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Towards Real-time Simulation of Aeroservoelastic Dynamics for a Flight

Vehicle from Subsonic to Hypersonic Regime

Patrick Hu*

Advanced Dynamics Corporation, KY 40511

Marc Bodson†

Department of Electrical and Computer Engineering, University of Utah, Salt Lake City, UT84112

and

Marty Brenner †∗

Dryden Flight Research Center, Edwards, CA 93523

Although modern super computers are very fast, they are still not powerful

enough to perform the near-real or real time simulation of aeroservoleastic

dynamics for a flight vehicle if full-order and full-coupled CFD/CSD approaches

are used. Fortunately, reduced order model (ROM) provides such an approach

(i.e. POD/ROM, Volterra/ROM, and ROM adaptation, etc.), which is basically

generated by a set of full-order and full-coupled CFD/CSD simulation data.

Once such a ROM model is constructed, the near-real or real time simulation of

aeroservoleastic dynamics may be performed on a PC (personal computer). This

paper will discuss the whole process of how to generate a ROM model from a

full-order and full-coupled CFD/CSD simulation data for a flight vehicle and

how to facilitate the near-real or real time simulation of aeroservoelastic

dynamics on a PC. Specifically, the full configuration aircraft model will be used

to demonstrate the construction process of a ROM model for a flight vehicle and

the details of the near-real or real time simulation of aeroservoleatsic dynamics,

along with the active flutter control system (AFS) design for aeroelastic

flutter/limit-cycle oscillation (LCO) suppression. The results will show the

innovations and unique features of our new methodology and the success of this

method which will bring a revolution in the near-real or real time simulation of

aeroservoelastic dynamics of a flight vehicle from subsonic to hypersonic

regime.

I. Introduction

Aeroservoelastic dynamics of an aircraft involves aerodynamics, structure dynamics and control,

therefore is a comprehensive and multidisciplinary. The analysis and evaluation of the aeroservoelastic

dynamics is very important for performance and stability analysis of aircrafts. Particularly, the near-real

or real time simulation of aeroservoelastic dynamics will provide the insight for the performance and

Patrick Hu *, President and Chairman, Senior Member of AIAA

Marc Bodson † , Professor, Senior Member of AIAA and Fellow of IEEE

Marty Brenner †∗

, Aerospace Engineer, Member of AIAA

AIAA Atmospheric Flight Mechanics Conference and Exhibit AIAA-2008- 6375

18 - 21 August 2008, Honolulu, Hawaii

Copyright © 2008 by Advanced Dynamics Corporation, Published by the American Institute of Aeronautics and Astronautic, Inc. with permission

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

AIAA 2008-6375

Copyright © 2008 by Advanced Dynamics Corporation. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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safety of aircrafts in the flight envelope before the flight test. However, If high-fidelity computational

fluid dynamics (CFD) and computational structure dynamics (CSD) approaches are used, the large

degree-of-freedom, nonlinear fluid and structural system may take days to weeks to finish the

computation and, thus prohibits the near-real or real time simulation of aeroservoelastic dynamics. A

reduced order model (ROM) that captures the dominant feature of the full system is highly desired and

extremely useful in practical simulation. Dowell and Hall 1 presents a comprehensive review of reduced

order models. In general, ROM involves several steps: (1) generation of training data (snapshots or time-

histories of loading excited by prescribed inputs) by conducting full-order coupled fluid/structure system

simulations, (2) generation of the ROM model by utilizing methods such as eigenmode based methods

and system identification methods, and (3) deployment of the ROM model for the full-order system

analysis. Different approaches have been extensively investigated in the last few decades for ROM of a

complex nonlinear system, including linearization about a nonlinear steady-state condition, linear model

fitting (such as the ARMA model), representation of the aeroelastic system in terms of its eigenmodes,

and linearized representation of a ROM for nonlinear aeroelastic/aeroservoelastic systems. The POD 2-8

and Volterra 9-10

ROM models are selected for use in present study.

The high-fidelity coupled fluid/structure system simulation and the ROM is the innovative idea of

variable-fidelity modeling and simulation. The ROM does not only facilitate the near-real or real time

simulation, but also can be used at the beginning of the design stage so as to achieve fast turn-around

time. On the other hand, the high-fidelity full-order and full-coupled simulation can be used at the final

design stage to verify and validate whether the design will meet the design objective and the mission

requirement. Advanced Dynamics Corporation recently has been focusing on developing such a

comprehensive variable-fidelity software toolset for simulation of the aeroservothermoelasticity and

propulsion effects of flight vehicles ranging from subsonic to hypersonic regimes, called ASTE-P,11,12

which serves as the computational test bed for the present study.

Although many studies have investigated aeroelastic phenomenon using POD and Voltrra ROM, few

have addressed the near-real or real time simulation, even for simple configuration and trajectory.

Therefore, in present study, we explored the POD and Volterra ROM approach towards real time

simulation of the aeroservoelastic dynamics and the design of an active controller/stability augmentation

system for flutter/LCO suppression of a flexible aircraft. In section II, the flexible aircraft dynamics

model will be established; in Section III, the ROM model for a flexible aircraft will be introduced; in

Section VI, the flight control system (FCS) and active flutter/LCO suppression system (AFS) will be

designed; and finally, in Section V, several testing case will be illustrated.

II. Flexible Aircraft Dynamics Model

When a modal description of a flexible aircraft is available, the elastic deformation at a position (x, y,

z) on the structure may be written as follows:

1

( , , ) ( , , )N

i i

i

d x y z x y z η=

= Φ∑

where i

η is the generalized coordinate of the structure and i

Φ (x, y, z) is a mode shape associated with the

ith generalized coordinate. When the number of mode shapes, N, is large enough, the Eq. 1 can be used to

accurately represent the structural deformation of a flexible aircraft.

(1)

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Define the inertial velocity components U, V, and W of the aircraft in the body-referenced (mean) axis,

and the three components of the angular velocity (p, q, r) projected in the body-fixed axis by the relation

pi qj rkω = + +�� �

(2)

where these components are related to the rates of change of the Euler angles by the following relation-

ships.

. . . .

. .

sin( ), cos( )sin( ) cos( )

cos( ) cos( ) sin( )

p q and

r

φ ψ θ ψ θ φ θ φ

ψ θ φ θ φ

= − = +

= −

(3)

Following Schmidt and Raney,13

a flexible aircraft model that incorporates the rigid body and the first

few structural modes can be mathematically described as bellow:

'

'

'

' ' ' 2 2

' ' ' 2 2

[ sin( )]

[ sin( ) cos( )]

[ cos( ) cos( )]

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

x

y

z

xx xy xz zz yy xy xz yz B

yy xy yz xx zz yz xy xz B

zz

M U rV qW g Q

M V pW rU g Q

M W qU PV g Q

I p I q I r I I qr I r I q p r q I Q

I q I p I r I I pr I p I r q p r I Q

I r

φ

θ

θ

φ θ

φ θ

− + + =

− + − =

− + − =

− + + − + − + − =

− + + − + − + − =

' ' ' 2 2( ) ( ) ( ) ( )xz yz yy xx xz yz xy BI p I q I I pq I q I p r q p I Qψ− + + − + − + − =

(4)

Now define the total aerodynamic and propulsive forces and moments components projected into the

mean axes as:

,

sin( ) cos( ) cos( ) cos( )sin( )

sin( ) cos( )

cos( ) sin( ) cos( ) sin( )sin( )

x

y

z

F Xi Yj Zk M Li Mj Nk

X L D S T

Y D S T

Z L D S T

α α β α β

β β

α α β α β

= + + = + +

= − + +

= − − +

= − − + +

� �� � � �

(5)

where L is lift, D is drag, S is the lateral force, L is rolling moment, M is the pitching moment, N is the

yawing moment, and T(.) is appropriate component of the propulsive thrust vector. Also, andα β are

the angles of attack and sideslip, respectively, of the mean axis relative to the wind vector.

The virtual work done by the aerodynamic and propulsive forces and moments can be written as:

( ( )]

[ ( )] [ ( )]

( , , ).

B

B B

i i

Area

W X x Y y Z z L yZ zY

M zX xZ N xY yX

P x y z dS

φ

θ ψ

φ η

∂ = ∂ + ∂ + ∂ + + − ∂ +

+ − ∂ + + − ∂ +

∂∑∫

(6)

Then Qx=X, Qy = Y, and Qz = Z, and ,B B BQ L Q M and Q Nθ ψΦ = = = , plus the generalized

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aerodynamic force:

( , , ). ( , , )i i

Area

Q P x y z x y z dSη = Φ∫ (7)

Now the development of the equations for aircraft motion is complete. The resulting state vector may be

written as bellow.

state vector = 1 2[ , , , , , , , ,...]TU V W p q r η η (8)

where thei

η ,i

η� are the responses of the structure which will be coupled to i

Qη through the aeroelastic

system equation and will be discussed in Section III. If we want to simulate the motion of the flexible

aircraft in near-real or real time, the unsteady general aerodynamics must be computed on-line. Of

course, we could use the quasi-steady theory to compute all of the unsteady generalized aerodynamic

coefficients, but for transonic flow the higher accurate method such as CFD is required. As we all know,

it cost too much for CFD-based approaches to solve the aeroelastic system equation, so it is impractical

for CFD to simulate the motion of flexible aircraft. Recently the reduced order model methods, such of

POD/ROM and Volterra/ROM, have made a progress in aeroelastic modeling and simulation. But in

order to achieve sufficient accuracy, the POD/ROM and Volterra/ROM can only be used to compute the

system response around the state at which the ROM is constructed. If they can be improved to simulate

aeroelastic system response in the state that is much different from the state at which the ROM is

constructed, we could simulate motion of the flexible aircraft with high accuracy in near-real or real time

using CFD-based approaches. ROM adaptation that will be discussed in Section III.B may serve as the

powerful tool to achieve this goal. Furthermore, it would also become an innovative and powerful tool

for flight control system design and continuous aeroservoelastic modeling and simulation of a flexible

aircraft.

Using the ROM model to compute the unsteady aerodynamics coefficient, the flexible vehicle model

can be translated into the following state space equation:

[ ]

.

.,

r rr re r r r

r e

er ee e e ee

x A A x B xu y C C Du

A A x B xx

= + = +

(9)

r

x represents the rigid body motion variables and e

x represents the structural dynamics variables and

unsteady aerodynamic states. There

A ander

A represent the interaction of rigid motion and aeroelastic

motion, referred as coupled terms. For conventional aircraft which is not very flexible, re

A and er

A are

much smaller than rr

A andee

A . So the coupled terms are often neglected and the flight control system

is usually designed based on the rigid model. But for flexible aircraft, the coupled terms should be

retained for the controller design.

III. ROM Model of a Flexible Aircraft

III.A. POD/ROM Method of Aeroelastic Modeling and Simulation

For a full-coupled nonlinear aeroelastic system of a flexible aircraft, the fluid flow over the aircraft

can be described as Euler or Navier-stokes equation in a finite volume as bellow:

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( )( ) ( ),

, , 0t

A u w F w u v+ =�� � � � �

(10)

where w�

is the conservative flow variables, F�

is the flux, A is a ，fluid cell volume u�

is the

structure general displacement, and v�

is the derivative of the structure general displacement.

Solving Eq. (10) to, we can obtain the following steady state solution:

( )0 0 0, , 0F w u v =� � � �

(11)

Supposing that ( , ,w u v∆ ∆ ∆� � �

) is a small disturbance around the steady state ( 0 0 0, ,w u v� � �

), we can

obtain the following linearized equation:

( )0 0A w Hw E C v Gu+ + + + =� (12)

( )

( )

( )

0 0 0

0 0 0

0

0 0 0

, ,

, ,

, ,

FH w u v

w

FG w u v

u

AE w

u

FC w u v

v

∂=∂

∂=∂

∂=∂

∂=∂

(13)

where 0A is the fluid cell volume in steady state. In order to simply the notation of the linearized

equation, , ,w u v is used to represent the perturbation , ,w u u∆ ∆ ∆� � �

, respectively.

Structural dynamic equation without damping can be written as follows:

( )int , ( , )extMv f u v f u w+ =� (14)

( ) ( )

int

0

t

0 0 0 0

( , )

( , ) , ,ext ext

ex

f u v K u

f ff u w u w u u w w

u w

= ∂ ∂ = + ∂ ∂

(15)

Suppose ( )0 0,ext

fP u w

w

∂=

∂, ( )0 0 0,

ext

s

fK K u w

u

∂= −

∂, the following fluid system equation can be

obtained:

[ ]

Tw Aw B v u

F Cw

= + =

� (16)

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1

0A A H−=− ， [ ]1

0B A E C G−=− + ，C P= (17)

Combine the structure and fluid system equations, the following full-coupled linearized aeroelastic

system equation can be obtained.

( )1 1 1

0 0 0

1 10

0 0

s

w A H A E C A G w

v M P M K v

u I u

− − −

− −

− − + − = −

�

�

�

(18)

CFD-based solution of Eq. (18) is too large for real time simulation of aeroservoelastic dynamics and

controller design for a flexible aircraft. Therefore, the full-order system has to be reduced. For

illustration purpose, the POD approach is used to reduce the aeroelastic system. Using one series data

{ },k k nx x ∈� in n-dimension spacen n×Φ∈ℜ , POD searches a m -dimension proper orthogonal

child spacen m×Ψ ∈ℜ to minimize the mapping errors from Φ toΨ :

1 1

min ,m m

k H k k H k H

k k

x x x x IΦ

= =

−ΦΦ = −ΨΨ Φ Φ=∑ ∑ (19)

Eq. (19) is equivalent to:

( ) ( )

2 2

2 21 1

, ,max ,

k km m

H

k k

x xI

Φ= =

Φ Ψ= Φ Φ=

Φ Ψ∑ ∑ (20)

Suppose that { }1 2 mX x x x= � is the snapshot matrix, then solving equation (20) is

equivalent to solving the equation of ( ) 0H

XX Iλ− Ψ = .Then the problem is transformed to find

the eigenvalue of POD kernelHK XX= . For high orders of

n nK

×∈ℜ , it is not easy to solve the

problem. Consider thatHXX and

HX X have the same eigenvalue, we can obtain Ψ as follows:

HX Xν ν= Λ (21)

1/ 2Xν −Ψ= Λ (22)

[ ]1 2 mψ ψ ψΨ= � , ( )1 2 mdiag λ λ λΛ= � , 1 2 mλ λ λ≥ ≥ ≥� . Truncate Ψ to r-order

vectors [ ]1 2r rψ ψ ψΨ = � , then the system represented by Eq. (18) is reduced to r-order

system:

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T T

r r r r r

r r

x A x Bu

y C x

= Ψ Ψ +Ψ = Ψ

� (23)

and the reduced r-order aeroelastic system can be obtained as follows:

( )

−−Ψ

Ψ−+Ψ−ΨΨ−

=

−−−

∞∞

−−−

u

v

w

I

KMCMPMV

GACEAHA

u

v

w r

Sr

T

r

T

rr

T

rr

002

1 1112

1

0

1

0

1

0

ρ

�

�

�

(24)

III.B. POD Adaptation Method

Although POD/ROM can greatly decrease computational time, constructing POD/ROM is still

computational intensive. Since POD/ROM is constructed by perturbations based upon flight parameters

(i.e. Mach number, Reynolds number and angle of attack) at a nonlinear steady state solution for the

coupled aeroelastic system, it is only accurate when the flight condition is sufficiently close to the

nominated steady state condition. Therefore, when the flighty condition changes, a new POD/ROM

needs to be re-constructed in order to better the approximation to the full-order system. To this end, we

may compute several POD/ROMs in a range of flight parameters with a specified interval, and then

obtain a new POD/ROM by interpolating two pre-computed POD/ROMs. This idea is very attractive,

and is particularly efficient for near-real or real time simulation of aeroservoelastic dynamics of a

flexible aircraft.

Lieu14

provided the numerical procedure to interpolate POD bases by means of subspace angle

interpolation. Golub and Van Loan 15

provided a definition for the distance between two equal-

dimensional subspaces that involves the notion of principal subspace angles 16

as follows:

Definition: The principal angles, ]2/,0[ πθ ∈k between two subspaces, F of dimension p and G

of dimension q with qp ≥ , of a unitary space are recursively defined for qk ,...,1= by

kkvu

k vuvu**maxmaxcos ==θ (25)

12

=u , 12

=v , subject to the constraints

0* =kjuu , 0* =kjvv , for 1,...,1 −= kj (26)

where the columns of the matrices ),...,( 1 quuU = , ),...,( 1 qvvV = are called the principal vectors of

the subspace pair.

Since the POD vectors form a unitary basis, the following theorem which was proven by Bjorck et

al.14

provides a means of computing the principal vectors and angles between two subspaces.

Theorem: Assume that the columns of AQ and BQ form unitary bases for two subspaces of a unitary

space mE . Furthermore, let singular value decomposition (SVD) of the qp × matrix,

*

A BQ Q , be

* *

A BZ=Q Q YΣ , ),...,diag( 1 qσσ=Σ (27)

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where qIZZZZYY ** === *. If the singular values are in descending order, then the principal

angles and principal vectors associated with this pair of subspaces are given by the following equations:

kk σθ =cos (28)

YQU A= , ZQV B= (29)

Thus, for two POD bases computed at different Mach numbers, )( 11 MΦΦ = and )( 22 MΦΦ = , the

principal angles and vectors are determined by computing the SVD of the correlation matrix, as 2

*

1ΦΦ ,

then as in Eq. (30) and Eq. (31),

* *

1 2 =Φ Φ YΣZ (30)

1YΦU = , 2ZΦV = (31)

Linearly interpolated principal angles, k

~

θ , can be computed according to the following relation.

kk

MM

MM θθ

12

~

)(−

= (32)

Each principal vector ku is then rotated towards its corresponding principal vector kv through the

interpolated rotation angle, k

~

θ , according to the following rotation formula to obtain the interpolated

vector, kw

k

kkkk

kkkkkkk

~

*

*~

sin)(

)(cos θθ

uuvv

uuvvuw

−

−+= (33)

And finally, the new interpolated POD basis is formed via the interpolated vector, { }kw , as follows:

( )M =Φ WZ , ),...,( 1 qww=W (34)

VI. Flight Control System and Active Control System

VI.A. Control System Design

The flight control system consists of an outer loop for flight path and attitude control and an inner

loop for stability augmentation and aeroelasticity control. Firstly the outer loop is designed, then the

inner loop is constructed according to aeroelastic model of the flexible aircraft. In present study, the

outer loop controller is supposed to have been determinate, and thus leave the task of the inner loop

controller as to track the aircraft flight trajectory command and to stabilize the aircraft.

There are two different types of inner loop controllers. Figure 1 presents the inner controller consisting

of attitude tracking controller and aeroelastic controller which use the same control surfaces to

accomplish the two tasks defined by its name. This inner controller will use the coupled rigid and

flexible motion model (e.g.,re

A and er

A are not zero). This inner controller may achieve good

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performance with high control gain for a very flexible aircraft on which the rigid body motion frequency

is very close to the structural vibration frequency. Figure 2 shows the cases for not very flexible aircraft

for which the active control system consists of two controllers. One is the ordinary stability

augmentation system for rigid aircraft, and the other is the active flutter/LCO suppression system (AFS).

When the aircraft flies beyond the flutter envelope, the AFS can be activated to suppress the structural

vibration. Most of the conventional aircrafts can apply this control strategy because the lowest structure

vibration frequency is far from the rigid body motion.

Figure 1. Integrated Flight Control System.

(FCS - Flight Control System, ACS - Active Control System = AFS)

Figure 2. Decoupled Flight Control System. (FCS – Flight Control System, SAS – Stability Augmentation

System, AFS – Active Flutter/LCO Suppression System)

The ACS may be realized through conventionally available control surfaces for flight control, such

as elevator, rudder and inner and outer ailerons. Symmetrically deflected inner and outer ailerons are

available as means of direct lift control and restricted to low authority aeroelastic control purposes.

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For rigid body flight dynamics, all states ( , , , , ,..., .)p q r etcα β are physical parameters and can be

directly measured. For flexible aircraft dynamics, the states of an aeroservoelastic system are defined in

generalized coordinates (defined as modal parameters), which do not have explicit physical

corresponds, and therefore not suitable for direct measurement. Velocity and acceleration of the

structure in physical degrees of freedom . . .

( , , )z z z are expressed in terms of modal coordinates by

. . .. ..

, ,z q z q z q= Φ = Φ = Φ (35)

VI.B. Outer Loop Controller

The outer loop controller design is usually based upon a decoupled model (the longitudinal and lateral

designs are decoupled), and on a coupled model (the longitudinal and lateral designs are coupled). We

focused on the design for coupled model, because the design based on the decoupled model is a

traditional approach that can be accessed in the literature. The outer loop controller uses elevator and

thrust for both flight path and speed control. Therefore, the outer loop control structure is assumed to be

determinate, while inner loop control is performed such as to satisfy the requirements for the outer loop

structure.

VI.C. Inner Loop Controller

The aeroelastic system model described by Eq. (36) with active controller model can be re-written as

bellow:

[ ]

asease

aseaseasease

xCy

qBxAx

=

+= δδ �� (36)

asex is state variables of the aeroservoelatic system include structure and aerodynamic

，states aseA 、aseB 、

aseC ，is state matrix, input matrix and output matrix. q is dynamics

pressure,δ ，is control surface deflection [ ]Tvuy = is structural displacement and velocity. The

order of the POD/ROM is still higher for controller design. Therefore, the POD/ROM must be further

reduced according to control theory. In this study, the balance truncation (BT) method was used to

further reduce the POD/ROM model, and output feedback control law is selected to stabilize the

aeroelastic system of aircraft.

1

1

N

i i

i

N

i i

i

K

k

δ ξ

δ ξ

=

=

=

=

∑

∑� �

(37)

iξ is the general displacement of the structure and ,

i iK k is the control gain. For different flight

conditions, the gain schedule method is used.

V. Simulation Cases

V.A. Flight Control System Test Using F-16 Fighter Plane

In this section, we present the results of a number of flight control system designs under various

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conditions, including different altitudes, wind turbulence, and model uncertainties. We will test the

effect of controller for gust turbulence, which may influence the longitudinal and lateral stability of the

aircraft. Wind turbulence was simulated by adding noise generated by the Dryden Wind Turbulence

Model block to the deflection of the rudder, which may result in deviations to both the yaw and roll

angles. Three different operating points, with varying altitudes and velocities, were chosen to represent

uncertainty models during vehicle flight (see Table 1). For each of these operating points, the pulling

up maneuver (altitude tracking) and coordinated turning were simulated.

Operating Points Altitude(feet) Velocity (feet/s)

1 15000 500

2 10000 350

3 30000 600

Table 1. Conditions of Three Operating Points.

To simulate the longitudinal dynamics of a F-16 fighter plane, a linearized F-16 model was

constructed for a speed of 500 ft/s and a height of 15000 ft. The control problem was posed to provide

robust performance with multiplicative plant uncertainty. The control variances include the deflection

of rudder eδ and throttle tδ ; thus, the dynamics equation of the system can be depicted as follows:

DuCxy

BuAxx

+=

+=� (38)

0 500 3.553 10 500 0 0 00 0 0 0 1 0 0

1.074 4 3.217 1 1.321 2 2.669 1.186 1.565 3 3.87 22.076 6 3.681 13 2.552 4 6.761 1 9.392 1 2.48 7 1.437 39.632 12 0 1.184 9 5.757 1 8.741 1 0 1.188 1

0 0 0 0 0 1 00 0

e

e e e e eA e e e e e e e

e e e e e

− −

− − + − − − − − −= − − − − − − − − − − − −

− − − − − − − − −−

0 0 0 0 2.02 1e

− +

0 00 00 00 00 00 00 2.02 1

B

e

= +

,

1 0 0 0 0 0 00 57.3 0 0 0 0 00 0 1 0 0 0 00 0 0 57.3 0 0 00 0 0 0 57.3 0 0

C

=

,

0 00 00 00 00 0

D

=

.

where the state variances are [ , , , , , , ]t e

X h v qθ α δ δ ′= , and the control variances are [ , ]t e

u δ δ ′= ,

outputs measurements are [ , , , , ]y h v qθ α ′= .

The first step was to transfer the actual flight control problem to the standard robust control problem.

Then, we compiled an algorithm, which uses the “Hinfsyn” function in the Matlab robust control

toolbox, to obtain the controller K :

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6.6353 66.8792 15.4958 43.0506 129.03893.0496 4.0258 8.9858 15.1149 9.6533

0 0 3.5028 0.6094 43.55930 0 0 3.4843 32.94290 0 0 0 0.2848

kA

− − − − − − − − −

= − − − − −

472.639 63.9101 73.5174 30.3486 39.600181.8371 10.7091 0.4404 4.8979 6.635624.3452 2.4045 26.7683 0.6758 1.489935.7647 5.2580 20.0907 2.7184 3.2580.0447 0.1471 4.8637 0.1439 0.0911

kB

− − − − − − − −

= − − − − − − − − − − − − −

0 0.0002 0.0054 0.0031 0.10310.0031 0.0088 0.0042 0.005 0.005kC

− − =− −

0 0 0 0 00 0 0 0 0kD =

Simulation 1: Effect of Gust Turbulence for the FCS Design

The simulation results for the coupled model, without and with the ∞H controller, are presented in

Figure 3 and Figure 4, respectively. With noise added to the rudder, Figure 3 shows that the roll angle

has reached the peak value of 6 degrees (see Figure 3a), and that the yaw angle has deviated from the

set course for about 15 degrees after 100s (see Figure 3b). In contrast, when the ∞H controller is

applied, the peak value of roll angle is reduced to about 0.7 degree (see Figure 4a); meanwhile the yaw

angle does not diverge and nearly maintains the set course (see Figure 4b). Figure 5 displays the control

process through visualization software Flight Gear.

(a) (b)

Figure 3. Simulation Result for the Coupled Model without Controller Applied to FCS

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(a) (b)

Figure 4. Simulation Result for the Coupled Model with ∞H Controller Applied to FCS.

Figure 5. Control Process Displayed by Flight Gear.

Simulation 2: Coupled Model at FL3 with Gust Turbulence and Model Uncertainties

Altitude Tracking. To examine model uncertainties, simulations for altitude tracking were used to

illustrate the robustness of the designed ∞H controller under different operating points. When the

altitude tracking command was given, the curves in Figure 6b reached the desired altitude rapidly and

accurately. Figures 6c-d show that the deviations from the trim values of roll angle and yaw angle are

tolerable and that the oscillating frequencies are acceptable.

Coordinated Turning (Roll Angle Tracking). The simulation for Coordinated Turning (Roll Angle

Tracking) was aimed to determine the performance of a coordinated turning by providing a desired roll

angle for the aircraft. Figure 7b shows that roll angle reached the desired value rapidly, and Figure 7a

and Figure 7d show that the AoA and pitch angle oscillated in a small range above and beneath zero,

respectively. The altitudes of the aircraft remained almost unchangeable during the simulation.

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Figure 6. Simulation for Altitude Tracking (Red Line for Operating Point 1, Green Line for

Operating Point 2, and Blue Line for Operating Point 3, and altitude in (b) is meter).

Figure 7. Simulation for Coordinated Turning (Roll Angle Tracking).

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V.B. The Active Flutter/LCO Suppression System (AFS) Control Test

In this section, we will use a full aircraft Advanced Fighter Aluminum (AFA) model to demonstrate the

effect of AFS for flutter/LCO suppression. The mode frequency of the first four elastic mode of AFA

model are 4.54Hz,8.19Hz,10.65 Hz, and 14.65Hz, as presented in Figure 8.

Figure 8. The First Four Elastic Modes for the Full Aircraft AFA Model.

The infinite plate spline (IPS) method and the beam spline method were used to generate the surface

mesh, which is shown in Figure 9. After the surface mesh is generated, the multiblock transfinite

interpolation (TFI) dynamic mesh method was used to re-mesh the whole transformed CFD grids

during the simulation.

Figure 9. CFD Surface Mesh of the First Four Modes for the AFA Model.

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The outboard control surface at the rear edge of wing is used to suppress the flutter/LCO of the

aircraft. The configuration and size of the wing and control flap is presented in Figure 10.

Figure 10. The Surface Mesh over the Wing and Control Flap.

We have computed the unsteady aerodynamics of given wing movements to construct the ROM

model for aeroservoelastic simulation and control design. Figure 11 shows: (a) a series of pressure con-

tour on the surface of the aircraft, and (b) the time history of the lift coefficient for the first mode.

t=0.05 t=0.10

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t=0.15 t=0.20

t=0.25 t=0.30

Figure 11 (a). Surface Contours on the aircraft at a Series of Time.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

CL

Time

Figure 11 (b). Unsteady Aerodynamics for Mode 1 Vibration.

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And finally, the discrete state-space form of the AFA model was constructed upon the ROM model. In

present study, a 125-order aeroelastic ROM model was constructed first and then was further reduced to

much lower order ROM using Balance Truncation for active flutter/LCO suppression system (AFS)

design. At the velocity V = 308 m/s, the AFA aircraft produces flutter. Figure 12 shows that the system

goes to divergence very quickly, and finally runs into LCO. The state graph of the AFA aircraft at this

state is presented in Figure 13. What next is to design AFS to reduce or eliminate the structural vibration

and thus enhance the stability of the aircraft.

Figure 12. Structure Response of the AFA Aircraft without Control at V = 308 m/s.

Figure 13. State Graph of the AFA Aircraft without Control at V = 308 m/s.

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Active Flutter/LCO Suppression System (AFS) Design

Figure 12 indicates that the instability of the AFA aircraft is caused primarily by the first two modes.

Thus, the simplest approach to stabilizing the system would be to weaken the coupling of the two modes

by adding a control flap moving according to the control law that is computed in flight. In this study, an

output feedback control law is used to stabilize the aircraft while avoiding the need of observation.

The control law may be represented as 2Kδ ξ= (K is the control gain and 2ξ is the general

displacement of the second structural mode of the AFA aircraft). From the simulation based upon the

ROM model of the AFA model, we selected -0.05 as the control gain. Similarly, we may select other

control gains according to the performance requirement.

The AFS was activated after the system ran into LCO. The control law restricts the deflection of the

control flap so that it does not exceed 10 degrees. This means that if the deflection is larger than 10 de-

grees, it will hold on 10 degrees. The general displacements of the first 3 structural modes are presented

in Figure 14. From Figure 14, it can be seen that the active control law based upon the ROM model has

successfully suppressed the LCO of the AFA aircraft.

Figure 14. Response of the AFA Aircraft with Active Control/Stability Augmentation System.

The Flight Trajectory Selection

In this section, a near-real time aeroservoelastic simulation of the AFA aircraft is investigated to

evaluate if the AFS system can suppress flutter. The vehicle flied from 0.7 Mach to 1.4 Mach in vertical

plane’s horizontal line trajectory with accelerating velocity in the altitude 3480 ft. The trajectory was

particularly selected to include the speed regimes between 0.9-0.96 Mach at which the aircraft will lead

to flutter as shown in Figure 12. The simulation results are presented in Figure 15. From Figure 15 it

can be seen that after 20s from the start, the pitch rate started oscillation at which the Mach number is

about 0.92 and the aircraft run into flutter. Almost at the same time as flutter started, the AFS system

was activated, then oscillation of the pitch rate was gradually reduced to zero, and thus no LCO

appeared. Although the aircraft flied beyond its safe envelope, it can still fly safely because AFS system

can suppress the flutter and thus LCO.

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VI. Conclusion

In the present study, the flight control system (FCS) and active flutter/LCO suppression system (AFS)

for full configuration aircraft have been established and demonstrated and the near-real time simulation

of aeroservoleatsic dynamics of aircraft has been implemented based on the ROM frame work. The

designed FCS was successful to track the pilot command without affection of disturbance and the AFS

system designed based upon ROM model can reduce or eliminate the flutter and thus LCO. Although

the selected trajectory is simple, the speed-up process through which the aircraft runs into flutter and the

suppression of flutter by AFS system were fully illustrated. Some of the basic elements for real time

simulation of aeroservoelasticity, i.e., the ROM, were pre-computed offline, and the flutter point of the

aircraft flight envelope was predicted offline as well. In order to conduct all computations for real time

aeroservoelastic simulation online with a PC, not only significant research needs to be continued in the

future, but also the computational speed of PC needs to make a breakthrough. A parallel computing

algorithm may need to be developed in the future work. Also it would be very interesting to see whether

the active flutter/LCO suppression system can be combined with the reconfigurable control law.

Figure 15. Real Time Aeroservoelatsic Simulation.

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Acknowledgement

This work was supported by National Aeronautics and Space Administration SBIR Phase II contract

NNX08CA58P. Any opinions, findings, conclusions, or recommendation expressed here are those of

the authors and do not necessarily reflect the views of the National Aeronautics and Space

Administration.

References

1. Dowell, E.H. and Hall, K.C., “Modeling of Fluid-Structure Interaction,” Annual Review of Fluid Mechanics,

Vol. 33, pp. 445–490, January 2001.

2. P. Holmes, J. L. Lumley, and G. Berkooz, “Turbulence, Coherent Structures, Dynamical Systems, and Symme-

try,” Cambridge University Press, New York, 1996

3. Sirovich, L. “Turbulence and the dynamics of coherent structures. Part 1: Coherent structures,” Quarterly of

Applied Mathematics. 1987, v45 i3. 561-571

4. Beran, P.S., “A Reduced Order Cyclic Method for Computation of Limit Cycles,” Nonlinear Dynamics, Vol.

39, No. 1-2, pp. 143-158, January 2005.

5. Hall, K.C., Thomas, J.P. and Dowell, E.H., “Proper Orthogonal Decomposition Technique for Transonic Un-

steady Aerodynamic Flows,” AIAA Journal, Vol. 38, No. 10, pp. 1853-1862, October 2000.

6. Thomas, J.P., Dowell, E.H., and Hall, K.C., “Three-Dimensional Transonic Aeroelasticity Using Proper Or-

thogonal Decomposition-Based Reduced-Order Models,” Journal of Aircraft, Vol. 40, No. 3, pp. 544-551,

May-Jun 2003.

7. Lucia, D.J., and Beran, P.S., “Reduced-Order Model Development Using Proper Orthogonal Decomposition

and Volterra Theory,” AIAA Journal, Vol. 42, No. 6, pp. 1181-1190, June 2004.

8. Lucia, D.J. and Beran, P.S., “Aeroelastic System Development Using Proper Orthogonal Decomposition and

Volterra Theory,” Journal of Aircraft, Vol. 42, No. 2, pp. 509-518, Mar-Apr 2005.

9. Silva, W. A. and Bartels, R. E., “Development of Reduced-Order Models for Aeroelastic Analysis and Flutter

Prediction Using the CFL3Dv6.0 Code,” AIAA Paper 2002- ．1596

10. Raveh, D.E., “Reduced Order Models for Nonlinear Unsteady Aerodynamics,” AIAA paper 2000- ．4785

11. Hu, G. and Xue, L., “Theoretical and Developer Manual of ASTE-P Software Tool Set,” Second Edition, Ad-

vanced Dynamics Inc, 2007.

12. Hu, G. and Xue, L., “Integrated Variable-Fidelity Tool Set for Modeling and Simulation of Aeroservothermoe-

lasticity - Propulsion (ASTE-P) Effects for Aerospace Vehicles Ranging from Subsonic to Hypersonic Flight,”

NASA SBIR Phase I Final Report, Advanced Dynamics Inc, 2007.

13. Schmidt, D. and Raney, D., “Modeling and simulation of Flexible Flight Vehicles,” AIAA Journal of Guid-

ance, Control, and Dynamics, Vol. 24, No. 3, May-June 2001, pp. 539-546.

14. Lieu, T. “Adaptation of reduced order models for applications in aeroelasticity,” Ph.D. thesis, University of

Colorado at Boulder, 2004.

15. Golub, G.H. and Van Loan, C.F. “Matrix Computations,” The John Hopkins University Press, second edition,

1989.

16. Bjorck, A. and Golub, G.H., “Numerical methods for computing angles between linear subspaces,” Mathe-

matics of Computation, 27(123):579-594, 1973.