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Nonlinear Adaptive Model Free Control of an Aeroelastic 2-D LiftingSurface

A. Behal

, P. Marzocca

, D.M. Dawson

, and A. Lonkar

Electrical & Computer Engg.,

Mechanical & Aeronautical Engg., Clarkson University, Potsdam, NY 13699

Department of Electrical and Computer Engineering, Clemson University, Clemson, SC 29634-0915

E-mail: abehal,pmarzocc@clarkson.edu; ddawson,alonkar@clemson.edu

Abstract

This paper concerns adaptive control of a nonlinear 2-D wing-flap system operating in anincompressible flowfield. A model free output feedback control law is implemented and its per-formance toward suppressing flutter and limit cycle oscillations (LCOs) as well as reducing thevibrational level in the subcritical flight speed range is demonstrated. The control law proposedhere is applicable to minimum phase systems and we provide conditions for stability of the zero

dynamics. The control objective is to design a control strategy to drive the pitch angle to asetpoint while adaptively compensating for uncertainties in all the aeroelastic model parameters.It is shown that all the states of the closed-loop system are asymptotically stable. Furthermore,an extension is presented in order to include flap actuator dynamics. Simulations have beenpresented in order to validate the efficacy of the proposed strategy. Pertinent conclusions havebeen outlined.

Nomenclature = Pitch angleb = Semi-chord of the airfoil = Flap anglecl, cm = Lift and moment coefficients per angle-of-attackcl, cm = Lift and moment coefficients per control surface deflectionch, c

= Structural damping coeffi

cients in plunge and pitch due to viscous dampingC(k) = Theodorsens functionh = Plunging displacementI = Mass moment of inertia of the airfoil about the elastic axiskh, k = Structural spring stiffness in plunge and pitchL,M = Lift and aerodynamic momentm = Mass of airfoil = Air densityx = Dimensionless distance from the elastic axis to the mid-chord, positive rearwardt = TimeT = Unsteady torque moment offlap springU = Freestream velocityx, y = Horizontal and vertical coordinates

1 Introduction

Flutter instability can jeopardize aircraft performance and dramatically affect its survivability. Although flutterboundaries for aircraft are known, due to certain events occurring during its operational life - such as escapemaneuvers - significant decays of the flutter speed are possible with dramatic implications for its structural integrity.Passive methods which have been used to address this problem include added structural stiffness, mass balancing,and speed restrictions [1]. However, all these attempts to enlarge the operational flight envelope and to enhancethe aeroelastic response result in significant weight penalties, or in unavoidable reduction of nominal performances.

Corresponding author

1

AIAA Guidance, Navigation, and Control Conference and Exhibit6 - 19 August 2004, Providence, Rhode Island

AIAA 2004-522

Copyright 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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All these facts fully underline the necessity of the implementation of an active control capability enabling oneto fulfil two basic objectives of a) enhanced subcritical aeroelastic response, in the sense of suppressing or evenalleviating the severity of the wing oscillations in the shortest possible time, and b) expanded flight envelope bysuppressing flutter instability, thereby, contributing to a significant increase of the allowable flight speed. Theinterest toward the development and implementation of active control technology was prompted by the new andsometimes contradictory requirements imposed on the design of the new generation of the flight vehicle thatmandated increasing structural flexibilities, high maneuverability, and at the same time, the ability to operatesafely in severe environmental conditions. In the last two decades, the advances of active control technology have

rendered the applications of active flutter suppression and active vibrations control systems feasible [2]-[9]. A greatdeal of research activity devoted to the aeroelastic active control and flutter suppression offlight vehicles has beenaccomplished. The state-of-the-art of advances in these areas is presented in [4, 5]. The reader is also referred toa sequence of articles [8] where a number of recent contributions related to the active control of aircraft wing arediscussed at length.

Conventional methods of examining aeroelastic behavior have relied on a linear approximation of the governingequations of the flowfield and the structure. However, aerospace systems inherently contain structural and aerody-namic nonlinearities [1, 27] and it is well known that with these nonlinearities present, an aeroelastic system mayexhibit a variety of responses that are typically associated with nonlinear regimes of response including Limit CycleOscillation (LCO), flutter, and even chaotic vibrations [28]. These nonlinearities result from unsteady aerodynamicsources, such as in transonic flow condition or at high angle of attack, large deflections, and partial loss of struc-tural or control integrity. Early studies have shown that the flutter instability can be postponed and consequentlythe flight envelope can be expanded via implementation of a linear feedback control capability. However, the con-

version of the catastrophic type of flutter boundary into a benign one requires the incorporation of a nonlinearfeedback capability given a nonlinear aeroelastic system. In recent years, several active linear and nonlinear controlcapabilities have been implemented. Digital adaptive control of a linear aeroservoelastic model [30], -methodfor robust aeroservoelastic stability analysis [29], gain scheduled controllers [31], neural and adaptive control oftransonic wind-tunnel model [32, 33] are only few of the latest developed active control methods. Linear controltheory, feedback linearizing technique, and adaptive control strategies have been derived to account for the effectof nonlinear structural stiffness [34]-[40]. A model reference variable structure adaptive control system for plungedisplacement and pitch angle control has been designed using bounds on uncertain functions, [37]. This approachyields a high gain feedback discontinuous control system. In [39], an adaptive design method for flutter suppressionhas been adopted while utilizing measurements of both the pitching and plunging variables.

In this paper, an adaptive model free control capability is implemented in order to rapidly regulate the pitchingdisplacement to a setpoint while utilizing measurements of only the pitching displacement variable. Specifically, abank offilters is designed to estimate the unmeasurable state variables. These filter variables are then utilized todesign a flap deflection control input in tandem with a gradient based estimation scheme via the use of a Lyapunovfunction. Since the controller is designed to be continuously differentiable, we are able to provide an extension inorder to include the effect of manipulator dynamics.

In Section 2, the system dynamics are introduced. In Section 3, we lay down the control objective. In Section4, the state estimation strategy is designed. Section 5 deals with control and adaptive update law design alongwith stability analysis and signal chasing. In Section 6, we introduce an extension to our control strategy in orderto account for the dynamics of the flap. Section 7 shows some simulation results. Conclusions are presented inSection 8.

2 Configuration of the 2-D Wing Structural Model

Figure 1 shows a schematic for a plunging-pitching typical wing-flap section that is considered in the present analysis.

This model has been widely used in aeroelastic analysis [15],[16]. The plunging (h) and pitching () displacementsare restrained by a pair of springs attached to the elastic axis of the airfoil (EA) with spring constants kh and k (),respectively. Here, k () denotes a continuous, linear parameterizable nonlinearity, i.e., the aeroelastic system hasa continuous nonlinear restoring moment in the pitch degree of freedom. Such continuous nonlinear models forstiffness result from a thin wing or propeller being subjected to large torsional amplitudes [1, 27]. Similar models[34, 35, 37, 38] have been examined and provide a basis for comparison. As can be clearly seen in (1), the aeroelasticsystem permits two degree-of-freedom motion, whereas an aerodynamically unbalanced control surface is attachedto the trailing edge. In addition to controlling the aerodynamic flow and providing increased maneuverability, theflap is used here to suppress instabilities. h denotes the plunge displacement (positive downward), the pitch angle(measured from the horizontal at the elastic axis of the airfoil, positive nose-up) and is the aileron deflection

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LE

TE

kh

k

b b

h

MC=CG

x

y

EAAC

L

M

x

U

Figure 1: Fig. 1: 2-D wing section aeroelastic model

(measured from the axis created by the airfoil at the control flap hinge, positive flap-down). The governing equations

of motion for the aeroelastic system under consideration are given by [34]-[35]m mxbmxb I

h

+

ch 00 c

h

+

kh 00 k ()

h

=

LM

(1)

where k () denotes the pitch spring constant a particular choice for which is presented in Section 7. The quasi-steady aerodynamic lift (L) and moment (M) can be modeled as [41]

L = U2bcl

+

h

U+

1

2 a

b

U

+ U2bcl

M = U2b2cm

+

h

U+

1

2 a

b

U

+ U2b2cm

(2)

One can transform the governing EOM of (1) into the following equivalent state space form as follows [35]z = f(z) + g(z)y = z2

(3)

where z (t) =

z1 (t) z2 (t) z3 (t) z4 (t)T

4 is a vector of system states that is defined as follows

z =

h h T

(4)

(t) 1 is a flap deflection control input, y (t) 1 denotes the designated output, while f(z), g (z) 1 assumethe following form

f(z) =

z3z4

k1z1

k2U2

+ p (z2)

z2 c1z3 c2z4k3z1

k4U2 + q(z2)

z2 c3z3 c4z4

g(z) =

00

g3U2

g4U2

, g4 = 0 (5)

where p (z2) , q(z2) 1 are continuous, linear parameterizable nonlinearities in the output variable resulting from

the nonlinear pitch spring constant k(). The auxiliary constants ki, ci i = 1...4 as well as g3, g4 are explicitlydefined in Appendix A. Motivated by our desire to rewrite the equations of (3) in a form that is amenable to outputfeedback control design, we apply Kalmans observability test to the pair

A, C

where A 44, C 14 are

explicitly defined as

A =

0 1 0 0k4U

2 c4 k3 c30 0 0 1k2U2 c2 k1 c1

, C =

1000

. (6)

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A sufficient condition for the system of (3) to be observable from the pitch angle output (t) is given as

dobs = k2

3 k3c3c1 + c2

3k1 = 0

which is easily satisfied by the nominal model parameters of (48) presented in Section 7. After a series of coordinatetransformations, we can then express the governing equations for the aeroelastic model in the following convenientstate space form

x = Ax +(y) + By = CTx = 1 0 0 0 x

(7)

where = U2 is an auxiliary control input, x (t) =

x1 (t) x2 (t) x3 (t) x4 (t)T 4 is a new vector of

system states, A 44, B 4 are explicitly defined as

A =

0 1 0 00 0 1 00 0 0 10 0 0 0

, B =

0234

(8)

where i i = 2, 3, 4 are constants that are explicitly defined in Appendix A. In (7) above, (y) 4 is a smooth

vector field that can be linearly parameterized as follows

(y) = W(y)=

pj=1

jWj(y) (9)

where p is a vector of constant unknowns, W(y) 4

p is a measurable, nonlinear regression matrix whilethe notation Wj () 4 denotes the j

th column of the regression matrix W() j = 1...p. For a particular choiceof the pitch spring nonlinearity k(), explicit expressions for in terms of model parameters are provided inAppendix A.

Remark 1 As mentioned in [34], the model of (3) is accurate for airfoils at low velocity and has been confirmed bywind tunnel experiments. Following [34], unsteady aerodynamic modeling can also be implemented. In that sense,the aerodynamics would be presented in terms of the Theodorsens model, whereas for the quasi-steady aerodynamicmodel presented here, we utilize C(k) = 1.

Remark 2 The proposed control strategy is predicated on the assumption that the system of (7) is minimum phase.In Appendix B, we study the zero dynamics of the system and provide conditions for stability of the zero dynamics.

3 Control ObjectiveProvided the structures of both the aeroelastic model and the pitch spring constant nonlinearity is known, ourcontrol objective is to design a control strategy to drive the pitch angle to a setpoint while adaptively compensatingfor uncertainties in all parameters of the model and the nonlinearity. We assume that the only measurementsavailable are those of the output variable y = while we compensate for the remaining states via use of stateestimators. Toward these goals, we begin by first defining the pitch angle setpoint error e1 (t)

1 as follows

e1 = y yd (10)

We also define a state estimation error x (t) 4 as follows

x = x x (11)

where x (t) is a state vector previously defined in (7) while x (t) 4 is yet to be designed. Motivated by our

subsequent control design, we defi

ne parameter estimation error signals 0 (t) p1

and 1 (t) p2

as follows0 = 0 0 1 = 1 1 (12)

where 0 p1 ,1

p2 are unknown constant vectors that will be subsequently explicitly defined while0 (t) , 1 (t) are their corresponding dynamic estimates that will be generated as part of a Lyapunov functionbased control design scheme. We note here that p1, p2 are non-negative integers whose values are determinedthrough an explicit choice for the nonlinearity in the structural model.

Remark 3 The regulation control objective can be further modified in order to design a tracking controller with adesired output trajectory yd (t) with a broad spectrum that allows for satisfaction of a mild persistence of excitation(PE) condition (See [45]). This would allow us to exactly identify the model and the nonlinearity in simulation aswell as controlled wind tunnel experiments.

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4 Estimation Strategy

In this section, we design the state estimation vector x (t) . Given the fact that both (y) and b of (7) containunmeasurable terms, we take a modular approach [43] to the state estimation problem. Ignoring the last two termsin the first equation of (7), we design the standard observation signal 0 (t)

4 as follows

0 = A0 + kCT(x 0) (13)

where k 4 is a gain vector chosen so that A0 = A kCT is Hurwitz. Given that A and CTx = y are measurable,

one can see that the design of (13) is readily implementable. Moving on to the second term in the first equation of(7), one can now exploit the linear parameterization of (9) and the fact that A0 is Hurwitz in order to design a setof implementable observers j (t)

4 for Wj(y) as follows

j = A0j + Wj(y) j = 1....p (14)

Moving on to the last term in the first equation of (7), the following facts are readily noticeable: (a) B is unmea-surable, and (b) B(t) can be conveniently rewritten as

B=4

j=2

jej (15)

where the notation ej represents the jth

standard basis vector on 4

. Motivated by the observation design of (14),it is easy to design the following implementable set of observers for (t) as follows

i = A0i + ei i = 2, 3, 4 (16)

Given (13),(14), and (16), we are now in a position to patch together an unmeasurable state estimate x (t) as follows

x = 0 +

pj=1

jj +4

j=2

jj (17)

By taking the derivative of (11) and substituting for the dynamics of (7) and (17), we can obtain the followingexponentially stable estimation error system

.

x= A0x (18)

Although the state estimate is unmeasurable, the worth of the design above lies in the linear parameterizability ofthe right hand side of the expression of (17)as well as the exponential stability of the estimation error as will beamply evident in Section 5.

5 Adaptive Control Design and Stability Analysis

After taking the derivative of the regulation error e1 (t) and substituting (7) for the system dynamics, we obtain

e1 = x2 +

pj=1

jWj,1 (y) (19)

Since x2 (t) is unmeasurable, we add and subtract (17) to the right hand side of the above equation in order toobtain the open loop error dynamics as follows

e1 = 0,2 +

pj=1

jj,2 + Wj,1 (y)

+

4j=2

jj,2 + x2 (20)

where (11) has been utilized where the notation j,i denotes the ith element in the jth column vector of . From(16), it is easy to notice that 2,2 (t) is only one derivative removed from the control input (t) and is a suitablecandidate for a virtual control input. By applying the backstepping methodology as in [43], we define an auxiliaryerror signal e2 (t)

1 as followse2 = 2,2 2,2,d (21)

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where 2,2d (t) 1 is a yet to be designed backstepping signal. After adding and subtracting 2,2d (t) as well as

a feedback term (c11 d11) e1 to the right hand side of (20) and rearranging the terms in a manner amenable tothe pursuant analysis, one obtains

e1 = 2

12 (c11 + d11) e1 + 12 02 +

pj=1

12 jj,2 + Wj,1 (y)

+

4j=3

12 jj,2

(22)

+22,2d (c11 + d11) e1 + 2e2 + x2

where c11, d11 are positive gain constants. It is easy to see that the terms inside the brackets in the above equationcan be linearly parameterized as T1 0 where 0,1 (t)

p1 denote, respectively, a vector of unknown constantsand a measurable regression vector and are explicitly written as follows

T0 =12

12

1

12

2

..... 12

p 1

23

1

24

(23)

T1 =

c11e1 + d11e1 + 0,2 1,2 + W1,1 (y) 2,2 + W2,1 (y) ..... p,2 + Wp,1 (y) 3,2 4,2

(24)

One can now succinctly rewrite the expression of (22) as follows

e1 = 2T1 0 + 2,2d

(c11 + d11) e1 + 2e2 + x2 (25)

Given the structure of (25) above, we design the desired backstepping signal 2,2d (t) as follows

2,2d = T1 0 (26)

After substituting this control input into the open loop expression of (25), one can obtain the following expression

e1 = 2T1 0 (c11 + d11) e1 + 2e2 + x2 (27)

where the definition of (12) has been utilized. At this stage, one can define a Lyapunov like function to perform apreliminary stability analysis as follows

V1 =1

2e21

+|2|

2T00 + d

1

11xTPx (28)

where P 44 is a positive-definite symmetric matrix such that P A0 + AT0

P = I. After taking the derivativeof V1 (t) along the system trajectories of (18) and (27) and rearranging the terms, we obtain

V1 = c11e2

1 + |2|T0 1e1sign (2) T0.

0+ 2e1e2 d11e21 + e1x2 d11xTx (29)From (29), the choice for a dynamic adaptive update law is made as follows

.

0= sign (2)1e1 (30)

After substituting (30) into (29) and performing some algebraic manipulation, an upperbound for V1 (t) can beobtained in the following manner

V1 c11e2

1 3

4d111

xTx + 2e1e2 (31)

where the first two terms are negative definite terms and the last term is an indefinite interconnection term whichis dealt with in the next stage of control design. Our next step involves design of our actual control input (t)which is easily reachable through differentiation of (21) along the dynamics of (16) as follows

e2 = k22,1 + 2,3 + 2,2d (32)

From (26) and (24), it is obvious that 2,2d () can be represented in the following manner

2,2d = 2,2d

y, yd, 0,2, 1,2, 2,2,...,p,2, 3,2, 4,2, 0

(33)

Since 3,2 (t) , 4,2 (t) are more than one derivative away from the control, potential singularities in the controldesign are avoided when the desired control signal 2,2d () is differentiated as follows

2,2d =2,2d

y

02 +

pj=1

jj,2 + Wj,1 (y)

+

4j=2

jj,2 + x2

+

2,2d

0[A00 + ky]

+pj=1

2,2d

j

A0j + Wj (y)

+

4j=3

2,2d

jA0j +

2,2d

0[sign (2)1e1]

(34)

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It is now possible to separate out (34) into its measurable and immeasurable components as follows

2,2d = 2,2dm + 2,2du +2,2d

yx2 (35)

where 2,2dm (t) 1 denotes the measurable components that is matched with the control input (t) and can

be done away via direct cancellation, 2,2du (t) 1 denotes unmeasurable components that are linearly parame-

terizable and dealt with via the design of an adaptive update law, while the last term can be damped out owing

to the exponential stability of (18) (For explicit expressions for

2,2dm (t) and

2,2du (t), we refer the reader to theAppendix A). We are now in a position to rewrite the open-loop dynamics for e2 (t) in the following manner

e2 = k22,1 + 2,3 + 2,2dm 2,2d

yx2 2e1 + [2e1 2,2du] (36)

where we have added and subtracted 2e1 to the right hand side of (32) and (35) has been utilized. It is importantto note that the bracketed term in the above expression can be parameterized as T2 1 where 1,1 (t)

p1

denote, respectively, a vector of unknown constants and a measurable regression vector and are explicitly writtenas follows

T1

=1

2

..... p 2 3 4

(37)

T2 =

2,2d

y 1,2 + W1,1 (y)

... 2,2d

y p,2 + Wp,1 (y)

2,2d

y2,2 + e1 3,2 4,2

(38)

Motivated by our preliminary stability analysis and the structure of (36), we design our control input (t) asfollows

= c22e2 d22

2,2d

y

2e2 2,3 + k22,1 + 2,2dm

T2 1 (39)

where c22, d22 are positive gain constants. After substituting this control input into (36), we obtain the closed loopdynamics for e2 (t) in the following manner

e2 = c22e2 d22

2,2d

y

2e2

2,2d

yx2 2e1 +

T21 (40)

From the structure of (40), one can design an update law for 1 (t) as follows

.

1= 2e2 (41)

Toward a final stability analysis, the function of (28) can be augmented as follows

V2 = V1 +1

2e22

+ T11 + d

1

22xTPx (42)

The time derivative of V2 (t) along the closed-loop trajectories of (40) and (41) yields the following upperbound

V2 c11e2

1 c22e2

2 3

4

d111

+ d122

xTx (43)

where we utilized the nonlinear damping argument [44]. Since V2 (t) is negative semi-definite and V2 (t) is positivedefinite, we can conclude that V2 (t) L. Therefore, from (28) and (42), it is easy to see that 0 (t) , 1 (t) L

and e1 (t) , e2 (t) , x (t) LL2. From the apriori boundedness of yd,0,1, we can say that y (t) , 0 (t) , 1 (t) L. The boundedness of y (t) guarantees that i (t) L i = 0....p. From (7) and the assumption of stable zerodynamics, the boundedness of 2,1 (t) , 3,1 (t) ,3,2 (t) , 4,1 (t) , 4,2 (t) ,4,3 (t) can be readily established. Hence,1 (t) ,2,2d (t) L from (24) and (26) which implies from the definition of (21) that 2,2 (t) L. From thisand the preceding assumptions, one can state that 2 (t) , (t) ,(t) L. From (16), one can guarantee theboundedness of i i = 2, 3, 4. Thus, all the states of the system are bounded in closed-loop operation. From(27) and (40), one can see that e1 (t) , e2 (t) L. Given all the preceding facts, we can now apply BarbalatsLemma [45] to state that lim

te1 (t) , e2 (t) = 0. From (18), the estimation error x (t) is exponentially regulated to

the origin.

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6 Inclusion of Actuator Dynamics

In order to model the control surface dynamics along with the quasi-steady equations of (1), we follow the approachof [46] and assume a second-order system as follows

+ p1+ p2= p2u p2 = 0 (44)

where (t) 1 denotes the actual control surface deflection while u (t) is the flap actuator output and the de

facto control input signal. In this Section, we will discuss 3 different strategies to include the effect of the dynamicsof the flap.

Foe the first strategy, let p1 and p2 be chosen such that the dynamics of (44) are faster than the dynamics of(1); then, similar to a cascaded control structure, we can neglect the coupling of the plunge and pitch motion ofthe wing section. The signal (t) introduced in (1) and designed subsequently in (39) can be treated as a desiredflap deflection d (t) and the control input u (t) can be simply chosen to be

u (t) = d + p1d + p2d

such that the closed-loop system for (44) becomes

+ p1 + p2 = 0 (45)

which is an exponentially stable system and where d

denotes the error between the desired and actualflap deflection. Thus, (t) tracks d (t) exponentially fast.

However, if the actuator has slow dynamics, then one must consider the interconnection between the dynamicsof (1) and (44). One must then rewrite (36) as follows

e2 = k22,1 + 2,3 + U2d 2,2dm

2,2d

yx2 2e1 +

T2 1 + U

2 (46)

Having designed d (t) as done previosuly in (39), it is easy to see that one is left with a U2 mismatched indefinite

term in the closed-loop dynamics for e2 (t) . One can now represent the time derivative of (t) as = v d where

v (t) (t) 1. We now define an auxiliary design variable v (t) 1 as follows

v = v vd

where vd (t) is a virtual control input that is chosen as follows

vd = d U2e2 + k

such that the closed dynamics for (t) are obtained as follows

= U2e2 k + v

After time differentiating v (t) and utilizing the actuator dynamics of (44), one can obtain

v = p1p2+ p2u vd

Motivated by the structure of the open-loop dynamics above, we can design the control input signal as follows

u (t) = p12

vd + p1+ p2 kv

(47)

which yields a closed-loop form for the actuator dynamics as follows

v = kv

We can now augment the function of (42) as V3 = V2+1

22+

1

22v which yields after differentiation along the system

closed-loop trajectories the following upperbound

V3 c11e2

1 c22e2

2 3

4

d111

+ d122

xTx k

2 k2v

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Again, a signal chasing argument would show that limt

e1 (t) , e2 (t) , x (t) , r (t) = 0. It is to be noted here that the

augmentation strategy involves two derivatives ofd (t) which certainly leads to measurements of more variablesthan was previously stated. With a great deal of extra work, one could work around making the extra measurements;however, that will not be pursued here in the interest of brevity. One must also note that we needed to utilizemeasurements of the state variables (t) , (t) as well as the model parameters p1 and p2. In order to mitigate thisdependence and design a true output feedback control strategy with measurements of only the pitch displacementas well as adaptively compensate for uncertainty in p1and p2, one would need to comprehensively redesign theestimation, adaptation, and control strategies. The methodology followed in this paper would still be valid but

the estimation strategy would have to include observers for the new states, i.e., 1 (t) and its time derivative.Additionally, the parameter update laws would need to be augmented to compensate for the effects of the actuatormodel in terms of p1 and p2.

7 Results and Discussion

The model described in (3) was simulated using the output feedback observation, control, and estimation algorithmof (13), (14), (16), (30), (39), and (41). The values for the model parameters were taken from [34] and are listedbelow

b = 0.135 [m] kh = 2844.4 [Nm1] ch = 27.43 [Nm

1s1]c = 0.036 [Ns] = 1.225 [kgm3] cl = 6.28cl = 3.358 cm = (0.5 + a) cl cm = 0.635

m = 12.387 [kg] I = 0.065 [kgm2

] x = [0.0873 (b + ab)] /ba = 0.4

(48)

The nonlinear pitch spring stiffness was represented by a quintic polynomial as follows

ky(y) =5

i=1

iyi1 (49)

where the unknown i extracted from experimental data [35] are given as

{i} =

2.8 62.3 3709.7 24195.6 48756.9T

The first simulation was run with freestream velocity U = 20 [ms1] while the initial conditions for the system

states, observed variables, and estimates were chosen as follows (0) = 0.1 [rad] (0) = 0 [rads1] h (0) = 0 [m]

h (0) = 0 [ms1] 0 (0) = 0 1 (0) = 00

(0) = 0 j (0) = 0 j (0) = 0(50)

Figure 2 shows the closed-loop plunging, pitching, and control surface deflection time-histories for the case whenthe actuation is on at time zero. In the presence of active control, the system states are regulated to the originfairly rapidly. It is interesting to note that the adaptive control strategy is able to quickly regulate the plungingand pitching response in the presence of large uncertainty in the parameters. The second simulation was run withfreestream velocity U = 15 [ms1] with the same initial conditions as (50). However, in this case, the system isallowed to evolve open-loop (i.e., (t) 0) for 15 [s] to observe the development of an LCO. During the first 15seconds, the estimation strategy is allowed to evolve according to the dynamics of (13), (14), (16). This is possiblebecause the estimation strategy is designed to be stable independent of the control methodology used. However,the estimation strategy of (30) and (41) are turned offbecause of their dependence on the control strategy of (39).At t = 15 [s], the control is turned on and the immediate attenuation of oscillations can be seen in Figures 3 and4. Again, the response is seen to be quick in the presence of large model uncertainty.

Remark 4 It is worth remarking that the methodology presented here can be extended to the compressible flightspeed regimes. In this sense, pertinent aerodynamics for the compressible subsonic, supersonic and hypersonicflightspeed regimes have to be applied [42]. However, the goal of this paper was restricted to the issue of the illustration ofthe methodology and to that of highlighting the importance of the implementation of the nonlinear adaptive controlon the lifting surfaces equipped with aflap.

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8 Conclusions

Results related to the adaptive control of a nonlinear plunging-pitching wing section operating in an incompressibleflight speed have been presented here. In this paper, a true output feedback strategy (that utilizes only measure-ments of the pitching variable ) has been obtained. A quasi-steady aerodynamic model is used and an unsteadyone is contemplated as future work. The performance of the adaptive model free control is analyzed. It clearlyappears that the model free control is able to efficiently control the LCO.

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Appendix AThe auxiliary constants ki, ci i = 1...4 as well as g3, g4 that were introduced in the model description of (5)

are explicitly defined as follows

k1 = Ikhd1 k2 = d

1

Ibcl + mxb3cm

k3 = mxbkhd

1 k4 = d1

mxb2cl mb2cm

c1 = d1

I (ch + Ubcl) + mxU b3cm

c2 = d

1

IU b2cla mxbc + mxU b4cma

c3 = d1 mxbch mxUb2cl mUb2cm c4 = d1 mc mxUb3cla mUb3cmag3 = d

1

Ibcl mxb3cm

g4 = d1

mxb2cl + mb2cm

d = m

I mx2b2

a = (0.5 a)

The elements i i = 2, 3, 4 introduced in (8) are explicitly defined in as follows

2 = g43 = g3c3 + c1g44 = k1g4 + k1c23g3k

1

3 c1g3c3 dobsg3k

1

3

(51)

Given the pitch spring constant of (49), explicit expression for W(y) and introduced in (9) are provided below

W(y) =

y 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 y 0 0 y2 0 0 y3 0 0 y4 0 0 y5 0 00 0 y 0 0 y2 0 0 y3 0 0 y4 0 0 y5 00 0 0 y 0 0 y2 0 0 y3 0 0 y4 0 0 y5

(52)

1 = 12 = 2 +

k4U

2 md11

3 = 3 + c3k2U2 c1k4U

2 mxb1c3d1 c1m1d

1

4 = k1k4U2 k1md

11 + kk2U

2 kmxbd115 = 2md

1

6 = 2(c3mxbd1 + c1md

1)7 = 2(k1md

1 + kmxbd1)

8

= 3md1

9 = 3(k1md1 + kmxbd

1)10 = 3(k1md

1 + kmxbd1)

11 = 4md1

12 = 4(k1md1 + kmxbd1)

13

= 4(k1md1 + kmxbd

1)14 = 5md

1

15 = 5(k1md1 + kmxbd1)

16 = 5(k1md1 + kmxbd

1)

(53)

where k, i are defined as follows

1 = c1 c42 = c2c3 k1 c1c43 = k

1

3

k1c4c5 k1c2c

23 c1c3c4k1 + c1c2c3k3 + dobsc2

k = k1

3

c1c3k3 k1c

23

+ dobs

We note here that it is simple to alter the size and definition of if it is desired that the freestream velocity U

should be a part of the measurable regression vector W() of (9).The explicit expressions for 2,2dm (t) and 2,2du (t) are obtained as follows

2,2dm =2,2d

y02 +

2,2d

0[A00 + ky] +

pj=1

2,2d

j

A0j + Wj (y)

+

4j=3

2,2d

jA0j +

2,2d

0[sign (2)1e1]

2,2du =2,2d

y

p

j=1

jj,2 + Wj,1 (y)

+

4j=2

jj,2

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Appendix BThe state space system of (7) can be expanded into the following form

y = x1 = x1 = x2 +1(y)x2 = x3 +2(y) + 2x3 = x4 +3(y) + 3

x4 = 4(y) + 4

(54)

Here, we study the stability of the zero dynamics for the case when the pitch displacement is regulated to theorigin. Mathematically, this implies that

y 0 y = x2 0 x2 0

which implies from the second equation of (54) that

u =1

2[x3 +2 (0)] (55)

From (52), it is plain to see that W(0) = 0 which implies that i(0) = 0 i = 1.....4. The zero dynamics of thesystem then reduce to the second order system given by

x3 = x4 + 3u

x4 = 4u

Substituting (55) into the above set of equations for u, we obtain the linear system of equations x = Ax where

x =

x3 x4T

and A given by

A =

3

21

4

20

the eigenvalues of which are given by

A

=

1

22

3

2

3 424

From the nominal system of (48) and the expressions of (51), we can compute 2 = 0.1655, 3 = 0.6275, 4 =50.4645 from which Re (A) = 1.8958 which makes for asymptotically stable zero dynamics. For the sake ofcompleteness, the eigenvalues for the zero dynamics turn out to be A = 1.8958j5.7458. It is to be noted herethat a more general analysis would analyze the stability of the zero dynamics for a range of values of yd.

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-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time, s

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Figure 2: Time evolution of the closed-loop plunging and pitching deflections and flap deflection for U = 20 [ms1];U = 1.6UFlutter

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0 5 10 15 20 25-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time, s

Before Actuation

LCOAfter Actuation

LCO Suppression

Figure 3: Time-history of pitching deflection for U =[15 ms1], U = 1.25UFlutter

-800

-600

-400

-200

0

200

400

600

800

(t), deg

.

Before Actuation

LCO

After Actuation

LCO Suppression

Figure 4: Phase-space of the pitching deflection for U = 15 [ms1], U = 1.25UFlutter