Aeroservoelastic Design Optimization of a Flexible Wing

Sohrab Haghighat∗

University of Toronto, Toronto, Ontario M5A 4L8, Canada

Joaquim R. R. A. Martins†

University of Michigan, Ann Arbor, Michigan 48109

and

Hugh H. T. Liu‡

University of Toronto, Toronto, Ontario M5A 4L8, Canada

DOI: 10.2514/1.C031344

In this article, a multidisciplinary design-optimization approach is developed that integrates wing aerostructural

design with control-system design to reduce the critical structural stresses through load alleviation. The equations of

motion are derived for a flexible aircraft and used to perform time-domain aeroservoelastic analysis. The

aerostructural model couples a beam element structural model of the wing spar with a vortex-lattice model of the

wing aerodynamics. The objective of thedesign-optimization problem is tomaximize the endurance of a high-altitude

long-endurance aircraft by varying wing planform parameters, structural sizing, and control gains simultaneously.

Structural stress and steady-state control-error constraints are enforced for two dynamic flight conditions: a gust

response and an altitude-change maneuver. The design-optimization results show that the aircraft that is optimized

with the load-alleviation control system has an endurance that is 21.7% higher than the aircraft that is optimized

without the control system.

Nomenclature

Ajk = area of panel �j; k�CD = aircraft drag coefficientCL = aircraft lift coefficientCbi = transfer matrix from inertial frame to body framedi = spar diameter at the ith spanwise sectiondv = infinitesimal volumeF = vector of rigid-body aerodynamic and propulsion

forcesfe = vector of generalized forcesg = vector of gravitational accelerationherr = steady-state altitude errorKGG = global stiffness matrixKee = modal stiffness matrixJ = rigid aircraft mass moment-of-inertia matrixM = vector of rigid-body aerodynamic and propulsion

momentsMee = modal mass matrixnjk = sector normal to panel �j; k�pjk = pressure at panel �j; k�qelastic = control-weighting parameter for elastic parametersqrigid = control-weighting parameter for rigid-body parametersR = absolute position of a point on aircraftr = vector of rigid mass distributionSref = wing reference areaS1, S2 = rigid-elastic coupling termsSU = matrix that relates the nodal displacement to

von Mises stressti = spar thickness at the ith section

u = vector of structural deformationVc = velocity vector of the body-frame originVs = stall speedx = state vector�i = twist angle of the ith spanwise panel� = vector of Euler angles� = vector of structural mode amplitudes� = density�KS = Kreisselmeier–Steinhauser function parameter�i = von Mises stress in the ith finite element� = matrix of structural modes! = body-frame rotational-velocity vector~� = cross-product operator

I. Introduction

W ITH the advent of multidisciplinary design optimization(MDO), designers have started to integrate various disci-

plines in the aircraft design process. The strong coupling betweenaerodynamics and structures has motivated aircraft designers to useMDO to solve aerostructural design problems. Previous work in thisarea has focused on minimizing the total drag or maximizing theaircraft range by simultaneously optimizing the structural andaerodynamic design variables [1–4]. To improve aircraft efficiency,researchers have also investigated nonplanar lifting surface config-urations, where aerostructural optimization are performed to findoptimal configurations, such as C-wings, joined wings, and winglets[5–7]. In all these efforts, the aircraft structures are designed to haveoptimum cruise performance while withstanding the critical loadscorresponding to other flight conditions.

Through the use of multiple control surfaces, active aeroelasticwing technology can be exploited to reduce thewing loading during aflight maneuver [8–12]. More than 25 years of extensive research inthe field of aeroservoelasticity has shown that aeroservoelasticanalysis and design should be considered as early as possible in theaircraft design process [13,14]. One of the earliest works in the fieldof aeroservoelastic optimization was performed by Suzuki [9], whominimized wing structural weight by designing the structure andcontrol system simultaneously. Control stability andmaximum stressdue to an atmospheric gust were the design constraints used inthis optimization work. Aerodynamic design, however, was not con-sidered in this work and the aircraft planform was kept unchanged.An aeroservoelastic design framework was also presented by Idan

Received 13December 2010; revision received 25October 2011; acceptedfor publication 26 October 2011. Copyright © 2011 by Sohrab Haghighat,Joaquim R. R. A. Martins, Hugh H. T. Liu. Published by the AmericanInstitute of Aeronautics and Astronautics, Inc., with permission. Copies ofthis paper may be made for personal or internal use, on condition that thecopier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc.,222 Rosewood Drive, Danvers, MA 01923; include the code 0021-8669/12and $10.00 in correspondence with the CCC.

∗Ph.D. Candidate, Institute for Aerospace Studies. Student MemberAIAA.

†Associate Professor, Department of Aerospace Engineering. SeniorMember AIAA.

‡Associate Professor, Institute for Aerospace Studies. Senior MemberAIAA.

JOURNAL OF AIRCRAFT

Vol. 49, No. 2, March–April 2012

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et al. [15], where the interaction between the aircraft structure andcontrol system was considered. First, a preliminary structural andcontrol optimization was performed separately. The resultingstructure and control system were then optimized consideringclosed-loop control margins and flutter. The same framework waslater used to perform simultaneous structural and control opti-mization in the presence of parameter uncertainty [16].

Zink et al. [17] addressed the design of structural parameters andgear ratios of a lightweight fighter performing symmetric andantisymmetric (rolling) maneuvers. The optimization formulationwas based on static aeroelastic equations. The integrated approachresults were compared to those obtained using a sequential approach;the former was shown to be more effective and converged to a lowerstructural weight. The same authors also considered maneuver loadinaccuracies and their effects on the optimal design [18].

The development of a detailed mathematical model that canintegrate flight dynamics with structures and aerodynamics isessential in active aeroelastic wing design. Some of the work thataddressed the active aeroelastic wing design adopted quasi-steadyflight models, where structural deflections are assumed to havereached their steady-state conditions [17,19,20]. Mathematicalformulations of flexible aircraft flight dynamics are mainly based onone of two approaches: the mean-axes method or the quasi-coordinate method. Waszak and Schmidt [21] and Schmidt andRaney [22] developed the equations of motion for a flexible aircraftusing the mean-axes method. The use of simple aerodynamic striptheory and the small structural deflections assumption are the mainlimitations of this formulation. Formulations using the quasi-coordinate method, where the axes are fixed to a specific point of theaircraft, have also been developed [23–27]. In this work, the mathe-matical formulations of aircraft flight dynamics is based on the quasi-coordinate method. Several models have been used to evaluate theaerodynamic forces and moments of a flexible aircraft. However,most researchers so far have used the aerodynamic strip theory.

In addition to the various approaches used to analyze flexibleaircraft, various design objectives have also been considered.Previous work on aeroservoelastic synthesis has focused either onachieving the requiredmaneuverability or on designing lighter wingsby using load alleviation. However, aeroservoelasticity can also beused to maximize the overall performance of an aircraft by opti-mizing an objective function, such as endurance, range, or fuelconsumption. Also, the design variables used in previous work havemostly been limited to structural and control-system parameters withfixed aerodynamic shape parameters. Nam et al. [10] conducted oneof the few investigations that considered integrated planform andcontrol-system design.

Because simultaneous optimization yields better results than thoseobtained through performing sequential optimization [1,17,28], thereal potential of including aeroservoelastic synthesis in the aircraftdesign process can only be realizedwith a simultaneous optimizationapproach. Using the traditional design process, the aircraft config-uration is designed first, and the control system is designed second.This traditional approach prevents the flight-control system fromaffecting the aircraft configuration. It has been shown that, bydesigning the aircraft configuration and the control system concur-rently, tradeoffs between these two design subspaces can improve theaircraft performance [29,30].

For flexible aircraft, it is particularly advantageous to include thecontrol-system design as one of the disciplines and couple it with theaerodynamics and structures. This approach has the potential toexploit synergies between the three disciplines and yield higherperforming aircraft than is currently possible with the traditionalapproach. Also, highly flexible aircraft can deform in unexpectedways under varying atmospheric conditions, as exemplified by theHelios aircraft incident. Among the technical recommendationsincluded in the investigation that followed the Helios crash, thedevelopment of more advanced multidisciplinary approaches thatinclude control systems and time-domain analysis methods appro-priate to flexible aircraft was emphasized [31].

The work presented in this paper integrates control-system designin an aerostructural design-optimization framework for flexible

aircraft. Both time-dependent maneuvers and gust excitations areconsidered in this optimization work. The main goal is to study theeffect of the control system on the optimal design of the wing and itsstructure. The objective is to maximize the endurance by using anactive control system and designing it concurrently with the aero-dynamic shape and structural sizing. Parameters such as wing area,, taper ratio, and wing twist distribution were considered in thedesign optimization. Design constraints are enforced to ensurestructural integrity, satisfactory stall behavior, and desirable flyingqualities. The endurance is calculated based on cruise performance.The structural stresses are computed for the duration of an altitudegain maneuver and for the response due to a discrete (1 � cos) gust.The aeroservoelastic model involves the nonlinear equations ofmotion for a flexible aircraft based on the quasi-coordinate method,which has been derived in previous work by the authors [32–34]. Avortex-lattice panel method is used to evaluate the aerodynamicforces and moments, and the wing structure is modeled with a beamfinite-element model.

In the remainder of this article, the aeroservoelastic formulation isfirst presented, and the general nonlinear form of the equations ofmotion for a deformable aircraft are derived. Based on theseequations of motion, we then derive the equations for theaerostructural analysis in Sec. III and the state-space form of thelinearized equations in Sec. IV. TheMDO framework and the design-optimization problems are presented in Sec. V. The optimizationresults are presented and discussed in Sec. VI, and Sec. VII reportsour conclusions.

II. Aeroservoelastic Formulation

As mentioned in the previous section, the quasi-coordinatemethod is adopted in this work to derive the aeroservoelasticequations of motion of a flexible aircraft. The quasi-coordinatemethod expresses the body movements with respect to a fixedreference frame. Figure 1 shows a deformable aircraft with itsoriginal and deformed wing.

The inertial frame is denoted by Fi, whereas Fb is used to denotethe body-fixed frame, which is not necessarily connected to thecenter of mass. The absolute position and velocity of a infinitesimalmass element, dm, on the aircraft is given by

R �Rc � r� u (1)

_R� Vc �! � �r� u� � _u� Vc �! � r� _u (2)

The structural deformations can be expressed in a discrete finite-element formulation, or it can be represented as a superposition of theaircraft structural vibration modes. Although the modal representa-tion is widely used to represent dynamic response and to performstability analysis, static aeroelastic (i.e., aerostructural) analysis isusually based on finite-element methods with thousands of degreesof freedom [35]. This is mainly due to the effect of concentratedforces, such as the weight of external stores, which the modal repre-sentation does not capture adequately, unless the structuralmodes arecalculated using large fictitious masses [36]. In this work, the

Fig. 1 Deformed aircraft axis system.

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following modal representation is used for both static and dynamicanalysis:

u �Xni�1�i�i ��� (3)

where �� �1; . . . ;�n and �� �1; . . . ; �nT .

A. Kinetic Energy

The kinetic energy of a deformable aircraft can be formulated asfollows:

T � 1

2

Zv

� _RT _R dv� 1

2mVT

cVc � VTc

�Zv

� ~rT dv

�!

� VTc

�Zv

��dv

�|�������{z�������}

S1

_�� 1

2!TJ!�!T

�Zv

�~r�dv

�|��������{z��������}

S2

_�

� 1

2_�TMee

_� (4)

where J is the mass moment-of-inertia matrix for the rigid aircraftandMee ��TMGG� is the modal elastic mass matrix. S1 and S2 arethe rigid-elastic coupling terms that represent the interaction be-tween the rigid and deformable dynamics. These terms are not theonly source of coupling between the rigid and deformable dynamics;interaction also occurs through the aerodynamic forces andmoments. If the origin of FB is placed at the undeformed aircraftcenter of gravity, then

Rv �~r

T dv� 0, and the kinetic energy is

T � 1

2mVT

cVc � VTc S1 _��

1

2!TJ!�!TS2 _��

1

2_�TMee

_� (5)

B. Potential Energy

The potential energy can be divided into two categories:gravitational energy and strain energy. The gravitational potentialenergy is

Ug ��Zv

��Rc � r� Cbiu�Tg dv��mRTcg � �TST1Cbig (6)

The strain energy of an elastic system can be written as

U� 1

2�TKee� (7)

where Kee ��TKGG� is the modal stiffness matrix.

C. Lagrange’s Equation

The generalized coordinates in this problem are Rc, �, and �. Inaircraft flight dynamics, it is usually more convenient to express thetranslational and angular velocities in the body frame [37,38], i.e.,

V c � Cbi _Rc; !�1 0 sin �0 cos� cos � sin�0 � sin� cos � cos�

" #|����������������������{z����������������������}

D

( _�_�_

)|{z}

_�

(8)

However, the body-frame velocities cannot be integrated. Therefore,the Lagrange equations in quasi-coordinates are used in this work[39,40],

d

dt

�@L

@Vc

�� ~!

@L

@Vc

� Cbi@L

@Rc

� F (9)

d

dt

�@L

@!

�� ~Vc

@L

@Vc

� ~!@L

@!� �D�T� @L

@��M (10)

d

dt

�@L

@ _�

�� @L@�� fe (11)

where L� T � V. Substituting the kinetic and potential energy intoEqs. (9–11) and rearranging them, the equations of motion for thedeformable aircraft become

M O3�3 S1

O3�3 J S2

ST1 ST2 Mee

264

3758>><>>:

_Vc

_!

��

9>>=>>;

��m ~! O3�3 ~!S1

O3�3 ~!J � ~VcS1 � ~!S2�On�3 On�3 On�n

264

3758>><>>:Vc

!

_�

9>>=>>;

�O6�6 O6�n

On�6 Kee

" #8>><>>:Rc

�

�

9>>=>>;�

mCbi

O3�3

ST1Cbi

264

375fgg �

8>><>>:

F

M

fe

9>>=>>; (12)

where F andM are the total forces and moments in the body frame,including propulsive and aerodynamic forces. The generalized forcesdue to elastic deformations fe are calculated by integrating theproduct of the modes and nodal aerodynamic pressures over thesurface, i.e.,

f e �Zs

�TP ds (13)

where, P represents the nodal aerodynamic pressure.

III. Aerostructural Analysis

The aerostructuralmodel used in thiswork is shown in Fig. 2.Nowthat the equations of motion for a flexible aircraft have been derived,the steady-state equations can be obtained. To derive the equationsfor rectilinear flight, all derivatives are set to zero, and so is !. Theequations of motion for a flexible aircraft in a steady flight are

F �mCbig� 0 (14)

M � 0 (15)

� Kee�� �S1�TCbig� fe � 0 (16)

The aforementioned equations are coupled through the aero-dynamic forces and moments and can be used to find the trimparameters ��; �e� as well as the wing deformation �. A Newton–Raphson method is used to solve the coupled system, with gradientscomputed using the complex-step derivative approximation [41]. Toaccelerate the algorithm, the gradient computations are parallelizedand distributed over multiple cores.

X

1214

16

Y

0

5

10

15

20

25

30

Z

0

2

4

Fig. 2 Aerostructural model.

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A. Structural Model

The aircraft fuselage and empennage are considered to be rigid.Grid finite elements (also known as torsional beams) are used tomodel the wing deformations. A grid element has two nodes, eachwith three degrees of freedom: displacement in the z direction androtations about the x and y axes. The wing box is approximated as ahollow tubewith variablewall thickness along the span. The stiffnessand mass matrices for a grid element can be found in Paz and Leight[42]. The stress value is computed from the modal deflections usingthe following formulation:

� � SU�� (17)

where SU is a constant matrix that can be formed when the structuralmesh is generated. During the course of the optimization, tomaintainaccuracy as the wing structure is changed, the structure modes arereconstructed. Karpel [36] suggests that in a minor design cycle, thestructural mode shapes can be kept constant while themodal stiffnessand mass matrices are being updated. In this work, due to the use of aswarm-based optimizer, the designvariables can change significantlyat each design cycle, preventing the use of sensitivity information toupdate stiffness and mass matrices. Therefore, the structural modeshapes are recalculated for each design cycle evaluation, and thestiffness and mass matrices are updated accordingly.

B. Aerodynamic Model

The vortex-lattice method is used to calculate the aerodynamicforces and moments. Figure 3 shows the vortex rings and wakevortices generated for the aerodynamic analysis of the aircraft. Thepanel and wake vorticities are calculated by enforcing the tangentialflow condition at each panel and enforcing the Kutta condition at thetrailing edge. The calculated vorticity at each panel is then used tocalculate the pressure on each panel. The rigid and elastic aero-dynamic forces and moments can then be calculated as follows:

F aero �Xnsj�1

Xnck�1

pjkn̂jk�Ajk (18)

M aero �Xnsj�1

Xnck�1

~r�pjkn̂jk��Ajk (19)

f ei �Xnsj�1

Xnck�1

�ijjk � pjkn̂jk�Ajk (20)

where pjk, Ajk, and n̂jk are the pressure, area, and normal vector forpanel j, k, respectively; and �ijjk is the deflection due to structuralmode i at panel j, k. The summation ranges, nc and ns, show thenumber of elements along the chordwise and spanwise directions,respectively.

To capture the unsteady aerodynamic loads, a wake roll-upcomputation would need to be added to the vortex-lattice method.This would increase the computational cost of the analysis

dramatically, unless reduced-order modeling techniques are used forthe unsteady aerodynamics [43–47]. In this work, because the mainfocus is on the demonstration of aeroservoelastic design at theconceptual level, only quasi-steady aerodynamic loads are computedwith the vortex-lattice method.

IV. Aeroservoelastic Control

Substituting the perturbed flight parameters into Eq. (12) andusing the first-order approximation, the linearized equations about asteady-state rectilinear flight condition become

� � _Vc

� _!��

��M�1

m ~Vc;0�! �m��Cbi�g� �F� ~Vc;0S1 _�� �M

Kee��� �fe � ST1 �Cbig

24

35 (21)

Performing sensitivity analysis on aerodynamic forces andmoments and substituting into Eq. (21), the state-space repre-sentation of an aeroservoelastic aircraft can be written as

x � fV ! � _� � h gT (22)

_x� Ax� Bu� Bwwg (23)

y � Cx (24)

z � Ex (25)

where� is a vector of Euler angles, f� � gT . The effect of a guston the aircraft is modeled by the matrixBw. The state-space matricesfor an aeroservoelastic aircraft are presented in Sec. A. Elevator andaileron deflections are the two control inputs used in this work. Theleft and the right ailerons deflect differentially for roll control and canalso deflect individually to perform load alleviation. The totalcontrol-surface deflection for each aileron is calculated by addingthese two deflections.

By analyzing the system matrices, it can be shown that the currentconfiguration is controllable. To make this system observable, thewing deflection has to be measured at least at two points. In thisrepresentation, y is the tracked output, which is the aircraft altitude,whereas z� f h �tip �mid gT is the measured output, whichincludes the altitude and wing vertical deflections at the tip andmidspan. Based on themeasured parameters, an observer is designedto provide all states to the controller.

The structural stresses in the wing are directly proportional to themodal displacements, as shown in Eq. (17). Therefore, by regulatingthem to zero, the instantaneous stress can be controlled to be closer tothe stress level at cruise.

Based on the aforementioned explanation, the state-space repre-sentation is used to design a control system that performs altitudetracking and regulates structural deformations to lower themaximumstress during themaneuver.We use a linear quadratic (LQ) controller,with the cost function

J�Z�xTQx� uTRu� dt (26)

The key for designing a successful LQ controller is the selection oftheweightingmatricesQ andR.We assemble theQmatrix using twocoefficients: qrigid and qelastic, yielding

Q� diag�qrigid; . . . ; qrigid|���������{z���������}6

; qelastic; . . . ; qelastic|�����������{z�����������}2n

; qrigid� (27)

These two variables are assigned to the optimizer as designvariables. To speed up the optimization process, the Rmatrix that isused to penalize the control deflections is fixed during theoptimization. A proper value for the Rmatrix is selected by trial anderror before starting the optimization.Fig. 3 Sample vortex-lattice model.

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The control input to the system is u��K�x � xref�, where K isthe gain matrix, and xref represents the desired values for the statevector. To design a realistic control system, control-surface deflec-tions are limited by upper and lower bounds. Two cases are presentednext to highlight the effect of the proposed control system on theaircraft structural loading; the obtained results are presented inSec. VI.

V. Problem Formulation

A. Design Variables

The objective of this framework is to surpass the current aircraftendurance limits through the use of an active load-alleviation systemthat is designed concurrently with the rest of the aircraft. In theoptimization problem solved here, design variables from all threedisciplines are included. The aerodynamic design variables are the, wing area Sref , and the wing spanwise twist distribution �i. Thestructural sizing is represented by the spar-wall thicknesses ti.Finally, the weighting parameters qrigid and qelastic, are the controldesign variables.

B. Flight Conditions

Two different flight conditions are analyzed here: an altitude-change maneuver and a flight through a sharp vertical gust. Bothconditions are selected to study the effect of inertial and aerodynamicloads on aircraft structure and the design optimization. It should benoted that both study cases represent longitudinal flight conditions.In the first maneuver, the aircraft is commanded to gain 1000 m in40 s. Unfortunately, there are no published performance require-ments for high-altitude long-endurance (HALE) very flexibleaircraft. However, Shearer and Cesnik [48] have considered a veryflexible aircraft as a large transport type aircraft (class III-L) and useda maximum climb rate of 10:16 m=s at sea level as a guideline forcontrol design. In this work, a fast maneuver that results in largewingloadings is performed. For the second flight condition, a (1 � cos)discrete gust profile is used, which also results large wing loadings.The gust profile is given by

wg ��wg2

�1 � cos

2t

Lg

�(28)

where �wg is the maximum gust velocity, and Lg is the gust length, interms of the time needed for a point in the aircraft to travel across itslength. In this work �wg is set to 4:575 m=s and a value of 0.5 s is usedfor gust length.

C. Constraints

Two constraints are enforced: one guarantees the structuralintegrity by constraining the maximum stress in the structure for theduration of the maneuver or gust, whereas the other ensuresmaneuvering capability by limiting the altitude error at 40 s after thestart of themaneuver. The aircraft is required to climb from2000m to3000m. The addition of the altitude constraint at 40 s ensures that theoptimizer achieves load-alleviation capability without sacrificing theaircraft maneuverability. A gust strength of 4:575 m=s (15 fps) isused for the optimization, which is the same as the gust strength usedfor the control-system simulation.

To reduce the number of structural constraints, the constraints areaggregated using the Kreisselmeier–Steinhauser (KS) function [49].For a vector of constraints gi � 0 (i� 1; . . . ; n), the KS function is adifferentiable function that has the form

KS �gi� �1

�KSlnXni�1

e�KSgi (29)

where�KS is a parameter that controls how close this function is to themaximum of the constraints, as we can infer from the inequalityproperty,

gmax � KS�gi� � gmax �ln �n��KS

(30)

Based on this property, all stress constraints can be enforced by thesingle constraint,

KS �g��i�� � 0 (31)

where

gi��i� ��i�yield

� 1 (32)

A more detailed analysis of the properties of KS function and itsapplication to optimization problems can be found in previous work[49,50].

In this optimization, the taper ratio is fixed. Therefore, increasingthe may decrease the wing chord, resulting in high sectional liftcoefficients near the wing tip. To avoid tip stall, the stall speed of thedesignedwing is required to be smaller than that of the baselinewing.To satisfy this condition, all sectional lift coefficients must be lowerthan Clmax

, the maximum sectional lift coefficient of the baselinewing, i.e.,

Vs � Vsref ) Cli jVsref � Clmax(33)

The same technique used for aggregating the structural constraintsis used to aggregate the stall constraints:

KS �g�Cli�� � 0 (34)

where

g�Cli � �CliClmax

� 1 (35)

In this design problem, the thickness-to-chord ratio is fixed for thewhole wing and, therefore, the spar diameter depends on the localchord. Ultimately, this means that the spar diameter decreases as the increases. The spar-wall thickness is bounded below by theminimum gauge thickness and above by the spar radius. Spar

Fig. 4 Aeroservoelastic optimization flow diagram.

X

0

10

20Y

-30

-20

-10

0

10

20 Z

02

Fig. 5 Three-dimensional geometry of a generic high- UAV.

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thicknesses andwing twist angles are varied at six locations along thespan, and the intermediate values are linearly interpolated.

D. Optimization Problem

The complete optimization problem is as follows:

min;Sref ;ti;�i;qrigid;qelastic

� CLCD

lnWinitial

Wempty

subject to

8>>>>>>>>>>>><>>>>>>>>>>>>:

KS�g��i�� � 0

KS�g�Cli�� � 0

herr � 0:05ti � di

2� 0

�10 � �i � 10

145 � Sref � 245

10 � � 20

10 � qrigid � 100

500 � qelastic � 1500

where �i� 1; . . . ; n�

(36)

The computation of drag includes both the induced drag and theviscous drag. The induced drag is calculated using the previouslydescribed vortex-lattice panel code, and the viscous drag is estimatedusing an assumed skin friction coefficient and the wetted area.

Although the structural and aerodynamic parameters directlyaffect the endurance through weight and aerodynamic performance,the control parameters have an indirect effect on the objective

function through load alleviation. A simple linear control theory canbe used to explain the effect of the control parameters on the finaldesign. A system’s steady-state error is dependent on the choice ofgain matrix, K. Therefore, different values of control-weightingparameters result in different gainmatrices that can affect the altitudeconstraint. However, maximum stress can be related to themaximumovershoot of the structural modes amplitudes, which is related to theeigenvalues of the system. The system’s eigenvalues depend on thegain matrix, which itself depends on the weighting parameters. Thebounds for qrigid and qflexible are selected through a trial-error process.Although the Rmatrix is selected to penalize the control deflectionsand avoid control saturation, large values of qrigid and qflexible canresult in control saturation over a long period of time, which candestabilize the aircraft. Therefore, an upper limit is used to avoidprolonged control saturation. However, as this type of aircraft ismarginally stable, a sharp atmospheric gust can result in excessiveoscillations. As a result, a lower bound is set in place for qrigid andqflexible to avoid low gain controllers that cannot damp out the gust-induced oscillations effectively. The flow diagram for theaeroservoelastic optimization is shown in Fig. 4.

An augmented Lagrange multiplier particle swarm optimizer isused to solve the optimization problem. This is a gradient-freealgorithm that tends to find global optima, and this particular versioncan solve constrained problems [51].

VI. Results

A. Baseline Model

A HALE UAV with a large is the focus of this study. Thegeometry of the baseline aircraft is shown in Fig. 5, and the geometricparameters are listed in Table 1.More detailed information about thisUAV can be found in Shearer and Cesnik [25,48].

The chordwise location of the spar is set to 45% of the chord. Thedesign cruise and stall speeds are set to 80 m=s and 60 m=s,respectively. A maximum airfoil lift coefficient of 1.6 is used for thestall calculations. The value of �KS in the KS function is set to 100.The initial and final altitudes for the altitude gain maneuver are set to2000 m and 3000 m, respectively.

B. Control-System Design

The proposed control system is integrated with the aircraftdynamics. First, a control system that performs an altitude gain

Table 1 Aircraft geometric properties

Property Value

Fuselage length 26.4 mWing span 58.6 mWing area 196 m2

Wing taper 0.48Horizontal tail span 18 mHorizontal tail area 53:5 m2

Horizontal tail taper 0.7Vertical tail span 4 mVertical tail area 8:9 m2

Vertical tail taper 0.81

Fig. 6 Altitude and bending-moment time history for climb maneuver.

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maneuver with and without load alleviation is designed. The resultshighlight the effect of the load-alleviation system on the maximum-stress value �max. As shown in Fig. 6, both control systems showsatisfactory climb performance. However, when the load-alleviationsystem is used, the same climb rate can be achieved with lowerstructural stresses, as seen in Fig. 6. This is due to the fact that whenload alleviation is used, the structural relaxation is performed byapplying the ailerons, as shown in Fig. 7. As expected, lower elevatordeflections are applied when the load alleviation is not used. Whenthe load alleviation is used, the structural stresses induced by a fastermaneuver (higher elevator deflection) are suppressed by applying theailerons symmetrically.

Another simulation was performed to show the gust load-alleviation capability of the control system. The results are shown in

Figs. 8 and 9. The load-alleviation system reduces the structuralstress significantly. However, when the system is off, the controllercannot stabilize the aircraft within the same time period. The absenceof the load-alleviation system causes excessive structural stress whenthe aircraft is passing through atmospheric turbulence.

C. Aeroservoelastic Optimization

Two design-optimization cases are considered: one with loadalleviation and the other without. The optimized endurance anddesignvariable values are summarized in Table 2 and Figs. 10 and 11.The constraint results are shown in Figs. 12 and 13, where thehorizontal lines represent the stall and the stress constraint values,respectively.

Fig. 7 Control effectors deflection time history for climb maneuver.

Fig. 8 Load factor, bending-moment, and maximum-stress time history due to gust excitation.

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The optimization results presented in Table 2 show that theoptimization of the load-alleviation system in the design procedureresults in awing structure that is 41.5% lighter than thewing that wasoptimized without the load-alleviation system. The optimizationresults plotted in Fig. 10 show that the spar thickness decreasesmonotonically when load alleviation is used. However, the optimalthickness distribution is different when load alleviation is notemployed: a significant amount of material is added near the tip.When the active load-alleviation system is not used, the optimizer isforced to explore passive design options. Increasing the wing tipmass can help stabilize the aircraft and lower the maximum stresswhen encountering an atmospheric turbulence. The increase of thewing tip thickness effectively adds mass to the wing tip.

Fig. 9 Control effectors deflection time history in response to gust excitation.

Table 2 Optimization results with and without

load-alleviation system

Load alleviation Off On

Sref , m2 219.18 191.47

13.98 14.03L=D 34.29 34.37qelastic 1499.95 1499.88qrigid 90.63 75.71Wing mass, kg 13,378 7817Endurance factor 31.90 38.83

Fig. 10 Spanwise distribution of spar-wall thickness and wing twist.

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It can be shown that a lower added tip mass would result in controleffector saturation and aircraft instability. In this work, only anupward wind gust is considered. However, increasing the wing tipmass would also be helpful in reducing the maximum stress of aflexible wing passing through a downward gust. To satisfy the stressconstraints without using the added mass at the tip, a much higherthickness near the wing root would be required, resulting in a farheavier aircraft.

The converged values of control-weighting parameters listed inTable 2, and the altitude and load factor time history (Fig. 13) showthat both the cases achieve comparable performance in the climbmaneuver.

Figure 12 shows the lift coefficient distribution at the referencestall speed of 60 m=s. The stall constraint is satisfied in both cases.

However, when load alleviation is not employed, the wing area is14.5% higher than thewing area when load alleviation is considered.This ismainly due to theweight of thewing structure, which is 71.1%higher than the weight of the wing that uses the load-alleviationsystem.

Finally, the results show that the aircraft endurance is increased by21.7% when an active load-alleviation system is used and optimizedconcurrently with the wing aerodynamics and structural sizing.

VII. Conclusions

In this work, the design optimization of an active load-alleviationcontrol system has been integrated with the design optimization ofthe aerodynamic shape and structural sizing of a UAV. The equations

Fig. 11 Spanwise distribution of vertical displacement, stress, and lift at the cruise condition.

Fig. 12 Spanwise Cl distribution at the minimum allowable speed.

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of motion for an aircraft with a flexiblewing were derived to performthe time-dependent analysis, which can handle large rigid-bodymotions with structural deformations. The analysis framework wasused to compare two design cases: one for which the control systemsimply worked towards achieving or maintaining a target altitude,and another where the control system was also performing loadalleviation. The use of the active load-alleviation system resulted in a21.7% improvement in the endurance relative to the optimum resultwithout load alleviation. The results show that the inclusion ofcontrol-system discipline along with other disciplines at the earlystages of aircraft design improves aircraft performance. It was alsoshown that structural stresses due to gust excitations can be bettercontrolled by the use of active structural control systems. In thefuture, we expect that the inclusion ofmore sophisticatedmaneuvers,including asymmetric ones, will show that the integration of theactive load-alleviation system designwith the design of thewingwillresult in further increases in performance.

Appendix A

The state-space matrices for an aeroservoelastic aircraft arepresented here. TheAmatrix represents the systemdynamics, and theB and C matrices are the control and output matrices, respectively.These matrices represent both longitudinal and lateral-directionaldynamics of a flexible aircraft.

A�

M�1 @F@V M�1 @F

@!� ~V0 M�1 @F

@�M�1 @F

@ _�

0 � cos �0 � gcos �0 � g 0

0 sin �0 � g

24

35

J�1 @M@V J�1 @M

@!J�1 @M

@�J�1 @M

@ _�� J�1 ~V0S1 O3�2

On�3 On�3 On�n In�n On�2

M�1ee@fe@V M�1ee

@fe@!

M�1ee

�@fe@�� Kee

�M�1ee

@fe@ _�

ST1

0 � cos �0 � gcos �0 � g 0

0 sin �0 � g

24

35

O2�3 I2�3 O2�n O2�n O2�2

26666666666664

37777777777775

B�

M�1 @F@�a

M�1 @F@�e

J�1 @M@�a

J�1 @M@�e

On�1 On�1M�1ee

@fe@�a

M�1ee@fe@�e

O2�1 O2�1

266664

377775

Bw �

M�1 @F@V M�1 @F

@!

J�1 @M@V J�1 @M

@!

On�3 On�3M�1ee

@fe@V M�1ee

@fe@!

O2�3 O2�3

266664

377775

Acknowledgment

The authors would like to thank the Natural Sciences andEngineering Research Council of Canada for providing financialsupport for this project.

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