Reducing Conservatism in Flutterometer Predictions Using Volterra Modeling with Modal Parameter Estimation

Transcription

1 JOURNAL OF AIRCRAFT Vol. 42, No. 4, Jly Agst 2005 Redcing Conservatism in Fltterometer Predictions Using Volterra Modeling with Modal Parameter Estimation Rick Lind and Joao Pedro Mortaga University of Florida, Gainesville, Florida Accrate prediction of fltter speeds is essential to efficient and safe flight testing for envelope expansion. Sch accracy is particlarly difficlt to obtain when analyzing flight data from speeds well below the critical speed at which fltter occrs. The fltterometer was introdced as a tool that cold predict the onset of fltter even at lowspeed conditions; however, the conservatism in those predictions redced testing efficiency. A method to agment the fltterometer to decrease the conservatism, and conseqently increase the accracy, of the predicted fltter speeds is presented. The method incorporates a scheme for model pdating that ensres consistency between the analytical dynamics and the flight data. The tool is sed to compte fltter speeds for the aerostrctres test wing. The speeds predicted with the model pdating scheme are very close to the actal fltter speed and demonstrate the benefits for improving efficiency of envelope expansion. Introdction THE fltterometer is a tool designed to increase the efficiency of flight fltter testing. 1 The concept of this tool is to combine model-based and data-based analysis to predict the onset of fltter dring envelope expansion. The procedre is to compte the worst-case stability margins for a theoretical model with respect to ncertainty as determined by flight data. The development for sch a procedre is based on µ-method analysis. 2 The tool has been sed to predict fltter speeds for a variety of aeroelastic systems. The fltterometer was tested, along with a variety of techniqes, for a simple system that will be addressed again in this paper. 3 Also, the techniqe has been sccessflly applied to the prediction of fltter for systems sch as a wind-tnnel model 4 and F-16 with stores. 5 Fltter speeds predicted by the fltterometer have an inherent level of conservatism reslting from the worst-case natre of the predictions. This conservatism provides an inherent level of safety; however, that conservatism can also adversely impact the efficiency of the flight test. Some aircraft have relatively small fltter margins, and so the envelope mst be determined as accrately as possible. Any conservatism in the prediction of this envelope cold reslt in time-consming flight testing. The conservatism in the fltterometer is directly related to the model and its associated ncertainty. The only way to remove the conservatism is to improve the accracy of the model. The formlation of a theoretically perfect model wold garantee the fltterometer has minimal conservatism; however, sch a formlation is not yet feasible. The isse of model pdating is, therefore, an important objective for aeroelasticity research. 6 This paper agments the fltterometer with a simple approach for model pdating. This approach actally bilds on previos research that sed Volterra kernels to analyze flight data and redce ncertainty. 7 The Volterra kernels are now copled with a parameter estimation algorithm to derive modal properties that are more Received 14 Jne 2003; revision received 22 Janary 2004; accepted for pblication 24 Janary Copyright c 2004 by Rick Lind. Pblished by the American Institte of Aeronatics and Astronatics, Inc., with permission. Copies of this paper may be made for personal or internal se, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; inclde the code /05 $10.00 in correspondence with the CCC. Assistant Professor, Department of Mechanical and Aerospace Engineering, 231 Aerospace Bilding; rick@mae.fl.ed. Senior Member AIAA. Visiting Researcher, Department of Mechanical and Aerospace Engineering. 998 indicative of the tre dynamics of the system. These properties are sed to pdate the model and associated ncertainty description. The reslting formlation is an pdated model that can represent the system dynamics with more accracy than the original model. Conseqently, the pdated fltterometer predictions are more accrate and have less conservatism than original fltterometer predictions. The fltterometer is sed to predict fltter speeds for the aerostrctres test wing (ATW). The inclsion of model pdating allows the tool to accont for a phase shift that was not indicated by preflight comptational analysis. The reslting predictions are highly accrate and, ths, indicate a redction in conservatism. Parameter Estimation Formlation The basic problem of parameter estimation is to analyze data and identify coefficients of an eqation of motion. This problem is, in essence, a strctred form of the more general problem of system identification. 8 Parameter estimation has been especially sccessfl when applied to rigid-body eqations of motion for an aircraft 9 ;however, it has had only limited sccess when applied to the flexible-body eqations of motion associated with aeroelasticity. The difficlty with direct application of parameter estimation for aeroelasticity arises from both the data and the eqations of motion. Flight systems have inherent difficlties generating aeroelastic data becase of isses sch as poor excitation at high freqencies, poor measrement of excitation force, and poor observability in dense modal spaces. Frthermore, the eqations of motion contain terms, sch as lags, that have only small contribtions to the response at most flight conditions. The problem is particlarly difficlt when an attempt is made to identify nsteady aerodynamics separately from strctral dynamics. Ths, the estimation of all parameters in the model from flight data is qestionable. This paper adopts a simple approach to parameter estimation that considers the aeroelastic dynamics. The concept is not to estimate specific parameters in either the strctral dynamics or nsteady aerodynamics; rather, estimates are compted for the combined aeroelastic dynamics. The correct parameters in the separated dynamics are obviosly of interest, bt they are not necessarily reqired for the comptation of fltter. The prediction of aeroelastic instability can be accomplished by the se of the correct parameters for the copled dynamics. The general formlation of the approach ses a state-space representation of the model. This representation ses two sets of matrices, {A, B, C, D}, and {Â, ˆB, Ĉ, ˆD},asthe realization of the theoretical model. This model, P, relates the predicted vale of a measrement

2 LIND AND MORTAGUA 999 Y in response to excitation given as : Y = P (1) [ ] A +  B + ˆB = C + Ĉ D + ˆD (2) Assme flight data have been recorded as the measrements of the aircraft in response to the excitation. A model P can be identified that relates these actal measrements Ȳ to the excitation by the se of another state-space realization Ȳ = P (3) [Ā ] B = C D (4) The objective of parameter estimation, as described in this paper, is to choose optimal vales of {Â, ˆB, Ĉ, ˆD} so that the predicted vale of Y is similar to Ȳ. This application is somewhat different from traditional applications of parameter estimation that attempt to identify specific parameters. In a sense, the approach discssed here is a type of model pdating that attempts to identify errors in specific parameters. Modal Parameter Estimation The aeroelastic system can always be written in modal form as a set of one-state convergences and two-state modes. The one-state convergences are associated with the lag terms, and these will not be pdated becase of observability difficlties. 10 The two-state modes are sally easy to observe, and so the parameter estimation will concentrate on these modal dynamics. In this case, the two-state modes are neither pre strctral or aerodynamic modes becase they actally represent a model of the copled aeroelastic system. Consider the model, inclding theoretical dynamics and pdate matrices, formlated in this specific modal form. In this formlation, R represents the real part of the eigenvales and I represents the imaginary part of the eigenvales associated with modal parameters: R + ˆR I + Î B 1 Y = I Î R + ˆR B 2 (5) = C 1 + Ĉ 1 C 2 + Ĉ 2 D + ˆD ( { D + ˆD + [C 1 + Ĉ 1 C 2 + Ĉ 2 ] ( [ ]) 1 [ ]}) R + ˆR I + Î B1 si I Î R + ˆR B 2 (6) [ = D + ˆD + (C 1 B 1 + C 2 B 2 + Ĉ 1 B 1 + Ĉ 2 B 2 )(s R ˆR) (s R ˆR) 2 + (I + Î ) 2 (7) + (C ] 1 B 2 C 2 B 1 + Ĉ 1 B 2 Ĉ 2 B 1 )(I + Î ) (s R ˆR) 2 + (I + Î ) 2 (8) Similarly, consider the identified model in this modal form: R Ī B 1 Ȳ = Ī R B 2 C 1 C 2 D { ( [ ]) 1 [ R Ī ]} B 1 = D + [ C 1 C 2 ] si Ī R B 2 [ s R = D + ( C 1 B 1 + C 2 B 2 ) (s R) 2 + Ī 2 ] Ī + ( C 1 B 2 C 2 B 1 ) (9) (s R) 2 + Ī 2 The se of this modal form allows direct formlation of the pdates needed to the theoretical model. Actally, the vales of the pdates can be chosen to match the models exactly and, conseqently, ensre that Y = Ȳ so that the predicted data match the measred data. ˆR = R R Î = Ī I ˆD = D D Ĉ 1 = ( C 1 B 1 + C 2 B 2 C 1 B 1 )B 1 + ( C 1 B 2 C 2 B 1 C 1 B 2 )B 2 B B2 2 Ĉ 2 = ( C 2 B 1 C 1 B 2 C 2 B 1 )B 1 + ( C 1 B 1 + C 2 B 2 C 2 B 2 )B 2 B B2 2 (10) Model Updating The fltterometer is a tool to be sed dring flight testing for envelope expansion. Ths, the isse of parameter estimation for model pdating mst be addressed in that context. The application of parameter estimation for model pdating is considered for the aircraft at the ith test point dring an envelope expansion. The procedre at the ith test point is to record flight data and identify a state-space model P that relates the measrements and excitations. The theoretical model P is compared to this identified model, and the optimal vales of modal estimates are compted as { ˆR i, Î i, ˆD i, Ĉ1 i, Ĉi 2 }. The objective is to find the vales of {Â, ˆB, Ĉ, ˆD} for model pdating. Most importantly, the models P and P are expressed in the modal form described earlier, so that the model pdating is straightforward. The easier, and most direct, approach to model pdating is to choose the model pdates sch that the theoretical model exactly matches the model identified from flight data. Correspondingly, the model pdates are chosen as the optimal vales from the modal parameter estimation made with the ith data set. Note that ˆB can be set to nll becase a single pdate to the observability matrix, given as Ĉ, issfficient to accont for the effects of errors in both inpt and otpt: [ ] ˆR i Î i  = (11) Î i ˆR i [ ] 0 ˆB = 0 Ĉ = [ ] Ĉ i 1 Ĉ i 2 (12) (13) ˆD = ˆD i (14) Sch an approach has safety implications that mst be considered for flight testing. Essentially, this simple approach completely replaces the theoretical dynamics with the identified dynamics. The safety concerns arise given that the identified model may have significant errors becase of properties associated with the flight data. Obviosly sch an erroneos model shold not be sed for fltter prediction, and so the model pdating mst try to avoid this sitation. A reasonable approach for model pdating can be chosen that ses the parameter estimates while still retaining some level of safety. The concept wold be to pdate the model to correct errors that show some consistency or pattern as the envelope is expanded. Consider a possible sitation for the natral freqency of a mode. The theoretical model shows an error of 1.0 Hz at a test point, bt

3 1000 LIND AND MORTAGUA shows an error of -1.0 Hz at another test point. The correct pdate to the model is not clear. The model may indeed have an error that is strongly dependent on the flight condition or, perhaps, the variations in the data may indicate random errors that are not realistic. Conversely, consider a different sitation. The theoretical model shows an error of 1.0 Hz at a test point and shows an error of 1.1 Hz at another test point. The estimate of this error may be incorrect at each test point; however, the consistency of the error indicates the model probably has an inherent error of roghly 1.0 Hz in the natral freqency. The approach for model pdating sed in this paper is to adopt a scheme based on rnning averages of errors. The errors at each test point, indicated by the optimal vales of modal parameter estimation, are accmlated in a vector. The actal pdates applied to the model are then the average vales of these vectors from the first to the ith test point. { [ ] [ ]} ˆR Â = mean 1 Î 1 ˆR i Î i,..., (15) Î 1 ˆR 1 Î i ˆR i Ĉ = mean {[ Ĉ 1 1 ˆB = [ ] 0 0 Ĉ 1 2],..., [Ĉi 1 Ĉ i 2 ]} (16) (17) ˆD = mean{ ˆD 1,..., ˆD i } (18) In this way, the pdating will only reflect consistent errors. Random errors, which are assmed to have an average near zero, will not be reflected in the model pdate. Consistent errors will have a nonzero average that will be reflected in the model pdate. Also, smoothly time-varying errors will be reflected in the averages, and, ths, these errors will reslt in model pdates. The safety level is not overly sacrificed with this approach becase the model is only pdated by the se of errors whose consistency provides confidence that the pdate is indeed valid. This approach may seem simplistic in comparison to the advanced techniqes being investigated for the general problem of parameter estimation; however, the simplicity presents distinct advantages for this application. As mentioned earlier, the direct estimation of strctral and aerodynamic parameters from flight data is not feasible, bt the simplistic estimation of aeroelastic modal parameters is straightforward. Also as mentioned earlier, the direct tilization of errors estimated at a single test point may lead to incorrect predictions of fltter speeds, bt a simplistic average will generate pdates that reflect consistent errors withot sacrificing safety. Fltterometer Model Formlation The initial process in the fltterometer is the formlation of a model. This model mst be a state-space description of the aeroelastic dynamics. In particlar, the model mst be parametrized arond flight condition and associated ncertainties. The model pdating, both for theoretical dynamics and ncertainties, is performed by comparison of theoretical and measred transfer fnctions at a flight condition. Ths, the theoretical model mst be formlated at a reference condition matching the test point. This formlation is easily done by the se of one of several methods, inclding the approaches sed standard software packages like NASTRAN and ZAERO; however, the model mst retain its parameterization arond flight condition and ncertainty. The procedre sed for model formlation in the fltterometer is actally straightforward. The preflight theoretical dynamics are generated at a reference condition V 0 with standard approaches. A state-space model P(V 0 ) is generated and parametrized arond flight condition δ V and ncertainty. This parameterization makes it trivial to formlate the model at any reference condition. Ths, the model associated with the velocity V i at the ith test point is generated by simple replacement of δ V with δ V + V i V 0. The reslting model can be expressed as P(V i ) with feedback signals relating these dynamics to δ V and. Also, the model needs to be expressed in modal form to facilitate the parameter estimation. This modal form is generated by standard rotines for state transformations. 10 The final model describes the aeroelastic dynamics at V i with 2 2 and 1 1 modal blocks that are parametrized arond flight condition and ncertainty. Data Filtering The next process in the fltterometer is the filtering of flight data. This step is vital becase aeroelastic flight data are typically of very poor qality. In general, these data do not satisfy linearity reqirements de to nonlinearities in the aircraft dynamics and high levels of noise in the measrements. The process of model pdating reqires data that accrately reflect the linear dynamics of the aeroelastic system. Ths, the data mst be filtered to provide the reqired information. The filtering process sed in this paper is based on Volterra modeling. The concept of Volterra modeling is to represent data as a set of signals that describe different orders. In other words, the data can be described by a linear component, pls a qadratic component, pls higher-order components. The dynamics of each component are given by a Volterra kernel. 11,12 Volterra kernels are identified whose inpt otpt characteristics match the flight data. 7 These kernels are identified by the se of a wavelet-based approach that makes se of mltiresoltion decomposition. 13 Volterra kernels have been sed to represent many aeroelastic systems and demonstrate that the linear, qadratic, and third-order terms often provide sfficient accracy. 14,15 Ths, this paper will only consider the identification of low-order terms to represent the aeroelastic dynamics. The data are filtered by the extraction of the linear component. This component is fond by comption of optimal vales of the Volterra kernels from the flight data. The first-order kernel represents the linear dynamics. The desired linear component of the data is, ths, generated by a simple convoltion of the excitation signal with the kernel. This type of filtering is only one approach from a mltitde of possibilities. Any filtering can be sed; however, previos experience has shown that Volterra-based filtering works particlarly well. Standard approaches, sch as low-pass filtering, or advanced approaches, sch as time-freqency featre filtering, may be sed, bt they are not necessarily optimal for this type of application. The Volterra filtering is appropriate becase it inherently considers transfer fnctions for data representation, which are exactly what is reqired by the fltterometer. Model Updating The process of model pdating makes direct se of the prodcts of model formlation and data filtering. Essentially, the parameters of the formlated model are altered based on information from the filtered data. An important isse to note is that model pdating actally refers to both the nominal dynamics and the associated ncertainty. These elements are both part of the ncertain model sed for µ-method analysis, and so they shold both be pdated. The first step of model pdating is to alter the theoretical dynamics. This alteration is done by pertrbation of the state-space matrices by the se of the modal parameter pdates. More precisely, the pdates are the rnning averages of the differences between the theoretical and estimated dynamics at each test point. The altered dynamics are now considered the nominal dynamics. The other step of model pdating is to compte the ncertainty associated with the nominal dynamics. This step is done via the test of a condition for model validation. Essentially, the effect of ncertainty is to allow the analytical dynamics to vary within a bonded range. The model pdating occrs by an increase in the size of ncertainty ntil the flight data lie within the range of dynamics. The actal data sed for this process may vary sch that consideration of a single test point will redce conservatism, bt also safety margins, whereas consideration of many test points will

4 LIND AND MORTAGUA 1001 increase conservatism, bt also safety margins, for the reslting fltter predictions. 1 In this way, the data filtered via Volterra kernels are actally sed mltiple times. This data are sed for parameter estimation becase they contain only the desired linear component. These data are also sed for model validation becase they relate a realistic measre of linear ncertainty. 7 Essentially, the filtered data contain only information abot the linear dynamics, and so they are optimal for se for parameter estimation and model validation of linear models. Robst Fltter Speeds The final process in the fltterometer is the comptation of a robst fltter speed. This process is a direct application of µ-method analysis to the model with optimal parameter and ncertainty pdates. The reslting analysis is a robst stability comptation that considers the worst-case fltter speeds of the model with respect to the ncertainty. Flight Test ATW The ATW was developed at NASA Dryden Flight Research Center. 16 This testbed was specifically designed for testing methods to predict the onset of fltter. The ATW was essentially a wing and boom assembly that was flown by sing an F-15 aircraft and associated flight-test fixtre. The ATW was monted horizontally to the fixtre and the reslting system attached to the ndercarriage of the F-15, as shown in Fig. 1. The ATW was flown to several test points dring envelope expansion. These points inclded altitdes of (6096 m); (4572 m); and ft (3048 m) and Mach nmbers of 0.50, 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, and The final flight ended with destrction of the ATW de to fltter at flight conditions of 10,000 ft and 0.83 for Mach nmber. Eqivalently, the flight conditions at which fltter occred correspond to 460 kn of eqivalent airspeed (KEAS). Fig. 2 mode, Measred modal dampings for bending mode, + and torsion. Envelope Expansion Estimates of modal parameters were compted from the flight data. At each test point, a standard freqency-domain method of system identification was sed to formlate a model whose magnitde and phase characteristics were similar to the transfer fnction. The modal parameters of that model were then extracted and sed as representative of the ATW parameters. The modal dampings that were extracted at each test point are given in Fig. 2. The fltter instability affecting the bending mode is clearly evident in the data trends. The modal freqencies for the ATW are given in Fig. 3. The actal mechanism of fltter is somewhat difficlt to discern from the flight data. The dampings in Fig. 2 seem to indicate the mechanism is a classical type of fltter sch that one mode is becoming less stable, whereas the other mode is becoming more stable. Conversely, the freqencies in Fig. 3 do not appear to coalesce, as expected for classical fltter, ntil a possible coalescence at the airspeed very close to the onset of fltter. Also note in Figs. 2 and 3 is that only 15 estimates are shown, even thogh the flights operated at 21 test points. The responses Fig. 1 Monting of the ATW. Fig. 3 mode, Measred modal freqencies for bending mode, + and torsion. from several of the test points were nable to present sfficient information abot the bending mode. The response levels were qite low at these test points, and so accrate modal estimates cold not be obtained. The reason for the poor data qality at some points was nclear bt possibilities inclded high levels of trblence and noise or nexplained high damping. Data Filtering The flight data are filtered to extract the linear component by the se of Volterra kernels. Previos analysis of the ATW has indicated that its dynamics are relatively linear; therefore, only first- and second-order terms in the Volterra expansion were sed to reflect the linear and qadratic dynamics. The actal comptation of kernels reqired several parameters, sch as memory length and resoltion, 7 to be chosen. Several sets of parameters were tested, and the reslts indicated a range within which the reslts were fairly consistent and reasonable. Conseqently, the kernels were identified with a 1-s dration and 256 points. These parameters indicate the kernel decayed within 1-s and comprised 256 wavelet coefficients. Figre 4 shows a measrement from the flight data that is representative of the measrements taken throghot the flight. These data show the response to a chirp excitation for the accelerometer located in the boom near the trailing-edge endcap. This chirp is generated by variation of the sinsoids, commanded to the excitation mechanism on the system, between 5 and 35 Hz (Ref. 16). The corresponding simlated data, as compted by the first-order Volterra kernel, are shown in Fig. 5. Clearly, the data filtering has redced a significant portion of the energy in the response. Some of this energy was random noise, whereas some of this energy reslted from excitation of nonlinearities. Either way, the second-order

5 1002 LIND AND MORTAGUA Fig. 4 Response of the trailing-edge boom accelerometer to a chirp excitation. Fig. 6 Error, + and rnning average of the error, in modal damping of the theoretical model for bending mode and error, and rnning average of the error, for torsion mode. Fig. 5 Simlated response to chirp inpt from first-order Volterra kernel. kernel was negligible, and so it is assmed that nonlinearities do not play a prevalent role in the dynamics and, ths, can be ignored for in the prediction of the onset of fltter. Clearly, the data filtering has redced a significant portion of the noise. Some of this noise may have actally reslted from nonlinearities that were not represented by the first-order kernel; however, the second-order kernel is negligible, and so nonlinearities can be ignored. Model Updating The filtered data were sed for model pdating. This pdating altered the theoretical dynamics to match properties estimated from the data. Specifically, the damping and natral freqencies, along with observability, are pdated for each mode. The pdates to the damping for each mode are of particlar interest becase damping is traditionally sed to indicate fltter. The differences, and rnning average of the differences, between the theoretical and measred damping are given in Fig. 6 for each test point. The errors in damping for the bending and torsion modes shown in Fig. 6 seem to show different trends. The error in damping for the bending mode is small at the low-speed test points bt generally increases as the envelope is expanded. Conversely, the error in damping for the torsion mode is small at the low-speed test points and actally decreases to even smaller levels as the envelope is expanded. These trends sggest that the theoretical model of the torsion mode is more accrate, at least in terms of damping, than the theoretical model of the bending mode. Sch behavior is extremely important for fltter prediction. The onset of fltter reslts in an nstable bending mode that is characterized by zero damping. The magnitde of damping for the bending, Fig. 7 Error, + and rnning average of the error, in natral freqency of the theoretical model for bending mode and error, and rnning average of the error, for torsion mode. as seen in Fig. 2, is decreasing as the envelope is expanded. Ths, the increasing error indicates that the theoretical model does not correctly predict the flight conditions at which fltter occrs and the bending mode becomes nstable. Also, the rnning averages in Fig. 6 show the pdates sed to alter the theoretical model. The rnning average is seen to be an excellent indicator of the error in the torsion mode and a reasonable indicator of the error in the bending mode. In particlar, the rnning average represents the error well for low- and high-speed test points and smooths ot the effects of the inconsistently large errors at some of the test points. The pdates to the natral freqency for each mode are of related interest becase natral freqency provides additional information that may be sed to predict fltter. The differences between the theoretical and measred natral freqencies for each mode are shown in Fig. 7 along with the corresponding rnning averages. The errors in natral freqency, like the errors in damping, show differing trends between the torsion and bending modes; however, the trends in Figs. 6 and 7 are actally opposite. Specifically, the bending mode shows an increasing error in damping and a decreasing error in natral freqency as the envelope is expanded. Conversely, the torsion mode shows a decreasing error in damping and an increasing error in natral freqency as the envelope is expanded. These trends are qite interesting and sggest freqency and damping were modeled with very different levels of accracy. The pdates to observability of each mode are shown in Fig. 8. The differing trends between errors in bending and torsion dynamics is frther exemplified by these reslts.

6 LIND AND MORTAGUA 1003 a) b) Fig. 8 Error, and average error, in observation of first state and error, + and average error, in observation of second state of the theoretical model for a) bending mode and b) torsion mode. Fig. 9 Uncertainty in pdated model for strctral stiffness and damping, ; aerodynamic forces, *; excitation, +; and measrement,. Fig. 10 Fltter speeds predicted by damping extrapolation, +; original fltterometer, *; and pdated fltterometer,. Figre 8 shows another significant difference between the modeling of the bending and torsion modes. The error in observability of the two states associated with the bending mode are changing signs. Conversely, the error in observability of the two states associated with the torsion mode remain the same sign. Sch a difference in sign for the bending mode implies a change of phase. Note the pdates to observability. The observability is directly related to mode shape and, conseqently, the fltter mechanism. The data in Fig. 8 indicate that the theoretical predictions of mode shape are distinctly different than the data indicate. Sch an error is probably the reason that comptational analysis was nable to predict the onset of fltter to within 50 KEAS. The ATW experienced a phase change in the aeroelastic dynamics that was not predicted by preflight modeling. Uncertainty Estimation The theoretical model is generated with an ncertainty description. This description allows for variation in the parameters of the strctral stiffness and strctral damping matrices. Also, the description incldes terms for magnitde and phase variations of the aerodynamic forces. Finally, the ncertainty description acconts for errors in magnitde and phase of the excitation energy and data measrements de to nmeasred qantities sch as wind gsts and noise. The ncertainty levels reqired to ensre the pdated models are not invalidated by the flight data are shown in Fig. 9. These ncertainty levels are compted atomatically by the fltterometer. The range of transfer fnctions from the model with these levels of ncertainty are sfficient to bond the transfer fnctions of the flight data. The maximm amont of ncertainty needed at any test point was 9.5% for the strctral stiffness and damping, 149% for the aerodynamic forces, 4.7% for the excitation, and 1.5% for the measrement. 16 The pdated models did not reqire mch ncertainty for most parameters; however, the ncertainty associated with the aerodynamic forces was qite large. This sitation was acceptable becase the fltter speeds were considerably more sensitive to strctral parameters than to the variations in aerodynamic forces. Ths, these ncertainty levels do not correspond to excessive conservatism in the model. Note that these ncertainty levels correspond to the model with pdating of the average pertrbations. If the models are pdated with the exact pertrbation needed to match theoretical and experimental transfer fnctions, then no ncertainty is needed for the model, bt the reslts wold be accepted with less confidence. As stated earlier, the averaging approach for model pdating redces local effects at a single test and, ths, increases confidence in the reslts by identifying errors trends at many test points. Fltter Prediction Fltter speeds are predicted for the ATW with several techniqes. One set of predictions reslts from standard extrapolation of damping parameters. Another set of predictions reslts from the original implementation of the fltterometer that only pdates ncertainty descriptions. The final set of predictions reslts from the new implementation of the fltterometer that ses Volterra modeling of the data and incldes modal parameter estimation. These predictions are shown in Fig. 10.

7 1004 LIND AND MORTAGUA The predictions from the new implementation of the fltterometer are clearly sperior to either of the other sets of predictions. Specifically, the fltterometer now estimates a fltter speed that is within 10 KEAS of the tre fltter speed when flight data from any test point of the envelope expansion are sed. The fltterometer with model pdating has clearly removed most of the conservatism in the predictions that were associated with the original fltterometer. The predictions are now able to accont for the flight data by estimation of parameters for the model for which less ncertainty is reqired to validate the flight data. The reslt of this pdating is a fltterometer that comptes accrate predictions of fltter speed. Conversely, the fltterometer with model pdating has not removed too mch conservatism or cased the predictions to be overly optimistic. The predictions in Fig. 10 show that the predictions remain very close to the tre fltter speed. The inclsion of model pdating has increased the accracy of the fltterometer withot sacrificing the safety provided by considering a worst-case fltter mechanism. The extensions to the fltterometer presented in this paper make the tool especially valable in comparison to the traditional method of extrapolation of damping trends. The predictions from damping are seen in Fig. 10 to be widely scattered and not very accrate ntil the envelope expansion has reached test points above 350 KEAS. The fltterometer now accrately predicts the fltter speed at both low- and high-speed test points. Conclsions This paper has introdced an approach to improve the accracy of the fltter speeds predicted by the fltterometer. Sch accracy reslts from agmention of the tool to inclde a scheme for model pdating based on modal parameter estimation by the se of Volterra kernels. The pdates are optimal in the sense that they reflect consistent errors between the theoretical dynamics and the linear dynamics associated with the first-order Volterra kernels that represent the flight data. Fltter speeds are compted for the ATW by the se of the new version of the fltterometer. The model pdates are able to correct a phase shift that was not predicted by comptational analysis. Conseqently, the fltterometer predicted highly accrate fltter speeds. References 1 Lind, R., and Brenner, M., Fltterometer: An On-Line Tool to Predict Robst Fltter Margins, Jornal of Aircraft, Vol. 37, No. 6, 2000, pp Lind, R., and Brenner, M., Robst Aeroservoelastic Stability Analysis, Springer-Verlag, London, 1999, pp Lind, R., Flight Testing with the Fltterometer, Jornal of Aircraft, Vol. 40, No. 3, 2003, pp Borglnd, D., Robst Aeroelastic Stability Analysis Considering Freqency-Domain Aerodynamic Uncertainty, Jornal of Aircraft, Vol. 40, No. 1, 2003, pp Beker, B., Bennani, S., van Stavern, J. W. and Meijer, J. J., Robst Fltter Analysis of an F-16A/B in Heavy Store Configration, AAIA Paper , Ag Brenner, M.J., Aeroservoelastic Model Uncertainty Bond Estimation from Flight Data, Jornal of Gidance, Control, and Dynamics, Vol. 25, No. 4, 2002, pp Prazenica, R. J., Lind, R., and Krdila, A. J., Uncertainty Estimation from Volterra Kernels for Robst Fltter Analysis, Jornal of Gidance, Control, and Dynamics, Vol. 26, No. 2, 2003, pp Ljng, L., System Identification, Prentice Hall, Englewood Cliffs, NJ, 1987, pp Iliff, K. W., Parameter Estimation for Flight Vehicles, Jornal of Gidance, Control, and Dynamics, Vol. 12, No. 5, 1989, pp Kailath, T., Linear Systems, Prentice Hall, Englewood Cliffs, NJ, 1980, pp Rgh, W. J., Nonlinear System Theory: The Volterra-Wiener Approach, Wiley, New York, 1980, pp Schetzen, M., The Volterra and Wiener Theories of Nonlinear Systems, Wiley, New York, 1980, pp Krdila, A. J., Prazenica, R. J., Rediniotis, O., and Strganac, T., Mltiresoltion Methods for Redced-Order Models for Dynamical Systems, Jornal of Gidance, Control, and Dynamics, Vol. 24, No. 2, 2001, pp Marzocca, P., Libresc, L., and Silva, W. A., Aeroelastic Response of Nonlinear Wing Sections Using a Fnctional Series Techniqe, AIAA Jornal, Vol. 40, No. 5, 2002, pp Marzocca, P., Libresc, L., and Silva, W. A., Volterra Series Approach for Nonlinear Aeroelastic Response of 2-D Lifting Srfaces, AAIA Paper , April Lind, R., Voracek, D. F., Trax, R., Doyle, T., Potter, S., and Brenner, M., A Flight Test to Demonstrate Fltter and Evalate the Fltterometer, Aeronatical Jornal, Vol. 107, No. 1076, 2003, pp