[American Institute of Aeronautics and Astronautics AIAA Guidance, Navigation, and Control Conference - Chicago, Illinois ()] AIAA Guidance, Navigation, and Control Conference - Synthesis

American Institute of Aeronautics and Astronautics

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Synthesis of an Hinf Controller for Fully Movable Aeroelastic Control Surfaces of Air Vehicles

Alper Akmeşe* and Mutlu D. Cömert† TÜBİTAK-SAGE

The Scientific and Technical Research Council of Turkey Defense Industries Research and Development Institute

Ankara, Turkey

and

Bülent E. Platin‡ Middle East Technical University, Ankara, Turkey

A controller synthesis method is proposed for fully movable aeroelastic control surfaces of air vehicles. The proposed synthesis method is based on the mu flutter analysis method. The performance of the method is compared with an alternative flutter suppression controller synthesis method available in literature. The methods are implemented to a theoretical aeroservoelastic control actuation system based on the aeroservoelastic test system constructed in TUBİTAK-SAGE for incompressible subsonic flow. The synthesis and analysis are performed by using the codes developed on MATLAB®. The airfoil is modeled via typical section model. The aerodynamics is modeled at distinct Mach numbers of 0.5, 0.6 and 0.7 by using the indicial functions. Hinf type controllers are synthesized by using the two synthesis methods. Aeroservoelastic analyses of the systems are performed considering the stability, performance and the effect of backlash type nonlinearity. Comparing the two methods, it is seen that the proposed method has better stability and performance characteristics.

Nomenclature Latin Script: a = Parameter for the position of elastic axis from mid cord, positive backwards [Ai] = Aerodynamic coefficient matrices (i=1,..,4) b = Half cord ch = Plunge damping coefficient of typical section wing cα = Torsion damping coefficient of typical section wing cθ = Equivalent damping coefficient of motor and transmission calculated at wing shaft clα = Aerodynamic lift coefficient cmα = Aerodynamic moment coefficient cji = Coefficients of indicial function (j:1..n, i: c, cM, cq, cMq) [C] = Damping matrix of typical section structural model [Co] = Nominal damping matrix of typical section structural model [Ceq] = Equivalent damping matrix {di} = Disturbance inputs (i: disturbance identifier) [DAi] = Aerodynamic coefficient matrix (i: c, cM, cq, cMq) {ei} = Error/performance outputs (i: error/performance identifier)

* Chief Research Engineer, Mechatronics Division, † Chief Research Engineer, ‡ Professor, Mechanical Engineering Department,

AIAA Guidance, Navigation, and Control Conference10 - 13 August 2009, Chicago, Illinois

AIAA 2009-6302

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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ess = steady state error [E1i] = Aerodynamic coefficient matrix (i: c, cM, cq, cMq) [E2i] = Aerodynamic coefficient matrix (i: c, cM, cq, cMq) [FAi] = Aerodynamic coefficient matrix (i: c, cM, cq, cMq) Fl([P],[K])= Lower linear fractional transformation Iα = Mass moment of inertia of wing about its elastic axis Im = Mass moment of inertia of motor and transmission calculated at wing shaft k = Reduced frequency kh = Plunge spring constant of typical section wing kα = Torsion spring constant of typical section wing kθ = Equivalent spring constant of motor and transmission calculated at wing shaft kθα = Equivalent spring constant obtained from serial connection of plunge spring and equivalent spring constant of motor and transmission calculated at wing shaft. kT = Motor torque constant [K] = Stiffness matrix of typical section structural model [Kc] = Transfer matrix of controller [Ko] = Nominal stiffness matrix of typical section structural model [Keq] = Equivalent stiffness matrix m = Mass of wing [M] = Mass matrix of typical section structural model M = Mach number Mp = Percentage of overshoot N = Transmission ratio p = Laplace variable [P] = Aeroservoelastic plant [Pae] = Aeroelastic plant qi = Generalized coordinates qα = Angle of attack of typical section qh = Plunge motion of elastic axis of typical section wing from undeflected position qθ = Angle of motor shaft kinematically amplified to wing shaft q = Dynamic pressure

0q = Nominal dynamic pressure flutq = Flutter pressure

{q} = Structural state vector QAL = Aerodynamic lift force QAM = Aerodynamic moment Qc = Control force {QA} = Aerodynamic force vector {Qext} = External force vector l = Wing span, distance from root cord to tip of wing s = Nondimensional time, distance travelled in terms of reference length, b S = Wing area Sα = First mass moment of the control surface about the elastic axis, m.xcg. tr%0.05 = rise time from %0.05 to 0.95 of command ts = settling time to %0.05 error Tcs = Motor continuous stall torque Tpeak = Allowable peak torque U = Free stream velocity

}{ qw = Perturbation to dynamic pressure {wc} = Perturbation to damping [Wi] = Weighting matrix for uncertainty/performance (i: identifier of weighting function) xcg = Position of center of gravity from elastic axis, positive backwards {XAi} = Aerodynamic state vector (i: c, cM, cq, cMq)

}{ qz = Additional states due to perturbation to dynamic pressure


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{zc} = Additional states due to perturbation to damping Greek Script: βi = Coefficients of Küsner and indicial functions δq = Norm bounded perturbation multiplier of dynamic pressure δαθ = deflection of torsional stiffness [ ]∆ = Norm bounded perturbation multiplier matrix

δ⎡ ⎤⎣ ⎦O

Oc = Norm bounded perturbation multiplier of damping φ(s) = Aerodynamic indicial function Γnom = Nominal flutter margin Γrob = Robust flutter margin ρ = Density of air κ i = Magnitude scaling constant of weighting functions (i: identifier of weighting function)

{ }ξ = State vector of aeroelastic/aeroservoelastic system Miscellaneous notation: a& = First time derivative of a a&& = Second time derivative of a [ ]mxn0 = n by m zero matrix [ ]nxnI = n by n identity matrix

[ ]OOA = Diagonal matrix

inf = infinity norm Abbreviations: AE = Aeroelastic

ASE = Aeroservoelastic

LFT = Linear Fractional Transformation

I. Introduction eroelasticity is the study of the effect of aerodynamic forces on elastic bodies”1. It is an old subject in which the studies were started before 1930’s. For many years, people were faced with aeroelastic problems and

solved them by passive solutions. These passive solutions are generally still the first step. On the other side, with the improving technology and mankind’s passion of obtaining the better, control technology has been introduced to the field of aeroelasticity. The new field that is the intersection of aeroelasticity and controlled structures technology is named as aeroservoelasticity. Its objective is to modify the aeroelastic behavior of a system by introducing calculated control forces. Aeroservoelastic studies have been performed since 1960’s. However, most of these studies were concentrated on aircraft wings. There exist only a few studies about the flutter suppression of fully movable control surfaces (fins). For the flutter suppression of aircraft wings, a controlled motion of a flap is used. However, the flutter suppression of the fin is achieved via controlling the angular position of the fin shaft. This fact differentiates the two systems.

In the controller synthesis studies for flutter suppression of airplane wings, first classical controllers2, then controllers similar to LQG/LQR type optimal controllers were used3-8. Although these controllers did suppress the flutter, due to their synthesis method the uncertainties of the system were not considered by them. In some recent studies, robust controller methods that consider the uncertainties of the system were used9-12. In this study, an H∞ controller method is used for the controller synthesis in order to take into account the system uncertainties.

Simply, the aeroelastic or aeroservoelastic instabilities happen with the change of aerodynamic loads, which change with the flight parameters. The result can be explained on a root locus plot; the change of aerodynamic loads moves the poles of the system from left hand plane to right hand plane. Hence, in the robust controller synthesis two different approaches can be used. The shift in the poles can be modeled as an uncertainty and a robust controller can be synthesized considering this uncertainty. This method was applied by Vipperman J.S., Barker J.M., Clark R.L.,

“A


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and Balas G.J.12 in 1999 to an airplane wing model. On the other hand, it is also possible to define uncertainties for flight parameters and a robust controller can be synthesized considering these uncertainties. Waszak M.R.10 used the dynamic pressure uncertainty in order to determine the nominal aeroelastic system model and the uncertainties of the model including the actuators. Then, he presented a robust controller synthesis method for the determined aeroelastic system model. Waszak also used an airplane wing model in his study.

In this study a different approach is devised, in which the perturbation to dynamic pressure is directly included in the robust controller synthesis method. The idea of this method is based on the µ flutter search algorithm which up to present was only used as an analysis tool. The controller synthesis is performed for a fin model. Alternatively, the method devised by Vipperman J.S., Barker J.M., Clark R.L., and Balas G.J.12 is applied to the fin model for the purpose of comparison. The synthesized controllers are analyzed and compared considering the stability and performance of the aeroservoelastic systems.

In the literature, there are studies13-22 on the flutter analysis of airplane wings and missile fins with several types of structural nonlinearities. Beyond the nonlinearities studied in literature, it is known by experience that the freeplay type nonlinearity usually exists in the missile fins. Hence, the effect of the freeplay type of nonlinearity on the aeroservoelastic system is also examined.

In order to observe the affect of the controllers fin is structurally modeled via typical section1 model. The aerodynamic forces are obtained by using the indicial functions derived by Mazelsky, B., and Drischler, J. A.23,24, , at 0.5, 0.6, and 0.7 Mach for thin airfoil.

This paper is devised from the PhD study of A.Akmeşe25, hence the further detailed explanations can be found from Ref. 25.

Numerical calculations are made by using MATLAB® (The MathWorks, Inc.) computer software package.

II. Typical Section Model The structural part of the fully movable aeroelastic control surface is modeled by a three degree of freedom

structural model, as given in Fig. 126. The three degree of freedom aeroservoelastic model given in Fig. 1 is obtained by adding a servo actuator to control the angular degree of freedom of the typical section wing, Fig. 21.

The equation of motion of the unconservative system given in Fig. 1, with disregarding the properties that

belongs to the actuator such as Im and cθ, is given in Eq. 1.

[ ]{ } [ ]{ } [ ]{ } { } { }A extM q C q K q Q Q+ + = +&& & (1)

QAM

QAL

qα

qh

kh, ch

kα, cα

Figure 2. Typical section model.

Figure 1. Three degree of freedom structuralmodel.


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where

[ ] α

α α

⎡ ⎤= ⎢ ⎥⎣ ⎦

pm SM

S I, [ ]

00hc

Ccα

⎡ ⎤= ⎢ ⎥⎣ ⎦ ,

[ ]0

0 αθ

⎡ ⎤= ⎢ ⎥⎣ ⎦

hkK

k

(2)

{ } hqq

qα

⎧ ⎫= ⎨ ⎬⎩ ⎭ ,

{ } ALA

AM

QQ

Q−⎧ ⎫

= ⎨ ⎬⎩ ⎭ ,

{ } extext

ext

FQ

M⎧ ⎫

= ⎨ ⎬⎩ ⎭ ,

. cgS m xα =

The above matrices are constructed according to the Aeroservoelastic Test Setup27 constructed in TÜBİTAK-SAGE. The mp term in [M] mass matrix is the total plunging mass, where as the m in Sα is the mass of the fin only. The aerodynamic lift and moment of the aeroelastic plant for non dimensional time are given in Eq. (3) and Eq. (4).

( ) ( ) ( ) ( ) ( ) ( )0 0

2 4 αα

σ σπ σ φ σ σ π φ σ σ

σ σ⎡ ⎤

= + − + −⎢ ⎥⎣ ⎦

∫ ∫& &s sh

AL c cq

q dqd bQ s qS q s d qS s dd U U d

(3)

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

0

0

2 2

4 2

α

α

σπ σ φ σ σ

σ

σπ φ σ σ

σ

⎡ ⎤= + −⎢ ⎥

⎣ ⎦

+ −

∫

∫

&

&

s hAM cM

s

cMq

qdQ s qS b q s dd U

dqbqS b s dU d

(4)

where s is the nondimensional time defined as given in Eq. (5).

Utsb

= (5)

where U is the air speed, t is the time, and b is the half chord length of the fin. q is the dynamic pressure as given in Eq. (6).

212

q Uρ= (6)

where ρ is the air density. The numerical values of the model are given in Table 1. The indicial functions, φc , φcM , φcq and, φcMq are obtained by using the indicial functions of Mazelsky, B., and

Drischler, J. A.23,24. The indicial functions of Mazelsky, B., and Drischler, J. A are converted to an applicable form for the Theodorsen’s notations by using the method given by Bisplinghoff R.L., Ashley H., and Halfman R.L.28. The orders of the indicial functions increase due to the conversion. In order to decrease the order of the model, the indicial functions are calculated numerically for the selected a and b values of the typical section wing and curve fitted by using least square algorithm to third order functions as given in Eq. (7).


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( ) 31 20 1 2 3

ss ss b b e b e b e ββ βφ −− −= + + + (7)

Implementing the Padé approximation and augmented states method, presented by Edwards J.W., Ashley H., and Breakwell J.V.5, to equations of motion of the typical section aeroelastic plant for unsteady compressible subsonic flow, the state space equations given in Eq. (8) can be obtained.

{ } { } { }{ } { } { }

ξ ξ

ξ

⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦

&AEsys AEsys

AEsys AEsys

A B v

e C D v (8)

where

{ } { } { } { } { } { } { }ξ ⎡ ⎤= ⎢ ⎥⎣ ⎦&

TT TT T T TAc Acq AcM AcMqq q x x x x (9)

{ } θ=v q (10)

[ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ]( ) [ ] [ ] [ ]( ) [ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

3 3 3 31 1 1 1 1 1

1 2

2 3 3 3 3 3 3

3 2 3 3 3 3 3 3

2 3 3 3 3 3 3

3 2 3 3 3 3 3 3

− − − − − −⎡ ⎤ ⎡ ⎤− − ⎣ ⎦ ⎣ ⎦

⎡ ⎤ =⎣ ⎦ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

⎡ ⎤⎣ ⎦

2x2 2x2 2x 2x 2x 2x

Ac Acq AcM AcMq

1Ac Ac Ac x x xAEsys

x2 Acq x Acq x x

1AcM AcM x x AcM x

x2 AcMq x x x

0 I 0 0 0 0

M A q K M A q C M D q M D q M D q M D q

E E F 0 0 0A

0 E 0 F 0 0

E E 0 0 F 0

0 E 0 0 0 F

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎡ ⎤ ⎥⎣ ⎦⎣ ⎦AcMq

(11)

Table 1. Numerical values of the model. Elastic axis a (-) -0.6 Half cord b (m) 0.15 Mass of fin m (kg) 9.83 Total plunging mass mp (kg) 28.7 Moment of inertia of fin Iα (kg.m2) 0.098 Plunge stiffness kh (N/m) 15,000 Total torsional stiffness kαθ (N.m/rad) 1,100 Plunge damping ch (N.s/m) 65.6 Pitch damping ca (N.m.s/rad) 0.476 Parameter of span xcg (m) 0.5 x b Motor torque constant kT (N.m/A) 2.22 Motor continuous stall torque Tcs (N.m) 3.53 Allowable peak torque Tpeak (N.m) 3.53 × 12 Motor and transmission inertia Im (kg.m2) 2.97e-4 Motor and transmission damping cm (N.m.s/rad) 1.24e-4 Transmission ratio N (-) 87


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{ }

[ ]

{ }{ }{ }{ }

2 1

1

2 1

2 1

2 1

2 1

00

0000

αθ

−

⎧ ⎫⎪ ⎪

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭⎪ ⎪⎡ ⎤ = ⎨ ⎬⎣ ⎦

⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

x

AEsys x

x

x

x

Mk

B (12)

[ ] [ ] [ ] [ ] [ ] [ ]2 2 2 2 2 2 2 2 2 2 2 20 0 0 0 0⎡ ⎤ ⎡ ⎤= ⎣ ⎦⎣ ⎦AEsys x x x x x xC I (13)

{ }2 10⎡ ⎤ =⎣ ⎦AEsys xD (14)

In Eq. (9), { }Acx , { }Acqx , { }AcMx , and { }AcMqx are the aerodynamic states. The aerodynamic coefficient matrices

in Eq. (11) are defined in Eqs. (15)-(20).

[ ] ( )0

10

02

0 2c

cM

cA S

b cπ

−⎡ ⎤= ⎢ ⎥

⎣ ⎦ (15)

[ ] 0 022

0 0

2122 4

c cq

cM cMq

c bcA S

bc b cUπ

−⎡ ⎤= ⎢ ⎥

⎣ ⎦ (16)

[ ] 3 2 120 0 0

c c cAc

c c cD Sπ

⎡ ⎤= − ⎢ ⎥

⎣ ⎦ (17)

3 2 140 0 0cq cq cq

Acq

c c cbD SU

π⎡ ⎤

⎡ ⎤ = ⎢ ⎥⎣ ⎦⎣ ⎦

(18)

[ ] ( )3 2 1

0 0 02 2AcM

cM cM cM

D S bc c c

π⎡ ⎤

= ⎢ ⎥⎣ ⎦

(19)

( )3 2 1

0 0 04 2AcMq

cMq cMq cMq

bD S bc c cU

π⎡ ⎤

⎡ ⎤ = ⎢ ⎥⎣ ⎦⎣ ⎦

(20)


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( ) ( ) ( )( )

( )( )

( )

( )

( )

0 0 1 2 3

1 1 1 2 2 3 3

2 2 3 2 3 1 3 1 3 1 2 1 2

2

3 1 2 3 1 2 3

4 1 2 3

2

5 1 2 2 3 1 3

3

6 1 2 3

β β β

β β β β β β

β β β

β β β

β β β β β β

β β β

= + + += − − −

= − + + + + +

⎛ ⎞= − + +⎜ ⎟⎝ ⎠

= + +

⎛ ⎞= + +⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠

u u u u u

u u u u u u u

u u u u u u u u u u u u u

u u u u u u u

u u u u

u u u u u u u

u u u u

c b b b bc b b b

Uc b b b b b bbUc b b bb

UcbUcb

Ucb

(21)

The c coefficients with different subscripts used in the aerodynamic coefficient matrices can be obtained by

using Eq. (21). In Eq. (21) the biu and βiu terms are the coefficients of the indicial functions of u being c, cq, cM, and cMq.

The [E] and [F] matrices in Eq. (11) are the matrices associated with the aerodynamic states, { }Acx , and { }Acqx in

Eq. (22) and Eq. (23). The [E] and [F] matrices for { }AcMx , and { }AcMqx can be obtained by changing the subscripts of Eq. (22) and Eq. (23) correspondingly.

[ ] [ ] [ ]1 2

6 5 4

0 1 0 0 0 0 00 0 1 , 0 0 , 0 0

1/ 00

⎡ ⎤⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= = =⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥− − −⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

Ac Ac Ac

c c c

F E Ec c c U b

b

(22)

2

6 5 4

0 1 0 0 00 0 1 , 0 0

0

⎡ ⎤⎢ ⎥⎡ ⎤⎢ ⎥⎢ ⎥⎡ ⎤ ⎡ ⎤= = ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎢ ⎥− − −⎣ ⎦ ⎢ ⎥⎣ ⎦

Acq Acq

cq cq cq

F Ec c c U

b

(23)

III. Controller Synthesis The fully movable control surfaces (fin) are by default a part of servo system. Hence, the synthesized controller

should have some performance requirements. In this study, the synthesized controllers considers performance requirements besides flutter suppression. The nominal aeroservoelastic plant model for the fin is given in Fig. 3.

_α cmdq is the command input for the angular position of the fin. The Sensor block is taken as unity for simplicity. The equations of the aeroelastic plant model are given in Eq. (8). The numerical values of the aeroservoelastic system are given in Table 1.


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A. q-Method In robust controller synthesis and analysis, the weightings are the one of the most important part of the problem.

All of the performance requirements, limitations, disturbances, and uncertainties are described by these weightings. Thus the result of the analysis or the controller synthesis is defined with these weightings. The placement of these blocks, the type of the weighting function and their numerical values are very important.

Flutter is aeroelastic instability that occurs when the aerodynamics transfers energy to the elastic structure. The µ-flutter analysis method29 searches for the aerodynamic parameters, in which the flutter occurs, by using µ algorithms. For this purpose, an uncertainty to dynamic pressure is defined in the aeroelastic plant. In q-method, which is used for controller synthesis, the uncertainty to dynamic pressure is used in order to define the change of flight conditions. Hence the synthesized controller with q-method stabilizes the despite the change of flight conditions. The interconnection structure of the aeroservoelastic plant for flutter controller synthesis via q-method is given in Fig. 4. The definitions of the signals are given in Table 2.

N

Controller

N

kT +

-21+m mI p c p 1/N

Aeroelastic Plant Model

+

-kαθ1/N

Sensor

_α cmdq qh qθ

qα Actuator dynamics

Figure 3. Aeroservoelastic plant model.

N

N

kT +

- 21+m mI p c p

1/N

Aeroelastic Plant Model

+

-kαθ1/N

Sensor

qθqα

Actuator dynamics

2qWWFd

{dFd}

Wcmd

W1sen W2sen

{wsen} {zsen}

+

+

[Wref_model]

+ - Wper

{eper}{dcmd} {qcmd}

{qmeas}

{u}

Controller

Wact

{eact}

Wn {dn}

1qW { }qw { }qz

Figure 4. Interconnection structure of aeroservoelastic plant for flutter suppression controller synthesis via uncertainty on dynamic pressure.


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The aeroelastic plant model is modified, considering the dynamic pressure uncertainty. The details of this modification can be found in References 26, and 29. In Fig. 5, the generalized control design structure of the system is given. The plant [P], in Fig. 5 is the state space form of the interconnection structure of the aeroservoelastic plant given in Fig. 4. ∆ is the uncertainty block with unit magnitude. {z}, {e}, {w}, {d}, {u} are the vectors containing the signals defined in Table 2, and {y}=[qcmd qmeas]T.

1. Reference model and Uncertainty descriptions

_⎡ ⎤⎣ ⎦ref modelW block defines the reference model. The performance of the main system is compared with the output

of this block. It is defined as a second order system with the natural frequency of 10 Hz and damping ratio of 1/ 2 as given in Eq. (24).

2

_ 2 2

(20 )20 2 (20 )

ππ π

=+ +ref modelW

p p (24)

where p is the Laplace variable. [ ]cmdW block defines the expected input commands. It is used to convert the unit disturbance input { }cmdd in to the physical value of expected commands in radians. This weighting function is defined as a low pass type as given in Eq. (25), in which the command is expected to be higher in low frequencies and lower in high frequencies.

4 400

18000 4π π

π+

=+cmd

pWp

(25)

[ ]nW is the weighting function of sensor noise. This weighting function converts the unit disturbance input { }nd in to sensor noise. Sensor noise is taken as a constant error as given in Eq. (26), with the magnitude of the smallest increment of the measurement.

16

22π

=noiseW (26)

[ ]FdW block is used to minimize the steady state error. Besides, it is also used to define the aerodynamic

disturbance forces. This weighting function scales the unit disturbance signal { }Fd in to the actual aerodynamic

Table 2. Input/output signals of plant. Signal Meaning { }qw Weighted uncertainty input for plunge and pitch motion {wsen} Weighted uncertainty input for measurement {wFd} Weighted aerodynamic disturbance force and moment {dn} Weighted sensor noise {dcmd} Weighted input command {u} Controller command { }qz Normalized uncertainty output for plunge and pitch motion {zsen} Normalized uncertainty output for measurement {eper} Normalized tracking error of pitch motion {eact} Normalized output torque of actuator {qcmd} Input command send to controller {qmeas} Measured value of pitch position

[P] Plant

∆ { }{ }⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

ze

[Kc]

Controller

{u} {y}

{ }{ }⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

wd

Figure 5. General control design structure.


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disturbance forces that is expected to occur. A low pass filter type weighting function is used for aerodynamic disturbance.

32 0 600

0 0.56 . 0.6ππ

⎡ ⎤ += ⎢ ⎥ +⎣ ⎦

Fd

N pWN m p

(27)

[ ]actW block is used to define the actuator limits. The [ ]actW block normalizes the torque output of the motor

torque constant block to unit output { }acte . Hence, the [ ]actW is as a punitive function, and it is proportional to the inverse of the actuator limits. On the other hand, there exists only “motor torque constant” block between the Wact block and the controller output. Hence, this weighting function also limits the controller output. Due to the characteristics of the actuators, a high pass type function is used for the [ ]actW normalization function, as given in Eq. (28).

100 200

20000ππ

+=

+actpeak

pWT p

(28)

⎡ ⎤⎣ ⎦perW block is used to penalize the tracking error of the system. This function normalizes the tracking error of pitch motion to one. In general, it is required from the aeroservoelastic system to track better in low frequencies and the system is permitted to be worse at higher frequencies. Due to the inverse characteristic of the normalization blocks, this block is modeled with a low pass type function, as given in Eq. (29).

180 1000080 100

ππ π

+=

+perpWp

(29)

1qW⎡ ⎤⎣ ⎦ is the normalization function that scales both channels of the output { }qz to unity. This function is taken

as a constant, as given in Eq. (30), and 1qW⎡ ⎤⎣ ⎦ is the inverse of the maximum of the expected { }qz value for each

channel. The units of { }qz is N/Pa for plunge channel and N.m/Pa for pitch channel. Introducing a dynamic pressure

perturbation term to dynamic pressure in aeroelastic system equations, { }qz can be derived based on unsteady compressible aerodynamics, as given in Eq (31).

1

6.25 00 400

⎡ ⎤= ⎢ ⎥⎣ ⎦

qW (30)

[ ] [ ] [ ] [ ] [ ] [ ][ ]{ }ξAcMqAcMAcqAcq DDDDAAz 21}{ = (31)

2qW⎡ ⎤⎣ ⎦ is the weighting function that defines the required change of aerodynamics for which the aeroservoelastic

system should be robust. The units of the out put of the 2qW⎡ ⎤⎣ ⎦ block is N for plunge motion and N.m for pitch

motion. Hence, 2qW⎡ ⎤⎣ ⎦ can be evaluated by multiplying inverse of 1qW⎡ ⎤⎣ ⎦ with the expected dynamic pressure change. Due to the physics of the problem, the dynamic pressure change will be effective up to a frequency level. Hence 2qW⎡ ⎤⎣ ⎦ weighting function can be modeled with a low pass type function, Eq. (32). Note that this weighting is


12

similar to [ ]FdW , however their effects are separated with the selected cut off frequencies, in which this weighting has higher gains at higher frequencies.

2

1 0 1000000 0.0156 . 100

ππ

⎡ ⎤ += ⎢ ⎥ +⎣ ⎦

q

N pWN m p

(32)

2. Controller Synthesis For purpose of controller synthesis, the one can take the linear fractional transformation of the generalized

controller design structure given in Fig. 5, as given in Eq. (33).

{ }{ } [ ] [ ]( ) { }

{ } [ ] [ ][ ] [ ] [ ][ ]( ) [ ] { }{ }

111 12 22 21,

−⎧ ⎫ ⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎡ ⎤= = + −⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ ⎩ ⎭l c c c

z w wF P K P P K I P K P

e d d (33)

where

[ ] [ ] [ ][ ] [ ]

11 12

21 22

⎡ ⎤= ⎢ ⎥⎣ ⎦

P PP

P P (34)

The H∞ synthesis problem is to find the admissible controller that minimizes the transfer function from { }{ }⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

wd

to

{ }{ }⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

ze

, in a different way;

[ ]

[ ] [ ]( )min ,∞c

l cKF P K (35)

For further information on H∞ synthesis and robust controllers, Ref. 30 is suggested. The H∞ controller is synthesized using the MATLAB® Robust Control Toolbox. The synthesized controller is at the order of plant [P], which is 28th order. The Bode plot of the controller is given in Fig. 6. The controller is reduced to 14th order via balance reduction method.


13

B. g-Method In order to perform this second method, the damping uncertainty is defined in the aeroelastic plant. Hence, a

similar procedure defined in Section A is used. The damping uncertainty is applied for the pitch degree of freedom only. The equation of the aeroelastic plant is reconstructed considering the damping uncertainty. The interconnection structure of aeroservoelastic plant for flutter suppression controller synthesis via uncertainty on damping is given in Fig. 7. The additional channels {zc} and {wc} are weighted using the weightings [Wc1] and [Wc2], respectively.

[Wc1] is the normalization function that scales the output {zc} to unity. This function is taken as a constant, as given in Eq. (36). [Wc1] is the inverse of the maximum of the expected {zc} value, where {zc} is equal to the velocity state vctor .

Wc1 = 0.912 sec/rad (36)

[Wc2] is the weighting function that defines the required change of damping for which the aeroservoelastic system should be robust. [Wc2] can be evaluated by multiplying inverse of [Wc1] with the expected damping change. Due to the physics of the flutter problem, the damping change will be effective up to a frequency level. Hence, [Wc2] weighting function can be modeled with a low pass type function, Eq. (37).

26.582 10000100 100

ππ

+=

+cpWp

(37)

Similar to q-method, the H∞ controller is synthesized using the MATLAB® Robust Control Toolbox. The synthesized controller is 27th order. The Bode plot of the controller is given in Fig. 8. The controller is reduced to 14th order via balance reduction method.

10-1 100 101 102 103 104 10510-2

100

102

103

Log

Mag

nitu

de

Frequency (radians/sec)

Full Order Hinf Controller

10-1 100 101 102 103 104 105-360

-180

0

180

Pha

se (d

egre

es)


Cmd to ContOutSensor to ContOut

Figure 6. H∞ controller synthesized via q-Method.

Frequency, rad/s

Frequency, rad/s

Phas

e, d

eg

Log,

A/d

eg

{qcmd} to {u} {qmeas} to {u}


14

IV. Results of Analyses

A. Stability Stability analyses are performed for the aeroelastic plant and the aeroservoelastic plant with two different

controllers. The results are given in Fig. 9. The analyses are performed via µ- flutter analysis method, with match point search iteration. The aerodynamic equations of compressible flow are valid for a corresponding Mach number, since the indicial functions are separately defined for each Mach numbers. In the atmosphere model, a physical point can be obtained by setting two parameters, for fixed temperature. Hence any two of three parameters, namely, Mach number, air density, and air speed, define the third one. The physical point at which these three parameters satisfy each other is named as a match point. In µ-method the plant [Pae] is established for a match point. The result of µ-method gives an air density of instability. Since one of the three parameters is altered, a second parameter should be changed to make this new point a match point. However, Mach number is fixed due to indicial functions of compressible flow, and the air speed is fixed due to undisturbed air speeds in µ-method equations. Hence the new point is an unphysical point. But the instability margin of air density is still a valuable data. Besides defining the stability of the initial point, it can be used to determine the actual flutter margin. A simple method to iteratively searches the match point of instability, with accuracy ε , is given below.

- Calculate the air density of instability ρj using µ-method. - Calculate the corresponding air speed Uj of the new match point, for the initial Mach number and the

calculated air density of instability ρj. - Calculate the percentage of the change of airspeed, %δUj=(Uj-Uj-1)/Uj-1. - Setup plant [Pae] for the new match point - j=j+1 - if %δUj>ε go to first step - match point of instability; Uflutter=Uj, ρflutter=ρj, initial Mach number.

10-1 100 101 102 103 104 105

100

102103

10-2

10-3

Log

Mag

nitu

de


Full Order Hinf Controller

10-1 100 101 102 103 104 105-180

0

180

360

Pha

se (d

egre

es)


Cmd to ContOutSensor to ContOut

Figure 8. H∞ controller synthesized via g-Method.

Frequency, rad/s

Frequency, rad/s

Phas

e, d

eg

Log,

A/d

eg

{qcmd} to {u} {qmeas} to {u}


15

From Fig. 9, it can be seen that both controllers enlarge the flutter flight envelope from aeroelastic stability limits. For 0.5 and 0.6 Mach numbers, the instability limits are enlarged below sea level. However, among the two controllers the q-controller enlarges the stability limit of the aeroservoelastic system to a larger envelope than the controller synthesized with g-method.

B. Performance The performance analyses are performed at five distinct aerodynamic points, both at frequency and time

domains. In frequency domain, the bandwidth of the system is analyzed. In time domain, the system performances are analyzed through step responses. The results are given in Table 3.

Considering the bandwidth requirement, it is seen that the q-controller provides a better bandwidth. Moreover, q-controller is affected from the flow variation less than the g-controller. The rise time performance of the q-controller is better than or the same as the g-controller in all cases. On the other hand, the settling time performances of the two controllers are comparable. In three out of five cases, the g-controller provides smaller settling times than the q-controller. Considering the overshoot, it is seen that the q-controller performs better in four out of five cases. However, considering the steady state error, the g-controller performs better in all cases. The two controllers perform comparably according to deformation of the torsional spring. The q-controller consumes less current than the g-controller nearly in all cases. As a sum up, considering the bandwidth, rise time, overshoot, and current consumption requirements, the q-controller performs better than the g-controller. However, the g-controller is better for the steady state error and partially better in settling time requirements.

140 150 160 170 180 190 200 210 220 230 240

-1

-0.5

0

0.5

1

1.5

2x 10

4

Nominal speed [m/s]

Alti

tude

[m]

AEASE q-MethodASE g-Method

Figure 9. Results of stability analysis.


16

C. Effect of Backlash In order to analyze the affect of backlash, the rotational version of the analogous system given in Fig. 10 is

introduced between actuator and fin shaft in time domain model.

x2

x1

k, c

F2

F1

2*bv

m2

m1

Figure 10. Physical model of backlash between two translational bodies.

Table 3. Performance analyses results of aeroservoelastic systems. Mach 0.5 0.5 0.6 0.7 0.7 Altitude [m] 0 20000 0 5000 10000 q [Pa] 17232 958 25534 18528 9067 U [m/s] 170 148 204 224 210 Si

mul

atio

n pa

ram

eter

s

ρ [kg/m3] 1.225 0.088 1.225 0.736 0.413 ωn [Hz] 9.87 7.77 10.86 10.45 9.95 tr%0.05 [sec] 0.043 0.048 0.026 0.028 0.041 ts [sec] 0.058 0.556 0.077 0.234 0.268 Mp % 1.42 14.90 20.35 16.78 7.66

-0.033 -0.026 -0.027 -0.025 -0.027 ess [deg]

±0.004 ±0.005 ±0.005 ±0.003 ±0.005

αθδ [deg] 0.75 0.20 1.10 0.95 0.41

cmax_step [A] 0.97 1.05 1.01 1.02 1.13 cmean_steady [A] 0.028 0.004 0.007 0.002 0.005 cstd_steady [A] 0.272 0.296 0.314 0.218 0.327

H∞&

q

cmax_steady [A] 0.916 0.709 0.893 0.560 0.837 ωn [Hz] 9.69 5.49 9.83 9.57 9.22 tr%0.05 [sec] 0.047 0.058 0.028 0.028 0.042 ts [sec] 0.061 0.456 0.223 0.199 0.192 Mp % 0.84 17.60 29.88 25.39 7.95

-0.024 -0.020 -0.021 -0.020 -0.021 ess [deg]

±0.003 ±0.003 ±0.003 ±0.004 ±0.004

αθδ [deg] 0.73 0.18 1.10 0.97 0.42

cmax_step [A] 1.54 1.54 1.54 1.54 1.54 cmean_steady [A] 0.029 0.003 0.007 0.002 0.005 cstd_steady [A] 0.252 0.303 0.327 0.245 0.369

H∞&

g

cmax_steady [A] 0.605 0.769 0.843 0.574 0.844


17

Nonlinear analysis results of the aeroservoelastic system with 0.05° and 0.2° backlash values, and two different controllers are presented in Figure 11. From these figures, it can be seen that the controllers are affected differently from the variation of the flow conditions. The performance of the q-controller enhances as the altitude decreases or the Mach number increases, which stands for an increase in the dynamic pressure. On the contrary, the performance of the g-controller enhances at the opposite cases. Considering the deformation of the spring, δαθ, that is the degree of freedom at which the backlash exist; the controllers perform comparably and the amplitude of the oscillations increases up to two or three times of the backlash value. However, considering the pitch motion of the typical section wing, qα, that is one of the performance merits defined in controller synthesis and also in the analysis, the q controller performs better nearly in all cases. This is in contrast with the steady state error results obtained without the backlash, in which the g controller performs better than the q controller. In fact, steady state error is the only evident advantage of the g controller that is seen in performance analyses.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

altitude [m]

osci

llatio

n m

agni

tude

[deg

]

q-Hinf bv=0.05q-Hinf bv=0.2g-Hinf bv=0.05g-Hinf bv=0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

altitude [m]

osci

llatio

n m

agni

tude

[deg

]


a. LCO amplitude of qα-qθ at 0.5 Mach b. LCO amplitude of qα at 0.5 Mach

0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mach #

osci

llatio

n m

agni

tude

[deg

]


0.5 0,6 0.70

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Mach #

osci

llatio

n m

agni

tude

[deg

]


c. LCO amplitude of qα-qθ at 10000 m altitude d. LCO amplitude of qα at 10000 m altitude Figure 11. Limit cycle analyses results of the aeroservoelastic systems.


18

In some cases that are encircled in Fig. 11, the time domain signals are not converged to a pure sinusoidal signal. In order to further investigate the system, phase plane plots are constructed and results for one case is presented in Fig. 12 (a), (b), and (c). As it can be seen from the figures, the phase plots are not composed of one closed circle as in an LCO case. By going one step forward a sensitivity analysis is performed. For this purpose, the amplitude of the aerodynamic force input, which is used in the backlash analysis as a pulse input, is increased by 1%. The analysis is performed and the time results of the two analyses are plotted on top of each other. The results are presented in Fig. 12 (j), (k), and (l). It is seen from the results that after 0.8 s, the results starts to separate from each other. With this information, it is concluded that the aeroservoelastic system with the q controller possibly acts chaotically at 0.5

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-50

-40

-30

-20

-10

0

10

20

30

40

50

position [deg]

velo

city

[deg

/s]

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-15

-10

-5

0

5

10

15

position [deg]

velo

city

[deg

/s]

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 10-3

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

position [m]

velo

city

[m/s

]

a. phase plot of qα-qθ b. phase plot of qα c. phase plot of qh

3.5 3.55 3.6 3.65 3.7 3.75 3.8

x 104

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

posi

tion

[deg

]

time [miliseconds]3.5 3.55 3.6 3.65 3.7 3.75 3.8

x 104

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

posi

tion

[deg

]

time [miliseconds]3.5 3.55 3.6 3.65 3.7 3.75 3.8

x 104

-1.5

-1

-0.5

0

0.5

1

1.5x 10

-3

posi

tion

[m]

time [miliseconds] d. time response plot of qα-qθ e. time response plot of qα f. time response plot of qh

0 10 20 30 40 50 60 70 80 90 10010

-2

10-1

100

101

102

103

104

0 10 20 30 40 50 60 70 80 90 10010

-2

10-1

100

101

102

103

104

0 10 20 30 40 50 60 70 80 90 10010

-6

10-5

10-4

10-3

10-2

10-1

100

101

g. FFT plot of qα-qθ h. FFT plot of qα i. FFT plot of qh

500 600 700 800 900 1000 1100 1200 1300 1400 150-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

posi

tion

[deg

]

time [miliseconds]500 600 700 800 900 1000 1100 1200 1300 1400 1500

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

posi

tion

[deg

]

time [miliseconds]500 600 700 800 900 1000 1100 1200 1300 1400 150

-1.5

-1

-0.5

0

0.5

1

1.5x 10-3

posi

tion

[m]

time [miliseconds] j. time response plots of qα-qθ k. time response plot of qα l. time response plot of qh for %1 different initial conditions for %1 different initial conditions for %1 different initial conditions

Figure 12. Limit cycle analyses results of the aeroservoelastic system with controller of q-Method at 0.5 Mach, 5000m altitude, for 0.2° backlash value.


19

Mach and 5,000 m altitude. These points are encircled in Fig. 11. Since a steady state condition is not satisfied, the presented values in Figure 11 are the maximum values of the corresponding signal.

V. Summary and Results The typical section wing that is used in the structural modeling part of aeroelastic system is a simple but a valid

model. It was used by many aeroelastic pioneers. But it has restrictions, such that the lifting surfaces should have a large aspect ratio, small sweep, and smoothly varying cross sectional characteristics across span. More complicated wings can be structurally modeled with different methods such as finite element method (by finite number of normal coordinates), Rayleigh–Ritz method (finite number of assumed mode shapes), or lumped mass (rigid segments) method. For the aerodynamics, it is also possible to use nonlinear aerodynamics obtained through CFD analysis. In practice, CFD was used in two ways; either a direct time domain flutter search can be performed or the generalized aerodynamic force matrix can be derived. Both usages, primarily the first one, are time consuming methods. These methods can be advantageous when analyzing a finished product. However, their utilization in synthesis steps of the aeroelastic system will not be effective. In this study, the lift and moment equations calculated with potential flow theory are used. This theory was widely used in flutter analysis, such that the commercial programs such as NASTRAN FLDS, UAI/ASTROS use simplified potential flow theories for unsteady aerodynamics in aeroelastic analysis. The main purpose of this study is to establish a method for the synthesis of a controller for a CAS, considering some performance specifications and also taking flutter suppression into account. Hence, using the lift and moment equations obtained by potential flow theory and the typical section wing model is considered to be adequate for this study.

Among the flutter search methods, µ-method is selected as the flutter search method in this study. This method is one of the recently developed methods; furthermore it can be extended to robust flutter search. Although it is not mentioned in the paper, some of the µ-method results are compared with the results of the p-method. It is seen that the results of the both methods are in agreement. Among these methods, the p-method performs a brute force approach, but provides larger information. In the p-method, the frequencies and the damping values of all modes of the system, and their variations with the dynamic pressure are derived. This information can be used to trace the effect of controller tuning. However, the accuracy of the p-method is limited by the minimum interval between the search points. Hence, increasing the accuracy results in an increased computation time. The p-method also requires some post-processing work to derive the solution from the results. It is better to use the p-method, in order to get an idea about the system rather than specifically obtaining the instability point. On the other hand, the µ-method directly calculates the dynamic pressure of instability and the frequency and damping value of the corresponding mode of the system, but does not provide any additional information about the system. Nevertheless, the µ-method is a much faster and accurate calculation method. Furthermore, the µ-method is appropriate for the instability match point calculations, which is an indispensable compressible flow analyses. In this study, the µ-method is adapted to the instability match point search by using a bisection search algorithm.

As already mentioned, flutter is an instability that occurs due to the interaction of inertial, elastic, and aerodynamic forces. Assuming that the inertial and the elastic properties of the structure are unchanged, the aerodynamics is left as the only varying parameter. This is the main assumption in the µ flutter search method. In this study, it is aimed to convert this search algorithm to a controller synthesis method. In the literature, the direct use of the µ-method for synthesis purposes does not exist. With this method, it is possible to directly set the stability margin that is required from the aeroservoelastic system. For the purpose of comparison, an alternative flutter suppression method available in the literature is used. The alternative method is adapted from the study of Vipperman et al. 12, in which an airplane wing was used and a parametric uncertainty was added to the first mode of the system to account for the variation of the real part of a complex-conjugate pole pair. In this study, the method presented by Vipperman is applied to the fin (missile control surface), and the parametric uncertainty is added to the damping parameter of the pitch degree of freedom of the fin.

Considering the above mentioned flutter suppression methods, H∞ type of controllers are synthesized in MATLAB® In the synthesis of H∞ controllers it is seen that the selection of upper and lower values of the γ limits, which is the search limit parameters of the MATLAB®’s hinfsyn command, affects the optimization algorithm used for the synthesis. The synthesis algorithm can sometimes converges to a local minimum. It is specifically advised to vary the limits of γ and re-run the synthesis algorithm if the γ norm of the system is slightly above one.

Through the case studies it is seen that: - The established q-method successfully suppresses the onset of flutter. Furthermore, a satisfactory

performance is obtained from the aeroservoelastic system by using the q-method controller. It is seen that the tough part of this method is to determine the values of the additional states due to perturbation to


20

dynamic pressure { }qz and the values of perturbation to dynamic pressure { }qw . These are the input and output vectors of the aeroservoelastic plant to the dynamic pressure uncertainty port. After these parameters and the rest of the uncertainty and performance parameters are set, two free parameters remain for the tuning of the synthesized controller. These two parameters are the dynamic pressure contq at which the controller is synthesized and the disturbance to dynamic pressure distq . It is seen that, these parameters affect the stability and performance of the system. The increase of contq increases the upper limit of the flutter flight envelope, however it reduces the lower limit. Hence, a tuning is required for the selection of contq . The second tuning parameter is the dynamic pressure of disturbance, which is used as the primary object for the flutter suppression. During the case studies it is seen that, increasing distq increases the stability margin from both upper and lower limits as expected. However, after a limit value of distq , the γ norm of the controller exceeds one, thus a proper controller cannot be obtained. In H∞ type controller synthesis it is seen that the upper limit of distq is well below the obtained nominal stability margin of the ASE system, which is due to the conservative property of the H∞ controller.

- The g-method is successfully implemented to the fin, and effective flutter suppression controllers are synthesized. The key point of this method is the derivation of the values of perturbation to damping {wc} and the values of additional states due to perturbation to damping {zc}, which corresponds to the velocity states of the pitch degree of freedom. After the uncertainty and the performance parameters of the system are set, the performance of the controller is tuned by means of the damping uncertainty, which is a parameter of {wc} .

- Two distinct case studies are performed considering the flutter suppression methods. It is seen that the established q-method and g-method successfully enlarge the stability envelope of the ASE system. Considering the performance analysis, both methods perform comparably in the step response analyses, however the q-method end up with a better bandwidth results in frequency response analyses. Similar to the step response performance comparison, the flutter suppression methods’ performances in LCO suppression are also comparable. Through this study it is seen that the established q-method is a successful flutter suppression method and its overall performance is comparable to the g-method.

- Further investigating the nonlinear system possible chaotic behavior of the aeroservoelastic system are seen in some aerodynamic conditions.

VI. Conclusions In this study a different controller synthesis method is devised for the flutter suppression of the fully movable

aeroelastic control surfaces. The devised algorithm uses the change of dynamic pressure as an uncertainty during the synthesis of the H∞ controller. In order to compare the performance of this method, an alternative method based on the study of Vipperman12 is also adopted for this aeroelastic model.

Performances of the controllers synthesized by both methods are compared considering the stability limits, bandwidths, step responses, and attitudes against backlash. It is seen that the controller synthesized with the devised controller synthesis method gives better results in most of the performance merits.

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21

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20 Price, S.J., Lee, B.H.K., and Alighanbari, H., “Postinstability Behaviour of a Two-Dimensional Airfoil with a Structural Nonlinearity,” Journal of Aircraft, Vol 31, No.6, 1994, pp. 1395-1401.

21 Brase, L.O., and Eversman, W., “Application of Transient Aerodynamics to the Structurally Nonlinear Flutter Problem,” Journal of Aircraft, Vol 25, No.11, 1988, pp. 1060-1068.

22 Lind, R., and Brenner, M.J., “Robust Flutter Margin Analysis That Incorporates Flight Data,” NASA/TP-1998-206543, 1998.

23 Mazelsky, B., “Determination of Indicial Lift and Moment of a Two-dimensional Pitching Airfoil at Subsonic Mach Numbers from Oscillatory Coefficients with Numerical Calculations for a Mach Number of 0.7,” NACA T.N. 2613, 1952.

24 Mazelsky, B., and Drischler, J. A., “Numerical Determination of Indicial Lift and Moment Functions for a Two-dimensional Sinking and Pitching Airfoil at Mach Numbers 0.5 and 0.6,” NACA T.N. 2739, 1952.

25 Akmeşe, A., “Aeroservoelastic Analysis and Robust Controller Synthesis for Flutter Suppression of Air Vehicle Control Actuation Systems,” Ph.D. Dissertation, Mechanical Engineering Dept., METU, Ankara, Turkey, 2006.

26 Akmeşe, A., Cömert, M.D., and Platin, B.E., “Aeroservoelastic Analysis of Missile Control Surfaces Via Robust Control Methods,” Proceedings of 16th IFAC Symposium on Automatic Control in Aerospace, edited by Alexander Nebylov, Vol. 2, St.Petersburg, Russia, 2004, pp. 159-163.

27 Ünal, U., “Design, Construction, and Performance Testing of an Aeroservoelastic Test Apparatus to be Used in AWT (Ankara Wind Tunnel),” M.Sc. Dissertation, Mechanical Engineering Dept., METU, Ankara, Turkey, 2005.

28 Bisplinghoff, R.L., Ashley, H., and Halfman, R.L., Aeroelasticity, corrected republication, Dover, New York, 1996, Chap. 6.

29 Lind, R., and Brenner, M., Robust Aeroservoelastic Stability Analysis, Springer-Verlag, New York, 1999. 30 Zhou, K., and Doyle, J.C., Essentials of Robust Control, Prentice Hall, New Jersey, 1998, Chap. 14.

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