerotor tiltbearing

wing

equivalent wingstiffness/damping

flight speed

only one rotor blade shown for clarity

Fig. 3 . Basic kinematics of the multibody model (excluding constant veloc-ity joint).

Fig. 4 . Wind tunnel model.

provide structural damping and their stiffness is chosen toyield the natural frequencies corresponding to the overallhalf wing assembly, including nacelle and rotor system, Ta-ble 2 . As these are structural values, they are given for thecase of a non-rotating rotor with a clamped gimbal joint.

An overall view of the multibody model is given in Fig. 2 .A detailed representation of the major bodies and articula-tions is shown in Fig. 3 , although only one rotor blade isshown for clarity and the constant velocity joint is not rep-resented. It should be noted that there is no individual flapor lead-lag articulation on the rotor blades. The rotation ofthe swashplate is synchronized with the mast by a virtualcoupler.

Model Validation

An in-house aeroelastic wind tunnel model whose dynamiccharacteristics are adjustable to a large range of different dy-namic configurations is used to determine open-loop stabil-ity behavior and to validate the numerical model. The windtunnel model possesses a gimballed stiff-in-plane rotor and aconstant velocity drive system. A picture of the model in theEurocopter wind tunnel is given in Fig. 4 . Although the val-idation was made on a configuration slightly different from

the ERICA Froude scaled model, it was possible to confirmthe overall validity of the approach. Especially the behav-ior of the uncontrolled system in terms of critical speed anddamping gradient has been found to be well represented af-ter some minor adjustments of structural damping. For moredetailed information, see Ref. 15 .

CONTROL ALGORITHMS

The study focuses on three different control algorithms: ASISO algorithm is considered first as it is a relatively simplemethod. Due to the nature of the algorithm, it is possibleto limit its effects to a narrow frequency band and henceto reduce negative effects on the system dynamics and thepilotability of the aircraft. The present SISO algorithm isstraightforward to implement as it consists of a fifth-orderfilter. Furthermore, its functional principle is clearly de-fined and the stability margins can be determined precisely,which makes it particularly suitable for industrial applica-tion. Only plant gain and phase-shift at the critical reso-nance frequency need to be known for the basic synthesisof the filter; however, knowledge of an analytical approx-imation of the eigen-mode allows a more detailed stabilityanalysis. In the present study, a second-order transfer func-

5

tion is adjusted to measured gain and phase data at severaldiscrete frequencies.

An optimal regulator with a Kalman-Bucy state estimatorand Loop Transfer Recovery is the first MIMO control al-gorithm to be evaluated. It increases damping on more thanone natural mode and therefore augments the aeroelastic sta-bility in a more comprehensive way. This LQG/LTR al-gorithm is considered since the technology is well estab-lished in the industrial environment. For the synthesis ofthe controller, a mathematical model of the plant needs tobe known. In the present study, this state-space model isobtained by system identification and the subsequent appli-cation of system realization techniques. An identificationalgorithm called Autoregressive with Exogenous Input, ARXis used in the time-domain. This method was chosen in or-der to evaluate its validity for the present phenomenon withthe objective of applying the same algorithms to a physi-cal wind-tunnel model. As LQG/LTR controller synthesisis performed off-line and results in a state-space model, itis easy to implement on real-time hardware for real-worldexperiments.

The second MIMO algorithm is Generalized PredictiveControl, which has been developed more recently and is di-rectly based on identified ARX coefficients.

Time Domain Identification

The mathematical plant model for the MIMO control algo-rithms is calculated from input/output time histories takenfrom the multibody simulations. The system identificationprocess is based on an ARX-model of a given order p.

In discrete time, the output at time step (k) can be written asa function of the previous time steps (k" i) in the followingform:

y(k) =p∑

i=0!i u(k" i)+

p∑

i=1+i y(k" i) (1)

This equation can be applied to SISO and MIMO systems,where in the latter case, the output vector y has m elementsand the input vector u has r elements where m is the numberof outputs and r is the number of inputs. The matrices + and! have m$m and m$ r elements respectively.

When applying Eq. (1) to a sequence of l previous timesteps, this yields l equations where the unknowns are theARX coefficients +i and !i . A unique set of ARX coef-ficients can then be calculated by solving these equationsin a least-squares sense. More information about the ARXmodel can be found in Refs. 11 and 16 .

SISO Feedback

A classic approach to control vibration is the direct feed-back of a measured quantity, like displacement or acceler-ation, to one of the control actuators. The principle of thecontrol algorithm of this study is to calculate a command

+

-

.

"%G

#

&

&s/p

#s/p

"%H

"%G

Fig. 5 . Schematic of the SISO control principle. Theresponse &s/p acts 180 # out of phase to the perturbation& that is to be attenuated.

that generates a system response that is 180 # out-of-phasewith the perturbation that is to be attenuated. Phase shift andamplification are depicted in the complex plane in Fig. 5 ,where # is the swashplate input and & the plant responseor output. The phase angle of this transfer function %G cor-responds to the angle between the two arrows. In order toobtain a system response &s/p which acts in opposition toa measured perturbation, the controller has to account forthis phase shift. Thus, the necessary phase angle %H of thecontroller is:

%H = "180# "%G (2)

The mathematical formulation of the control law consistsof a band-pass H1, an all-pass H2 and a proportional gaing. The band-pass filter reduces the disturbance of aircraftdynamics and handling characteristics by filtering out thedynamics that are not associated with the critical mode:

H1 =s2

s2+2"H'hps+'2hp︸ ︷︷ ︸

high pass

·'2l p

s2+2"H'l ps+'2l p︸ ︷︷ ︸

low pass

(3)

The cut-off frequencies 'hp and 'l p are chosen in the vicin-ity of the natural frequency of the critical mode. A highdamping ration "H needs to be used in order to yield lowphase variation with frequency and therefore good robust-ness margins with respect to variations in modal frequency.The adjustment of the phase shift of the filter is done withan all-pass transfer function of the following form:

H2 ="'ds+1'ds+1

(4)

Here, the parameter 'd allows variation of phase shift whilemaintaining a constant gain. The complete filter including astatic gain g is therefore:

H = g ·H1 ·H2 (5)

A block diagram of the feedback loop is given in Fig. 6 .

6

+

+

# &#t = 0#

multibody model

phase-shift band-pass

g H2 H1

G

Fig. 6 . Block diagram of the SISO feedback loop.

Optimal Control

A well proven algorithm for MIMO control is the LinearQuadratic Regulator. It consists of calculating the optimalgain matrix for feedback of the system states to the plantinputs. If applied alone, the Quadratic Regulator possessesexcellent performance and robustness characteristics but itis rather impractical to measure all system states. Therefore,the Quadratic Regulator is often accompanied by a Kalman-Bucy filter that estimates the states of the system on the basisof a few output measures. The so-formed Linear QuadraticGaussian Control is the first MIMO control algorithm of thiswork.

Basic Problem. The starting point for the development ofthe Linear Quadratic Gaussian State Feedback Regulator isthe linear time-invariant state-space model with noise on thesystem states and the output measures:

x= Ax+Bu+Mwy=Cx+ v

(6)

In this equation, x is the state vector, u the input vector, ythe output vector, A the system matrix, B the input gain ma-trix, C the output gain matrix, and M is a constant matrixthat is weighting the disturbances. The process noise vec-tor w and the measurement noise vector v are non-correlatedwhite noise with zero means and the constant, symmetric co-variance matricesW (positive semi-definite) and V (positivedefinite) that are defined as follows:

E[w(t)w(/)t

]=W, (t" /)

E [v(t)v(/)t] =V, (t" /)

E [w(t)v(/)t] = 0

(7)

Here, E denotes the error function, , the Dirac or delta func-tion, and t the transpose.

The LQG synthesis consists of finding a control law that sta-bilizes the system and minimizes the quadratic performanceindex

J = limT%0

E[∫ T

0

(xtQx+utRu

)dt

](8)

where Q is a constant, symmetric and positive semi-definitematrix and R is constant, symmetric and positive definite.These weighting matrices give the relative importance ofstate distortion and control power. It is useful to setQ=CtCwhich allows the distortions to be weighted in terms of the

physical output measurements. In the present study, the in-puts as well as the output measurements are angles of thesame order of magnitude. The weighting matrix R is there-fore set to the identity matrix I, which gives an equal impor-tance to output distortion angles and command angles.

State Estimator, Kalman-Bucy filter. The first step of thesynthesis of an LQG controller is to search for the maximumlikelihood / minimum variance estimate x of the state vector.

The state estimator dynamics is given by

˙x= Ax+Bu+Kf (y"Cx) (9)

where the estimation error y"Cx needs to tend to zeroasymptotically.

The optimal estimator (or Kalman) gain matrix isKf = Pf CtV"1 where Pf is the constant, symmetric, posi-tive definite solution of the following algebraic matrix Ric-cati equation:

Pf At+APf "Pf CtV"1CPf +MWMt = 0 (10)

Here,Wx =MWMt can be interpreted as the covariance ma-trix of the state noise that can expressed in terms of the morephysical input noise by setting Wx = BBt. The covarianceof the measurement noise is taken to be equal for all anglemeasurements and can therefore be set to V = $I with theparameter $ for adjusting the convergence speed of the esti-mator dynamics.

Linear Quadratic Gaussian Regulator. By using the es-timated state vector, a linear quadratic regulator can nowbe designed. It consists of finding a gain matrix Kc so thatthe feedback u(t) = "Kc x(t) stabilizes the system and min-imizes the quadratic performance index Eq. (8).

This optimal controller gain is given by

Kc = R"1BtPc (11)

where Pc is the constant, symmetric, positive definite matrixsolution of the algebraic Riccati equation:

PcA+AtPc"PcBR"1BtPc+Q= 0 (12)

Loop Transfer Recovery. The LQR and the Kalman-Bucyfilter have excellent individual robustness characteristics.However, when used in combination, the resulting LQGhas much smaller stability margins. The objective of LoopTransfer Recovery, LTR is to chose the matrices Wx and Vappropriately in order to recover asymptotically the robust-ness of either the LQR or the Kalman-Bucy filter. Therefore,the stochastic nature of the process environment modeling isdiscarded in favor of robustness and performance criteria.

The objective is to recover the open-loop transfer function ofthe LQR "Kc (sI"A)"1B via the open-loop transfer func-tion K(s) ·G(s) of the LQG regulator. In the present case

7

and under the hypothesis of a minimum phase transfer func-tion G, this can be achieved when the factor $ tends to zero.On the other hand, it is important not to choose too small avalue for $ in order to avoid very fast estimator dynamics.An overly fast estimator would degrade the complementaryoutput sensitivity function at high frequency and thereforereduce robustness to non-structured uncertainty.

Generalized Predictive Control

GPC is a receding-horizon LQ control law for which the fu-ture control sequence is recalculated at each time step. Themethod used in the present work is based on the ARXmodelas described in Ref. 17 .

A key feature of GPC is the calculation of control actionaccording to the predicted reaction of the system. Thus,the algorithm allows the choice of the controller speed in awide range between the extreme deadbeat (minimum-time)behavior with high control effort, and an approximation ofthe minimum control energy solution.

Multi-Step Output Prediction. The ARX equation Eq. (1)can be interpreted as a one-step prediction of system re-sponse y into the future:

y(k) = ! (0)0 u(k)+

p∑

i=1! (0)i u(k" i)+

p∑

i=1+(0)i y(k" i) (13)

Here the superscript (0) denotes the current time step.

When recursively applying this equation for successive pre-diction time steps, the predicted output at time step j in thefuture can be written as follows:

y(k+ j)=j∑

i=0! (i)0 u(k+ j" i)

+p∑

i=1! ( j)i u(k" i)+

p∑

i=1+( j)i y(k" i)

(14)

where all ARX coefficients beyond the model order are as-sumed to be equal to zero. The remaining coefficients +( j)

iand ! ( j)

i are calculated in a recursive manner according to:

! ( j)i = ! ( j"1)

i+1 ++( j"1)1 ! (0)

i

+( j)i = +( j"1)

i+1 ++( j"1)1 +(0)

i(15)

In these equations, the coefficients at j = 0 , +(0)i and ! (0)

i ,are the observer Markov parameters +i and !i which havebeen identified by means of the ARX model, Eq. (1).

When grouping the future inputs and outputs according to

uhc =

u(k)u(k+1)

...u(k+hc"1)

, yhp =

y(k)y(k+1)

...y(k+hp"1)

(16)

and the past inputs and outputs according to

up =

u(k"1)u(k"2)

...u(k" p)

, yp =

y(k"1)y(k"2)

...y(k" p)

(17)

the multi-step output prediction equation Eq. (14) can be putinto matrix form:

yhp = Tp uhc +Bp up+Ap yp (18)

with the matrices Ap, Bp and Tp being composed of the co-efficients +(i)

j and ! (i)j according to Eq. (14) and Eq. (15).

Here, the future inputs and outputs are assumed to be zerobeyond the respective control horizon hc and predictionhorizon hp where hc& hp. The matrices are therefore knownat each time step and the only unknowns are the future out-puts yhp as functions of the future inputs uhc .

Control Law. The objective of the GPC algorithm is to cal-culate the future inputs in order to achieve the desired out-put. The regulation error can therefore be formulated usingEq. (18) as the difference of system response yhp from thetarget response yt :

1 = yt " yhp = yt "Tp uhc "Bp up"Ap yp (19)

For the present regulator problem, the target response yt isequal to zero.

A cost function that is quadratic in the error as well as in thefuture control inputs can be formulated similar to the LQRproblem of Eq. (8):

J = 1 tQ1 +uthcRuhc (20)

The weighting matrix Q is constant, symmetric and positivesemi-definite and R is constant, symmetric and positive def-inite. Similar observations as in the LQR problem suggestdiagonal matrices with equal coefficients for the r controlinputs and the m system outputs. Here the matrices are setto Q = I and R = wr I with the parameter wr to be tunedduring closed-loop simulations.

Minimizing J with respect to uhc and solving for this vectoryields

uhc = "(Ttp Q Tp+R

)†Ttp Q ("yT +Bp up+Ap yp) (21)

which gives the future control input up to the control hori-zon. Only the first time step is used as control input at thepresent time step and a new control sequence is calculatedat the next time step. The pseudo-inverse ()† gives the solu-tion in a least-squares sense and is computed using singularvalue decomposition.

8

Choice of the Controller Parameters. The ARX modelorder p can be considered as the speed of the implicit ob-server. It is to be chosen so that p ·m' n where n is the as-sumed number of significant natural modes and m the num-ber of outputs. This only holds if disturbance states are notto be accounted for.

The speed of the implicit controller is governed by the pre-diction horizon hp that needs to be greater than or equal top. Higher values move the controller away from the dead-beat solution towards a low control energy solution. Here,the prediction horizon is set equal to the model order andthe control energy is adjusted using the weighting matricesin the cost function J. The control horizon hc needs to beless than or equal to the prediction horizon hp; for this studyhc = hp.

SIMULATION PARAMETERS

Simulation Procedure

The simulations are carried out in the time domain and canbe divided into three phases: First, the aircraft is acceleratedto its cruise speed, starting with rotor and aircraft at rest.During this phase, the rotor is driven by the aerodynamicforces and the collective blade pitch is set by a Proportional-Integral (PI) feedback controller in order to adjust the rotornominal rotation speed. This feedback loop is deactivatedas soon as a steady flight condition is reached to avoid anydisturbance of the stability measurements. During the accel-eration phase, the wing is clamped by means of high springstiffnesses. Once the nominal flight speed is reached, thewing clamp is removed and the controller is activated. Inorder to allow the rotor to reach a steady flight condition,several rotor revolutions are simulated without any externaldisturbances. Next, the model is excited by adequate swash-plate input at the frequency of the mode of interest, and thetime history is recorded. The free decay response of the os-cillations is analyzed using Fast Fourier Transforms and amoving block method in order to extract modal frequencyand normalized damping.

The sampling frequency of the simulations is set to 200Hz(( 12.6/rev) in order to have a reasonable number of timesteps per rotor revolution. This choice also provides a goodresolution of the natural modes (frequencies up to 35Hzhave been found), which are well beneath the Nyquist fre-quency.

Controller action is calculated on-line with dedicated controlsoftware. Data exchange between the dynamics simulationand the controller is done for a number of measured valuesof plant inputs and outputs at each simulation time step. Themain plant outputs are the three wing degrees of freedom,the gimbal flapping angles and rotor rpm. On the input side,there are mainly three swashplate actuator commands thatare calculated from the input commands #0, #1c and #1s.

ARX Identification

For system identification, the plant model needs to be ex-cited by a signal with a large frequency content. In thisstudy, the excitation signal is white noise that is band-passfiltered between 1Hz and 75Hz. This choice is mainly forpractical reasons as white noise can be easily generated dur-ing the subsequent wind tunnel tests.

In the present work, the system is identified once and thecontrollers are synthesized off-line based on this data. Thisis done for a number of advance ratios at equal incrementsof 0.14 up to µ = 0.83 , which is just beneath the uncon-trolled stability limit. During closed-loop simulations, thecontroller corresponding to the current aircraft speed is ap-plied up to µ = 0.83 . Above this advance ratio, the samecontroller is maintained. The controller parameters are opti-mized at µ = 0.83 and the same parameters are used in theentire velocity range.

The major parameters of the identification algorithm aremodel order and length of the identification time histories.For both MIMO control applications, a number of l = 1000time steps is taken into account which covers about 20 peri-ods of the lowest natural frequency. The state-space modelsof the LQG synthesis are identified at an order of p = 10whereas the GPC controller is designed with a lower orderof p= 5 .

Controller ParametersSISO Control Parameters. The primary design objectiveof the controllers is to augment the aeroelastic stability ofthe aircraft. Basically, only one axis can be stabilized withthe present SISO controller and it is therefore designed toaugment the damping of the first mode to go unstable. Here,this is the wing beamwise bending mode qb. Therefore thedisplacement measurement &b of the associated degree offreedom is taken as plant output. The analysis of the dynam-ics of the uncontrolled system indicated a high-gain transferfunction between the longitudinal cyclic input #1c and &bwhich leads to the choice of #1c as input command.

The resonance frequency at µ = 0.83 is situated at 4.6Hzand plant phase shift is about %G|4.6Hz ="60#. The optimalcontroller phase shift would therefore be %H |4.6Hz ="120#.During the studies it was found that the plant resonancefrequency decreases slightly with increasing flight speed.Consequently, the phase angle of the controller decreasesand the optimal phase shift is no longer provided. In or-der to compensate for this phenomenon and therefore havea more robust design, a slightly higher filter phasing of%H |4.6Hz ="150# is chosen. This is a good compromise be-tween maximum damping at the design speed and good ro-bustness at higher flight speed. Furthermore, the cut-off fre-quencies 'hp and 'l p of the band-pass filter Eq. (3) are cho-sen at about ±0.5Hz of the dominating natural frequency,and the damping ratio is set to "H ( 0.7 . In order to opti-mize the controller gain g with respect to system damping,

9

0 1 2 3 4 5 6 7 8 9 10−60

−40

−20

0

20

0 1 2 3 4 5 6 7 8 9 10−180

−90

0

90

f , Hz

g,dB

%,deg.

SISOw/o controller

Fig. 7 . Transfer function from #1c to &b with and withoutSISO controller.

standard root-locus techniques have been applied. This re-sults in an optimal gain close to g = 1 . For this filter, theslope of the phase angle curve at the resonance frequency is"4.1deg./0.1Hz which is sufficient for good parameter robust-ness.

A Bode diagram of the closed-loop behavior in comparisonto the uncontrolled system is given in Fig. 7 . The associ-ated gain and phase margins are 12dB and 59#, which is areasonable trade-off between bandwidth and stability.

Optimal Control Parameters. In this study, the main con-cern is the aeroelastic stability of the wing modes whichhave low damping. The three degrees of freedom to be stabi-lized are wing beamwise bending, wing chordwise bendingand wing torsion. Therefore, the corresponding angles &b,&c, and &t are taken as plant output. When analyzing plantdynamics, it turns out that all three swashplate inputs havea large influence on the plant outputs, albeit there are privi-leged transfers of high gain as for example the transfers #1cto &b, #1c to &t and #0 to &c. All three plant inputs are there-fore controlled.

As described above, the main parameter of the optimal con-troller is the weighting factor $ , which determines the rel-ative covariance of the measurement noise with respect tothe state noise. A value of $ = 10"4 is appropriate as theopen-loop singular values of the LQG well recover the sin-gular values of the open-loop LQR in a frequency rangebetween approximately 1Hz and 30Hz, which encloses thethree main natural modes. The chosen value resulted in con-siderable damping augmentation on the modes and good ro-bustness with respect to flight speed.

Generalized Predictive Control Parameters. The mainparameters of the GPC controller are the model order, theprediction and control horizons and the weighting parame-ter wr. There are five significant natural modes in the plantmodel, and the three plant outputs. An order of p = 5 is

therefore sufficient when there is no need to take into ac-count any disturbance states. In this case, a rather quickcontroller can be designed which results in a choice of theprediction and control horizons of hc = hp = 5 . Under theseconditions, the weighting factor for good system dampingwith adequate control energy is wr = 0.1 . The plant inputsand outputs have been chosen identical to LQG control.

RESULTS

Controller Performance

The performance of all three controllers with respect todamping augmentation is shown in Fig. 8 . Here, the eigen-values of the principal modes are shown as a function ofµ in the complex plane. The progressive gimbal mode !+is not much affected by the controller and as its eigenvalueis higher than those of the other modes, it is not shown inthe graphs. Each sub-figure includes the principal naturalmodes of the uncontrolled system for reference. The criticaladvance ratio of the uncontrolled system is µ = 0.94 . Asthe regressive gimbal mode !" of the uncontrolled systemhas very high damping, it does not appear on the charts.

All three controllers considerably augment stability andonly slightly influence model frequencies. Flutter speedis µcrit = 1.11 for SISO control, µcrit = 1.21 for LQG andµcrit = 1.15 for GPC, which constitutes an increase of 18%,29% and 22% respectively. In either case it is the !" modethat goes unstable first – see also Ref. 10 for a similar behav-ior. As this mode is not considered in the controller design,the critical speed is largely independent of the controllertype but the overall better performance of the MIMO con-trollers is also favorable to this mode. The results give suf-ficient perspectives for practical applications as they showconsiderable stability augmentation up to the flutter speedof the uncontrolled system and slightly beyond. Therefore,stabilization of the !" mode has not been considered in thiswork.

The SISO controller, Fig. 8 a.) , considerably stabilizes themost critical qb mode: Near the natural flutter speed, thedamping ratio rises to about 10% and exhibits a positive gra-dient. Furthermore, a slight stabilization of the qt mode canbe observed while there is almost no impact on the qc mode.Generally, there is little effect on damping at low speed,which was expected as the controller is tuned at µ = 0.83 .

Concerning the LQG controller it is noted from Fig. 8 b.)that the achieved damping on the qb mode is about twice thevalue obtained with the SISO controller. In addition, the qcmode is increasing its damping to 12% at the uncontrolledflutter speed whereas, comparable to the SISO case, the tor-sion mode is not much affected.

The GPC controller, Fig. 8 c.) , achieves slightly better re-sults on the qb mode than LQG and is the only controllerthat considerably augments the damping of the qt mode.

10

a.)

−8 −6 −4 −2 00

20

40

60

80

100

0.25 0.5 0.75 1 1.25

unstable

-(2 ) , rads

.(2

),rad s

µ

w/o control

qb

qt

qc

!"

SISO

µcrit = 1,11

b.)

−8 −6 −4 −2 00

20

40

60

80

100

0.25 0.5 0.75 1 1.25unstable

-(2 ) , rads

.(2

),rad s

µ

w/o control

qb

qt

qc

!"

LQG

µcrit = 1,21

c.)

−8 −6 −4 −2 00

20

40

60

80

100

0.25 0.5 0.75 1 1.25

unstable

-(2 ) , rads

.(2

),rad s

µ

w/o control

qb

qt

qc

!"

GPC

µcrit = 1,15

Fig. 8 . Root loci of the main natural modes as a functionof advance ratio µ: a.) SISO control, b.) LQG, c.) GPC.

However, this occurs at the expense of the !" mode whichyields a lower flutter speed compared to the LQG controller.Results on the other modes are only slightly different from

a.)

0.8 0.9 1 1.1 1.20

10

20

30

fbfb,0

",%critical

0.9 0.95 1 1.05 1.10

10

20

30

ftft,0

",%critical qb

qcqt

b.)

0.8 0.9 1 1.1 1.20

10

20

30

fbfb,0

",%critical

0.9 0.95 1 1.05 1.10

10

20

30

ftft,0

",%critical

c.)

0.8 0.9 1 1.1 1.20

10

20

30

fbfb,0

",%critical

0.9 0.95 1 1.05 1.10

10

20

30

ftft,0

",%critical

Fig. 9 . Robustness of the controllers at µ = 0.83:a.) SISO control, b.) LQG, c.) GPC.

LQG control.

Robustness

Robustness of the controllers is analyzed with respect tovariations in beamwise bending ( fb), chordwise bending ( fc)and torsion ( ft) natural frequencies. The influence of si-multaneous changes in structural damping ("s) of all wingmodes, rotor rotational frequency ((), altitude (h) and whitemeasurement noise is also assessed. The results for fb andft are presented in Fig. 9 a.) to c.) for the three controllers.The baseline value of each parameter is denoted by the indexzero.

In closed-loop, the wing beamwise bending mode experi-ences the highest damping augmentation. Variations of itsnatural frequency ( fb) considerably influence the controllerperformance with regard to this mode. The SISO controlleryields a higher stability gain at lower frequencies fb whereasLQG performs better at high frequencies. GPC control is de-grading for either deviation from the nominal value. Modi-fications of the torsion frequency tend to decrease beamwisedamping at higher frequency, but very low frequencies alsodegrade the behavior of both MIMO controllers. The impacton other modes is quite small.

It has also been observed that structural damping has lit-tle influence on the stability gain due to SISO and LQG,

11

0 5 10 15 20

−2

−1

0

1

2

t , s

& b,deg.

offoff onon

0 5 10 15 20−2−1012

t , s

# 1c,deg.

Fig. 10 . Efficiency of LQG controller and control effort,wing beamwise bending mode, µ = 0.97 .

and it only has a small effect on GPC. Furthermore, aerody-namic damping as well as controller generated shear forcesdecrease with flight altitude h. This leads to a decrease inoverall damping but there is no negative effect on controllerperformance. Corresponding graphs of the robustness withrespect to structural damping and flight altitude are thereforenot given here.

The rotor rotation speed did not have a notable influenceon controller behavior. Variations in chordwise bending fre-quency also have very limited influence on the three con-trollers as the associated natural mode is almost decoupledfrom the other natural modes. The SISO and the LQG con-trollers were very robust with respect to measurement noise;only GPC was not able to handle noisy measurement sig-nals. In the latter case, a noise level of 5% of the maximumwing deformation angle destabilized the controller. In gen-eral, all three controllers performed quite well for simulta-neous variations of the parameters.

Control Power, Flutter Mechanism

In the current study, the controller commands are transmit-ted to the rotor via the swashplate. The command angles aretherefore a measure for the control effort. Figure 10 gives asimulation time history with LQG controller in the loop foran advance ratio of µ = 0.97 that is slightly above the natu-ral stability limit but stable in closed-loop. For this simula-tion, the model is excited by white noise of an amplitude ofabout 0.5# on the collective and 1# on the cyclic inputs. Themain swashplate input #1c is also represented and it can beseen that only moderate input commands of well below 1#are necessary. The amplitudes on the other inputs are evensmaller. However, the abrupt activation of the controller re-sulted in a control command with a high bandwidth and anamplitude of about 1.9#.

Figure 11 shows an enlargement of the wing deformationmeasure from Fig. 10 and the corresponding rotor longitudi-nal H-force in the vicinity of t = 15s when the controller is

14 14.5 15 15.5 16−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

14 14.5 15 15.5 16−200

−150

−100

−50

0

50

100

150

200

t , s

& b,deg.

H,N

off on

&bH

Fig. 11 . H-force acting on the qb mode, µ = 0.97 .

turned on. Both signals are zero-phase band-pass filtered be-tween 3Hz and 5.5Hz. The H-force presented in the graphis oriented vertically in the airplane configuration and actsdirectly on the qb and the qt mode. It is seen to have a slightphase lead with respect to the wing deformation &b. By thismechanism, the rotor aerodynamic force is injecting energyinto the wing mode that can ultimately drive the system un-stable. Starting from t = 15s the controller is turned on andthe phase lead of the H-force is transformed into a phase lagwhich is stabilizing.

CONCLUSIONS

This work presents the development and the evaluation ofthree different controllers designed to augment whirl flut-ter stability of tiltrotor aircraft in high speed level flight. Asimulation model of the wing/pylon/rotor system was set upusing a general purpose multibody simulation tool. Timedomain simulations were performed with a controller in theloop. The primary conclusions are:

1) SISO control yields 17% increase in flutter speedwhereas GPC achieves 21% and LQG almost 29% .

2) At high flight speeds, the regressive gimbal mode be-comes unstable while the wing modes are well damped.

3) The most critical natural mode, the beamwise bend-ing mode, is well stabilized by all controllers. In addition,the damping of the chordwise bending mode is increased bybothMIMO controllers and the damping of the torsion modeonly by GPC.

4) In a steady flight condition, low control effort of con-siderably less than 1# of swashplate angle is necessary tostabilize the system.

5) The controllers are generally very robust with respectto variations of natural frequency, structural damping, flight

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altitude, rotor rotation speed, measurement noise and flightspeed. However, a wing bending natural frequency well bel-low 90% of the nominal value can destabilize the LQG/LTRcontrol loop. Furthermore, the GPC is less tolerant to vari-ations in wing torsion frequency and it has very low robust-ness with respect to noisy measurement signals.

6) The controllers stabilize the wing beamwise bendingmode by generating rotor in-plane forces of adequate mag-nitude and phase angle.

From this study it can be concluded that the LQG/LTR al-gorithm is the most promising for subsequent wind tunneltests. It shows good performance and robustness character-istics when an identified plant model can be provided. Espe-cially its good robustness with respect to measurement noisegives it an advantage over the GPC controller. For industrialapplication, the SISO controller may be more appropriate asits dynamics are well defined and its influence on aircraftbehavior can be precisely predicted.

ACKNOWLEDGEMENTS

The work presented in this paper has been co-funded by aMarie Curie Scholarship (5th Research Framework Programof the European Commission) and Eurocopter, Marignane,France. It was partially carried out at ENSICA, Toulouse,France. The contributions of ENSICA and Eurocopter staffare largely acknowledged.

REFERENCES1 Hall, W. E., Jr.: Prop-Rotor Stability at High AdvanceRatios. Journal of the American Helicopter Society,11(3):11–26. June 1966.

2 Nixon, M.: Parametric Studies for Tiltrotor AeroelasticStability in Highspeed Flight. Journal of the AmericanHelicopter Society, 38(4):71–79. October 1993.

3 Hathaway, E. L. and Gandhi, F. S.: Design Optimizationfor Improved Tiltrotor Whirl Flutter Stability. In 29thEuropean Rotorcraft Forum. Friedrichshafen, Germany.September 16-18, 2003.

4 Ghiringhelli, G. L.; Masarati, P.; Mantegazza, P. andNixon, M. W.: Multi-Body Analysis of an Active Con-trol for a Tiltrotor. In CEAS Intl. Forum on Aeroelas-ticity and Structural Dynamics, pp. 149–158. Williams-burg, Virgina. June 22–25, 1999.

5 Manimala, B.; Padfield, G. D.; Walker, D.; Naddei, M.;Verde, L.; Ciniglio, U.; Rollet, P. and Sandri, F.: LoadAlleviation in Tilt Rotor Aircraft through Active Con-trol; Modelling and Control Concepts. In American He-licopter Society, 59th Annual Forum. Phoenix, Arizona.May 6–8, 2003.

6 Komatsuzaki, T.: An Analytical Study of Tilt Propro-tor Aircraft Dynamics in Airplane Cruise Configura-tion Including the Effects of Fuselage Longitudinal RigidBody Motion. Ph.D. thesis, Princeton University. January1980.

7 Nasu, K.: Tilt-Rotor Flutter Control in Cruise Flight.TM 88315, NASA. December 1986.

8 van Aken, J. M.: Alleviation of Whirl-Flutter on Tilt-Rotor Aircraft using Active Controls. In American He-licopter Society, 47th Annual Forum. Phoenix, Arizona.May 6–8, 1991.

9 Vorwald, J. G. and Chopra, I.: Stabilizing PylonWhirl Flutter on a Tilt-Rotor Aircraft. In 32ndAIAA/ASME/ASCE/AHS/ASC Structures, Structural Dy-namics, and Materials Conference. Baltimore, Mary-land. April 8–10, 1991. Article AIAA-91-1259-CP.

10 Wang, J. C.: A Tilt-Rotor Aircraft Whirl-Flutter Allevi-ation Using Active Controls. In J. Clark et al., eds., Ac-tive control of noise and vibration., vol. 38, pp. 139–148.American Society of Mechanical Engineers, New York.1992.

11 Kvaternik, R. G.; Piatak, D. J.; Nixon, M. W.; Langston,C. W.; Singleton, J. D.; Bennett, R. L. and Brown, R. K.:An Experimental Evaluation of Generalized PredictiveControl for Tiltrotor Aeroelastic Stability Augmentationin Airplane Mode of Flight. Journal of the American He-licopter Society, 47(3):198–208. July 2002.

12 Nixon, M.; Langston, C.; Singleton, J.; Piatak, D.;Kvaternik, R.; Corso, L. and Brown, R.: AeroelasticStability of a Four-Bladed Semi-Articulated Soft-InplaneTiltrotor Model. In American Helicopter Society, 59thAnnual Forum. Phoenix, Arizona. May 6–8, 2003.

13 Hathaway, E. L. and Gandhi, F. S.: Modeling Refine-ments in Simple Tiltrotor Whirl Flutter Analyses. Jour-nal of the American Helicopter Society, 48(3):186–198.July 2003.

14 Nannoni, F.; Giancamilli, G. and Cicale, M.: ERICA:The European Advanced Tiltrotor. In 27th EuropeanRotorcraft Forum. Moscow, Russia. September 11–14,2001.

15 Mueller, J. P.: Methodologie d’OptimisationAeroelastique Active d’une Voilure de Convertible– Flottement Gyroscopique. Ph.D. thesis, UniversitePaul Sabatier (Toulouse III), Toulouse, France. October2004.

16 Juang, J.-N.: Applied System Identification. PTRPrentice-Hall, Englewood Cliffs, New Jersey. 1994.

17 Phan, M. Q. and Juang, J.-N.: Predictive Controllersfor Feedback Stabilization. Journal of Guidance, Con-trol, and Dynamics, 21(5):747–753. September–October1998.

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