Paper No. 8 Advances In Experiment Design For High .

2,THEORETICALDEVELOPMENTAirplane dynamics can be describedlinear model equations:I yk(t)where/.tj(2)y(t)- Cx(t) Ou(t)v(i)(3)i 1,2,E{v(i)}noise Ov(i) is assumed. N (4)andGaussian(5)Eqs. (1)-(5) can be used to characterize bare airframedynamics, where the inputs are control surfacedeflections and the outputs can include air data(V, o ,fl), body axis angularanglesvelocities( , 0, IV), and translational( a x, ay, a z ). The same generalaccelerationsmodelstructurecanVtj l,2 . ni(8)u(t) dt ECramtr-Raolower bounds for the parameter standarderrors are computed as the square root of the diagonalelements of the dispersionmatrix 1912jsa9.Thedispersion matrix is defined as the inverse of theinformationmatrix M, the latter being a measure ofthe informationcontent of the data from anexperiment.The expressions for these matrices areNM E S(i)T R-Is(i)i l(9)D M -1Constraintsarising from practical flight testconsiderationscan be representedas limits on all inputamplitudes and selected output amplitudes.Inputamplitudes are limited by mechanicalstops, flightcontrol software limiters, rate limits, or linear controleffectiveness.Selected output amplitudes must belimited to avoid departure from the desired flight testcondition and to ensure validity of the assumed linearmodel structure.In addition, constraints may berequired on aircraft attitude angles for flight testoperational considerations,such as flight safety andmaintainingline of sight from the downlink antennaaboard the aircraft to the ground station. Theseconstraints are specified byjconstants.When estimating model parameter values frommeasured data, the minimum achievable parameterstandard errors using an asymptoticallyunbiased andefficient estimator (such as maximum likelihood) arecalled the Cramtr-Raolower bounds 12'js'lg. The(p, q, r), Eulerbe used to characterize closed loop dynamics, wherethe inputs are pilot stick and rudder deflections and theoutputs are selected from the same list as before. Forclosed loop modeling, the input includes a pure timedelay z, called the equivalent time delay, to account forphase lag effects from sources such as high ordercontrol system dynamics, digital sampling delay, andactuator dynamics.For either bare airframe or closedloop modeling, longitudinaland lateral cases aretreated separately, with the linear model structureshown above resulting from the usual smallperturbationassumptions.luj(t)l .lt(7)where E is some fixed value of the allowable inputenergy, chosen by experienceor intuition.Thisconstraint is intended to limit input and outputamplitudes,but it is also chosen for convenienceinthe optimization.Input energy is typically introducedas a constraint on the input form, while the costfunction quantifying achievable model parameteraccuracy based on the data is optimized usingvariational calculus to arrive at an optimal inputdesign. In practice, there is no direct constraint on theamount of input energy which can be applied duringthe flight test, since neither the pilot nor the controlsystem have inherent energy limitations.Thepractical flight test situation dictates that theconstraints be directly on the amplitudes of both theinput and the output variables, as given by Eqs. (6)and (7), respectively.The constraint in Eq. (8) limitsthe input and output amplitudes indirectly with anintegral expression.withEIv(i)v(j)T RSiJtJand {k are positiveTu(t)TLinear models are used in Eqs. (1) and (3) because ofthe common practice of estimating stability andcontrol derivatives from flight test data collected at achosen flight condition.Elements of the systemmatrices A, B, C, and D contain stability and controlderivatives, which are the unknown model parametersto be estimated from flight test data. If the maneuveris designed for small perturbationsof the inputs andoutputs about a chosen flight condition, the stabilityand control derivatives can be assumed constant.Measurementk e (1, 2. n o)Some researchers have implementedpractical flighttest constraints using an energy constraint on theinput,(I)x(O) x oy(i) Vtby the following:t(t) Ax(t) nu(t)z(i) l kwhere S(i) is the matrix(10)of output sensitivitiesto theparameters,(11)S(i) Oy(i)30 band 0 denotes the parameter vectorestimate.The informationmatrix can be loosely interpreted assignal-to-noiseratio for multiple output, multipleparameter linear systems.In this interpretation,thesignal is the sensitivityof the outputs to theparameters.If these sensitivitiesare large relative tothe noise level (3 to 1 ratio or greater) and are(6)Q"1

uncorrelatedwith one another, then the outputdependence on the parameters is strong and distinct foreach parameter.Parameter values can then beestimated with high accuracy when adjusting each ofthe parameters so that model outputs match measuredoutputs in a least squares sense. Elements of theinformationmatrix also depend on the measurementsampling rate and the measurementnoisecharacteristics,which are determined when specifyingthe instrumentationsystem.Similarly, the dispersion matrix g dependsnonlinearly on the states, which are often the same asthe outputs.Therefore, output amplitudes must becomparableif an input design comparisonis to befocused only on the merits of the input forms. Forthis reason, as well as to ensure validity of theassumed model structure, the inputs should bedesigned to produce comparable output amplitudes.Ifthe maneuver duration, input amplitudes, and outputamplitudes are not the same for all input designs beingcompared, it is possible to arrange matters so thatalmost any chosen input form will appear to be thebest, based on a criterion function that depends on g .The output sensitivitiesfor thejth parameter appear asthe jth column of the sensitivity matrix, and arecomputed fromt A xaoj uaojFor the global optimal square wave input design, theflight test maneuver duration T NAt might be fixed(12)a priori due to practical time constraintsof the flighttest or an analysis of the rate of decrease of theCramtr-Raobounds with increasing maneuver timeusing the optimized input. For the flight testexamples included in this work, the cost function tobe minimized for a fixed maneuver duration was thesum of squares of the Cramtr-Raobounds for theparameter standard errors,(13)ayaxacaDaojaojEqs. (12)-(14)followoojfor j 1,2, . np.differentiatingEqs. (1)-(3) with respect(14)npfromJ ay TdM-l] Tr[j lto Oj,combined with the assumed analyticity ofx in themodel equations.Note that it is necessary to havenominal (a priori) values for the model parameters tosolve Eqs. (12)-(14).The output sensitivitiesS(i)can also be computed using finite differenceperturbationsfrom the nominal parameter values and] foragivenT(15)Another formulation of the cost can be defined todesign the input for minimum flight test time toachieve specific goals for the Cram&-Raobounds 12.This is a minimum time problem, so that the cost isgiven byEqs. (l)-(3).J TFrom Eqs. (9)-(14), it is clear that the informationmatrix elements (and therefore the Cramtr-Raobounds) depend on the input through the sensitivityequations (12)-(14).The input u influences thesensitivitiesboth directly as a forcing function in thesensitivity equations and indirectly as an influence onthe states, which also force the sensitivity equations.The dependence of the Cramtr-Raobounds on theinput is nonlinear in the input amplitude, regardless ofwhether or not the system equations (1) and (3) arelinear, because of the nonlinear character of Eqs. (9)and (10).whenO'k kVk l,2. np(16)For the flight test examples included here, the optimalinput applied to the dynamic system described byEqs. (1)-(5) minimized the cost function in Eq. (15),subject to the constraints in Eqs. (6) and (7).3.OPTIMALINPUTSOLUTIONThe optimization problem posed in the last section isdifficult to solve in general. For the particularproblem of optimal input design for aircraft parameterestimation, there are good reasons to restrict theallowable input form to full amplitude square wavesonly. Among these are analytical work on a similarproblem 6, which indicated that the optimal inputshould be "bang-bang"(i.e., a full amplitudeswitching input). Square wave inputs are simple toimplement for either an onboard computer or the pilot.Finally, several flight test evaluations { ,t4,16,17 havedemonstrated that square wave inputs were superior tosinusoidal and doublet inputs for parameter estimationexperiments, largely due to richer frequency spectra.Eq. (9) is a discrete approximationto a time integralover the maneuver duration T NAt.Therefore,when comparing the effectivenessof various inputdesigns using some function of the dispersion matrix9 as the criterion for comparison,the input designsbeing compared should have the same maneuverduration, and in light of the last paragraph, also thesame allowable maximum input amplitude.Thisapproach contrasts with comparisons presented inpreviousworks9'10"11'17, which were based on constantinput energy. If only constant input energy isimposed on all inputs, a comparisonamong the inputsusing a criterion which is a function of 19 isinherently unfair because a wide range of values formaximum allowable input amplitude and maneuverduration can give the same input energy.For the above reasons, and to make the optimizationproblem tractable, input forms were limited to fullamplitude square waves only; i.e., only full positive,full negative, or zero amplitude was allowed for anyinput at any time. Full input amplitudewas used inR zl.

order to excite the system as much as possible.Choice of the pulse timing and having zero amplitudeavailable gave the optimizer the ability to use fullinput amplitudes without exceeding output amplitudeconstraints.With the above restrictionson the inputform, the problem becomes a high order combinatorialproblem involving output amplitude constraints,which is well-suited to solution by the method ofdynamic programming.Dynamic programmingis essentially a very efficientmethod for doing a global exhaustive search.Arbitrary dynamics such as control surface actuatordynamics, feedback control, and general nonlinearmodels can therefore be included inside theoptimizationwithout difficulty.The result obtained isa globally optimal square wave input obtained in asingle pass solution. The technique includesprovisionsto adjust the input possibilitiesat certaintimes in order to account for practical limitations onfrequency content of the input, such as avoidingstructural resonance frequencies.The dynamicprogrammingsolution smoothly handles the multipleinput problem, since this just changes the number ofsquare wave input possibilities.Keeping the systemresponses within the output space for which theassumed model structure is valid can be handleddirectly with dynamic programmingby discarding anyinput sequence whose output trajectory exceeds theconstraint limits. More details on the dynamicprogrammingsolution method can be found in Refs.[12] and [13].4,AIRCRAFTANDTESTPROCEDURESfeedback control system still operating. The pilot heldstick and rudder deflections constant at the trimmedvalues until the maneuver was complete.Themaneuver could be disengagedmanually by the pilottoggling the engage/disengagebutton, orautomaticallyby the research flight control system,based on g-limits, etc. The pre-programmedsquarewave perturbationinputs were standard 3-2-1-1 inputs,doublets, or square waves obtained from the optimalinput design technique described above.VariousFor square wave inputs implementedby the OBES,the pilot first selected a pre-programmedmaneuverusing buttons on a Digital Display Interface (DDI)inside the cockpit.The aircraft was then brought tothe desired trimmed flight condition and anengage/disengagebutton on the DDI was pressed toinitiate the maneuver.Square wave perturbationinputs from the OBES were added directly to theappropriate control surface actuator commands (forbare airframe modeling) or to pilot stick and ruddercommands(for closed loop modeling), with thedata transmissionrates wereoffsets from the center of gravity, and the angle ofattack measurementwas corrected for upwash. Datacompatibilityanalysis 21 revealed the need for a scalefactor correction on the angle of attack and sideslipangle measurementsfrom the wing tip vane, and smallbias error corrections on the measurementsfrom therate gyros and accelerometers. ,FLIGHTTESTRESULTSFor bare airframe short period longitudinaldynamics,the state vector x, input vector u, and output vector yin Eqs. (1)-(4) are defined byx [otq]TSystemmatricesThe F- 18 High Alpha Research Vehicle (HARV) is amodified F/A-I 8 fighter 2 . The flight test inputs wereimplementedby the pilot, and also by a computercontrolled On-Board Excitation .ystem (OBES).The pilot initiated each maneuver by first trimmingthe aircraft at the specified flight condition.For thepiloted square wave inputs, the pilot signaled theground controllers to send the square wave inputsequence via the uplink from a computer on theground. The maneuver was described by required stickand rudder deflections, represented to the pilot asmovementsof a cockpit indicator in real time.Accurate implementationof the input was achieved ifthe pilot accurately tracked the indicator movementwith his own stick and rudder inputs. Using thisprocedure, the pilot was able to produce a high fidelityrealization of the desired square wave inputs.downlinkemployed on the F-18 HARV aircraft, but all of thedata used for analysis was converted to a commonsampling rate of 40 Hz. Correctionswere applied tothe angle of attack, sideslip angle, and linearaccelerometermeasurementsto account for sensoru [t sI]Tcontainingy [aqthe modelaz]Tparameters(17)are:(18)A IZa M aMqJ LMtsMo00azFor bare airframex [flplateral-directionalrO]Ty [flpOt a1]T(20)(21)ay] rSystem matrices A, B, C, and D containparameters;A odynamics,u [t rr(19)the modelYflYp sint oYr - cos otocos 0 oLflLpLr0NflNpNr001tan 0 o0Vo(22)

Y6r a11o"L ,L8 aLol rN8 0VovPsu [rlr(t- )aPsdynamicsrsYprsin(26a)d?]T(26b)TLp(27)dPay] TYr - 1gcos0oLrL (28)ANfl0010000100001000000000000(30)(31)A priori linear models used for the input design casesincluded here were derived from a nonlinear batchsimulation of the F-18 HARV 22, which uses a windtunnel database for the aerodynamics.Noise varianceestimates for the a priori models were obtained fromprevious flight test data records using an optimalFourier smoothing technique 23. The models used forparameter estimation from flight test data wereidentical in structure to the a priori models, except thatthe apriori models did not include linear accelerometeroutputs.VoLfl0"lateral-directional model equations because the controllaw used bank angle feedback for gravity compensationto coordinatestability axis rolls. Closed loop modelparameters are in general different from the bareairframe parameters, because the closed loop modelparameters include the dynamics of the control systemin addition to the bare airframe. Model equationscould also be written using non-dimensionalparametersls.aYorla(t- )1]Y [fl0The L 0 and N O terms are present in the closed loopFor closed loop lateral-directionalstability axes, the model isX [fl0(25)VovT- Y#C INpNrcost osin r ocosOocosOoN0All flight test data analysis was done using outputerror maximum likelihood parameterestimation 1s,19,24. For the closed loop modeling, theequivalent time delays were estimated as the pure timedelay from pilot input to control surface deflection.The equivalent time delay can be estimated veryaccurately this way because the signals involved havevery low noise levels and the pilot inputs were squarewaves. Equivalent time delay was then held fixed atthis estimated value during the maximum likelihoodestimation. The Cram6r-Raobounds for the parameterstandard errors were the square root of the diagonalelements of the dispersion matrix 19 computed fromEq. (10). In the time domain, a correction for coloredoutput residuals from maximum likelihood estimationis necessary if the Cram6r-Raobounds are toaccurately represent the error in the parameter0- Y'orYrlaYo"LnrLTI,Lo(29)BNOrNrl aNo00PoR-6

sultswereunaffected by it, and alsobecause the Cramtr-Raobounds used in the optimalinput design assumed white Gaussian measurementnoise. The same white noise assumptionis made incomputing the Cramtr-Raobounds from flight testdata using output error maximum likelihoodestimation.For flight test data analysis in thefrequency domain, the correction is not necessary.Refs. [19] and [25] address this issue in detail. Modelstructure was held constant for the comparedmaneuvers, so that the number of parameters estimatedfrom each data record was identical. All data analysisand parameter estimation was done using anglemeasurementsin radians, but the plots were madeusing degrees.The first input design was a bare airframelateral-directionalcase using the OBES to implementsequential rudder and aileron inputs. The flightcondition was 5 deg angle of attack, Much 0.6, andaltitude of approximately25,000 ft. The model wasgiven by Eqs. (1)-(5) and (20)-(25).Perturbationinput and output amplitude constraintsresulting fromvarious practical flight test constraints were:la, l 4.0deglaa 1 2.5deg(32 )It I - 5.0 degI O ] 32. 0 deg(32b)The 3-2-1-1 input form has been shown to be veryeffective for aircraft parameter estimation in previousflight test investigations 9,1 , so this input was chosento compare with the globally optimal square waveinput design. Standard 3-2-1-1 inputs and globallyoptimal square wave inputs were designed using thesame input amplitude constraintsin (32a), the samemaneuver duration, the same a priori model, and thesame output amplitude constraintsin (32b).The 3-2-1-1 inputs were designed by matching thefrequency of the "2" pulse to the frequency of thedominant oscillatory mode for the a priori model, andadjusting amplitudes and control sequence timing sothat the chosen output amplitude constraints weresatisfied.Optimal inputs were designed with acomputer program that implementedthe optimal inputdesign procedure described above 12. The duration ofeach maneuver was 24 seconds.Figures l and 2 show the input and output timehistories measured in flight for the OBES lateraldirectional 3-2-1-1 and optimal inputs at 5 degreesangle of attack. The solid lines on the left side ofFigures 1 and 2 are the commandedinputs from theOBES, and the dashed lines are the actual measuredcontrol surface positions.The desired input formswere distorted by the feedback control system, as canbe seen in the figures. The distortion of the inputforms by the lateral-directionalfeedback controlsystem was not accounted for in the design process foreither input design.Figures 1 and 2 show that themaximum input and output amplitudes for these twomaneuvers were very nearly the same, and the lengthof each maneuver was the same. The maneuvers wererun in immediate succession on the same flight. Withthe model structure held fixed for the data analysis oneach maneuver, any differences in the resulting modelparameter accuracies can be attributed to effect of theinput form.Parameterestimationresultsfor the OBESlateral-directional3-2-1-1 and optimal inputs at5 degrees angle of attack are given in Table I.Column 1 in Table 1 lists the model parameters,column 2 contains the a priori values of theparameters used for the input design, column 3contains parameter estimates and Cramtr-Raolowerbounds for the parameter standard errors using the3-2-1-1 input. Column 4 contains the correspondingresults for the optimal square wave input. The dashedlines on the right side of Figures 1 and 2 are the modelresponses computed using the measured inputs and theestimated model parameters from columns 3 and 4 ofTable 1. The match is very good in both cases.Values in column 5 of Table 1 are the percent changein the Cram r-Raobound for each model parameterstandard error for the optimal input maneuvercompared to the 3-2-1-1 maneuver, based on the3-2-1-1 value. The optimal input reduced parameterstandard errors (equivalendy,increased parameteraccuracy) by an average 20%, with lower parameterstandard errors for every estimated parameter.Parameter estimates in columns 3 and 4 of Table 1 aregenerally in good agreement.The percent error of the a priori parameter valuesrelative to the parameter values estimated from flighttest data (computed as the average of values incolumns 3 and 4 of Table 1) varied from 4.2% to65.1%, with an average value of 24.2%.Nevertheless,both input design methods based on thea priori model produced experimental data withexcellent informationcontent, as evidenced by the lowstandard error bounds in Table 1.Symmetric stabilator input designs implementedbyOBES for longitudinalmodel identificationare shownin Figures 3 and 4. In this case, the distor

ADVANCES IN EXPERIMENT DESIGN FOR HIGH PERFORMANCE AIRCRAFT Eugene A. Morelli Dynamics and Control Branch MS 132, NASA Langley Research Center Hampton, Virginia 23681- 2199 USA SUMMARY A general overview and summary of recent advances in experiment design for high performance aircraft