Adaptive Mode Suppression Scheme for an AeroelasticAirbreathing Hypersonic Cruise Vehicle

Jason Levin∗ and Petros A. Ioannou†

University of Southern California, Los Angeles, CA 90089

Maj Dean Mirmirani‡

Embry-Riddle Aeronautical University, Daytona Beach, FL 32174

This paper presents an adaptive mode suppression scheme that is simulated on a longitudinal model of ageneric airbreathing hypersonic flight vehicle. An aerodynamically optimized vehicle model is used which hasa long slender configuration and flexible modes that are slow enough to get excited during normal operationand therefore pose a control issue. These elastic modes maybe cause adverse aeroservoelastic effects whichcan degrade the performance or even cause instabilities. The frequency and damping ratio of the flexiblemodes can be uncertain and also vary during flight due to atmospheric heating. A control scheme is presentedwhich incorporates an adaptive notch filter which can accurately track and suppress an unknown or changingflexible mode online. The adaptive scheme is then compared to a non-adaptive scheme in simulations using anaeroelastic airbreathing hypersonic cruise vehicle model that has been previously developed.

Nomenclature

c Mean aerodynamic chord, ftD Drag, lbfh Vehicle altitude, ftIyy Vehicle y-axis inertia per unit width, slug.ft2

L Lift, lbfM Vehicle flight Mach numbersm Vehicle mass, slugMyy Pitching moment, lb.ftq Vehicle pitch rate, rad/sr Distance from the Earth’s center, ftS Reference area, ft2

T Thrust, lbfV Vehicle velocity, ft/s

Symbols

α Effective angle of attack, radαe Elastic angle of attack, radαr Rigid body angle of attack, radδe Elevon control surface deflection, radδT Throttle settingηi ith generalized elastic coordinateγ Flight path angle, radµ Gravitational constant∗Graduate Student, Electrical Engineering, Student Member AIAA.†Professor, Electrical Engineering, Member AIAA.‡Professor and Dean, College of Engineering, Senior Member AIAA.

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AIAA 2008-7137

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction

IN recent years there has been a considerable research effort devoted to hypersonic flight vehicles that use airbreathingpropulsion systems. These hypersonic aircraft have application in a variety of missions which include reconnais-

sance, tactical, or low cost space access. The vehicles themselves are generally modeled with an integrated engine-airframe which can make control difficult.1, 2 Studies also indicate that there could be coupling between the structuraland rigid-body dynamics.3–5 The aerodynamically optimized full scale vehicle should have a long slender configu-ration with flexible modes that are slow enough to pose a control issue. These flexible modes may cause adverseaeroservoelastic effects degrading performance, cause potential structural damage and instabilities. What makes thecontrol problem even more challenging is that these modes are uncertain and may even change in flight. The modalfrequencies and damping ratios may change based on aircraft configuration, cg location, payload for a required mis-sion, or due to atmospheric heating.6–10 The flexible modal frequencies may drop by as much as 5-30% due to theaerothermoelastic effects.

A robust control scheme for a hypersonic aircraft must adequately deal with the flexible structural modes. Modesuppression in hypersonic vehicles has been studied and reported in the literature.11–13 A linear parameter-varying(LPV) synthesis approach to account for the changing flexible mode due to atmospheric heating has been studied inRef. 6,7. Mode suppression schemes for conventional aircraft usually incorporate notch filters to suppress the flexiblemodes.14–17 The notch filters are added to a rigid-body controller for a complete aircraft control scheme. However,there are also integrated techniques, which do not explicitly use any structural filters. This is accomplished with arobust controller design using µ-synthesis techniques,18 dynamics inversion,19 or adaptive dynamic inversion.20Anadaptive notch filter may be used when the frequencies and damping ratios of the structural modes are unknown ormay change during flight. The adaptive notch filter has been studied in research21–23 as well as other applications, suchas the HDD,24, 25 launch vehicles,26 aircraft,27 and space structures.28

This paper presents an adaptive notch filter based on plant parametrization which does not require a probe signal.The adaptive notch filter will be applied to a longitudinal model of an aeroelastic airbreathing hypersonic vehiclewhere the structural modes change during flight. This model has been used previously for control design,29, 30 butsuppression of changing or uncertain structural modes have not been taken into consideration. The control schemepresented here will incorporate a robust online observer that will track the frequency of the elastic mode(s) which willthen be used as the center frequency for a notch filter. This adaptive notch filter is combined with a robust rigid-bodycontroller for a complete adaptive mode suppression scheme. The scheme is simulated on the unstable nonlinearmulti-input/multi-out (MIMO) vehicle model. The results demonstrate the scheme’s ability to adapt online and fullysuppress a changing structural mode.

II. CSULA-GHV Mathematical Model

A longitudinal model of a full-scale generic airbreathing hypersonic vehicle developed at the MultidisciplinaryFlight Dynamics and Control Laboratory (MFDCLab) at the California State University, Los Angeles (CSULA), willbe used as the simulation platform for the adaptive notch filter.5, 31 The CFD-based CSULA-GHV has an integratedairframe-propulsion system using a scramjet engine whose geometry is shown in Fig. 1. The complete rigid-bodyaerodynamic and propulsion data are stored in lookup tables. It should be noted that the CSULA-GHV has a variableengine cowl geometry. The engine cowl moves so as to capture the shockwave from the bow at the optimum point tomaximize mass flow rate through the engine and ensure flow inside the engine is axial and no shock train is generatedinside the engine. The cowl is allowed two-degree of freedom motion and the optimum location is calculated basedon Mach number and angle of attack.

The vehicle model then has elastic modes added to the rigid-body dynamics. These flexible modes account forfuselage deformations at the nose and tail, which are then treated as changes in the angle of attack and elevon de-flection, respectively, resulting in an integrated flexible vehicle model, where the flexible modes interact with therigid-body aerodynamics and propulsion. The equations of motion of the GHV are

V =Tcos(αr)−D

m− µsin(γ)

r2(1)

γ =L + Tsin(αr)

mV− (µ− V 2r)cos(γ)

V r2(2)

h = V sin(γ) (3)

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Variable

50 110

31 m

1.82 m

2.38 m

Figure 1. Overall geometry of the CSULA-GHV used for simulation.

αr = q − γ (4)

q =Myy

Iyy(5)

ηi = −2ζiωiηi − ω2i + kNδe (6)

δe = δe,r + δe,e (7)

α = αr + αe (8)

L =12ρV 2SCL(M,α, δT , δe) (9)

D =12ρV 2SCD(M, α, δT , δe) (10)

T =12ρV 2SCT (M,α, δT , δe) (11)

Myy =12ρV 2ScCM (M, α, δT , δe). (12)

In the above equations the coefficients CL, CD, CT , and CM are computed using lookup tables with M, α, δT , andδe as the index parameters. The δe and α are the effective angle of attack and elevon deflection. These values are thesum of the rigid body deflection and the elastic deflection. The ζi and ωi are the damping and natural frequency of theith mode, and kN is the effective normal force at the elevon. The natural frequencies and mode shapes are computedusing finite element analysis. The first three elastic modes are included which have natural frequencies 20.33 rad/s,58.62 rad/s, and 116.18 rad/s. The damping of all three is set to ζi = 0.02. This damping is smaller than the dampingused in Ref. 5 but more in agreement with the damping used in Ref. 3, 4, 6, 11, 14. This smaller damping makes theflexible modes more pronounced and requires them to be suppressed in the control design.

The elastic deflections are calculated

δe,e =3∑

i=1

τt,iηi (13)

αe =3∑

i=1

τn,iηi (14)

where τt,i and τn,i are the tail and nose scaling factor, respectively, which is based on the mode shape. As theseequations show, an elastic deflection at the nose will add to the rigid angle of attack to create an effective angle ofattack. Similarly, a deflection at the tail will account to an effective elevon deflection. These effective values are thenused in the lookup tables for the aerodynamic and propulsion data. Therefore, in this model, the engine cowl locationwill remain optimized even when an elastic deformation is present. This may not be physically accurate and is oneof the drawbacks of this GHV model. This then creates a flexible GHV model which has two control inputs: elevondeflection δe,r and throttle δT , and five outputs: velocity V , flight-path angle γ, altitude h, angle of attack αr, andpitch rate q. The control inputs to the model must be limited as the lookup tables only exist for a range of inputs,similar to actuators having limits. These limits are δe ∈ [−0.3491, 0.3491] (rad) and δT ∈ [0, 1].

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CSULA-GHV

r

AdaptiveNotch Filter

x

Robust OnlineEstimator

Rigid-BodyController

h

T

e,r

Figure 2. Overall diagram of the adaptive mode suppression scheme for the generic hypersonic vehicle. Here x is all the measurable outputs from the GHV, his the altitude, and r if the reference input.

III. Adaptive Mode Suppression Scheme

The adaptive mode suppression scheme to control the CSULA-GHV will be presented in the following subsections.An overview of the control scheme is shown in Fig. 2. A robust online estimator tracks the modal frequency. Theestimate of the modal frequency is then used as the center frequency for the adaptive notch filter. The adaptive notchfilter, which only acts on the elevon control input then suppresses the uncertain or changing flexible modes. Onlythe elevon input, δe,r, is filtered since this is the input that drives the flexible modes in the GHV model. However,the adaptive notch filter presented could also be used for the throttle input, δT . The rigid-body controller is designedindependently to control the rigid-body dynamics. In this paper, for simplicity of design and simulation, a LQ designincluding an integral action to create a tracking controller is used for rigid body control. Each of these systems will befurther described below.

III.A. Rigid body controller

The rigid-body controller is independent of the proposed mode suppression scheme and can be any controller whichrobustly stabilizes the rigid-body dynamics of the GHV and meets performance requirements. For ease of design andimplementation in simulation, a LQ tracking controller used previously for the control of a GHV30 and a F-1632 hasbeen used for this study. A summary of the rigid-body LQ design is presented for completeness and use for simulation.

The nonlinear equations of motion for the GHV is linearized at a specific velocity and altitude. Only the rigid-bodystates are used for the rigid-body control design, since the adaptive notch filter suppresses the structural mode(s). Thelinearized model can be expressed as

x = Ax + Bu (15)

y = Cx (16)

where x = [ V γ h αr q ]T , u = [ δT δe,r ], and y = [ V h ]. This y is chosen since a velocityand altitude tracking problem is considered in the simulations presented in this paper. The objective is to track a

commanded velocity and altitude, Vcom and hcom respectively, which makes a command vector[

Vcom hcom

]T

.First Eq. (15) and Eq. (16) are augmented with new states, w, representing the integral of the tracking errors:

w =

[Vcom − V

hcom − h

]. (17)

So that the augmented system of equations becomes

xaug = Aaugxaug + Baugu +

[ycom

0

](18)

y =[

0 C]xaug (19)

xaug =

[w

x

](20)

Aaug =

[0 −C

0 A

](21)

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Baug =

[0B

]. (22)

Using the model described by Eqs. (18 - 22) the tracking problem is reduced to the standard regulator problem towhich the LQ solution can be applied. The deviations of the states xaug and control inputs u from their steady-statevalues are computed and denoted as xaug and u, respectively. The LQ cost function, assuming the augmented systemis stabilizable and detectible, is

J =∫ ∞

0

(xTaugQxaug + uT Ru)dt (23)

where Q > 0 and R > 0 are the chosen weights. The weights are chosen to limit the control bandwidth so as not toexcite the flexible modes or other unmodeled dynamics in the high frequency range. However the bandwidth is closeto the natural frequency of the slowest flexible mode therefore creating the need for a structural, or notch, filter. Thecontrol input becomes u = −Kxaug where K = R−1BT

augP , and P is the solution of the algebraic Ricatti equation.In the simulations it is assumed that all the original states, x = [ V γ h αr q ]T , can be measured and w canbe computed by simply integrating the tracking error.

As shown in Ref. 30, 32 this control scheme can either be gain-scheduled for the flight envelope or converted toan adaptive LQ design where the original system matrices A and B in Eq. (15) are estimated online. In this paper thenonlinear vehicle model is linearized at various points in the two-dimensional flight envelope, based on velocity andaltitude, and an optimal gain matrix K is calculated and stored in a database where the best controller gain is chosenby using two-dimensional linear interpolation.

III.B. Robust Online Estimator

The robust online estimator presented here is for the case of a single unknown flexible mode. However it can be easilyexpanded to estimate more unknown flexible modes. In the case of the hypersonic vehicle model, the flexible modewith the slowest natural frequency, the first bending mode, will have the greatest impact on the rigid-body controllerbandwidth. Therefore this mode which has a natural frequency of 20.33 rad/s has been modeled in simulations asthe uncertain and/or changing mode. The robust online estimator should be able to return an estimate of this naturalfrequency. The flexible equations of motion for the GHV, Eqs. (1 - 5), are nonlinear, but can be linearized around onepoint in the flight envelope. This will lead to the linear model for the altitude

h(t) = Ge(s)δe(t) + GT (s)δT (t). (24)

Which can then be rewritten as

h(t) =Ne(s)De(s)

δe(t) +NT (s)DT (s)

δT (t). (25)

Bode plots of the transfer functions Ge(s) and GT (s) can be seen in Fig. 3 and Fig. 4, respectively. The CSULA-GHVmodel flexible modes, which are driven by elevon deflections, do not influence the transfer function from δT to altitudeappreciably as seen in Fig. 4. So GT (s) in Eq. (24) can be treated as strictly a rigid-body model, and Ge(s) as havingboth rigid and flexible dynamics. It is therefore assumed that the rigid body linearized transfer function of the GHV forδe and δT to altitude is known. Also the damping and natural frequency of the two faster flexible modes are assumedto be known. The only unknown part is the damping and natural frequency of the slowest flexible mode, which hasthe largest effect on the stability margins and controller bandwidth. With these assumptions, Eq. (25) becomes

h(t) =Ne,k(s)(s2 + 2ζNωNs + ω2

N )De,k(s)(s2 + 2ζDωDs + ω2

D)δe(t) +

NT (s)DT (s)

δT (t) (26)

where Ne,k(s) and De,k(s) are the known rigid and flexible parts of the numerator and denominator, respectively,of the linearized transfer function from δe to altitude. It should be noted that De,k(s) does not equal DT (s), sinceDe,k(s) contains both known rigid and flexible dynamics and DT (s) is only rigid body dynamics.

The modal frequency that is changed in the mathematical model is the ωi in Eq. (6). This is not however thefrequency that the online estimator is designed to estimate. The frequency estimated, and therefore used for theadaptive notch filter center frequency, is ωd which is the frequency of the flexible mode in the denominator of theGe(s) transfer function. Which is the frequency of the complex conjugate poles that create the peak in the bodeplot which must be suppressed. Due to the interaction of the flexible and the rigid-body dynamics ωd may not be

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-100

-50

0

50

100

150

200

Mag

nitu

de (

dB)

10-4

10-2

100

102

-90

-45

0

45

90

135

180

225P

hase

(de

g)

Bode Diagram

Frequency (rad/sec)

Figure 3. Bode plot of the transfer function from the elevon to altitude, Ge(s).

-100

-50

0

50

100

150

200

Mag

nitu

de (

dB)

10-4

10-2

100

102

-225

-180

-135

-90

-45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Figure 4. Bode plot of the transfer function from the throttle to altitude, GT (s).

necessarily the same as ωi. The goal is to estimate ωd to be used as the center frequency of the adaptive notch filter.Now Eq. (26) is placed in the form of a continuous time parametric equation

z(t) = θ∗T φ(t) (27)

z(t) = zh(t)− ze(t)− zT (t) (28)

zh(t) =s2De,k(s)DT (s)

Λ(s)h(t) (29)

ze(t) =s2Ne,k(s)DT (s)

Λ(s)δe(t) (30)

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zT (t) =s2De,k(s)NT (s)

Λ(s)δT (t) (31)

φ(t) =[

φ1(t) φ2(t) φ3(t) φ4(t)]T

(32)

φ1(t) =−2De,k(s)DT (s)s

Λ(s)h(t) +

2De,k(s)NT (s)sΛ(s)

δT (t) (33)

φ2(t) =De,k(s)DT (s)

Λ(s)h(t) +

De,k(s)NT (s)Λ(s)

δT (t) (34)

φ3(t) =2Ne,k(s)DT (s)s

Λ(s)δe(t) (35)

φ4(t) =Ne,k(s)DT (s)

Λ(s)δT (t) (36)

θ∗ =[

ζDωD ωD2 ζNωN ωN

2]T

. (37)

Here Λ(s) is a polynomial added to make proper transfer functions, and takes the form Λ(s) = (s + λ)n. Here λ andn are design parameters which will determine the speed of the filter 1

Λ(s) .The unknown parameters are updated online using the following algorithm

˙θ = −Γ(Rθ + Q) (38)

θ = θ min(

1,M

|θ|

)(39)

R = −βR +φφT

m2(40)

Q = −βQ− zφT

m2(41)

m2 = 1 + ns + ms (42)

ns(t) = CsφT φ (43)

nd = −δ0nd + δ1(h2 + δ2e + δ2

T ). (44)

The online estimator uses a gradient algorithm with an integral cost and robustness modifications.33 For robustnessthe update term is normalized using a dynamic term which is calculated in Eq. (44). Parameter projection is used inEq. (39), to take advantage of the known region of the parameters, |θ∗| ≤ M for some known M > 0, just based ona priori knowledge of the vehicle dynamics. The adaptation gain Γ is a design parameter that satisfies, Γ = ΓT > 0.The Cs term and the forgetting factor β are both design parameters that must be greater than zero.

This online estimator must also be gain-scheduled for the full two-dimensional flight envelope since the vehicledynamics are nonlinear. Again, as in the LQ design, the model is linearized around various points and the knowntransfer functions Ne,k(s), De,k(s), NT (s), and DT (s) are computed. State space representations of the transferfunctions are calculated and the state space matrices are stored in a database. Online, the correct transfer functions arebe chosen using two-dimensional linear interpolation.

III.C. Adaptive Notch Filter

The adaptive notch filter is implemented to suppress the slowest flexible mode of the GHV. The frequency of theflexible mode may, for various reasons given earlier, be uncertain or change. The notch filters center frequency willuse the estimate ωD from the robust online estimator. Since the notch filter will accurately follow the flexible mode itcan therefore be designed with a narrower bandwidth, providing a smaller phase lag at lower frequencies. This has theadded benefit of enabling the design of higher bandwidth rigid-body controllers. The notch filter has the form

F (s) =s2 − 2ζN0ωDs + ω2

D

s2 − 2ζD0ωDs + ω2D

, (45)

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ζD0 > ζN0 are design parameters that define the width and depth of the filter. The ωD term is the estimate of themodal frequency of the plant which is computed online as ωD =

√θ2 where θ2 is the second element in the θ vector.

The θ vector is the online estimate of θ∗ in Eq. (37).The adaptive notch filter scheme presented differs from schemes previously reported in the literature for various

reasons. The adaptive notch filter presented in26 is used on the model of a booster from the Advanced Launch System(ALS) program. The least squares estimator in the publication uses a simple undamped resonator as the model andfunctions well since the resonant mode is very pronounced. However in the GHV, and other applications, full plantparameterizations is necessary as the flexible mode may not be as significant. Another strategy for the estimation ofthe center frequency can be found in,24 where frequency weighting functions are used. The downside is there areseveral failure modes that are known and avoidance requires some modal information a priori. A stochastic state spacealgorithm for mode frequency estimation is presented in;27 however it relies on the injection of a probe signal which isnot needed in the scheme presented here. The indirect adaptive compensation (IAC) scheme in34 also requires a probesignal to complete the estimation. The adaptive mode suppression scheme in28 uses a LMS algorithm to update filtercoefficients and then the modal parameters are extracted from the filter. This is opposite as to what is being presentedin this paper, where the modal parameters are first estimated and then used in the adaptive notch filter. The estimationscheme also allows for the identification of multiple modes simultaneously if the model in Eq. (26) is expanded toinclude more unknown flexible modes.

IV. Simulations

The simulations presented are performed on the full aero-elastic nonlinear CSULA-GHV model presented ear-lier. The initial set of simulations shows how the adaptive notch filter is necessary to fully suppress the uncertain orchanging flexible mode. The slowest flexible mode, which occurs at 20.33 rad/s is modeled as changing. The changein modal frequency could be due to thermal heating of the fuselage, which could lower the frequencies of the elasticmodes by as much as 30%.7 A non-adaptive notch filter cannot deal with a changing flexible mode. However theadaptive notch filter is able to track the flexible mode’s frequency. For this simulation the model is linearized arounda similar nominal cruise point M = 10 and h = 100, 000 ft, the same point as used in Ref. 29.

The GHV is required to track a 1, 000 ft/s velocity and 10, 000 ft altitude step command. This command is passedthrough a prefilter

P (s) =(0.05)2

s2 + 2(0.05)s + (0.05)2, (46)

again are the same as in Ref. 29. The modal frequency at 20.33 rad/s is subjected to a filtered step change to 15.33rad/s beginning at 5 seconds. This assumed change of the flexible mode frequency is much faster than an actual modechange due to thermal heating of the fuselage in a real flight situation. However, the example demonstrates the abilityof the online estimator to track the fast changing modal frequency. This mode change is input into the model as achange in ω1, which occurs in the mathematical model in Eq. (6).

As seen in Fig. 5, the commanded outputs are tracked closely, even with the slowly changing flexible modefrequency. This change can be more easily seen in the control inputs in Fig. 6. The modal frequency begins toappear in the control inputs after about 25 seconds, but then is suppressed as the adaptive notch filter tracks the modalfrequency. The center frequency for the adaptive notch filter is plotted in Fig. 7. It is seen that the frequency estimatedoes not immediately pickup the change at 5 seconds; rather it begins to adapt around 26 seconds and convergesaround 36 seconds. The change starts taking place only after the incorrect notch filter center frequency allows thecontrol inputs to excite the flexible frequency. Creating enough system excitation to allow the online estimator toestimate the modal frequency. In this case, this slow change is acceptable since the system remains stable. Althoughthe incorrect center frequency is input after 5 seconds, the flexible mode does not get excited immediately to create alarge enough excitation to require adaptation.

However, to more clearly see the effect of the flexible mode on the GHV, the normal acceleration at the nose isplotted in Fig. 8. As the control inputs excite the flexible modes, the nose of vehicle begins to vibrate at this flexiblefrequency. This vibration, in a real system, will affect the bow shock placement on the engine cowl since a variablegeometry cowl may not be able to compensate for the elastic deformations of the nose. In turn, this shock placementwill greatly affect the thrust and aerodynamics forces on the vehicle. However in this GHV model, as mentioned earlieras a drawback, the engine cowl is able to maximize performance given the elastic deformation. The vibration of thenose could, in addition to affecting the propulsion system performance, cause deterioration of the structure over timeand therefore should be suppressed quickly. This is accomplished using the adaptive notch filter, since the adaptivenotch filter tracks the modal frequency and eventually suppresses the flexible frequency. The vibrations damp out to a

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0 10 20 30 40 50 60-200

0

200

400

600

800

Time (s)

Vel

ocity

cha

nge

(ft/s

)

0 10 20 30 40 50 600

2000

4000

6000

8000

10000

Time (s)

Alti

tude

cha

nge

(ft)

Figure 5. Velocity and altitude for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is on. Red line: Command response.Blue line: Actual response.

0 10 20 30 40 50 600.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (s)

δ T

0 10 20 30 40 50 60-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Time (s)

δ e,r (

rad)

Figure 6. Control inputs for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is on.

negligible level after 40 seconds. The control inputs and normal acceleration at the nose for the case when adaptationis turned off are seen in Fig. 9 and Fig. 10. The flexible mode appears as large fluctuations in the control inputs andthe normal acceleration at the nose. The system appears stable, however it is saturating the elevon input and the verylarge fluctuations would surely exceed the structural limits of the flight vehicle thereby causing a crash.

V. Conclusion

The design of an adaptive notch filter for suppression of structural modes in an aeroelastic airbreathing hypersoniccruise vehicle model is presented. A robust online estimator tracks the frequency of the slowest flexible mode, whichmay change due to different aircraft configurations or heating of the fuselage. The estimation is accomplished byparameterizing the linearized version of the nonlinear flexible aircraft model. The notch filter then uses the estimatedmode frequency as the center for the notch to adequately suppress any flexible effects. Simulations for the adaptive

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0 10 20 30 40 50 6015

16

17

18

19

20

21

Time (s)

Not

ch c

ente

r (r

ad/s

)

0 10 20 30 40 50 6014

16

18

20

22

Time (s)

Fle

xibl

e m

ode

ω1 (

rad/

s)

Figure 7. Notch center frequency and modal frequency ω1, for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is on.

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

Time (s)

Nor

mal

acc

lera

tion

at n

ose

(g)

Figure 8. Normal acceleration at the nose for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is on.

notch filter and non-adaptive notch filter are shown and discussed. The adaptive scheme is able to suppress thechanging flexible mode before the oscillations become large enough to cause structural damage. Whereas simulationsusing a non-adaptive filter result in large oscillations in the control inputs and acceleration at the nose.

Acknowledgments

The authors would like to thank the hard work and support of Dr. Chivey Wu, Andrew Clark, Sangbum Choi,Matthew Kuipers, and Moataz Samir in developing the CSULA-GHV model. This research has been supported inpart by NASA Grant No. NCC4-158 and in part by the National Science Foundation under Grant No. 0510921. Anyopinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and donot necessarily reflect the views of the National Science Foundation.

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0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

Time (s)

δ T

0 10 20 30 40 50 60-0.2

0

0.2

0.4

0.6

Time (s)

δ e (ra

d)

Figure 9. Control inputs for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is off.

0 10 20 30 40 50 60-10

-5

0

5

10

15

Time (s)

Nor

mal

acc

lera

tion

at n

ose

(g)

Figure 10. Normal acceleration at the nose for case where modal frequency is shifted from 20.33 rad/s to 15.33 rad/s and adaptation is off.

References1Schmidt, D. K., “Dynamics and Control of Hypersonic Aeropropulsive/Aeroelastic Vehicles,” AIAA Paper 92-4326, 1992.2Schmidt, D. K., “Integrated Control of Hypersonic Vehicles - A Necessity Not Just a Possibility,” AIAA Paper 93-3761, 1993.3Bolender, M. A. and Doman, D. B., “Nonlinear Longitudinal Dynamical Model of an Air-Breathing Hypersonic Vehicle,” Journal of

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