DAMPING AUGMENTATION OF FLEXIBLE STRUCTURES— A ROBUST STATE SPACE APPROACH —

Udo B. CARLMarcus H. GOJNY

Technical University of Hamburg–HarburgInstitute of Aircraft Systems Engineering (2.08)

21071 HamburgGermany

Email: gojny@tu–harburg.de

The present paper addresses the regulation of weakly damped aeroelastic wing structures by means of the primary

flight controls (PFC). Synthesizing a sufficient low–order linear multi model system of the aggregate aeroservoe-

lastic plant requires the design of a robust state feedback. This ensures a remarkably augmented damping ratio in

comparison with the original aeroelastic system. Moreover, the influence of the actuation system performance on

the aeroservoelastic damping augmentation feedback is investigated. The suitability of the resulting controller is

verified by simulation as well as validated by real tests.

Keywords: aeroservoelasticity, flexible aircraft, analytical modelling, structural vibrations, typical

section, primary flight control actuation system, hydraulic actuator, uncertain paramters, robust

control, parameter space design

1 INTRODUCTION

Aeroelastic control can be more challenging than conventional controlled structures prob-

lem, in that the dynamics of the system change dramatically with the flight conditions. It

holds the promise of significant improvements in performance: reducing the ambient vibration

level, increasing the maneuver responsiveness and stabilizing an otherwise unstable system [4].

Considering the usual mission of a large commercial aircraft, aerodynamic forces and

moments entail a substantial deformation of the elastic structures of fuselage, tailplane and

wing. Each incremental change of the structural shape yields a new aerodynamic state which

causes aeroelastic interaction. Due to the increasing sizes of transport aircrafts, the spectral

gap between flightmechanical motion and internal structural modes decreases continuously.

Hence, dynamically coupled modes of the flexible aircraft arise, which can be excited by

gust loads as well as manoeuvres and flight mechanical stability augmentation functions.

These oscillatory modes are usually characterized by weak damping without supplementary

measures. The resultant vibrations reduce the fatigue life of the structure and may lead, under

worst conditions, to a complete loss of the aircraft’s controllability. In order to solve this

problem for light weight structures, new perspectives aim at an additional functionality of

the primary flight control surfaces, to ensure an active modal damping augmentation. Due to

changing flight operation conditions, as e.g. speed, and masses of the dynamic system by e.g.

fuel in the wing, the physical parameters of the present plant vary considerably. Moreover,

the required control circuit consists of a substantial nonlinear electrohydraulic servo actuation

system. Although the complete system is characterized by high order, nonlinearities and

significant parameter uncertainties, a linear controller of low order is aspired. In order to

outline a general approach, the basic plant consists of a quasi–stationary formulation of the

aerodynamics linked together with a mechanical airfoil model, which is connected with the

model of the electrohydraulic actuation system. Adapting its essential overall properties to a

seventh order linear multiple model system, allows the design of a robust static state feedback

which provides the desired additional active damping function.

The paper is organized as follows: The basic aeroservoelastic plant is presented in section 2.

The performance specifications and principals of the applied controller design procedure are

outlined in section 3. Section 4 presents the practical validation of the closed loop system as

well as current PFC actuation system aspects. Concluding remarks finalize this paper.

2 MODEL AGGREGATION

2.1 Aeroelastic model

Although, in practice the dynamic instability of an elastic body in an airstream usually de-

scribed as flutter, is a complicated phenomenon, this approach is based on a quite simple

aeroelastic system. It exhibits some of the dominat properties, which can be met at a current

flexible wing configuration in more formidable guise. With regard to an experimental variable

camber wing of large aspect ratio, the incisive reduction leads to a two–dimensional structure,

which is characterized by the center of gravityC, the elastic axisE, the aerodynamic centerA

and the hinge lineG (figure 1). Immediately, this structure can be adapted to therepresentative

(a) Swept, twisted wing

� �� �

(b) Uniform cantilever wing

Figure 1: Mechanical structure

� �

�

�

� �

� � �

� �

�

(a) Representative section [6]

� �

��

� �

�� � � � �� � � � � �� �� � � � �

(b) Quasistationary aerodynamics

Figure 2: 3DOF aeroelastic model

section[2], a rigid airfoil section, suspended in an airstream and with degrees of freedom in

bending and torsion by suitable suspension from two sets of springs. Assigning the geomet-

ric and inertial properties of this system to a wing cross section at three–quarters of the wing

span yields a basic mechanical representative dynamic model [6]. Hence, the set of generalized

coordinates is denoted by a downward vertical displacementh of the line of attachment (E),

a leading edge up angular rotationθ about this line and the trailing edge downward aileron

deflectionζ

q =

hh θ ζ

iT: (1)

In further discussion the latter will be ommitted by inserting a suitable actuator representative.

With the total stiffnesskh; kθ, the total massm, static unbalanceSPθ and mass moment of inertia

J Pθ about the generalized reference pointP 2 fA; E;C; Gg, the sum of potential energy of

strainV and the kinetic energyT can be expressed as function of the generalized coordinates

and its derivatives. By assumption that small deflections are to be expected, the application of

Lagrange’s Equations

�ddt

�∂ (T �V )

∂ ˙q

�+

�∂ (T �V )

∂ q

�+Q = 0 (2)

on this lumped parameter system in the presence of airstream [4] results in a non conservative

linear system. The mechanical equation of motion reads:26664

m SEθ SG

f

SEθ J E

θ (xθ+aζ)SGf +J G

f

SGf (xθ+aζ)SG

f +J Gf J G

f

37775

| {z }MS

¨q+ � � �

� � �+

26664

dh 0 0

0 dθ 0

0 0 �

37775

| {z }BS

˙q+

26664

kh 0 0

0 kθ 0

0 0 �

37775

| {z }KS

q =

26664

�L

M+eL

MH

37775 : (3)

In this case the (�) elements in (3) will be determined by the electrohydraulic actuation system

model. The right hand side of (3) denotes the liftL, the summarised aerodynamic momentum

M and the hinge momentumMH .

The essential features of the aerodynamic physics are described by a quasi–stationary

model which reflects incompressible wing–aileron lift characteristics [4]. The effective angle

of attackα(t) then results as sum of a steady state initial value and additional terms from wing

torsion and bending

α(t) = α0+θ(t)+h(t)U

: (4)

The coupling between the aerodynamic forces and the generalized mechanical motion leads to

a set of static gain matrices26664

�L

M+eL

MH

37775 = U

26664

�ρ2 S cLα 0 0

ρ2 S (ccMα +ecLα ) 0 0

ρ2 S f c f cHα 0 0

37775

| {z }BA

˙q+ � � � (5)

� � �+U 2

26664

0 �ρ2 ScLα �ρScLζ

0 ρ2 S (ccMα +ecLα ) ρS

�ccMζ +ecLζ

�

0 ρ2 S f c f cHα ρS f c f cHζ

37775

| {z }KA

q+U 2

26664

�ρ2 ScLα

ρ2 S (ccMα +ecLα )

ρ2 S f c f cHα

37775

| {z }FA

α0 :

Assembling (3) and (5), results in the aeroelastic equation where the homogeneous aerody-

namic terms have been moved to the left hand side�MS s2

+

�BS�U BA

�s+

�KS�U 2 KA

��q(s) = U 2 FA α0 : (6)

Bearing in mind that the aileron deflection is due to the control law of the electrohydraulic

PFC actuation system,ζ truely does not represent a third degree of freedom (DOF) within the

present system. Thus, the 3DOF system (3) must be reduced to a 2DOF plant (q ! q). The

third column elements of (6), which describe the inertia coupling between aileron motion and

wing, then must be considered as additional input. Whereas the equation represented by the

third row elements of (6) describe the load input of the PFC actuation system, the third column

�

�� ����

� � � � � � � �

��

��

����

Figure 3: General operating domain

contains the inertia effects of a control surface deflection on the wing. Therefore, regarding

the aeroelastic subsystem, the number of generalised coordinates (1) decrease. Defining the

structural state and subsystem input vector

xS =

hqT qT

i=

hh θ h θ

iT; uS =

hζ ζ α0

iT(7)

and considering the resulting 2–by–2 mass, damping and stiffness matrices yield the equivalent

state–space realisation

xS =

24 0 I

�M �1S

�KS�U 2KA

��M �1

S (BS�U BA)

35xS + � � � (8)

� � � +

2666664

0 0 0

0 0 0

�ρU 2 ScLζ SGf �

ρ2 U 2 ScLα

ρU 2 S�

ccMζ + ecLζ

� �xθ +aζ

�SG

f + JGf

ρ2 U 2 S (ccMα + ecLα )

3777775uS

yS = C xS +D uS ; xS(0) = 0 :

−5 −4 −3 −2 −1 0 1 25

10

15

20

25

bending

torsion U →

← U

U ∈ [0, 330] m s−1

Re {s} →

Im {s

} →

(a) Pole zero map, root loci

0 100 200 300

0

0.02

0.04

0.06

U [m s−1] →

σ [−

] →

0 100 200 3000

5

10

15

20

25

U [m s−1] →

ω [r

ad s

−1]

→

(b) Damping and eigenfrequency

Figure 4: Structural eigenmodes vs. true airspeed

Scrutinizing the parameter dependencies of (8) leads to a significant insight. Obviously, the

eigenvalues of (8) are severely effected by a few uncertain parameters:

� airstream velocityU 2 fU�

;U+g and squared valueU 2, respectively

� massm 2 fm�

; m+g, mainly effected by the varying fuel mass.

Figure 3 outlines the corresponding operating domain. For reasons of simplicity, the latter un-

certain parameterm is initially assumed to be fixed, because it varies quite slowly due to fuel

consumption. This leads to two dominant parameter dependent root locus branches within the

complex plane (figure 4(a)). When the torsion branch crosses Refsg = 0, the allocated air-

speed readsU =Ucrit. The damping of this eigenmode shifts its sign and the system becomes

instable. Simultaneously, the absolute values of the poles tend to converge (figure 4(b)). As

damping decreases rapidly with speed near the critical speedUcrit, the certification regulations

require a significant margin betweenUcrit and the maximum dive speedUD of the aircraft, say

Ucrit � 1:15UD.

2.2 Primary flight control actuation system

Exhaustive investigations have been made to improve the performance of the electrohydraulic

actuation system and realization aspects [11, 12, 13]. Based on robust state control and esti-

mation techniques the bandwidth of a current system could be trebled with simultaneous im-

provements in damping characteristic and minimal phase lag. Additionally, the resulting closed

actuator control loop ensures robust properties regarding uncertain system parameters caused

by e.g. fluid temperature. Further details outline [10, 12] et al.. Assuming low–level signal range

operation, a linear third order model represents the PFC actuation system. With the demanded

control surface deflectionζc(t), the external loadF(t) as input and the actual deflectionζ(t) or

its derivatives as output, respectively, it leads to a multiple input single output description

Z(s) = Gc(s)Zc(s)+Gd(s)F(s) =

hGc(s) Gd(s)

i| {z }

GA(s)

24 Zc(s)

F(s)

35: (9)

This characteristic yields a 1–by–2 transfer function matrixGA(s), whereGc(s) describes the

command transfer characteristic andGd(s) is synonymous to a complex stiffness [19]. Figure

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � �

� � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � �

� � � � � � � � � � � � �

� � � � � � � � � � � � � � � �

Figure 5: Electrohydraulic PFC actuation system (test rig)

0.1 0.2 0.3 1 2 3 10 13 20−10

−8

−6

−4

−2

0

2

mag

nitu

de [d

B]

frequency [Hz]

measurementsimulation

0.1 0.2 0.3 1 2 3 10 13 20 45

0

−45

−90

−135

−180

phas

e [d

eg]

measurementsimulation

(a) command transfer function

0.1 0.2 0.3 1 2 3 10 13 20−200

−190

−180

−170

−160

mag

nit

ude

[dB

m/N

]

frequency [Hz]

measurementsimulation

0.1 0.2 0.3 1 2 3 10 13 20 90

45

0

−45

−90

−135

phas

e [d

eg]

(b) disturbance transfer function

Figure 6: Measured PFC actuation system frequency responses

6 shows the Bode plots of both transfer functions comparing analysis and experimental results.

Measurement results are based on a conventional inboard aileron actuation system of theAir-bus A330/340installed on a test rig at the TUHH Institute of Aircraft Systems Engineering

(figure 5). The inertia of control surfaces can be imitated by the exchangeable disk mass and

actuator loads are controlled by a dynamic load simulation. Torsion stiffness of the aileron spar

and flexible adjustment of the actuators to the wing box can be simulated by shaft flexibility

and proper adjustment. The test rig equipment comprises a real–timehardware–in–the–loopsimulation software which allows coupling to large–scale simulation models of flexible aircraft

structures. The transfer function matrix of the real system reads approximately

GA(s) �1

a3s3+a2s2+a1s+a0

hb0 f1 s+ f0

i: (10)

The coupling of aeroelastic and actuation system can be deduced from (3). Therefore, the actu-

ator load leads to

F(t) =MH(t)

r?

�SGf h�

��xθ+aζ

�SG

f + J Gf

�θ ; (11)

with r?

representing the control surface lever.

2.3 Aggregate aeroservoelastic model

Finally, the entire aeroservoelastic model comprises a fourth order wing model, aerodynamics

with a constant feedthrough characteristic and a reduced, third order closed loop actuation

��

�

�� �

� � �

� � � � � � �

�

�

�

�

�

�� �

�

�

� � � �

� � � � � � �

�

�

� �

� ���

� �

� � � � � � � � �

� � � � � � � ��

�

� �

�

��

��

�� �

�� �

Figure 7: Blockdiagramm of the assembled model

system model. The resulting state, input and output vector reads

x =

hh θ ζ h θ ζ ζ

iT; u =

hζc α0

iT; y =

hh θ h θ

iT: (12)

Assuming the sensors to be ideal grants the availability of necessary state information: vertical

displacement and rate, torsion angle and rate respectively. Thus, the aeroservoelastic plant can

be summarized in a seventh order linear multi model system

x(t) = A(p)x(t)+B(p)u(t)

y(t) = C(p)x(t)+D(p)u(t) ; x(0) = x0 ; (13)

wherep= [p1; p2] =�U; U 2

�denotes the present uncertain parameter vector [1]. The interact-

ing quantities within the model (13) and the interdependencies of the submodels are illustrated

by the block diagram shown in figure 7. Assuming the angle of attackα0 = constant the fre-

quency domain input–output modelling yields a 4–by–1 transfer function matrix

Y(s) =

hGh(p; s) Gθ(p; s) Gh(p; s) Gθ(p; s)

iT

| {z }G(p;s)

Zc(s) ; (14)

whereG(p; s) denotes the ”polynomial plant family” [1]. Analysing the eigenvalues of the ag-

gregate model yields the aeroelastic modes maintaining their dominant character. As the actu-

ation system mainly operates as low–pass filter embedded within the forward path of the entire

aeroservoelastic system and has significant faster eigenvalues compared to the aeroelastic struc-

ture, the objective of the subsequent section focusses on a significant damping augmentation of

the aeroelastic modes: bending and torsion (figure 4).

� � �

� �

�

� �

�

� �

�� �

�

Figure 8: Damping augmentation control loop

3 VIBRATION CONTROL

In order to accomplish the outlined goals, the uncertain plant family requires a parametric

robust control law. Keeping pragmatic realisation aspects in mind, a low order compensator is

preferred, because an increase in order results in additional closed–loop eigenvalues that must

be robustly stabilised, too [1]. The preceding analytical modelling of the aeroservoelastic plant

favours the application of the Parameter Space Design method [1]. Aiming at a minimum order

controller the basic approach consists of a static state feedback (figure 8)

K(s) �! k =

hkh kθ kh kθ

i: (15)

3.1 Assumptions

With regard to the aeroelastic eigenmodes (figure 4), the dynamics of the state controlled ac-

tuation system reveal a very high bandwidthfB � 12 Hz. Considering the third basic rule of

robust control [1]:

Be a pessimist in analysis, then you can afford to be an optimist in design,

an incisive simplification only concerning the controller design process can be postulated:

GA(s) �! GA :=h

1 0i: (16)

If any robust stabilisingk exists, subsequent examination must focus on the effect of the ac-

tuation systems command and disturbance performance. Moreover it is assumed, that the state

quantities of the simplified aeroelastic plant are completely measurable. This implies no re-

striction, because the corresponding states can be derived from integration of the accelerometer

signals. The principle structure of the damping augmentation feedback control loop displays

figure 8. The input of the actuation system is composed of the basic aileron deflection com-

mand signalζc and the weighed state feedback�ky (12), which acts as a compensator relating

to the critical eigenmodes of the wing structure (figure 4).

3.2 Parameter boundaries

The uncertainty domain is comprised by a lower and upper boundary�

p�i ; p+i; i = 1; 2.

Even though the entire operating domain of the flexible aircraft coversU 2 f0;UDg, the

� �

�

� � ��

� �

�

� � � � � � � � � � � � � �

Figure 9: Modified operating domain of dependent parametersp

main effect of the feedback controller is desired to augment the damping at the interval

U�

< UD < Ucrit < U+;U �

> 0. As the open loop system indicates a stable characteristic

for all valuesU < Ucrit, the lower boundary is raised fromU�

= 0 m/s to some higher value,

e.g.U�

= 200 m/s, whereas the upper boundary must cover a safety margin with respect to

the critical speed:U+= 330 m/s. It should be emphasized, that the resulting operating domain

contains stable and unstable representatives (figure 4). The general representation in figure 3

degrades to an unequivocal assignmentp2= f (p1), because of the declaration ofp. This results

in an one–dimensional parameter spaceQ. Due to practical considerations thep–dependency

will be included by a finite number of operating pointsn distributed betweenfp�1 ; p�2 g and

fp+1 ; p+2 g along the outlined parameter trace (figure 9). Hence the problem reads: simultane-

ous stabilizing of a finite plant family [1]. The parameter effects on the open loop eigenvalue

characteristic are illustrated in figure 4(b).

3.3 Performance specifications

Generally, the demands on the closed loop time response of a linear time–invariant system are

formulated indirectly by eigenvalue specifications. With regard to the root locus design of a

single–input system pole placement yields a unique solution for state feedback. Unfortuately,

this unique solution refers to a single nominal plant. Assuming an uncertain plant family it

is a reasonable requirement that all eigenvalues are located within a specified regionΓ. The

pole region in figure 10(a) is bounded by a circular arc and its damping by a hyperbola that

guarantees damping according to the asymptotes and negative real part [1].

Bearing in mind that the real system also obtains an electrohydraulic actuation system

within the control loop, the requirements are shaped according to the second basic rule of

robust control [1]:

When you close a loop with actuator constraints, leave a slow system slow and

leave a fast system fast.

Moreover, the quantitative description of the desiredΓ–region is dominated essentially by the

demand on the augmented damping characteristic. For the analytical modelling a structural

damping factorDS � 0:02 in the absence of any airstream is assumed [6]. Hence, a moderate

minimal damping heuristically readsD�

= 0:1 covering the entire operating domain (figure

10(a)). The maximum permissible dynamic of this loop is prescribed by the bandwidth of the

PFC actuation system. In order to follow the statement of separated design, the latter should

be three to four times higher, than the maximum bandwidth of this loop withfmax� 3:5 Hz.

Preserving unnecessary high loop gain the negative real part denotes

2π fS = mini=1:::n

(jsij)D�

� 0:18 Hz; n = 10 :

−25 −20 −15 −10 −5 0

−20

−10

0

10

20 U = 200

U = 200

Γ

Re{s} →

Im{s

}

→

(a) Pole zero map, open loop

10 20 30 40−400

−300

−200

−100

0

100

200

300

κB [−] →

κA [−

]

→

(b) Invariance plane, complex (—) and real (��) mar-gins

−25 −20 −15 −10 −5 0

−20

−10

0

10

20 U = 200

U = 200

Γ

Re{s} →

Im{s

}

→

(c) Pole zero map, closed loop

Figure 10: SimultaneousΓ stabilisation

Finally, the completeΓ–pole region is outlined in figure 10(a), it is bounded by the edge∂Γ and

overlapped by the open loop aeroelastic root loci.

3.4 SimultaneousΓ–stabilisation

The core of the Parameter Space Design Method represents the derivation of a controller, which

shifts the entire set of open loop eigenvalues into the desiredΓ–region. Assuming any static

state feedbackk ((15) and figure 8 respectively) the objective is to determine the setk 2 KΓ for

which the polynomial family in closed loop configuration

G(s;p( j);k) ; j 2 [1;n] is Γ–stable:

Considering theBoundary Crossing Theoremit suffices to map∂Γ of each representativep( j)

into the new parameter space, thek–space [1]. The set of robustΓ–stabilising controllers re-

sult from a superposition of all related mapsK ( j)Γ . Unfortunately, theBoundary Representation

Theoremaccomplishes this mapping only for two controller coefficientski; i2 [1;4] simultane-

ously. Since a reduction of free controller parameters is not possible, a two–dimensional cross

section plane is selected in the state feedback gain space, where the boundaries are displayed.

This invariance planeapproach becomes a sequential procedure, identifying the most critical

eigenvalues, here the torsional motionλ(p+1 ; p+2 ) and successively shifting two eigenvalues

into the desired region. Decomposing the pre–image of theΓ–region boundary yields four sub-

sets (figure 10(b)): Two dashed straight lines, the real margins, which represent Imf∂Γg = 0

and two solid graphs, the complex margins, which relate to the hyperbola and circle branches

Imf∂Γg 6= 0. Thus, selecting[κB; κA] from the intersection of all∂Γ images effects a shift of

these eigenvalues, while the rest remain fixed [1]. Finally, the simultaneousΓ–stabilising con-

troller is determined indirectly by choosing an operating point from the admissable set within

the invariance plane. The total state feedback gain vector therefore results from the summation

of N iterative design steps

k :=N

∑m=1

km ; where km =

(km 2 KΓ

�����n\

j=1

κ ( j)Γ

); n = 10 : (17)

3.5 Verification

Bearing in mind that the outlined design procedure only represents a necessary condition

requires an adjacent verification of the closed loop performance. Selectingk = k � ((+) shown

in figure 10(b)) as representative candidate yields the closed loop eigenvalues in figure 10(c).

Although the pole zero map reveals thatG(s;p( j);k �) ; j 2 [1;n] meets the requirements, a

nonlinear simulation is essential for verification, including the actuation system characteristics.

Figure 11 shows the system response following a step inputα0. ForU = Ucrit the open loop

system executes coupled bending/torsion oscillations as expected. By means of the robust state

feedback controller a well damped time response is achieved.

0 1 2 3 4 5−1

0

1

2

3

U = Ucrit

bendingtorsionailerondeflection

t [s] →

θ, ζ

[ o

], h

[10

−1 m

]

→

(a) open loop

0 1 2 3 4 5−2

−1.5

−1

−0.5

0

0.5

1

U = Ucrit

bendingtorsionailerondeflection

t [s] →

θ, ζ

[ o

], h

[10

−1 m

]

→

(b) closed loop

Figure 11: Simulatedα0 step response

Returning to the preliminary assumption (16) raises the question to what extent the ro-

bust aeroservoelastic control loop characteristic depends on the PFC actuation system

performance. While keeping the outer loop gain vector constantk = k �, the controller of the

actuation system depends on the adjusted bandwidthfB 2 [3; 5; 8; 12; 15] Hz. To demonstrate

the effect of actuator bandwidth regulated by a robust state controller and for comparison with

a proportional controller offB = 3 Hz, figure 12 shows the root loci of the closed loop system.

Although the low–pass characteristic ofGA(s) increases with decreasing bandwidth, a robust

state feedback within the inner actuator control loop allows to decreasefB half to one of todays

robust state space controllers for adequate performance. Certainly, the prompter the actuation

system response the smaller is the phase lag of the compensating aileron deflection. Thus,

a high actuation system bandwidth is advanageous by increasing the damping of the torsion

!

� �

� !� �� !� � ��

!

�

� !

� � � " � ! � # $

� � � " � � � # $

� � � " � % � # $

� � � " � ! � # $

� � � " � & � # $ � � ' �

(�

)�*

+ � ) � *

� � � � � , � & & � � � � � � �

� � � � �

� � � � � �

' - . � � � � / � � � � /

�

Figure 12: Influence of PFC actuation system performance

trajectory with increasing actuator performance. For the proportional controlled actuation

system the actuator bandwidth is similar to the torsional eigenfrequency of the aeroelastic

structure. In this case an aeroelastic feedback controller in unable to stabilize the system for

the whole flight regime considered.

4 VALIDATION

To demonstrate the feasibility of the previously developped damping augmentation control

loop, an implementation of the outlined control law has been performed on the test rig. This in-

cluded the wing structure, sensors and aerodynamics. The PFC actuation system test rig (figure

5) has been upgraded to a real–time system. Two usual personal computers withIntel Pentium

III processors atfT = 450 MHz, which are linked together in target–host configuration, provide

the universal real–time simulation platform based onMatlab/Simulink. The robust state con-

trolled actuation system operateshardware–in–the–loopwithin a virtual dynamic wing struc-

ture simulation in the presence of quasistationary aerodynamics. A modular architecture of the

simulation model simplifies the transfer to the test set–up, as only the actuator model has to be

replaced by the test rig I/O module. The actuator control law as well as the damping augmenta-

� � & 0 ! 1

� � � � �

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(a) robust state controlled PFC actuation system

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(b) current PFC actuation system, proportional SISO controller

Figure 13: Measuredα0 step response atU �Ucrit.

tion feedback is realized within the simulation environment. The validation focusses especially

on two aspects: practical functionality, that means confirmation of the simulation results with

physical actuators in the loop. Beyond that, further examination of the interaction of a propor-

tional SISO controlled actuator and the presented damping augmentation control loop should

be done. In order to ensure comparability to the simulation results (figure 11), the measured

time response applies to the initial step in angle of attack att = 1 s. Figure 13 illustrates the

superior effect of damping augmentation by the robust state controlled actuation system. By

fast actuation system dynamics the compensation signalζ(t) effectively operates as an active

oscillation damper, whereas the significant phase lag of a conventional controlled PFC actua-

tion system causes further exitation of the coupled oscillatory motionh; θ. These experimental

results confirm the conclusions already gained from linear analysis (figure 12).

5 CONCLUDING REMARKS

It has been demonstrated that active damping augmentation by primary flight control surfaces

could efficiently effect the weakly damped aeroelastic wing motions. The deduction of the

aeroservoelastic synthesis model from principles of physics led to an analytical plant descrip-

tion. The resulting uncertain seventh order multi–model–system comprised originally stable

and unstable representatives. As physically motivated boundaries could be determined for the

uncertain parameters,Ackermann’s Parameter Space Design method yielded a set of robust sta-

bilizing state feedback vectors. A significant damping augmentation was verified by simulation

with the actuation systemhardware–in–the–loop. Practical investigations revealed the consider-

able influence of the actuation systems performance. Especially, the common SISO controller

of current electrohydraulic actuation systems caused an unsufficient phase lag, which led to

instability of the entire system. The aeroelastic compensation feedback grants a fast rejection

of exogeneous perturbations in connection with a double bandwidth than applied in actual

actuation systems. Finally, it has been demonstrated that the implemented controller design

method was capable of shaping a suitable actuator controller as well as an aeroelastic feedback

which simultaneously provides damping augmentation throughout the entire uncertain parame-

ter set. The ongoing investigations focus on elaborating enhanced models of current flexible

wing structures and examinating optimal sensor locations. Future work will also aime at ap-

plying suitable order reduction methods to preserve a state controller which mainly effects the

critical modes of the aircraft and can be transferred to an equivalent output controller.

ACKNOWLEDGEMENT

The author thanksDaimlerChrysler Aerospace Airbus GmbHfor promoting and supporting the

projectAktuatorregelung in aeroelastischer Umgebung.

NOTATIONS AND ABBREVIATIONS

Notationsα [� ] dynamic angle of attackα0 [� ] initial angle of attackaζ [m] distanceE GA [�] system matrix — state space representationB [�] input matrix — state space representationB [�] damping matrixc [m] distance from leading to trailing edge, hinge line to trailing edgecL; cM ; cH [�] aerodynamic coefficientsC [�] output matrix — state space representatione [m] distanceAED [�] damping coefficientD [�] feedthrough matrix — state space representation∂Γ [�] stability boundaryf [Hz] frequencyΓ [�] stability regionG [�] SISO transfer functionG [�] transfer function matrixθ [� ] leading edge up angular rotationh [�] downward vertical displacementJ [kg m2] moment of inertiaκA;B [�] basis of the invariance planek [kg/s2] mechanical stiffnessk [�] vector of feedback gainsK [�] stiffness matrixL [N] aerodynamic liftM [N m] aerodynamic momentMH [N m] hinge momentM [�] mass matrixm [m] total massp [�] uncertain parameter representativep [�] vector of uncertain parametersq [�] vector of generalized coordinates (2DOF)q [�] vector of generalized coordinates (3DOF)Q [�] vector of unconservative forcesρ [kg/m3] air densityr? [m] equivalent control levers [rad/s] complex frequencys = σ� jωS [kg m] static unbalanceS [m2] equivalent wing or aileron aera (vertical projection)T [Nm] kinetic energyU [m/s] equivalent airspeedUD [m/s] maximum dive speedu [�] input vectorV [Nm] potential energy of strainxθ [m] distanceE Cx [�] state vectory [�] output vectorζ [� ] trailing edge downward aileron deflectionZ [�] Laplacian ofζ

Indices(Note: The listed symbols may be used as lower as well as upper index.)

� [�] lower boundary+ [�] upper boundaryα [�] angle of attackA [�] aerodynamicB [�] bandwidthc [�] commandcrit [�] criticald [�] disturbancef [�] flap, aileronθ [�] angular rotationh [�] vertical displacementmax [�] maximummin [�] minimumP [�] reference pointP 2 fA;E;C;GgS [�] structuralζ [�] aileron deflection

AbbreviationsA [�] aerodynamic centerC [�] center of gravityE [�] elastic axisG [�] hinge lineDOF [�] degrees of freedomI/O [�] input–outputPFC [�] primary flight controlSISO [�] single input single outputTUHH [�] Technical University of Hamburg–Harburg

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