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Aeroelasticity
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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
REFERENCES
[1] Ackermann, J. et. al.:Robust Control — Systems with Uncertain Physical Parameters. London: Springer, 1993.[2] Bisplinghoff, R.L.; Ashley, H.; Halfman, R.L.:Aeroelasticity. Mineola: Dover Publications, 1996.[3] Bossche, D.; El Sherif, F. et. al.: Airbus A330/340 Primary Flight Control System — Aileron Actuation.SAE Committee A6. At-
lanta/USA, 1991.
[4] Dowell, E.H.(Ed.):A Modern Course in Aeroelasticity. 3rd edition. Dordrecht: Kluwer, 1995.[5] Foersching, H.W.: Einfluss servomechanischer Steuerungs– und Stabilitaetssysteme auf das Flatterverhalten von Flugzeugen.Z. Flug-
wiss.21 (1973), S.22–31, 1973.[6] Foersching, H.W.:Grundlagen der Aeroelastik. Berlin: Springer, 1974.[7] Gelb, A.; Vander Velde, W.E.:Multiple–Input Describing Functions and Nonlinear system Design. New York: McGraw Hill, 1968.[8] Green, W.L.:Aircraft Hydraulic Systems. Chichester: J. Wiley & Sons, 1985.[9] Guillon, M.: Hydraulic Servo systems — Analysis and Design. London: Butterworths, 1968.
[10] Kliffken, M.G.; Kruse, U.: Robust Control of Electro Hydraulic Actuators in Primary Flight Controls.at—Automatisierungstechnik11/45, 1997.
[11] Kiffken, M.G.: Nichtlineare strukturelle Regelung, angewandt auf Stellsysteme der Flugsteuerung. PhD thesis, Technical University ofHamburg–Harburg. Fortschritt–Rep. No.737, Series 8. Duesseldorf: VDI-Verlag, 1999.
[12] Kliffken, M.G.; Gojny, M.H.: A Unified Control Strategy for Flight Control Actuators.Recent Advances in Aerospace Hydraulics.Toulouse/France, 1998.
[13] Kliffken, M.G.; Gojny, M.H.: Numerical Robust Implementation of Sampled–Data Controllers for Flight Control Actuators.at—Automatisierungstechnik9/47, 1999.
[14] MacKinnon, A; Stollery, J.L..: Variable Camber Wing. COA Rep.9304, Cranfield University, 1993.[15] Lind, R; Brenner M.:Robust Aeroservoelastic Stability Analysis. London: Springer, 1999.[16] Middleton, R.H.; Goodwin G.C.:Digital Control and Estimation – A Unified Approach. Englewood Cliffs: Prentice–Hall, 1990.[17] N.N.: Donnees Aerodynamiques — Aileron Hinge Moments. Aerospatiale, Toulouse, 1991. Note Aerodynamique 445.343.[18] Raymond, E.T.; Chenoweth, C.C.:Aircraft Flight Control Actuation System Design. Warrendale: SAE, 1993.[19] Vaughan, N.D.; Raval, R.: The Use of an Impedance Analysis of an Hydraulic Servo to Investigate Control Compensation for Flutter
Suppression.Recent Advances in Aerospace Hydraulics. Toulouse/France, 1998.
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