JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICSVol. 15, No. 4, July-August 1992

Integrated Aeroservoelastic Wing Synthesis by NonlinearProgramming/Approximation Concepts

E. Livne*University of Washington, Seattle, Washington 98195

andP. P. Friedmannt and L. A. SchmitJ

University of California, Los Angeles, California 90024

Nonlinear programming/approximation concepts optimization techniques developed for structural synthesisare adapted to the integrated aeroservoelastic wing synthesis problem, in which constraints from severaldisciplines are taken into account simultaneously and the design space includes structural and control systemdesign variables. Structural, aerodynamic, and control system mathematical models that are suitable for thepreliminary design of airplanes are used in an integrated synthesis of wings and their active control systems. Theeffectiveness and efficiency of the new capability are studied using mathematical models of a remotely pilotedvehicle as well as a more complex YF16-type airplane model. Simplified handling quality constraints are addedto the set of design requirements. The performance of several complex eigenvalue approximations is examined.Effects of control law structure on the weight and robustness of the resulting aeroservoelastic designs providenew insights into the complex multidisciplinary interactions involved.

Introduction

T HE application of composite materials technology andhigh authority control systems to the design of actively

controlled, light weight, flexible wings1'2 has made the interac-tions among the various disciplines in airplane design morecomplex and important. Failure to take these interactions intoaccount can lead to potentially disastrous consequences.3 Onthe other hand, taking advantage of them early in the designprocess offers greater design flexibility.4'5 The integrated treat-ment of structures, control systems, flight mechanics, perfor-mance, and dynamic aeroelasticity has thus become a necessity.6'9

The synthesis of active control systems for elastic airplanesto guarantee flight mechanics and aeroelastic stability whileproviding desirable handling qualities and acceptable dynamicstresses is still considered to be a challenge to control systemsdesigners.10"13 The simultaneous synthesis of active controlsystems and the flexible structures that they control is evenmore complex. Many types of constraints on the behavior ofthe combined system have to be met. Stresses due to loads indifferent maneuvers have to be below an allowed level to pre-vent structural failure in yielding, buckling, or fatigue. Aero-servoelastic stability must be guaranteed. Control systempower requirements must be within the power limits of actua-tors if the actuators are preassigned. If the actuators are de-signed as a part of the overall system, their required power may beminimized or limited to some desirable value. The mass of thestructure may be minimized or limited to be below some de-sirable target value. The aeroelastic deformation of a wing hasto be tailored so as to minimize drag or prevent a drag in-crease. From a control systems point of view, this is a problemof designing the controls and the plant simultaneously. At theoutset the plant, the controls, and which constraints will drive

Received May 15, 1991; revision received Nov. 11, 1991; acceptedfor publication Dec. 22, 1991. Copyright © 1991 by E. Livne, P. P.Friedmann, and L. A. Schmit. Published by the American Institute ofAeronautics and Astronautics, Inc., with permission.

*Assistant Professor, Department of Aeronautics and Astronau-tics, FS-10. Senior Member AIAA.

fProfessor, Mechanical, Aerospace, and Nuclear Engineering De-partment. Fellow AIAA.

JRockwell Professor of Aerospace Engineering, Emeritus, Mechan-ical, Aerospace, and Nuclear Engineering Department. Fellow AIAA.

the design are unknown. Intuition provides little help andactually can be misleading in these circumstances.

Optimization has always been a part of aerospace systemsdesign14'18 at all levels and each phase of the design process.Yet in many cases, its usefulness has been limited to compo-nent design and to relatively simple problems with a smallnumber of constraints and design variables. In the structuresarea, it was an integration of numerical optimization and fi-nite element techniques that has brought structural synthesisto maturity18'22 so that it can deal with complex realistic prob-lems. The most general approach to structural synthesis isbased on the integration of nonlinear programming and ap-proximation concepts (NLP/AC). Nonlinear programming al-gorithms will find local minima of general constrained opti-mization problems starting from feasible or infeasible initialdesigns. No intuition is therefore needed to identify activeconstraints in advance. Approximation concepts are used tosave computation resources by replacing the computationallyintensive detailed analysis by an approximate, computation-ally cheap analysis and gradient calculation. General struc-tures modeled by thousands of finite elements can be sizedwith NLP/AC using current computing capabilities to meetstress, deflection, buckling, and natural frequency constraintswhile reducing weight to a minimum. Lacking the eleganceand rigor of classical analytical optimization methods (seeRef. 18, Chap. 2), the mathematical programming based tech-niques are more suited for fully automated digital computa-tion involving complex models and large numbers of designvariables and constraints. These techniques have been used foraeronautical applications of control system design23'24 andhave been recognized by the control systems community aswell.25'26 It is recognized that a major change in control sys-tems design over the next years will be the increasing use ofparameter optimization.25'38 Numerical optimization has beenused for quite some time for the integrated synthesis of controlaugmented space structures.39"47 The computationally inten-sive multidisciplinary control/aeroelastic design problem withits different types of design variables and constraints and itslarge scale is a natural candidate for NLP/AC.

In a recent series of publications,48'53 a new synthesis capa-bility for actively controlled fiber composite wings has beendescribed. It is based on balanced design and analysis modelsthat capture essential behavior characteristics, without making

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986 LIVNE, FRIEDMANN, AND SCHMIT: AEROSERVOELASTIC WING SYNTHESIS

the integrated multidisciplinary design optimization task in-tractable. This balanced approach combines high-quality, ap-proximate, but computationally efficient, analyses for thestructural, aerodynamic, and aeroservoelastic behavior ofcomposite wings at the preliminary design stage. The entireoptimization problem may be treated at one level without theneed for multilevel decomposition.

Exploratory studies in actively controlled wing multidisci-plinary optimization were presented in Ref. 50. They focusedon the interaction between the aluminum structure of a re-motely piloted vehicle (RPV) wing and the active flutter sup-pression/gust alleviation system and on the integration of con-trols and structure in aeroservoelastic design optimization.Structural and control system sizing-type design variables49'51

were included in the design space. Tradeoffs between struc-tural mass and required control system power were studied.The results of more comprehensive multidisciplinary wing syn-thesis studies made possible by the new synthesis capabilitywere reported in Ref. 52. Structural design freedom was in-creased by introducing fiber composite materials for wing skinconstruction. The design space was opened up by addingaerodynamic design variables to determine the jig shape (ini-tial camber) of the wing. Drag constraints were added to theset of constraints. Thus, the design space spanned three disci-plines, namely, structures, control, and aerodynamics, and theblend of behavior constraints covered strength, minimumgauge, flutter, gust response, and performance (induced drag).The same RPV model as used in Ref. 50 served as a test case.

The design studies in Refs. 50-53 contribute to the state ofthe art in design optimization by developing and gaining expe-rience with synthesis techniques for multidisciplinary complexproblems involving a very rich blend of constraints. They alsooffer better understanding and a fresh insight into the interac-tions between the various disciplines in wing design as well asnumerical results that may serve as a basis for design trade-offs.

The present paper adds to the growing experience in aero-servoelastic wing optimization by focusing on active controlaspects of the design problem. The design space for the nu-merical examples reported here is made up of sizing-typestructural and control system design variables.49 The presentcapability can handle aerodynamic design variables as well butthey are not used in the present studies. The RPV50~53 is usedto study locally optimal designs achieved with different con-trol law structures and the resulting gain and phase margins ofthese optimized control augmented composite wings. The per-formance of different complex eigenvalue approximations inthe context of NLP/AC technology is also examined. The newmultidisciplinary wing synthesis capability is then applied tothe aeroservoelastic synthesis of a realistic fighter-type air-plane in order to demonstrate its power and generality andassess computational efficiency in dealing with more complexaeroelastic and control system configurations. Simplified han-dling qualities requirements are also added to the set of con-straints.

Analytical Modeling TechniquesA unique integration of analytical modeling techniques

makes it possible to bridge the gap between those that areoversimplified (like beam/aerodynamic strip models) and thedetailed techniques that require too much computer time (likedetailed finite element/computational fluid dynamics). In thestructures area, an equivalent plate analysis is used.54'55 It isintegrated with the piecewise continuous kernel function methodfor lifting surface unsteady aerodynamics.56 The method ofRoger57 is used to generate finite-dimensional state-space ap-proximations for the unsteady aerodynamic loads. The inte-grated aeroservoelastic system is modeled as a linear timeinvariant (LTI) system. Thus, any automated modern controlsystem design methodology can be integrated and used. At thepresent stage of development, however, the control system iscompletely described by the location of sensors and control

surfaces and by the transfer functions of the sensors, controllaws, and actuators (Fig. 1). Realistic fiber composite wingsand flight/flutter suppression control systems can be modeledwith accuracy that is sufficient for preliminary design of airplanes.Several types of analysis can be carried out simultaneously:

1) Static aeroelastic maneuver loads analysis in which thewing is attached to a free-free airplane and the loads dependon the flight conditions and take aeroelastic load redistribu-tion into account. The rigid-body and elastic displacements aswell as control surface deflections needed for trim are ob-tained from a solution of the linear system of equations

[K] 0)

where the generalized stiffness matrix [K] contains structuraland aerodynamic contributions and the loads { P } depend onthe maneuver parameters (load factor, roll rate, Mach num-ber, altitude), aerodynamic design variables (initial camber),and the structural design variables (through inertial loads).

2) Natural frequency and mode shape analysis

= (0) (2)

where K and M are the stiffness and mass matrices, respec-tively.

3) Aeroservoelastic stability analysis

(3)

where [U] and [V] depend on structural and control systemdesign variables and X is an aeroservoelastic pole.

4) A gust response analysis for the state covariance matrix16,58

[A] [X] + [X] (A]T = - \B][QW] \B]T (4)

where [A] and [B] depend on structural and control systemdesign variables, and [Qw] is the intensity matrix of a Gaussianwhite noise driving a gust filter that generates either Dryden orapproximate von Karman gust excitation power spectral den-sity. Root mean square (rms) values of accelerations, measure-ments, and control surface activity in terms of rotation androtation rate are calculated based on [X],

Static given-loads structural analysis (in which the wing iscantilevered and subject to predetermined load distributions)and induced drag analysis51 are also available, but are not usedin the studies reported here.

Control Law

3 4 5 6 7 8 9

~3~1 +76>jc2 + 77>2"-4-

Aspect Ratio = 6.8

Thickness Ratio t/c = 10% (Biconvex Airfoil)

Fig. 1 Actively controlled remotely piloted vehicle.

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LIVNE, FRIEDMANN, AND SCHMIT: AEROSERVOELASTIC WING SYNTHESIS 987

Analytical derivatives of all behavior functions with respectto all design variables are obtained by implicit differentiationof the analysis equations.18'51

Multidisciplinary Synthesis MethodologyApproach to Optimization

Combining nonlinear programming with approximationconcepts has proven to be successful in solving structural op-timization problems.19'22 NLP is general. No a priori knowl-edge of the active constraints that drive a certain design isneeded. The behavior function approximations are used inorder to overcome the computationally intensive nature of theproblem. Only a small number of detailed analyses are carriedout during optimization. Each detailed analysis and the associ-ated behavior sensitivity analysis serve as a basis for construct-ing explicit approximations to the objective and constraintfunctions in terms of the design variables. Then a series ofapproximate optimization problems is solved until conver-gence to an optimal solution is achieved. Each step, consistingof a detailed analysis, behavior sensitivity analysis, approxi-mate problem generation and a solution of the optimizationproblem (using approximate objective and constraints), is con-sidered one optimization cycle. Experience in structural syn-thesis shows that for problems involving stress, deformation,and natural frequency constraints, convergence of the synthe-sis process is achieved in 10-20 optimization cycles. The opti-mizer, of course, needs many more function evaluations, butthese are based on the approximations and are cheap compu-tationally. In order to prevent the approximate problem frombecoming too inaccurate as the design wanders away from thepoint in design space where detailed analysis was carried out,additional constraints are added to the approximate optimiza-tion problem in the form of move limits on the design vari-ables. For the NLP/AC approach to be practical, it is crucialto avoid too many detailed analyses for function evaluationsand sensitivity calculations. This depends on making the approxi-mations accurate yet simple enough for efficient solution.

Intuition and experience with structural systems led to thedevelopment of successful approximations for displacement,stress, and natural frequency constraints in terms of the struc-tural design variables.18 No similar experience exists for aero-servoelastic system design. It is thus necessary to studyalternative approximations of aeroservoelastic poles and gustresponse and evaluate their effectiveness.

Behavior sensitivities of objective and constraint functionsare needed throughout the optimization for gradient calcula-tion and the construction of approximations. If finite differ-ences are used for behavior sensitivities, optimization of thelarge multidisciplinary system can become extremely costly. Inthe presence of design variables from different disciplines withdifferent effects on the overall behavior of the system, stepsize selection for the finite differences can also pose a prob-lem. In the present work, behavior sensitivity analysis is basedon analytic derivatives of all behavior functions with respectto all design variables.51 The nonlinear programming al-gorithm used for constrained function minimization through-out this work was the method of feasible directions as imple-mented in the CONMIN code.59

Design VariablesPreassigned parameters for the optimization include wing

planform and depth (airfoil thickness) distribution, materialproperties, and structural layout of the wing (number of sparsand ribs and their locations). Control system structure is alsopreassigned. Thus, the number of sensors and actuators andtheir locations are given along with the number of control lawstransforming given combinations of sensor outputs into con-trol commands. It is also assumed that the general form of thetransfer functions of sensors and actuators are given and can-not be changed during optimization. To take advantage of multi-disciplinary interactions, the design space includes structural,control system, and aerodynamic design variables simulta-

neously. Structural design variables include polynomial coeffi-cients in the series describing skin layer thickness distributionover the wing. In the case of composite skins, a separate seriesis assigned to the thickness in each fiber direction:

t(x,y) = Jj TiXmiyni

i= 1(5)

Additional structural design variables include spar/rib capareas, concentrated masses, and spring constants (for thesprings representing stiffness of actuator and backup structureconnecting control surfaces to the wing box or canard to thefuselage). Control system design variables include polynomialcoefficients in the transfer functions representing controllaws:

y(s)r(s)

bnsn+(6)

Aerodynamic design variables include coefficients in the poly-nomial series for wing initial (jig) shape

(7)

Reference 49 describes a hierarchy of design variables (DVs)that includes sizing-type, shape-type and topological-type DVsin an increasing level of complexity. Because of the ex-ploratory nature of the present work, only sizing-type DVs areused at the present stage across the disciplines. The controlaugmented structural synthesis problem formulated in thiswork is a sizing problem for the three disciplines. Thus, thebalanced treatment of these disciplines (controls, aerodynam-ics, and structures) is also retained in formulating the opti-mization problem as in the analysis modeling.

Objective Functions and Behavior ConstraintsThe wing can be synthesized to minimize mass or gust re-

sponse or maximize performance with constraints on stresses,aeroservoelastic stability, aircraft performance in terms of rollrate, drag, or drag polar specification, and control systemperformance in terms of activity in gusts and limits on controlsurface travel and hinge moment. The objective function canbe chosen to be mass, drag, rms value of any response toatmospheric turbulence (to be minimized), or steady roll rateor lift to drag ratio (to be maximized) or a combination ofthese. Constraints are imposed to meet a combined stress cri-terion for composite skin layers and a unidirectional stress cri-terion for spar/rib caps. The aeroservoelastic system poles areforced to reside in the left half of the complex plane to guar-antee dynamic stability.60 If not included as part of the objec-tive function, the drag, lift/drag, mass, or roll rate can beconstrained to ensure acceptable performance.

Test Cases and Results:Remotely Piloted Vehicle Wing

DescriptionOptimization studies presented here deal first with a small

RPV similar to the NASA DAST research vehicle.61'62 Itsplanform geometry is shown in Fig. 1. The wing control sur-faces are used only for active flutter control. An accelerometeris placed on the wing strip containing the control surface. It islocated in the middle (spanwise) and at the 0.65-chord point ofthe strip. Its measurement yS£ serves as an input to a controllaw that, in turn, generates an input command d to the actua-tor of the wing control surface. The RPV structure is modeledas an assembly of four equivalent plates. A flexible wing isattached to a rigid fuselage and rigid control surfaces. Themain wing box structure, extending from root to tip spanwiseand to 80% chordwise, is the structure to be synthesized. Apair of 2.5-kg concentrated masses is attached to the wingtipto simulate a wingtip pod. The weight of fuselage, control sur-faces, and nonstructural wing mass is 308 kg for a half air-

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988 LIVNE, FRIEDMANN, AND SCHMIT: AEROSERVOELASTIC WING SYNTHESIS

plane. A Dryden gust model with a scale length of 518.16 mand a vertical gust rms velocity of 1.06 m/s is used. The set ofthree load conditions for wing stress calculations consists of3-g symmetric pullups at sea level, 10,000, and 20,000 ft. Inthe maneuver load calculations, the airplane is trimmed usingthe elevator. All stress constraints reflect a 1.5 safety factor.Flutter, gust, and aeroservoelastic stability calculations,though, are carried out at sea level, Mach 0.9 for thecantilevered wing. This is done in order to first examine fluttersuppression/ structural optimization using a realistic but sim-ple example without flight mechanics interactions. The RPVwing box skins are made of glass/epoxy laminates.62 Fiberdirections are 0, 90, +45, and -45 deg relative to a linepassing through the midchord points of the wing box. Theskin is then modeled as made of four unidirectional lamina.The thickness distribution of each of these lamina is describedby a nine-term polynomial in x and.y as in Fig. 1. Thus, thereare 36 structural design variables.

Structural DesignsMimimum weight designs with structural design variables

only subject to minimum gauge and stress constraints (stressdesign) or gauge, stress, and flutter constraints (flutter design)are synthesized first (Fig. 2). The stress design is unstable. Theflutter constraints are in the form of a 2% lower bound onviscous damping in five modes corresponding to the lowestfrequencies. Move limits of 40% were used, and convergencewas achieved within 15 optimization cycles.

Structure/Control DesignsThe control system is now incorporated, and wing mass is

minimized subject to gauge, stress and flutter constraintswhile the design space includes both structural and controlsystem design variables. Following its successful application tothe all aluminum wing studied in Ref. 50, the first control lawused for this study is the localized damping type transferfunction (LDTTF).24'63 This second-order control law pro-vides damping locally in the range of frequencies where damp-ing is needed. Its form is

Skin Mass (Kgm)5.00-1 flutter design

_5_ysE cc

(8)

where y$E is the accelerometer measurement and acybc, and ccare control system design variables. The denominator coeffi-cients can be associated with equivalent damping fc and natu-ral frequency coc of the control law. Thus, cc and bc determinethe center frequency and gain peak width of the control lawtransfer function, whereas ac determines the effective gain.The active control system is assumed to have no weight in thecalculations performed in this study. In the case of the allaluminum wing,50 when the second-order control system wasadded to the the problem and design synthesis started with thestress/gauge constrained (unstable) design, aeroservoelastic

Skin Mass (Kgm) 4 Skin Layers (0, 45, -45 and 90 deg.)

36 Structural DV's

1 2 3 4- 5 6 7 8 9 10 11 12 13 14

Number of full analyses

Fig. 2 Skin mass histories—stress and flutter designs—RPV com-posite wing.

x-x

stress design

••X- No qc.sqc RMS constraints

-*- RMS(qc)<0.50. RMS(sac)<20.

-» RMS(qc)<0.25. RMS(sac)<10.

4 layer composite skin

second order control \i

0 2 4. 6 8 10 12 U 16 18 20 22 24 26

Number of full analyses

qc - aileron rotation (deg)

s qc - aileron rotation rate (deg/sec)

Fig. 3 Skin mass convergence histories for structure/control synthe-sis of the composite RPV wing (quadratic control law).

mass minimization made it possible to reduce the skin mass toits stress design value while the active control system preventedflutter. In the case of the composite wing, however, when thedesign space was opened up to include the three control systemdesign variables as well as the 36 structural design variables,convergence could not be achieved when starting with thestress design. When synthesis starts with the feasible flutterdesign, convergence is achieved with 10% move limits, butonly about 60% of the weight penalty needed for flutterprevention is recovered by the addition of the control system(Fig. 3). Limiting the control system power by including con-straints on the aileron activity in atmospheric turbulence yieldsthe same trend that was found for the aluminum wing: Themore limited the control system is, the higher the structuralweight penalty needed for aeroservoelastic stability. But thefact that, when the control system was unlimited in power itcould not take care of the flutter problem called for furtherstudy.

Careful examination of the iteration histories starting withthe stress design revealed complex eigenvalue approximationsthat were extremely sensitive to design changes. Linear64

ax

or reciprocal

A = x0 + £

(A - A)/)

J_

AM

(9)

(10)

eigenvalue approximations were used first, where p/ is a typ-ical design variable. The accuracy of these approximations isevaluated when constraint values obtained by a full analysis ata new design point are compared to approximations to thesevalues constructed based on full analysis at the previous de-sign. In the design synthesis cases that failed to converge, ap-proximation accuracy for complex eigenvalues could changedramatically from iteration to iteration, yielding very goodapproximations in some instances and substantial errors inothers. The Rayleigh quotient approximations (RQA),46

(11)[based on fixed direct </>0 and adjoint \I/Q eigenvectors of Eq. (3)evaluated at the reference point] improved approximation ac-curacy but did not solve the convergence problem. A closerlook at the poles of the stress design composite wing revealedtwo flutter mechanisms. It appears that the second-order con-trol law could not stabilize both simultaneously because of the

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LIVNE, FRIEDMANN, AND SCHMIT: AEROSERVOELASTIC WING SYNTHESIS 989

narrow range of frequencies for which it is effective. Indeed,the design does not converge since it fluctuates between thesetwo instabilities.

The control law was subsequently changed to a first-orderlow-pass filter of the form

(12)

This control law is expected to have a wider bandwidth and,thus, be more effective in controlling the two instabilities.Indeed, the first-order control law makes it possible to regainmost (85%) of the flutter weight penalty. The rms of controlsurface rotation and rotation rate, however, are much highercompared with the second-order control law.

A third control law was also studied. It is a fourth-order lawmade up from the sum of two second-order filters. This con-trol law is expected to be more effective than the single second-order law in suppressing the flutter of the stress design. Itsform allows tuning a second-order filter to each of the twoinstabilities:

J_ysE

dc

ecs+fc(13)

This control law has six design variables. The denominatorcoefficients can be associated with equivalent damping andnatural frequency of the control law for each filter.

It should be re-emphasized at this point that the capabilitydescribed here can be used to synthesize any control lawsdescribed by rational transfer functions. The selection of spe-cific control laws in the present study was influenced by theaerodynamic energy approach.24'63 The thrust of the work atthe present stage, though, is not to identify preferable controllaws but rather to examine the convergence characteristics and

Skin Mass (Kgm)

RPV composite wing

structural + control system DV's

gage+stress+flutter constraints

flutter design

stress design

2 4 6 8 10 12 U 16Number of full analyses

Fig. 4 Skin mass histories with different control laws (no gust re-sponse constraints).

RMS aileron rotation (deg)

Unstable.No gust responsecalculations.

RPV composite wing

fixed structure (at the stress design)

variable control gains

control law type:

1 2 3 4 5 6 7 8 9 10 11 12 13

Full AnalysesFig. 5 History of aileron gust rms rotation minimization.

performance and gain experience with NLP/AC when appliedto the multidisciplinary wing design problem.

The performance of the three control laws is compared inFig. 4 in terms of the minimum skin mass achieved when nogust response constraints are imposed on the control system.Indeed, the fourth-order control law makes it possible to sta-bilize the wing without any mass penalty over the stress design.Actually, it even reduces the mass slightly with respect to thestress design, but this is associated with the convergence crite-ria used. The stress design mass could be slightly reduced bytightening the percent change in objective function used in adiminishing return convergence criterion (see Ref. 59, pp.100-101). A relative change in objective of less than 1% inthree consecutive optimization cycles was used in the examplesshown. The final fourth order law is

1421.7 21,224.6s2 + 20.65 + 9596.8 s2 + 120. Is + 20752.3deg/g (14)

The double quadratic control law is thus tuned to frequen-cies of 15.6 and 23.0 cps with equivalent damping ratios of10.5 and 41.1% in these two frequencies, respectively.

Freezing the stress design of the composite RPV wing andlooking for some "best" control system to stabilize it using afourth-order control law leads to the design history in Fig. 5.Six control system design variables are now used in an effortto minimize the rms of aileron rotation due to atmosphericgusts. Move limits of 10% are used to converge to a controllaw

2289 50,900s2 + 17.2s + 9209.7 s2 + 3055 + 23,380deg/g (15)

yielding a minimum rms rotation of 1.9 deg and associatedrms of rotation rate of 249 deg/s. When, during the optimiza-tion, stability is lost, the gust response calculation is bypassed.This shows up as gaps in the design history of Fig. 5. If betterperformance in terms of rms of aileron activity is required, thedesign space has to include structural design variables in addi-tion to the control system DVs. A feasible minimum skinweight design was found satisfying the gauge, stress, flutter,and additional constraints on the rms aileron activity (thatlimit it to half the values achieved in Fig. 5). Some structuralweight, however, had to be traded in this case, and the result-ing skin weight was about 22% heavier than the stress design.

Of course, it might be argued that control system powertranslates in the end to added mass, and that for the tradeoffstudies to be more realistic, one needs to include the weight ofthe control system in the objective function. The current capa-bility can take this into account by linking the values of certainconcentrated masses to the rms aileron rotation and rotationrate needed. This was not carried out, however, in the exam-ples given here. In any case, as with the aluminum wing, thecomplex tradeoff between structural weight and control sys-tem power (or the resulting weight of the control system) isevident.

Another issue of extreme importance in control system syn-thesis is that of stability robustness.25'37'65'66 Robustness re-quirements are not included directly in the set of behaviorfunctions in this study. In light of the fact that control systemsynthesis here is carried out for a plant that is changing duringthe synthesis process, it is interesting to examine stability ro-bustness of the resulting synthesized control systems.

All of the examples up to this point involve a single-inputsingle-output control system. Gain and phase margins of thiscontrol system can be studied by examining the Nyquist plotsof the open-loop system. Figure 6 shows Nyquist plots foroptimized control augmented wing designs subject to gauge,stress, and flutter constraints for the first-order quadratic andfourth-order control laws. In studying them, it should beremembered that they reflect control laws applied to different

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990 LIVNE, FRIEDMANN, AND SCHMIT: AEROSERVOELASTIC WING SYNTHESIS

Imaginary

.10 .20 .30

a) First-order control law

-1.60 -

REAL

b) Second-order control lawImaginary

-1.20 V

c) Fourth-order control law

Fig. 6 Nyquist plots for control augmented RPV minimum mass de-signs (no gust response constraints).

final plants (wing structure). The second-order control law hasthe highest gain and phase margins (with the highest penalty interms of wing weight, see Fig. 4). Leading to smaller structuralweight, the first- and fourth-order laws have small gain andphase margins. It was found that a considerable improvementin gain margin of the fourth-order control law design could beachieved by including (in addition to gauge, stress, and flutterconstraints) gust response constraints on control surface activ-ity. In the present case, this resulted in a higher weight andhigher damping in the aeroservoelastic poles. Of course, othercontrol laws may be more robust than the ones studied here.66

In any case, it is clear that it is essential to include constraintsthat will guarantee robustness of the resulting design in theintegrated optimization problem formulation.

YF16-Type Airplane ModelThe studies presented thus far focused on a very simple air-

plane configuration (RPV) and a simple single-input, single-output control system. In order to demonstrate the power ofthe present capability in synthesizing more complex configura-tions with more complex control systems, a more realisticmodel is considered next. It is similar to a YF16 fighter air-plane and is shown in Fig. 7. A flexible wing/flaperon combi-

nation is attached to a rigid fuselage/elevator combination.The wing box skin thickness distribution is to be synthesized.The airplane is initially statically stable and weighs 4234 kg perhalf airplane (not including the skin mass). Minimum gaugeand stress constraints are imposed on an array of 5 x 5 gridpoints over the skin. The two maneuver conditions consideredfor stress calculations are a 7.33-g symmetric pullup and asteady 160-deg/s roll, both at sea level, M = 0.9. Stress (stress+ gauge constraints) and flutter (stress + gauge + flutter) de-sign histories converged in less than 14 optimization cycles forthe composite graphite/epoxy wing with structural designvariables only. Minimum gauge is 0.0127 mm (0.005 in.) foreach laminate (0,90, +45, -45 deg) in the graphite/epoxyskin. Aeroelastic stability is examined for symmetric and anti-symmetric vibrations of the free-free airplane at sea level, M= 0.9. A minimum of 5% damping (f) is required in the firstfour symmetric and three antisymmetric poles (the first sym-metric pole is a short-period pole). Damping of 1% is requiredfor higher-frequency poles. When only structural design vari-ables were considered, move limits of 40% were used. Thestress design skin weight was 30.93 kg and the flutter designskin weight was 100.5 kg. The penalty in terms of weight paidfor meeting the flutter constraints is clearly evident.

iHORIZONTflLTflll

Fig. 7 Lightweight fighter model—no tip missiles (wing box skinthickness synthesized).

Composite skin

gage+stress+flutter constraints

Skin Mass (Kgm) constraint on rigid body pole location

flutter design

stress design

1 2 3 4 5 6 7 8 9 1 0 1 1

Number of full analyses

36 structural DV's

(4 fiber directions x 9 thickness coefficients)

14 control system DV's

Fig. 8 Skin mass minimization history for the control augmentedlightweight fighter model.

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A multi-input, multi-output (MIMO) control system is nowadded to the model. It is structured after the actual YF16control system67 with an addition for flutter suppression. An-gle of attack, pitch rate, and normal acceleration are measuredat the center fuselage and used by control laws that take careof static stability and handling qualities of the airplane in sym-metric motion. Roll rate measurement at the fuselage is usedfor the roll channel to control rolling performance. Wingtipaccelerometers (at the tip leading edge) of each wing are usedfor flutter suppression in combination with the fuselage nor-mal acceleration reading. The sum of the tip accelerations isused for flutter suppression in symmetric motion. Their differ-ence is used for antisymmetric vibration stabilization. Assum-ing perfect sensors and using the transfer functions of theactual YF16 actuators,67 six control laws are synthesized si-multaneously to ensure 5% damping (f) in poles associatedwith elastic modes in symmetric and antisymmetric motion. A35% minimum damping constraint and constraints on short-period frequency that limit it to the range of 0.25-1.00 cps10 aswell as a 35% minimum damping requirement on the controlaugmented roll pole are a simple way to introduce handlingquality considerations. The interactions between a flight con-trol system and an active flutter suppression system can thusbe taken into acount in the early design stages.

Figure 8 shows a skin mass design iteration history for thecontrol augmented YF16-type airplane model. Move limits of

angle of attack(deg) -

pitch rate (deg/sec)

root acceleration (g's).

damped frequency (cps)

tip acceleration

(right wing) (g's)

tip acceleration(left wing) (g's)

T//* \

755+90

755+90 *

roll rate (deg./sec) -13 + Mil

'symmetric motion**anti symmetric motion

Fig. 9 Lightweight fighter model—control system at start ofsynthesis.

SENSOR 14 977

1

-.01348

SENSOR IJJI5J+I3-J5

1 f , , ™ 0 1 )i*.i6n" s ) + ACTUATOR1

5e (deg)

Sojdeg)(aileron)

Fig. 10 Lightweight fighter model—control system at the end ofsynthesis.

SENSOR 20JJ4S + 20J34 .00237

-.4-50 -.350 -.250 -.150 -.050 .050 .100

damping measure (Eq. 16)

Fig. 11 Control augmented lightweight fighter poles—symmetricvibrations.

Skin Mass (Kgm)65

—&— Taylor Series - Reciprocal—B— Taylor Series - Direct—e— Rayleigh Quotient

0 5 1 0Number of full analyses

Fig. 12 Skin mass minimization histories with different aeroservoe-lastic pole approximations.

10% were used for all design variables. A total of 36 designvariables are used for the thickness distribution of a wing boxskin consisting of four composite laminates, and 14 designvariables are used in the control system. Gauge, stress, flutter,and handling quality constraints are included. The design con-verges to the stress design weight in 11 full analysis/optimiza-tion cycles. The initial and final control systems are shown inFigs. 9 and 10.

It is interesting to notice the change in sign in the antisym-metric flutter suppression control law, the bandwidth increasein the two flutter suppression laws, and the practical elimina-tion of the symmetric flutter suppression law in the final con-trol system compared with the initial design. Locations of thecomplex poles of the symmetric control augmented YF16-typemodel are shown in Fig. 11 (actuator poles, aerodynamicpoles, and integrator poles are not shown). The damping mea-sure is defined as

(16)

where a and co are the real and imaginary parts of the complexpoles, respectively. Thus, a positive damping measure indi-cates instability.

The synthesis of the control augmented YF 16-type modelwing involved 820 constraints and 50 design variables. It took18 min of CPU time on the UCLA IBM 3090 Model 600J.

Before concluding this section, it is interesting to examinethe effect of different complex eigenvalue approximations onsynthesis results. Composite wing skin mass histories for astructural/control design subject to stress, minimum gauge,and flutter constraints are shown in Fig. 12. Move limits of40% were used for the structural design variables and 10% forthe control design variables. Taylor series direct and Rayleighquotient approximations lead to essentially the same result,

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992 LIVNE, FRIEDMANN, AND SCHMIT: AEROSERVOELASTIC WING SYNTHESIS

and their performance in terms of the rate of convergence issimilar. Taylor series reciprocal approximations lead to fasterconvergence but to a slightly higher final weight.

ConclusionsUsing current supercomputers, unique integration of analy-

sis techniques, analytic sensitivities, and optimization basedon approximation concepts methodology, the single-levelmultidisciplinary synthesis of realistic actively controlled com-posite wings is both practical and feasible. However, the intro-duction of control system design variables in addition to struc-tural design variables presents a challenge to current approx-imate optimization problem generation. Smaller move limitsare necessary when constraints on pole location and gust re-sponse are included. This slows the convergence of the opti-mization process and makes it harder to find the feasibleregion when starting with infeasible designs.

The importance of NLP based synthesis technology is alsoevident because the multidisciplinary interactions and con-straints that will drive the design are unknown at the outsetand are very hard to predict. It has also been shown that anintegrated approach to multidisciplinary wing synthesis cansuccessfully generate designs that satisfy a rich blend of multi-disciplinary constraints and possible objectives.

AcknowledgmentThis research was supported by Air Force Office of Scien-

tific Research Contract F49620-87-K0003.

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