Aiaa-2000-4565 Reconfigurable Flight Control Using Nonlinear Dynamic .
AIAA-2000-4565RECONFIGURABLEFLIGHT CONTROLUSING NONLINEARDYNAMICSPECIAL ACCELEROMETERIMPLEMENTATIONBarton J. Bacon and AaronNASA Langley ResearchHampton, VAan approachto on-line controldesign for aircraft that have suffered either actuatorfailure, missing effector surfaces, surface damage, orany combination.The approach is based on a modifiedversion of nonlinear dynamic inversion.The approachdoes not require a model of the baseline vehicle(effectors at zero deflection),but does require feedbackof accelerationsand effector positions.Implementationissues are addressed and the method is demonstratedonan advancedtailless aircraft.simulation analysisnonlinear system'sWITHAJ. OstroffCentersatisfy handling qualities.Unfortunately,both conceptstend to be sensitive to onboard model error.AbstractThis paper presentsINVERSIONresponse producing responses that were moredecoupledand closer to the desired ones.Some challenges from SDFC persist for RHO.An experimentaltool is used to directlystability n flight concerns the ability of thefeedback system to redesign itself, in the presence ofactuator failure, missing surfaces, and wing damage, tomaintain stability and some acceptable level of handlingqualities. A reconfigurable control law is generally theintegration of at least three important elements:l)failure detection and isolation to determine whichsurface is no longer useful, 2)on-line parameteridentificationto provide a model of the damagedvehicle, and 3)on-line control design which utilizes theinformationfrom the other two elements to reestablishcontrol. This paper deals exclusivelywith the thirdelement:specifically,developingan on-line controldesign methodologythat is robust to information errorsobtained from the other two elements.Over the past decade, the literature concerning theon-line design portion of reconfigurablecontrols hasbeen dominated by two concepts: dynamic inversion(DI) 1 and receding horizon optimal (RHO) 2 control.This is not surprising since both can be readily modifiedto handle changing nonlinear dynamics with updates tothe control's required onboard model. Also, bothconcepts include provisions for obtaining desiredclosed-loopdynamics (either explicitly or implicitly)toCopyright 2000 by the American Institute of AeronauticsandAstronautics, Inc. No copyright is assertedin the United States underTitle 17, U.S.Code. The U.S. Government has a royalty-free licenseto exercise all rights under the copyright claimed herein forGovernmental Purposes. All other rights are reserved by the copyrightownerAmericanInstituteRHO was used in the Self-DesigningFlightControl (SDFC) 3 which matured in Lockheed-Martin'sapproach to the ReconfigurableControl For TaillessFighter Aircraft (RESTORE)program 4. In SDFC, theLQ control stability robustnessguarantees of RHO wereexploited to successfully reconfigurethe VISTA/F-16on landing subject to a simulated missing left tailsurface. In RESTORE,the full predictiveadvantage ofRHO was utilized to better anticipate the desired plantSpecifically,the implementedLevel 1 models did notalways result in Level 1 flying qualities.This problemwas not observed in ground-basedsimulation.Contributingfactors could be sensitivityto onboardmodel errors in flight and the lack of guidelines forsetting or adjusting on-line the algorithm'scost functionweightings,penalty terms, and finite time horizon.These factors all influence how well the control followsthe reference model. In contrast, DI avoids the problemof manipulatingcost functions, but not necessarily theproblem of sensitivity due to onboard model error.The DI-based, Self-RepairingFlight ControlSystem (SRFC) s, for example, achieved on-linecontrol redesign using onboard models of both thenominal and the current/damagedvehicle to detectaccelerationmismatch.This mismatch drove a pseudoinverse allocator to recover the commandedaccelerationof the nominalflight controller.Augmentedto an existing F-15 controller, the flightexperimentdemonstratedthat this form of dynamicinversion was sensitive to model errors not only in thecurrent model but in the nominal model which wasbased on a mature aerodynamicdatabase of the F-15.A similar issue surfaced when the Intelligent FlightControl (IFC) used DI 6'7'8 as an explicit model follower.In the IFC structure, dynamic inversion was not appliedto the actual vehicle but to an onboard model. Plantinput and desired output reference signals from thisonboard system were fed to a closed-loopfeedbacksystem involving the actual vehicle to robustly producethe desired response.IFC was not robust enough,however, in that it failed to yield the desired1of Aeronauticsand Astronautics
longitudinalresponseon theACTIVE/FI5due toonboard model error produced by the control's neuralnet. The net was trained on a database for a vehiclewith square thrust vectoring nozzles. The test vehiclehad been recently upgraded to round nozzles.The issue of model error and dynamic inversion inon-line control design was revisited in Boeing'sapproach to RESTORE 9. A different explicit modelfollowing control structure was selected where dynamicof dynamic inversion is developedfor on-line controldesign.Implementationissues associatedwithobtaining the required accelerations,generatingthevehicles control derivatives,and promoting the desiredflying qualities follow. Several examples are thenoffered to demonstratethe efficacy of this approach toreconfiguration.Lastly, an experimentalsimulationanalysis tool is used to directly evaluate the nonlinearsystem's stability robustness.inversion was applied directly to the vehicle.An online neural network, based on the work of Kim andCalise I , adaptively regulated the error in the plantinversion to yield the desired rate response of selectcontrol variables.In truth, the net was a short termmemory device that modeled the inversion error. If theoptimal fit (not necessarilyobtained on-line) satisfied aLyapunov-basedconstraint m m2, the controller couldstabilize the vehicle without immediate interventionfrom parameteridentification.The optimal fit isdeterminedby the net's structure, i.e. the number/typeof basis functions and weightings.For the failuresconsideredin RESTORE,the number of weightingsrequired in the pitch, roll, and yaw channels were 18,72, and 72 respectively 13. For failures not considered,no mechanismwas offered to adapt the structure, i.e.complexity,of the neural net to model different sets ofinversion errors. This is one approach to desensitizedynamicapproachdynamicmodel.inversion to model uncertainty.Another is thetaken in this paper which reformulatesinversion to be less dependent on the onboardDynamicInversionfor On-lineIn this section, a general developmentof the online control design is offered. Unlike most of the otherdynamic inversion-basedon-line design techniquesmentioned above, the method pursued here is implicit.The desired dynamics of the closed-loopsystem do notreside in some explicit model to be followed but resultwhen the feedback loops are closed. Implicit dynamicinversion has been used previouslyin DI-based flightcontrol systems) aT'j8 Due to the demands placed on theon-line control design (i.e., unforeseenrapidly changingplant dynamics due to failure occurring possibly duringcomplex maneuvering)the form developed here will bemuch less dependenton the control'sonboard model.For this discussion, let x denote the state of theaircraft whose motion is governedJc F(x,t ) f(x) g(x,g)where the dim(g)obtain 3.have some desiredbehaviorfailed conditions.The desiredaircraft (no control deflections)and the requiredfeedback of the accelerationsand effector positions.Differencesinclude a proposed strategy for dealing withnonaffine controls--noinverse mapping via neural netsrequired l . Here, a special implementationofcommandedYc is typicallyInstituteobjectiveY [YlonYlatinputhere is tocontrolto the longitudinal,Ydiff"lateral,and h(x)(2)under both nominalbehaviordefinedandof y to someby fdes(Y' Yc)yielding low-orderaxes and compliant(3)responses that are decoupled alongwith military specifications19.A dynamic inversion approach to this probleminvolves constructinga control of the form k(x, Yaes)failure. The proposed control strategy is most closelyaligned with SRFC but without the required onboardmodels.The proposed control is an implicit modelfollowing control requiring only the system's controlderivatives.Robustnessresults, using an advancedtailless aircraft with multiple innovative controleffectors 1 :6, will show that the accuracy of thesecontrol derivatives are not stringent.To provide a clear exposition of the method, thepaper is organized as follows.First, a modified versionAmerican(1.)The controlreconfiguration,various response types, and theincorporationof handling qualities.Similar featuresinclude the absence of an onboard model of the baselinewithbyg such that three state-dependentvariables correspondingdirectional axes,the required accelerationsto accommodatesensorControlOesi nThe proposed approach to desensitize dynamicinversion for on-line control design is an extension ofSmith's approach for standard flight control 4 to yit,Modified(4)to realize some desiredcontrol J',les when substitutedequationequation2of Aeronautics2. To realize3, the control1 subjecttoofYc))"(5)If the input mappingand Astronauticsratedesired closed-loopdynamicsin its implicit form is6 k(x, fdes(h(x),g(x, 6) G(x)8,variableinto equationg(x, t ) can be expressedas(6)
the dynamic8rW-tSinversioncontrol,subjectfrom the recent, is(Ydes - hxf)hx Oh/Ox and W is a diagonalmatrix. The dependencynotational convenience.(7)state and controlThe correspondingstate rateforupdated in flight.Due to the model's form, an updaterequires more informationthan just the stability andcontrol derivatives identified online. 20of G.The elementsof fwith partialsevaluatedstate/controlF(x,6)at pointstrajectory, )c ---5co Ao(X-5, Oh(x)x(Xo,6o)Xo) BoA6inversion2 hx(2oa minimum'J'des J subjectHere, x is the actual state (not a perturbation)and theelements of A and G, are the stability and controlBod6).norm solutionprovidesto minimizingdependsome haveto dynamic inversion that eliminatesall the problemscited.To develop the modified form, consider theof motion as they actuallycontrol mappingF(x,6)A standardfollowing f(x) Taylorg(x,6)approximationand 6 in the neighborhoodF(x,6) f(xo) Ix x.,8 8.formulationA6of [Xo,6 o](16)x approachesxo(Ydes - hx:Co)(17)It is importantto note that in this)co is based on the effectorchangeto Ydes" Hence,positionto that positionsensedpositions(11)6 o andinof theeffectors are a must. The modified dynamic inversioninner loop is shown in figure 1. Note, the existing 6 othat correspondsto ko is taken from the outputThe weightingW consistsdynamicsdefinedof theof the effector'sin theby equation3 alongwith equation 17 define the proposed online controldesign method.Note, the control requires the vehicle'sderivatives,Bo , either in table lookupsome functional approximationform.not require f(x) or g(x,6).ChangesreflectedAmericanupdate rate,is the commandedresponsecontrolfor x(6 - 6o)(Ydes-hxBoW(hxBo)The desiredthe6))lx xo,8 8 (x-xo) control 6 o A6.actuator.(10)of F(x,6)sobecomesA6 W(hxBo)with aprovidesa A6A6Tw-IA6rate limit emphasizingthe faster controlsminimum norm solution.g(Xo,6 o) O xx(f(x) g(x,-- - (g(x,6))appear 5:.series expansionfirst-orderand the controlwith 6proposed neural nets x to get around this problem, thereis a much easier procedure using a modified approachequationsnonlinear(15)onAnother problem with equation 7 for on-linecontrol design concerns the system's actual controlmapping. The required mapping,G6, is actually thenonlinear control mappingg(x, g) rendering the pseudoWhileand(hxJc o hxA o (x - x o )))With a sufficientA key difficulty for dynamicstorage of the nominal bias andin (7) inappropriate.y h(x)l(9)typicallyA6 6 - 6 o . Ao(x-Xo) (8)f Ax bias.which(14)control,A6 W(hxBo)T[hxBoW(hxBo)Tthe required bias correctionsall the elements of x .on theawithderivatives,respectively.inversion is the onboard( 13b)of [Xo,6 o] whereWith regard to dynamicare definedalso identifies(13a)Bo -- (g (x, g)) IX Xo,8 8oAs before,if the on-line parameter identificationbias 3, or a model of the form. Ax G8 bias) g(x, 6))Ix xo,, g in the neighborhoodThe problem is that the stability derivatives do notdefine the elements of f.Note, the control derivatives(12)linear definition,Ao x(f(xa model was required for standard flight controls) 7asFor on-line control design, the model would have to beinverse solutionko g(Xo, 60)"Using the standardweightingon x has been dropped. o - f(Xo)The control in equation 7 is clearly dependent on theonboard nonlinear model described by f and G. Suchare the elementspast.some previoussatisfiesE8 W(hxG) T hxGW(hxG)whereLet xo and 6 o denoteto minimizingin the measurementidentificationderivatives.form orThe control doesin f(x) areof Xo The onlineneed only supply the corrected controlThe bias and the stability derivativesinequation 9 are not required.The problem of applyingDI to a system with a nonaffine control mapping hasInstitute3of Aeronauticsand Astronautics
AIRCRAFT ACTUATORDYNAMICSYdes ]lw,,,o,TE,,w,,,,TJ-IEfailure flag,control derivativeonce the failure detection and isolation (FDI) algorithmdetermines which effector has failed, that effector maybe easily taken off line and the remaining controlsreallocated by simply setting the correspondingweighting to zero. The control does requireof Xo and d;o . In the coming5co will be constructedaccelerometerinterestfrom a specialmeasurements.section,controlk o and the controlvariablesqb, and /'b are the respectiveTwo sets ofUsing a standard x-y-z body axes coordinate frame,the equations of motion for a flat-earth, rigid body,symmetricalairplane 21 can be expressed asaccelerationscorrespondingto the sum of inertial andgravitationalforces as sensed by instrumentationmounted at the cg. The following auxiliary equationsmust be added to the equations of (18)-man,cg m[ fv pb v - qb u - g cosOcosO]baselineaerodynamics,componentLet X6t denotethat portionvectoring.Consultingequations Zdiscussed- Ix,r) qbrblxz Nthe linear velocitiesthe respective(18)(ft-lb);and products(ft/s);lxx,Iyy,of inertiav, and wtJ, , and w lection.of X t due to thrust18 and 19, let ko correspondin the next section.Pb,19. Some economycan be realizedk o since only those elementscolumnsof h x are neededinmultiplyingin the control.which determineg(x, 6) used todefine the control derivatives is given.Solving for the linear and angularaccelerationsinequation 18, with the decompositionused in equation20, and noting the part due exclusivelyto control yieldsqb, andInstitute(21)0 and 6 are obtainedPrior to selecting control variableshx , the nonlinear control mappingthe total aerodynamicand N represent thelinear accelerationsincludesto the state vectori'bIzz - Pblxz pbqb(lyyrepresentX t is the axial forcethe axial force due to aerodynamicthe nonzerou,beof axial force due to theeffects of pitch and yaw thrustconstructingm is the mass (slugs);can generallydue to engine thrust whichfrom equation(slug-ft2);(19b)(20)X a is the componentqblyy Pbrb(lxx - lzz) r21xz Pb2 Ixz Mand lxz are the moments .The linear and angular accelerationswill be obtainedfrom a special grouping of accelerometersto beiJb lx.x - rbl xz qb rb( l zz - l yy ) - p bq b t xz LIzz,(19a)x [u, w, qb, O,v, Pb, rb, ] T may,cg re[f: rbu - PbW - g cos 0 sin 0] Yand thrust momentsrb sin The total forces and momentsdecomposedin the formmax, cg m[ti qb w -- rbv g sin O] Xtotal aerodynamicaccelerations(rad/s2); and ax, cg , ay, cg , and an,cg are thewherewill be considered.Here X, Y, and Z representand thrust forces (Ib), L, M,,bb ,X X a X t X&Controlderivatives.angular(b pb qbtanOsin rbtanOcosclean signals of Xo whenDI Flightthe roll, pitch, and yaw rates (rad/s); qb COS -This section includes key equations for assemblinga reconfigurablecontrol law using the modified form ofdynamic inversion.Specifically,details are given forobtainingrb representset ofthe accelerometersare subjected to noise, failure,biases, and contaminationdue to structural flexibility.Reconfi urableDI Inner-LoopIt is a topic of currenton how to construct g(x, 6)updateFigure 1. Modifiedalso been eliminated.Hence, the minimum normsolution for the inverse control is viable. Moreover,measurements6o 8 f(x):Co :C4of Aeronauticsand Astronautics
(X& X&)/m(Z& Z&) / m.'tacg{(M& M &)/ lyy}180/lr0(ve r ) / mg(x, #) ."-I, "' " ::.In equation 22, all angularto units of deg]s 2.accelerations2. Accelerometerthree orthogonalarelocatedy h(x) [qb, Ps, fl - O'2rs "to produceFigureare convertedIn this paper, two sets of control variablesconsideredfor reconfiguration.In one set,manipulateda3z[N& N&J0both the forces and moments(23)of the effectorsthe desired dynamics.areHere,stabilityrequiredattack,the sideslip,and Ps and rs representisin Ps and rs and the ( u, v, w ) dependenceof ft.a distance(24)of gravity(cg).sensor(25)angularof body-axisratesangularaccelerations(/ b, qb, ib ), and F//is a positionvectorthe distances( x i , Yi, zi ) to the cg. Body-axisangularaccelerations(/b b , qb,/'b ) are obtainedgroupsbetweenby takingany two linear accelerometerthesensor- j) - cox cox A j(26)the desireddynamics (not the same desired dynamics as the firstset). The control resulting from these two sets ofcontrol variables will be referred to as the 'Force andwhereMoment' (F&M), and 'Moment',(M), approaches toreconfiguration,respectively.Details for constructingthe cg, is needed for the angular accelerationsolution.A total of nine equations can be generated fromequation 24. In generic form these arehx can be foundin referenceofasx AT// j @are used to produce( axi, ay i , azi )from the centeris a vector of body-axisdifferenceIn the second set,only the momentsConfiguration.linear accelerometers( Pb, qb, rb ), " is a vectory h(x) [qb, Pb, rb "Groupa i acg co x r/ (o xco x riflaxis roll and yaw rates. Force manipulationdue to the ( u, w ) dependenceof angle ofa,SensorThe general accelerometerequation, excludingnoise, bias, and airplane flexibility effects, iswhererepresentsX.2.(22)vlyy J.A jrepresents{i, j 1,2,3; i * j}.distance22.betweenthe distanceOne importantsensorgroups,r/aspectrj,is that theand not the distanceto-i'b AYi j itbAZi j Ai jAccelerometerMeasurementsDeterminationi'bZlXij --pbZlZi j Bi jof the state accelerationvectorko- tbZlXi j PbAyi j Ci jis an important factor in the proposed modified DIcontrol law approach.Potential measurementsourcesfor the elements of this vector include linearaccelerometersand numericallydifferentiatedangularwhere0 1,2,3)are shown,Bi j (ay,i -ay,jI -- Pb rbAZi-j) - pbqbAxi j(Pb2 r# )AYi-J -qbrbAZi-JCi j -(an, i -an,j-qbrbAyi jwith the last equationacceleration.aiInstitute r# )Axi j(28) There are multiplewhere each sensor group consistAmericanthe right hand side is- PbqbAyi-jThis accelerometerimplementationcould allow asignificant amount of sensor redundancy and noisereduction, although these aspects have not beenexplored at this time. In addition, linear accelerometersare needed for the force and moment (F&M) controlapproach that requires state accelerations.In figure 2 three arbitrary sensor groupsi, j 1,2,3; i jai j (ax, i - ax, j ) (q;rates. Both sources have been chosen to getindependentdata. Angular accelerationdata is derivedfrom a unique implementationof linear accelerometermeasurements,in addition to the differentiatedrates.(27)of) - pbrbAxi j (P; q;)AZi-jin (28) writtensolutionsin termsof normalfor ,bb , 00, and i"b .Only five of the nine accelerometersare requiredthe stipulation that at least one accelerometer5of Aeronauticsand Astronauticswith
measurementgroups. Sensor redundancyhas notthis paper, but it appears that manyused for FDI. In addition, by usingthe effects of sensor noise and biasbeen explored incombinationscan bemultiple solutionsmay be reduced bysome type of averaging technique.In this paper, (/ b, qb,/'b ) are derivedfrom a least-squares solution of 12 equations, 9 from equationand 3 equations from the numerically differentiatedgyro signals. All equations are weighted equally,an FDI system was available equations associatedfailed accelerometerscould be weighted to zero.assumedThe tables are also useful when variousmust come from each of the three sensorthat the rate gyros would(27)ratebut ifwithIt ishave a separateFDIsystem.The next step is to calculatewhich requiresknowledge-ffcg using equationof .All control25,3 equationseach for ax,cg,State accelerationsapproachare requiredand are calculatednondimensionalMach, altitude,whereand 8." is dynamicreferenceequation( c i ) that are nonlinearvariables,for exampleRepresentativea,of theX r,,, c, ,x' Sto get thefor the F&Mby rearrangingcoefficientsof many airplaneaerodynamicforces and moments due to controls,consider only the axial force componentay.cg, and an,cg.The 3 equations along each axis are averagedlinear accelerations.force andin tables that aregenerated from wind tunnel and flight data. Thesetables, which require interpolation,usually containfl,use models that are based upon some fixed cg location,so this restriction is not unique. Nine equations aresolved,this paper.In the typical airplane simulation,moment aerodynamicdata are definedfunctionsdesignsuncertaintiesand time delays are injected into the control to quantify.worst case scenarios regarding the combined FDI andparameter identificationprocesses.Here, controlderivatives are required for both the aerodynamicandthrust vectoring controls.Thrust vectoring controlderivatives result analyticallyfrom partials taken offunctional expressionsin reference 22. Only theaerodynamiccontrol derivativeswill be consideredin18(30)pressure(Ib/ft2),area (ft2). Aerodynamicof lb. The term,Sforceis the wingX&has unitsc&, x , is the nondimensionalas ax,cg - g sin 0 vr b - wqb(29a)f ay,cg g sin cos 0 - ur b wpb(29b)(v -an,cg(29c) g cos4 cos/9 uqb-vPbwith units in ft/s 2. The state accelerationvector- 0 isset up with eight states in the order shown inequation 21.When the M control approach is used only threeaccelerationsare requiredspecifically( qb,/Tb'/ b)andbe noted that c&, x representsseveraltables includingtablesdue to controls.Ita sum fromof interferencecoefficients representingthe effect of one effectordeflecting upstream from another effector.The C,ya,xterms are also responsiblemapping.Consultingfor the nonlinearequation3g(x, 6)/Ot a is constructedcontrol22, the first row offrom the partialsOc&,x / 8 a . In the simulation,the aerodynamicdatatables are linearly interpolated.The derivatives,then,are constants between the breakpointsof the tables(actually up to some e of the breakpoint).TheControlgenerated derivative tables have been constructedthis in mind. In testing the on-line control design1.2.3.DerivativesThe modified dynamic inversion control mustcarry a representationof this control derivativedatabase.Problems with the proposed control redesignInstitutewithalgorithm, the derivative tables are interpolatedalongwith the original aerodynamicdatabase tables using thesimulation'sproprietary interpolationscheme.Flyinmethodologyare separated from potential problemswith real time system identificationmethodology.Tables provide a baseline for the real time systemidentificationmethod.American/axial force coefficientshouldequation 29 is not used. The linear accelerationsat thecg are still used, as discussed in the Flying Qualitiessection.For this study, complete tables of controlderivatives have been generated from the aerodynamicdatabase.There are at least three reasons for doing this.iIaerodynamicOualitiesIn this section, equation 3 defining the desireddynamic behavior for the control variables (CV) isconsidered.Referred to as a 'commandmodel',equation3 establishesthe flying qualitiesof the closed-loop system. The three control variables and theirrespective command models are consideredseparately6of Aeronauticsand Astronautics
since theoreticallycoupled integratorthe inner DI loop has producedblock relating Ydes to y.When a slowervariableto relateaxis a secondin figure 3) is wrappedPs,c to controlYot is to be controlled,outer loop (as illustratedaroundbank anglethe first outer loop for0 duringbatch simulationruns. The two outer loops together create a combinedtransfer function that has the form of equation 33. Withanother outer loop can be wrapped around the innerloops as illustrated in figure 3. Here the outer loopbandwidth is defined as COol. A nonlinear function,NL, is includedIn the laterala de-the second outer-loopfollowingYol,des to Yc" The referencefrequencyrelationshipsdefinedcan be derivedastheCOphi,as(34)COlat 2 equivalentCOequivalentsignal is used in the simulation sectionwith the desired system response.for comparisondOpen-loopv [CommandYot,c* fI--1whereModelsot,det- sl.IDynamics[--momentYdesControliyollIA rcraftIthat follows.bandwidthbank-anglefunctionsselectedfor the commandrateParameterfor approximatingOctes and is definedcoph i is thethe desiredasOdes C phi(Oc-( )"A nonlinearequationapproximately(36)is insertedrelateor the momentapproach.and MomentThe longitudinal,command-modelsinto the controlloop toOdes to Ps,c as(F&M)CommandModelsY latKlonCOlon(S S 2 2 lonCOlonYlat,c(31)COn,Ion)(32)COlatS COlatSimulationsa pitch-ratecommandsdirectionalcommandaxis, a stability-axis( Ps,c ) for the lateralstability-axisPb,c O)dir2for the F&M approachpaper includelongitudinal 2 dirCOdirSyaw-rateto the idealizedresponseusinggive asecond-orderCOequivalentand"(M) CommandModelsqb and rb with transfer )des -(qbThe longitudinalillustratedin thisft.( qc ) for theroll-ratefunctionsParametersbandwidthaxis, and a linear combination( rs,c ) and sideslip-anglevariables.ofp) tan0.(38)CV is a and the directionalCV iscoa and :nil are the outer-loopfrequenciestides respectively( tic )desfor approximatinggtdes andInstituteas coa(arc -co)(39)tides coil(tic - fl ) "The nonlinearbetweenAmericansin tp rb coscommandfor the directional axis. The lateral andaxis command models illustrate the mixingof linear and angular(37)all of the control loops. The only difference in thelateral loop is the nonlinear equation for body-axis rollrate command(33)COdirs0)coswill show that these equationsreferenceare used for Pb,2Ydir,ccosO)tansimilar to equation 32. A second outer-loopsimilar tothe one discussed for the lateral axis above is used fors COlon22Y dirresultsO rbThe moment approach only uses angular bodyrates for the DI loop and only three angular controlderivatives are required.First-order command models2Ylon,copen-loopMomentlateral, and directional axesfor the F&M approach aresinvery close bSimulationmodels are typically found in the militaryspecifications1 . Command models in this paper varywith selection of either the force-and moment approachfapproachouter-loopPs,c/7the natural3. Outer loop configurationTransferForceand equivalent representCOequivalentfrequency and damping ratio of the combined 2 -ordertransfer function.These equations are also used in OequivalentCOphi 27of Aeronauticsexpressionstides andand Astronauticsrb,c are(40)between&des and qb,c and
effectorsV cos a cos fl(41)- tan ot tan flfldes Pb,c tan flCOS/2'tidesg(ay,cgr 'c -- - cos----aof two real poles with transfer COSOSin )V cos a cos fl(42) Pb,c tan a.In the next section, examples illustrate theproposed online control design in various failurescenarios for both sets of control variables. Later, anexperimentalsimulation tool is used to determine thestability robustness of the proposed nonlinear closed-Ta AircraftLOLEF,ModelSome of the advantagesof a tailless aircraftare reduced radar signature and reduced weight anddrag. Analysis of the iinearized aircraft model showsthat it is unstable in the lateral-directionalaxes at highspeed. Table 1 shows the longitudinaland lateraldirectional eigenvalues(unstable eigenvaluesitalicized)for three Mach numbers at 25000 ft.Table 1. Eigenvaluesof Linearized Airplane 12 j0.0894-0.05113 j0.1609-0.53050.7-0.00328 j0.04584-0.6621 jl.783-0.2176 j0.7351&026880.86250.90.1017-0.1140-0.9073 j0.47280.0094771.446-0.9452Eleven('Oa2 (Oa2 j roachthe first outer loop for Ps to controlbatch simulationbank angleruns.All command models, correspondingequations andvariables are discussed in the section on FlyingQualities.Values for the commandmodel parametersare shown in table 3 with units of rad/s used for allfrequencies.Table 3. Command ModelParameterValueParameters,F&M ApproachParameterValuelon0.7O)tat2O)lo n5dir0.7O)n,lo n3O)di r3Klo n1/3O) ph i0.75In all of the examples shown below, the controllersampling-frequencyis 100 hz and the controleffectivenessmatrix is updated from tables everyiteration. The nonlinear simulationincludes the actuatordynamics describedand position limits.Martin(F&M)is stability axis roll rate, and the directionalCV is alinear combinationof sideslip and stability axis yawrate. In the lateral axis, a second outer loop is wrappedduring2) right elevon (RE), 3) symmetricpitch flap (SPF),4) all-moving tip (AMT), 5) spoiler-slot-deflector(SSD), 6) left outboard leading edge flap (LOLEF),7) right outboard leading edge flap (ROLEF), 8) pitchvectoring (PV), and 9) yaw vectoring (YV). The AMTand the SSD controls are unilateral with controlof Lockheed18, 10037, 4137, 41a previous section of this paper. Three control variable
analysis tool is used to directly evaluate the nonlinear system's stability robustness. Modified Dynamic Inversion for On-line Control Oesi_n In this section, a general development of the on-line control design is offered. Unlike most of the other dynamic inversion-based on-line design techniques mentioned above, the method pursued here is .
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For permission to copy or to republish, contact the copyright owner named on the first page. For AIAA-held copyright, write to AIAA Permissions Department, 1801 Alexander Bell Drive, Suite 500, Reston, VA, 20191-4344 AIAA 2005 - 1042 COMPUTATIONAL FLUID DYNAMICS STUDY OF TURBULANCE MODELING FOR AN OGIVE USING COBALT FLOW SOLVER AND A 2N TREE-BASED
AIAA Atmospheric Flight Mechanics Conference 2 - 5 August 2010, Toronto, Ontario Canada AIAA 2010-7930 Hypersonic Vehicle Flight Dynamics with Coupled Aerodynamics and Reduced-order Propulsive Models Derek J. Dalle , Scott G. V. Frendreisy, James F. Driscollz, Carlos E. S. Cesnikx . Aeronautics and Astronautics, Inc., with permission.
AIAA SciTech, 52nd AIAA Aerospace Sciences Meeting, January 2014, National Harbor, AIAA 2014{0414 Hybrid feedback design for subsonic and transonic airfoils and wings M. Zhang 1, C. Wang , A. Rizzi1, R. Nangia2 1Royal Institute of Technology (KTH), 10044 Stockholm, Sweden 2Nangia Aero Research Associates, Bristol, UK A hybrid inverse/optimization
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Institute of Aeronautics and Astronautics (AIAA) Standard S-120A-2015, November 23, 2015, to provide an effective space vehicle (SV) program technical baseline for mass properties control and mission success. 1.2 Application . This document is intended for use in acquisition and study contracts for SVs. The ANSI/AIAA standard
Flight Operation Quality Assurance (FOQA) programs are today customary among major . EASA European Aviation Safety Agency F/D Flight Director . FAF Final Approach and Fix point FCOM Flight Crew Operating Manual FCTM Flight Crew Training Manual FDAP Flight Data Analysis Program FDM Flight Data Monitoring FLCH Flight Level Change FMC Flight .
(ANSI) A300 standards of limitation on the amount of meristematic tissue (number of buds) removed during any one annual cycle (in general, removing no more than 25% on a young tree). The third circle is the top circle – the reason the other circles exist. We grow and maintain trees for aesthetic and functional values, and pruning properly for structure and biological health helps us achieve .
