American Institute of Aeronautics and Astronautics

System Identification of Large Flexible Transport Aircraft

Colin R. Theodore* and Christina M. Ivler†

San Jose State University Foundation, Moffett Field, CA, 94035

Mark B. Tischler‡

US Army Aeroflightdynamics Directorate (AMRDEC), Moffett Field, CA, 94035

Edmund J. Field§

The Boeing Company, Huntington Beach, CA, 92647

and

Randall L. Neville¶ and Heather P. Ross¶

The Boeing Company, Seattle, WA, 98124

This paper presents results and lessons learned from a proof-of-concept study that usedfrequency-domain system identification to extract models of the lateral-directional flightdynamics of a large transport aircraft. These identified models are intended to be used forsimulation model validation and upgrade, and for flight control law development andvalidation. Both of these applications require models that are accurate over a broadfrequency range. For large transport aircraft identifying such models can be challenginggiven that: 1) the lower-frequency rigid-body dynamics modes are often lightly damped anddifficult to identify accurately, and 2) the flexibility of the aircraft structure can significantlyaffect the response of the aircraft in the frequency range that is important for flight controland therefore must be accounted for in the dynamics models. A model structure isdeveloped that is used to identify simultaneously the rigid-body aircraft response dynamicsand the structural flexibility modes of the aircraft from flight data. Comparisons betweenthe model responses and time history data collected during flight-testing show that themodels are highly accurate in predicting both the rigid-body and structural responses. Thepaper presents details of the flight test procedures, frequency response and dynamics modelidentification, and the coupled rigid-body / structural flexibility model structure.

I. IntroductionHIS paper presents results and lessons learned of a proof-of-concept study to identify flight dynamics models ofa large fixed-wing transport aircraft from flight test data. The intended applications of these identified models

are simulation model validation and update, and flight control law validation and development. Both of theseapplications require flight dynamics models that are accurate over a broad frequency range. For large transportaircraft, identifying such models from flight test data can be challenging given that the lower-frequency rigid-bodydynamics modes are often lightly damped and can be difficult to identify accurately.

* Senior Research Engineer, M/S T12B-2; currently Research Engineer, Aeromechanics Branch, Flight VehicleResearch and Technology Division, M/S 243-11, NASA Ames Research Center, AIAA Senior Member.† Senior Research Engineer, M/S T12B-2; currently Senior Research Engineer, Aeroflightdynamics Directorate USArmy Aviation and Missile Research, Development and Engineering Center, M/S T12-B.‡ Flight Control Group Lead, Aeroflightdynamics Directorate, US Army Aviation and Missile Research,Development and Engineering Center, MS T12B-2, AIAA Associate Fellow.§ Associate Technical Fellow, Stability, Control & Flying Qualities Group, 5301 Bolsa Avenue, MC H45N-E407,AIAA Associate Fellow.¶ Experimental Test Pilot, Boeing Commercial Airplanes, PO Box 3707, MC 14-HA.

T

AIAA Atmospheric Flight Mechanics Conference and Exhibit18 - 21 August 2008, Honolulu, Hawaii

AIAA 2008-6894

Copyright © 2008 by The Boeing Company. All rights reserved. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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Another challenge comes from the structural flexibility of the aircraft that can have a significant effect on theaircraft response in the frequency range that is important for flight control application and therefore must beaccounted for in the identified dynamics models. Also as transport aircraft become larger the frequencies of thestructural modes generally decrease resulting in flexibility of the fuselage and wing structures having a greaterinfluence on the aircraft dynamic response. In addition, the desire to improve overall aircraft performance drives thedesign of lighter and more optimized structures that often use composites for weight reduction. These designs canlead to more flexible structures with lower structural frequencies, increasing their impact on the aircraft dynamicresponse. Finally, structural modes typically have low damping and their effects can be seen at frequenciessignificantly below the natural frequencies of the modes themselves.

At the same time that structural frequencies are decreasing, the requirements for increased performance and gustalleviation are pushing the flight control frequencies higher. As the flight control frequencies extend closer to thestructural frequencies, it becomes increasingly more important to account for the fuselage and wing structuralflexibility in the design of the flight control system, which requires models that include the structural effects.

The first section of this paper summarizes the system identification flight-testing performed in this study,including pre-flight training, matrix of system identification maneuvers, instrumentation, and lessons learned fromthe flight test. Next is a discussion of the frequency response identification from the data collected during the flighttest. The particular frequency-domain system identification method used is implemented in the CIFER©1

(Comprehensive Identification from FrEquency Responses) software tool. This tool has been used extensively inthe aerospace industry to identify dynamics models from flight data for both fixed-wing, rotary-wing1 andunmanned aerial vehicles2, and for flying qualities applications3. The frequency response identification is followedby a description of the state-space model structure used to identify simultaneously the lateral-directional rigid-bodyand structural dynamics. This model structure developed for this study is based on the work of Wasnak et. al.4,Meirovitch and Tuzcu5 and McLean6. Results from the model identification and model verification for a large wide-body transport aircraft are presented. Finally some conclusions and lessons learned from this study are presented.

II. System Identification Flight TestsThe flight test data used for this study was collected during a single flight on a large wide-body transport aircraft

in a nominal low speed cruise flight condition (clean configuration with landing gear and flaps stowed). Figure 1shows a simplified block diagram of the bare-airframe control input paths from the pilot controls to the actualsurface measurements and aircraft response.

The test matrix for the system identification flight included a series of manual frequency sweeps and doublets ofthe pilot wheel and rudder pedal controls, and automated sweeps of the individual aileron, flaperon and ruddercontrol surfaces. Table 1 lists the maneuvers flown and the number of repeats of each maneuver. For the automatedfrequency sweeps, two repeats were flown for each of the maneuvers to ensure that there were sufficient data forfrequency-domain identification. Three repeat maneuvers were flown for the piloted frequency sweep maneuvers toensure there were a minimum of two good sweeps for each of the piloted control inputs.

Example time histories of the different types of maneuver are shown in Figure 2. Figure 2 (a) shows the wheelposition time history for a piloted wheel frequency sweep. The frequency range for the piloted sweeps wasnominally about 0.05 to 2.0 Hz. Figure 2 (b) shows an automated sweep input directly into the aileron commandchannel (refer to Figure 1). The frequency range for the automated sweeps was selected to capture the low-frequency rigid-body dynamics and to capture the important wing and fuselage structural bending modes. Figure 2(c) shows an example piloted wheel doublet that was used for time-domain model verification. Two repeat doublets

Figure 1. Simplified block diagram showing lateral-directional control input paths to the bare-airframe.

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were flown in each axis, one with an initial positive control input, and the other with an initial negative controlinput.

Table 2 lists the instrumentation channels that were available for use in the model identification. Figure 3 showsan example of the wing-tip differential z-accelerometer measurements (∆az−wt

) that proved to be adequate to

identify the characteristics of the first and second wing asymmetric bending modes, which are clearly visible in thetime history.

A. Testing ConstraintsThe two biggest constraints on the testing were the need to use sufficiently large inputs to generate a response

with sufficiently high signal to noise ratio, and the requirement to limit the input magnitude at certain frequencies(basically the lightly damped rigid-body and structural mode frequencies) in order to avoid excessive structuralloads induced by the continually reversing control inputs. Naturally, these two requirements conflict with oneanother.

In order to constrain the structural loads, limits were placed on the permissible oscillatory control inputmagnitudes as a function of frequency. For the automated sweeps, with their pre-defined frequency ranges, it waseasy to define the largest input magnitudes that remain within these limits. However, for the manual sweeps real-time monitoring was required. Test engineers in the main cabin of the aircraft monitored the input magnitudes onstrip charts, and provided real-time feedback to the pilots. The pre-defined control input magnitude limits allowed

Table 1. Test matrix for lateral-directional dynamicsidentification.

ManeuversNumber of Repeat

ManeuversManual wheel sweeps 3

Automated wheel sweeps 2Automated aileron sweeps 2

Automated flaperon sweeps 2Manual wheel doublets 2Manual pedal sweeps 3

Automated rudder sweeps 2Manual pedal doublets 2

Wing-tip Differential Accelerometer

Time

Figure 3. Wing-tip differential accelerometermeasurement from an automated aileron sweep.

(a). Pilot Wheel (sweep)

Time

(b). Aileron Deflection

Time

(c). Pilot Wheel (doublet)

Time

Figure 2. Example control input time histories for:(a) piloted wheel frequency sweep, (b) automated

aileron sweep, and (c) piloted wheel doublet.

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larger inputs at lower frequencies, reducing at higherfrequencies close to the structural modes. The testengineers provided feedback to the pilots of their currentcontrol magnitudes and the maximum magnitudespermitted at that frequency. As they approached thehigher frequencies the test engineers directed the pilotsto reduce their input magnitudes. This process resultedin high quality data with sufficient signal to noise at allfrequencies.

B. TrainingThe day before the system identification flight a

training session was conducted for the test crew. Due toother commitments, one of the two pilots was unable toattend. The session commenced with a briefing thatstressed (from Tischler and Remple1):

- The sweeps should start and end in trim.- The need for a smooth progression from low

to high frequency.- Appropriate use of on-axis and off-axis

controls to remain close to the trim condition.- The use of timing and other call-outs from the test engineers.- That constant amplitude, exact sinusoidal shape, exact frequency progression and exact repeatability

were not important.

The briefing was followed by a session in a fixed base simulator where test techniques and procedures werepracticed and refined. Core to this was establishing the most beneficial callouts to guide the pilot smoothly throughthe frequency range in the target sweep duration. Also stressed was the importance of smoothness of the inputs,rather than in the aircraft response.

C. FlightThe pre-flight briefing reviewed the test procedures practiced the previous day in the simulator, and introduced

them to those who were unable to attend that session. The various maneuvers had been carefully sequenced toensure that those that would be analyzed as a data set would be flown in a continuous block.

Generally one pilot would fly a block of maneuvers to maintain consistency with that particular data set. Thepilot who had attended the simulator session flew the maneuvers first. The second pilot observed the technique andflew them during a subsequent block. Based on the coaching by the first pilot and observation, the second pilot wasquickly able to absorb the concept and readily duplicate the maneuvers very successfully. The session in thesimulator the previous day had resulted in a robust technique.

The test engineers in the cabin provided timing guidance for the sweeps to aid in a gradual progression offrequencies. Engineers monitoring strip charts provided feedback on input and response magnitudes, frequencyprogression, input shape and trim conditions. Where appropriate real-time guidance to alter input shape, frequency,magnitude and how best to return closer to trim conditions.

The success of the careful definition of the testing, training, and crew coordination was reflected in the highquality data collected and the fact that only one manual sweep had to be repeated out of the total of 22 flown by bothpilots.

III. Frequency Response IdentificationThe frequency-domain system identification for this study was performed using the Comprehensive

Identification from FrEquency Responses (CIFER©) software tool1. CIFER© provides a comprehensive set ofsystem identification tools to obtain dynamics response models from a set of test data. In frequency-domain systemidentification, the extraction and analysis of dynamics models first involves generating a set of non-parametricfrequency responses from the set of time history data and then fitting transfer functions and state-space models tothe frequency responses. The goal of this system identification process is to achieve the best possible fit betweenthe test data and a model that is consistent with the physical knowledge of the vehicle dynamics. The steps involved

Table 2. Instrumentation list for systemidentification.

Measurement Description Symbol

Pilot/automated wheel position δwhl

Pilot/automated pedal position δ ped

Aileron command and surface position δailc , δail

Flaperon command and surface position δ flpc , δ flp

Rudder command and surface position δrudc , δrud

Aircraft angular rates p, q, r

Aircraft attitudes and heading φ, θ, ψAngles of attack and sideslip α, β

C.G. accelerometers ax , ay , az

Wing-tip differential z-accels ∆az−wt

Fuselage differential y-accels ∆ay− fs

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in taking a set of flight data through the system identification procedure to a final verified state-space model are(details can be found in Tischler and Remple1):

1. Frequency response calculation from time history data.2. Multi-input frequency response conditioning to remove the effects of any off-axis control inputs.3. Multi-window averaging of the frequency response for different Fast Fourier Transform window sizes.4. State-space model identification from the set of frequency responses.5. Time domain verification of the state-space model using time-domain doublet data.One application of the frequency responses identified from flight data is for simulation model validation and

improvement by making a direct comparison between the flight test and simulation frequency responses. Anexample is shown in Figure 4 that compares the frequency responses of roll rate due to aileron input from the flighttest data and from a simulation model. For this particular case, the simulation model includes only rigid-bodydynamics and is a good match to the flight data in the region of rigid-body dynamics. At higher frequencies, theaccuracy of the simulation model degrades since the response from flight is increasingly influenced by the wingasymmetric bending dynamics, which are not included in the simulation model. Figure 4 also shows that the effectof the wing flexibility is to increase the magnitude and decrease the phase of the response when compared with thesimulation model (rigid-body only). This increase in magnitude and decrease in phase would have a negativeimpact on the stability margins of the control system. In fact, if the rigid-body simulation model were used todesign the flight control system for this aircraft the predicted stability margins would be higher than those measuredin-flight due directly to structural flexibility.

Figure 5 shows the input- and output-spectra, and frequency responses of sideslip angle output for rudder inputidentified from the piloted and automated rudder pedal frequency sweep data. The automated rudder inputs wereextended to higher frequencies than the piloted rudder pedal inputs, which are reflected in the input spectrum (top ofFig. 5) that shows more input power with the automated inputs than the piloted inputs at high frequencies. At thelower frequency end, the piloted frequency response shows a higher accuracy (as indicated by the coherencefunction) than the automated sweep because the piloted frequency sweeps started at lower frequency.

Figure 5. Comparison between automated andpiloted rudder pedal frequency responses.

Figure 4. Comparison between flight test andsimulation model frequency responses.

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Figure 5 also shows the presence of the Dutch Rollmode as a peak in magnitude and a 180-degree phasechange near the middle of the frequency range of theresponse. The accurate identification of lightly dampedmodes (such as the Dutch Roll mode) is sensitive to thelength of the Fast Fourier Transform (FFT) window usedto extract the frequency response from the flight data.To illustrate this, the Dutch Roll mode characteristics areidentified by fitting transfer function models to thesideslip angle due to rudder input frequency responsesgenerated using different FFT window lengths. Thetransfer function form used to identify the Dutch Rollmode is a 1st/2nd-order model with a time delay7. Thistransfer function takes the form:

βcg

δrud

=Nδ rud s + 1 / Tβ1( )

[ζdr ,ω dr ]e−τ eqs (1)

where ωdr and ζdr and are the Dutch Roll mode frequency and damping, Nδrud is the control derivative and τeq is anequivalent time delay to account for measurement delay and phase lag caused by unmodeled higher-frequencydynamics modes.

Figure 6 shows an example of the β/δrud frequency response along with an identified transfer function of theabove form. The excellent overall fit of the transfer function to the flight data in the rigid-body frequency range,with a good representation of the magnitude peak and the phase change associated with the Dutch Roll mode, is anindication that the transfer function accurately identifies the Dutch Roll mode dynamics.

Table 3. Identified Dutch Roll mode damping andfrequency characteristics as a function of FFT

window size.

FFT WindowSize (seconds)

Dutch Roll ModeDamping Ratio

Dutch Roll ModeFrequency

50 +12.6 % +0.71 %60 +8.6 % +0.34 %70 +6.8 % +0.16 %80 +4.8 % +0.29 %90 +3.7 % -0.03 %

100 +2.9 % +0.14 %110 +1.1 % -0.04 %120 +0.8 % -0.03 %130 -0.3 % -0.06 %140 0% - reference 0% - reference

Figure 7. Dutch Roll mode damping and frequencyvalues for different FFT window sizes.

Figure 6. Comparison between automated andpiloted rudder pedal frequency sweeps.

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Table 3 shows the sensitivity of the identified Dutch Roll mode damping and frequency characteristics to the sizeof the FFT window used to generate the frequency response. The frequency and damping values are normalized bythe identified frequency and damping values for the longest window (140 seconds), which is considered the mostaccurate. Figure 7 displays these values graphically. These results show that for larger FFT windows (above about80-90 seconds) the Dutch Roll mode frequency and damping values converge. For windows shorter than 80seconds, the modal damping is over-predicted, even though the modal frequency is correct. Since the FFT windowcannot be larger than any of the sweep records, the sweep record lengths place an upper-limit on the FFT windowsthat can be used.

These data agree with the rule proposed by Tischler and Remple1 for the minimum FFT window size for theidentification of lightly damped modes:

Twin ≥11

ζ ω(2)

Using this equation, the Dutch Roll mode location for the current aircraft indicates that a minimum FFT windowsize of about 90 seconds is required. The results in Table 3 and Figure 7 show that with a FFT window size of 90seconds or more (highlighted in Figure 7) the frequency is within 0.2% of the final asymptotic value, and thedamping is within 4% of the final asymptotic value for the 140-second FFT window.

IV. State-Space Model IdentificationIt has been shown previously in Figs. 4 and 5 that the effects of structural flexibility can be seen at frequencies

only slightly above the frequencies of the rigid-body modes. Therefore, to represent the aircraft dynamicsaccurately, the model structure must be formulated to include the effects of structural flexibility in addition to thedynamics of the rigid-body. This section describes the formulation of a lateral-directional state-space model thatincludes rigid-body and structural dynamics, and shows results for a model identified from flight data. The modelstructure developed here is based on the work of Wasnak et. al.4, Meirovitch and Tuzcu5 and McLean6.

The general form of the state-space model is:

&x = Fx + Gu (3)

where x is the vector of states, u is the vector of controls, F is the stability matrix, and G is the control matrix.The form of the F and G matrices with structural flexibility effects included is:

(4)

where the ‘Rigid Body Terms’ and ‘Rigid Body Control Derivatives’ are the rigid-body portions of the model, the‘Structural Flexibility Terms’ are the dynamics of the structural modes included in the model, the ‘Structural ModeControl Derivatives’ allow the control deflections to drive the structural modes, and the ‘Aeroelastic CoupingTerms’ and ‘Rigid Body Coupling Terms’ are the coupling between the rigid-body and the structural flexibility ofthe aircraft.

The ‘Aeroelastic Coupling Terms’ allow the structural modes to drive or influence rigid-body dynamics of theaircraft. For the model formulation presented in this paper, it is assumed that the excitation of the aircraft structuralmodes does not change or affect the rigid-body dynamics of the aircraft, which allows the ‘Aeroelastic CouplingTerms’ to be dropped. The excellent response prediction will be shown to justify this assumption.

The ‘Rigid Body Coupling Terms’ allow the rigid-body response of the aircraft to drive or excite the structuralmodes. These terms are retained for this model formulation and allow the structural modes to be excited by the

Rigid BodyControl

Derivatives

StructuralMode Control

Derivatives

G =

Rigid BodyTerms

AeroelasticCoupling Terms

StructuralFlexibility Terms

Rigid BodyCoupling Terms

F =

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aircraft rigid-body response to disturbances (including gusts and turbulence) in addition to control surfacedeflections. Without these terms, wind gusts and turbulence would excite the model rigid-body response, but not themodel structural response.

The lateral-directional state-space model is represented in the form:

x =

′β′p

′r′φ

η11 = &λ1

η12 = λ1

η21 = &λ2

η22 = λ2

u = δ ail δ flp δ rud (5)

where β is the sideslip angle, p is the roll rate, r is the yaw rate, φ is the roll attitude, η and λ are the states for thestructural flexibility modes, and δail, δflp, δrud are the control surface deflections. The prime symbols β´, p´, r´ and φ´denote the rigid-body portions of the total β, p, r and φ responses that are given without primes in Eq. (7).Additional structural modes can be included in this model structure by adding two additional η states for each modeto represent the structural mode second-order system.

F =

Yv* Yp

* + W0 /U0 Yr* −1 (g /U0 )cosθ0 0 0 0 0

Lβ Lp Lr 0 0 0 0 0

Nβ N p Nr 0 0 0 0 0

0 1 tanθ0 0 0 0 0 0

0 0 0 0 0 1 0 0

0 µwb− p 0 0 −ωwb2 −2ζwbωwb 0 0

0 0 0 0 0 0 0 1

µ fb−β 0 µ fb− r 0 0 0 −ω fb2 −2ζ fbω fb

G =

Yδail* Yδ flp

* Yδrud*

Lδail Lδ flp Lδrud

Nδail Nδ flp Nδrud

0 0 0

0 0 0

µwb−δail µwb−δ flp 0

0 0 0

0 0 µ fb−δ rud

(6)

where the µ parameters in the G matrix allow the aileron, flaperon and rudder controls to excite the structuralflexibility modes, the µ parameters in the F matrix couple the rigid-body and structural dynamics by allowing theaircraft rigid-body response to excite the structural response, ωwb and ζwb are the frequency and damping of thewing-asymmetric bending mode, and ωfb and ζfb are the frequency and damping of the fuselage lateral bendingmode. The starred derivatives in the top equation arise from the use of β = v U0

rather than v as the lateral motion

parameter. In general, the starred derivatives are: Yλ* = Yλ U0

.

For the model formulation presented above and used in this paper, it is assumed that: 1) the structural modes areuncoupled from each other; 2) the wing-asymmetric bending mode is excited by the roll rate rigid-body responseand by the aileron and flaperon inputs; and 3) the fuselage lateral bending mode is excited by the sideslip angle andyaw rate rigid-body responses and by the rudder inputs. This model can be extended to include additional structuralmodes by adding two additional states for each mode and corresponding coefficients in the F and G matrices. Thestructural modes can be coupled together by adding additional off-diagonal blocks in the lower right side of the Fmatrix. Also the structural modes can be excited by more controls and rigid-body responses by adding theappropriate µ parameters to the F and G matrices.

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The measured or total outputs from this state-space model formulation are the sum of the rigid-body responseplus a contribution from the structural modes. The output equations are:

βtotal = ′β + Φβ2η22

ptotal = ′p + Φ p1η11

rtotal = ′r + Φr2η21

φtotal = ′φ + Φφ1η12

aytotal = U0& ′β + U0 ′r −W0 ′p − (g cosθ0 ) ′φ + Φay2

&&λ2 (= &η21)

∆az−wt = Φazwt1&&λ1(= &η11)

∆ay− fs = Φayf 2&&λ2 (= &η21)

(7)

where the Φ terms are Bending Mode Displacement coefficients that define how much the structural modescontribute to each of the total outputs. These coefficients can be obtained from graphs of bending mode deflectionversus body station from static experiment or from numerical analyses. For this particular study, these coefficientvalues are identified as part of the state-space model identification.

Figures 8 and 9 show that the identified model captures the effects of the wing asymmetric bending and fuselagelateral bending modes on all of the model outputs as well as the lower frequency rigid-body dynamics. In additionto the rigid-body states, this also includes the wing-tip differential accelerometer measurements and the fuselagedifferential lateral accelerometer measurements. This identified model is able to predict the aircraft responsesaccurately to frequencies beyond those of the structural modes and significantly higher than the capability of a rigid-body model.

Figures 10 and 11 show time domain validation results for wheel and pedal doublets. These results demonstratethat the identified model is very accurate even in the presence of large control inputs and aircraft responses. For thiscase, the peak roll attitudes are of the order of 30-40 degrees. This is an indication that the model has very goodpredictive capability even for large perturbations away from the trim condition at which the model was identified.

Figures 12 and 13 show a verification of the identified model in the time domain using data for automated wheeland rudder sweeps. The automated sweeps were used for this time-domain verification because the piloted sweepsand doublets did not sufficiently excite the structural modes to verify the model. Figure 12 shows, for an automatedwheel sweep, that the model accurately predicts the primary roll rate and wing-tip differential accelerometerresponses that are dominated by the wing asymmetric bending mode. There are, however, some differences towardsthe end of the lateral acceleration and yaw rate responses due to an excitation of the fuselage lateral bending modeseen in the flight data. This excitation results from a coupling between the wing and fuselage structural modes onthe actual aircraft that is not captured in the simulation model since it is assumed that the structural modes areuncoupled. This verification could be improved by modifying the state-space model to include a coupling betweenthe two structural modes that would allow the aileron and flaperon control inputs to drive the fuselage lateralbending mode.

Finally, Figure 13 shows, for an automated rudder sweep, that the primary responses of lateral-acceleration, yawrate and differential fuselage lateral accelerometer show a good match between the measured data and modelresponses, and that there is very little excitation of the wing-differential bending structural mode.

The rigid-body and structural modes that are identified for this aircraft are shown in Table 4. This table indicatesthat the frequency of the wing asymmetric bending structural mode is 7.1 times higher than the pole location of therigid-body roll mode, and the frequency of the fuselage lateral bending mode is about 18 times higher than thefrequency of the Dutch Roll mode. This large frequency separation between the rigid-body and structural modesvalidates the assumption that the ‘Aeroelastic Coupling Terms’ can be dropped where the structural response of theaircraft is assumed to not change the rigid-body response.

The identified model can also be used in the development of the flight control laws to examine the stability ofthe structural flexibility modes with feedback gain. As a simple example, consider the Single-Input Single-Output(SISO) transfer function for roll rate due to aileron deflection that is extracted from the full lateral-directional state-space model. This transfer is of the form:

p

δail

(s) =Lδail [ζφ ,ωφ ][ζφ−wab ,ωφ−wab ]e−τail s

(1 / Ts )(1 / Tr )[ζdr ,ωdr ][ζwab ,ωwab ](8)

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Figure 8. Comparison between identified state-space model frequency responsesand flight data frequency responses for aileron inputs.

Figure 9. Comparison between identified state-space model frequency responses and flight datafrequency responses for rudder inputs.

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This transfer function includes only the wing asymmetric bending mode since the model structure assumes thatthe structural modes are uncoupled and the roll rate response is affected only by the wing asymmetric bending mode.A similar transfer function for sideslip angle due to rudder deflection includes only the fuselage lateral bendingflexible mode, as follows:

βcg

δrud

(s) =Nδ rud (1 / Tβ1)(1 / Tβ 2 )(1 / Tβ 3)[ζβ− flb ,ωβ− flb ]e−τ rud s

(1 / Ts )(1 / Tr )[ζdr ,ωdr ][ζ flb ,ω flb ](9)

The root locus for this single-input single-output (SISO) system with simple roll rate feedback is shown inFigure 14. This simple analysis shows that the roll rate feedback decreases the stability of the wing asymmetricbending mode as the value of the gain is increased. This simple analysis also shows that the stability boundarycould theoretically be reached with a roll rate gain that is sufficiently high. This simple example could be extendedfurther to include the full state-space model for flight control law design.

Figure 11. Time history verification of identifiedstate-space model for a piloted rudder pedal doublet.

Figure 10. Time history verification of identifiedstate-space model for a piloted wheel doublet.

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V. Discussion and ConclusionsThis paper presents results of a proof-of-concept study that used frequency domain system identification to

extract accurate flight dynamics models of a large transport aircraft from flight test data. The identified flightdynamics models have application to simulation model validation and update, and flight control law development,which often require models that are valid over a broad frequency range. The data described in this paper are for awide-body transport aircraft in a low speed cruise configuration with the gear and flaps up. The main conclusionsfrom this study are:

Figure 13. Time domain verification of structuralresponse for an automated rudder sweep exiting the

fuselage lateral bending structural mode.

Figure 12. Time domain verification of structuralresponse for automated wheel sweep exciting the

wing-asymmetric bending structural mode.

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1) The accurate identification of lightly-damped low frequency rigid-body response modes requires a carefulselection of the frequency sweep length and the FFT window size. This is illustrated for the Dutch Rollmode where the modal damping is over-predicted for FFT windows that are shorter than the guideline ofTwin > 11 /ζω . Since the FFT window length cannot be longer than any individual sweep records, the

guideline also places a lower limit on the length of the frequency sweep records.

2) Large transport aircraft have structural bending and torsion modes that can affect the dynamic response ofthe aircraft in the flight control frequency region. This is true for the aircraft dynamics used in this studywhere the effects of wing asymmetric bending and fuselage lateral bending affect the lateral-directionaldynamic response at frequencies close to the rigid-body response modes. It is then important to take thestructural flexibility into account when developing the flight control laws for the aircraft, particularly forhigher bandwidth flight control systems.

3) A model structure was developed that incorporates coupled rigid-body and structural dynamics. Thismodel structure was used to identify a model of the lateral-directional dynamics that is accurate tofrequencies above the wing asymmetric bending and fuselage lateral bending modes. A time domainverification of the model using doublet and sweep data shows that the model accurately predicts the lowfrequency rigid-body response (including the Dutch Roll mode) as well as the effects of the structuralflexibility.

References1Tischler, M. B., and Remple, R. K., Aircraft and Rotorcraft System Identification: Engineering Methods with Flight Test

Examples, AIAA Education Series, AIAA, Reston, VA, 2006.2Theodore, C. R., Tischler, M. B., and Colbourne, J. D., “Rapid Frequency-Domain Modeling Methods for Unmanned Aerial

Vehicle Flight Control Applications,” AIAA Journal of Aircraft, Vol. 41, No. 4, 2004.3Field, E. J., Rossitto, K. F., and Hodgkinson, J., “Flying Qualities Applications of Frequency Responses Identified from

Flight Data,” AIAA Journal of Aircraft, Vol. 41, No. 4, 2004.4Waszak, M. R., Buttrill, C. S., and Schmidt, D. K., “Modeling and Model Simplification of Aeroelastic Vehicles: An

Overview,” NASA TM-107691, 1992.5Meirovitch, L., and Tuzcu, I., “Integrated Approach to the Dynamics and Control of Maneuvering Flexible Aircraft,” NASA

CR-2003-211748, 2003.6McLean, D., Automatic Flight Control Systems, Prentice-Hall International, Hertfordshire, UK, 1990.7Hodgkinson, J., Aircraft Handling Qualities, AIAA Education Series, AIAA, Reston, VA, 1999.

Table 4. Rigid-body and structural modes of theidentified model.

Rigid-Body ModeDamping Ratio

ζModal Frequency

ωSpiral -- (1 / Ts )

Dutch Roll ζdr ω dr

Roll -- (1 / Tr )

Wing AsymmetricBending

ζwab ωwab *Tr = 7.1

Fuselage LateralBending

ζ flb ω flb /ωdr = 18.0

Figure 14. Root-locus generated with roll ratefeedback based on a SISO transfer function. Thestability of the wing asymmetric bending mode is

reduced with increasing roll rate feedback.