Matlab Linear State Space Control Systems

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Aerospace Science and Technology 7 (2003) 23–31 www.elsevier.com/locate/aescte

State feedback control of an aeroelastic system with structural nonlinearity

Sahjendra N. Singh, Woosoon Yim∗

Howard R. Hughes College of Engineering, University of Nevada, Las Vegas, 4505 Maryland Parkway, Las Vegas, NV 89154-4027, USA

Received 29 November 2001; received in revised form 21 June 2002; accepted 27 September 2002

Abstract

The paper treats the question of state variable feedback control of prototypical aeroelastic wing sections with structural nonlinearity. This type of model has been traditionally used for the theoretical as well as experimental analyses of two-dimensional aeroelastic behavior. The chosen dynamic model describes the nonlinear plunge and pitch motion of a wing. A single control surface is used for the purpose of control. A control law is designed based on the state-dependent Riccati equation technique. Unlike feedback linearizing control systems reported in literature, this approach is applicable to minimum as well as nonminimum phase aeroelastic models. The closed-loop system is asymptotically stable. Simulation results are presented which show that in the closed-loop system, flutter suppression is accomplished.  2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

Keywords: Aeroelasticity and control; Nonlinear flutter control; Nonlinear system; Suboptimal control

1. Introduction

Aeroelastic systems exhibit a variety of phenomena in- cluding instability, limit cycle, and even chaotic vibration [1–3]. Active control as a remedy against aeroelastic in- stability is an important task. Researchers have analyzed the stability properties of aeroelastic systems and designed controllers for flutter suppression. Digital adaptive control of a linear autoregressive moving average aeroservoelas- tic model has been considered [4]. At the NASA Lang- ley Research Center, a benchmark active control technique (BACT) wind-tunnel model has been designed and con- trol algorithms for flutter suppression have been developed [5–10]. References [6] and [7] describe unsteady aerody- namic data and flutter instability for the BACT project model. A robust flutter-suppression control law using classi- cal and minmax method has been derived [8]. Robust passifi- cation techniques have been used in [9] for control. Two lin- ear parameter-varying gain scheduled controllers have been designed in [10]. A computational two-dimensional aeroser- voelasticity study has been done [11]. Active control of tran- sonic wind-tunnel model using neural networks has been considered [12]. For an aeroelastic apparatus, tests have been

* Corresponding author. E-mail address: [email protected] (W. Yim).

performed in a wind tunnel to examine the effect of non- linear structural stiffness and control systems have been de- signed using linear control theory, feedback linearizing tech- nique, and adaptive control strategies [13–19]. In [14] the unsteady aerodynamics are modeled with an approximation to Theodorsen’s theory and linear control laws are derived. Adaptive control of an aeroelastic system using feedback of both the plunge displacement and pitch angle has been considered [17]. Based on modeling error compensation ap- proach, an output feedback adaptive controller has been de- signed in [19]. However, it uses a high-gain observer, which is sensitive to measurement noise. The Euler–Lagrange the- ory for controlling the aeroelastic model using two control surfaces has been used in [20].

The nonlinear control systems of [15–19] are essentially based on feedback linearization technique [21,22]. However, it must be noted that the results of these papers are based on the assumption that the zero dynamics [21,22] of the aeroelastic system are stable. That is, the residual dynamics of the system are asymptotically stable, if the chosen output variable (the pitch angle or the plunge displacement) is identically zero. The flow velocityU and the elastic axis locationa are the two important parameters in the aeroelastic model, and the stability of the zero dynamics depends on their values. Researchers have examined the stability of the zero dynamics only for the variations inU anda, and shown

24 S.N. Singh, W. Yim / Aerospace Science and Technology 7 (2003) 23–31

Nomenclature

A,B system matrices Ac closed-loop matrix a nondimensionalized distance from the midchord

to the elastic axis α pitch angle β flap deflection b semichord of the wing C(x) controllability matrix ch structural damping coefficient in plunge due to

viscous damping cα structural damping coefficient in pitch due to

viscous damping clα,cmα lift and moment coefficients per angle of attack clβ ,cmβ lift and moment coefficients per control surface

deflection H Hamiltonian h plunge displacement

Iα mass moment of inertia of the wing about the elastic axis

J performance index kh,ch structural spring and damping constants in

plunge kα,cα structural spring and damping constants in pitch L,M aerodynamic force and moment Λ Lagrange multiplier m mass P(x) positive definite symmetric matrix ρ density of air Q,R weighting matrices sp wing span U free stream velocity x states of the aeroelastic system xα nondimensionalized distance measured from the

elastic axis to center of mass

that the system is nonminimum phase (yielding unstable zero dynamics) for a set of values of the flow velocity and the elastic axis location [15–19]. Moreover, for the choice of plunge displacement as an output, one obtains an extremely nonlinear zero dynamics which have multiple equilibrium points and thus have complex behavior. Apparently, there is a need to design a control system for the flutter suppression of aeroelastic systems which may be nonminimum phase (systems with unstable zero dynamics). Furthermore, even for aeroelastic systems which are minimum phase, the residual motion evolves according to their governing zero dynamics in a natural way, which may have poor damping. As such it is interesting to design control systems which enhances the response characteristics of the pitch as well as plunge motion.

The contribution of this paper lies in the design of a con- trol law for the flutter control of nonlinear aeroelastic sys- tems by state variable feedback. The model represents a prototypical aeroelastic wing section which has been tradi- tionally used for the theoretical and experimental study of two-dimensional aeroelastic behavior. A single trailing-edge control surface is used for the control of the system. The control system design is based on the state-dependent Ric- cati equation method. This approach has been advanced in a series of interesting papers [23–26] and applied to vari- ety of aerospace control problems. Interestingly, minimum as well as nonminimum phase systems can be controlled us- ing this approach. Moreover, this method provides flexibil- ity in shaping the response characteristics of the pitch an- gle and as well as the plunge displacement trajectories. The closed-loop aeroelastic system is locally asymptotically sta- ble about the equilibrium state with zero deflections, there- fore, the pitch angle and the plunge displacement converge

to zero. Simulation results are presented which show that the control system accomplishes flutter suppression.

2. Aeroelastic model and control problem

The prototypical aeroelastic wing section is shown in Fig. 1. The governing equations of motion are provided in [15,16] which are given by

Fig. 1. Aeroelastic model.

S.N. Singh, W. Yim / Aerospace Science and Technology 7 (2003) 23–31 25

[ m mxαb

mxαb Iα

] [ ḧ

α̈

] +

[ ch 0 0 cα

] [ ḣ

α̇

]

+ [ kh 0 0 kα(α)

] [ h

] =

[ −L M

] , (1)

whereh is the plunge displacement andα is the pitch angle. In (1), m is the mass of the wing;b is the semichord of the wing; Iα is the moment of inertia;xα is the nondimensional- ized distance of the center of mass from the elastic axis;cα andch are the pitch and plunge damping coefficients, respec- tively; andM andL are the aerodynamic lift and moment. It is assumed that the quasi-steady aerodynamic force and moment are of the form

L = ρU2spbclα [ α + (ḣ/U) +

( 1

2 − a

) b(α̇/U)

]

+ ρU2spbclβ β, (2)

M = ρU2spb2cmα [ α + (ḣ/U) +

( 1

2 − a

) b(α̇/U)

]

+ ρU2spb2cmβ β, (3) wherea is the nondimensionalized distance from the mid- chord to the elastic axis,sp is the wing span,clα andcmα are the lift and moment coefficients per angle of attack, andclβ andcmβ are lift and moment coefficients per control surface deflectionβ. The model (1) has a diagonal damping matrix, however it should be noted that the SDRE method is applica- ble to models of larger dimensions which have nonlinear damping matrices with nonzero off-diagonal elements. Al- though, other forms of nonlinear spring stiffness associated with the pitch motion can be considered, for purposes of il- lustration, the functionkα(α) is considered as a polynomial nonlinearity given by [13,15,16]

kα = 2.82 ( 1 − 22.1α + 1315.5α2 − 8580α3 + 17289.7α4)

.= kα0 + kα1α + kα2α2 + kα3α3 + kα4α4. The nonlinear stiffness term has been obtained by curve fitting the measured displacement moment data for the nonlinear spring (see [13] for the details).

It is noted that the expressions forL and M given in (2) and (3) represent quasi steady aerodynamics and the control surface aerodynamics ignore the derivatives of control surface deflection. However these simplifications are in no way essential, and are used only to make the example more tractable. Indeed based on Theodorsen’s theory [14], state variables for unsteady aerodynamics representation that depend upon velocity and acceleration as well, could be included in the model. Then the SDRE design can be completed using the augmented state variable model of larger dimension.

Defining the state vectorx = (α,h,α̇,ḣ)T ∈ R4, one obtains a state variable representation of (1)–(3) in the form

ẋ = [

02×2 M1

I2×2 M2

] x +

[ 02×1 p0α

] knα(α) +

[ 02×1 b0

] β, (4)

wherekα = kα0 + knα , knα = kα1α + kα2α2 + kα3α3 + kα4α4, kαi are constants,p0 is a constant vector,b0 = (b01,b02)T (T denotes transposition), 0i×j and Ii×j denote null and identity matrices of appropriate dimensions, and

M1 = [ −(k4U2 + md−1kα0) −k3 −(k2U2 − mxαbd−1kα0) −k1

] ,

M2 = [−c41 −c31 −c21 −c11

] .

The expressions for the parameterski, d, cij , b0k, andp0 are given in Appendix A.

DefineB = [01×2, bT0 ]T and

Mn(α) = [ p0knα(α) 02×1

] ,

A(α) = [

02×2 I2×2 M1 + Mn(α) M2

] . (5)

Then writing (3) in a compact form gives

ẋ = A(α)x + Bβ. (6) The matrixA is a nonlinear function of the pitch angle. Thus x = 04×1 is an equilibrium point of the system with zero input (β = 0).

We are interested in designing a state variable feedback control law such that in the closed-loop system, both the pitch angle and the plunge displacement asymptotically converge to zero.

3. State variable feedback control law

In this section, a nonlinear control law based on the state-dependent Riccati equation (SDRE) method [23–25] for the flutter control is designed. This design procedure is applicable even if bothA and B matrices in (6) are nonlinear functions of the state vectorx. Obviously, hereB is a constant vector.

Consider an optimal control (infinite-horizon regulator) problem in which for the nonlinear system (5), the perfor- mance index of the form

J = (1/2) ∞∫

( x

T Q(x)x + R(x)β2) dt (7)

is to be minimized, whereQ(x) is a positive definite symmetric matrix andR(x) > 0 for all x ∈ R4. The weighting matrixQ and the scalar functionR(x) are chosen properly for obtaining desirable responses in the closed-loop system. Instead of deriving an optimal control law (which is extremely difficult for nonlinear systems), for simplicity, a suboptimal control law is designed using the SDRE method.

Consider a regionΩ ⊂ R4 of the state space surrounding the origin x = 0. For the existence of a solution using the SDRE method, the following assumption is made.

26 S.N. Singh, W. Yim / Aerospace Science and Technology 7 (2003) 23–31

Fig. 2. Determinant of controllability matrix as a function ofU andα.

Assumption 1. The pair{A(x),B} is pointwise controllable at eachx ∈ Ω. That is, the controllability matrix C(x) = [B,A(x)B,A2(x)B,A3(x)B] (8) has rank 4 for allx ∈ Ω ⊂ R4.

The controllability matrix is function ofU, α, and a. For the values of parameters given in Appendix A, the matrix C(x) is nonsingular at the originx = 0. Therefore, it follows that there exists an open neighborhood ofx = 0 in which the controllability matrixC(x) is nonsingular. The determinant of the controllability matrix is a polynomial function in α. Using the parameters given in Appendix A, the controllability matrix has been computed for several discrete values ofU ∈ [10, 30] (m/s), α ∈ [−60, 60] (deg), and a ∈ [−1, 0] using the step size'U = 1 (m/s), α = 0.1 (rad), and'a = 0.2, respectively and it has been found that its determinant is always negative. The plot of the determinant as a function ofU andα only for a = −0.6 is shown in Fig. 2. It is seen that the determinant is negative and its maximum value is−9.86 × 108. For the discrete values ofU ∈ [10, 30] (m/s) andα ∈ [−60, 60] (deg), the maximum values of the determinant fora = 0, −0.2, −0.4, −0.8, and−1.0 are−2.7 × 109, −1.67× 108, −3.8 × 107, −6.9× 1010, and−3.6× 1010, respectively. Thus the system is controllable for eachα for the set of values ofU anda in the parameter space of interest.

Now for obtaining a suboptimal solution using the SDRE method, one solves the state-dependent Riccati equation given by

A T (x)P(x) + P(x)A(x) − P(x)BR−1(x)BTP(x) + Q(x) = 0 (9)

to obtain a symmetric positive definite solution forP(x). Then the nonlinear feedback control law is given by

β(x) = −R−1(x)BTP(x)x. (10) Readers may refer to [24] for the properties and capabil-

ities of the SDRE method. It is interesting to note that the suboptimal law (9) satisfies

∂H(x,λ)/∂β = 0, (11) where the Hamiltonian of the nonlinear optimal control problem (6) and (7) is

H(x,λ) = (1/2)[xTQ(x)x + R(x)β2] + λT[A(x)x + Bβ]

andλ ∈ R4 is the co-state or the Lagrange multiplier. Substituting the control law (10) in (6) gives the closed-

loop system

ẋ = [A(x) − BR−1BTP(x)]x .= Ac(x)x. (12) The closed-loop matrixAc(x) is guaranteed to be Hurwitz at every x ∈ Ω from Riccati equation theory. Since the elements ofA(x) are smooth functions, expandingAc(x) aboutx = 0, and using mean value theorem, one can show that the the equilibrium pointx = 0 of (12) is asymptotically stable. The performance of the closed-loop system depends on the the matrixA(x) and the weighting matricesQ(x) andR(x).

4. Simulation results

In this section, numerical results for the control of the aeroelastic system are obtained. The parameters of the system given in [15] are collected in Appendix A.

Case 0. Simulation is performed for the open-loop system (β = 0) with the initial conditionsα(0) = 0.2 (rad) (11.44 deg),h(0) = 0.02 (m), α̇(0) = 0 (rad/s), and ḣ(0) = 0 and for the values ofa = −0.6, U = 15 m/s. The matrix A(0) has its two eigenvalues in the right half plane at 3.21+ j12.28 and 3.21− j12.28 and the remaining eigenvalues are in the left half plane. Thus the originx = 0 is locally unstable. As shown in Fig. 3, for the open-loop system persistent periodic oscillations (limit cycles) in the pitch angle and plunge displacement responses exist.

Case 1. Now the closed-loop system (12) including the nonlinear control law (10) is simulated. The parametersU anda and the initial conditions of case 0 are retained. The weighting matrix and the scalar function in the performance index are selected asQ = diag(1, 10, 1, 10) andR = 1000, respectively. As usual in the design of optimal control law, Q and R have been selected after several trials and by observing simulated responses. It has been shown in [15] that the transfer function relating the pitch angleα as an output and the inputβ has one zero in the right half plane for the selected value ofU and a, and the

S.N. Singh, W. Yim / Aerospace Science and Technology 7 (2003) 23–31 27

Fig. 3. Open-loop responses. (a) Pitch angle (deg). (b) Phase plane ploth (m)–ḣ (m/s).

Fig. 4. Nonlinear flutter control:a = −0.6, U = 15 (m/s). (a) Pitch angleα (deg). (b) Plunge displacementh (m). (c) Control inputβ (deg).

system is nonminimum phase. Apparently, the design based on feedback linearization [15–19] is not applicable for the trajectory tracking ofα, since the zero dynamics (the residual plunge motion whenα = 0) diverges.

We observe that in the closed-loop system including con- trol law (10), the pitch angle and the plunge displacement converge to zero (Fig. 4). The settling time for the stabiliza-

tion of both the pitch angle and the plunge displacement is of the order of two seconds, which is fast. The maximum con- trol magnitude for stabilization is 27 (deg). The peaks in the pitch angle and plunge displacement do not exceed their ini- tial values of 0.2 rad and 0.02 (m), respectively. It is found that the weighting parametersQ and R play an important role in shaping the response characteristics. The choice of

28 S.N. Singh, W. Yim / Aerospace Science and Technology 7 (2003) 23–31

Fig. 5. Linear feedback control:a = −0.6, U = 15 (m/s). (a) Pitch angle (deg). (b) Phase plane ploth (m)–ḣ (m/s).

Fig. 6. Nonlinear flutter control:a = −0.6, U = 20 (m/s). (a) Pitch angleα (deg). (b) Plunge displacementh (m). (c) Control inputβ (deg).

largerQ reduces the peaks in the state variables and gives faster convergence, but requires larger control input.

Case 1a. It is interesting to examine the advantage of nonlinear controller designed using SDRE technique compared to a linear optimal control system.

For this purpose, a linear control lawβ = −Kx (K is a constant vector) is designed using the linearized aeroelastic

model aboutx = 0 using the performance index of case 1. Note that in this case, solution of the Riccati equation (8) yields a constant matrixP since A is a constant matrix (obtained by settingMn(α) = 0 in (4)). Selected responses using the linear controller are shown in Fig. 5. We observe that the linear optimal controller fails to stabilize the system and the pitch angle and the plunge displacement diverge

S.N. Singh, W. Yim / Aerospace Science and Technology 7 (2003) 23–31 29

Fig. 7. Nonlinear flutter control:a = −0.4, U = 15 (m/s). (a) Pitch angleα (deg). (b) Plunge displacementh (m). (c) Control inputβ (deg).

rapidly to large values for the same operating parameters and initial conditions used in case 0 and case 1.

Case 2. Simulation is performed for a larger flow velocity (U = 20 (m/s)), but the remaining parameters of the model and initial conditions of case 0 and case 1 are retained. Although one can tune the values ofQ andR for reshaping the responses, their values of case 1 and case 1a are retained. We observe thatα and h asymptotically converge to zero (Fig. 6). A faster response time of the order of 1.3 seconds is obtained and the maximum control magnitude is less than 26 (deg). The subsequent peak values of bothα andh are below their initial values. It is observed that compared to case 1, for the same weighting matrices, faster responses with smaller peaks in the control input are obtained since the control surface is more effective at higher flow velocity.

Case 3. The closed-loop system for the model with different value ofa = −0.4, but with flow velocityU = 15 (m/s) as in case 1 is simulated. In this case, it is found that Q andR of case 1 give larger peaks in the pitch angle and plunge trajectories in the transient period. As such, a different set of weighting matrixQ = diag(10, 100, 10, 100) is selected, butR is not changed. Note that in order to reduce the peaks in the responses, the elements ofQ have been increased by a factor of 10 compared to case 1. We observe

in Fig. 7 that system state asymptotically converges to zero. Smaller control magnitude (about 24 (deg)) is required, but larger settling time of the order of 3 (seconds) is obtained. The subsequent peaks inα and h are lower in magnitude than the given initial conditions.

Extensive simulation has been performed. These results show that the nonlinear control system designed using the SDRE method accomplishes flutter suppression in spite of large perturbations in the initial state for different values of the flow velocity and the elastic axis location.

5. Conclusions

In this paper, stabilization of an aeroelastic system with structural nonlinearity was considered. This model has been traditionally used for the experimental and theoretical analy- ses of two-dimensional aeroelastic behavior. A single con- trol surface was used for the flutter suppression. A subopti- mal control law using the state-dependent Riccati equation method was designed. This nonlinear control law accom- plishes asymptotic regulation of the pitch and plunge mo- tion to the system equilibrium at zero deflections. It is seen that the linear control law based on the linearized model of

30 S.N. Singh, W. Yim / Aerospace Science and Technology 7 (2003) 23–31

the aeroelastic model fails to stabilize the nonlinear system. But the nonlinear control system using SDRE method yields at least some region of stability surrounding the origin in the state space. Simulation results for various flow velocities and elastic axis locations were presented. These results show that in the closed-loop system, the designed controller is ef- fective in flutter suppression. Unlike the feedback lineariz- ing controllers designed in literature, SDRE method yields a controller which stabilizes minimum as well as nonmini- mum phase systems.

Acknowledgements

Authors wish to thank Dr. Thomas W. Strganac of Texas A&M University for his comments on the aeroelastic model used in the paper and reviewers for his/her valuable comments and helpful suggestions.

Appendix A

System parameters

b = 0.135 m, cα = 0.036 Ns, clβ = 3.358, m = 12.387 kg, sp = 0.6 m, kh = 2844.4 N/m, ρ = 1.225 kg/m3, cmα = (0.5 + a)clα, Iα = 0.065 kgm2, α = 2.82(1 − 22.1α + 1315.5α2

− 8580α3 + 17289.7α4) N·m/rad, ch = 27.43 Ns/m, clα = 6.28, cmβ = −0.635, xα = [0.0873− (b + ab)]/b. System variables are given by

d = m(Iα − mx2αb2 ) ,

k1 = Iαkh/d, k2 =

( Iαρbclα + mxαb3ρcmα

) /d,

k3 = −mxαbkh/d, k4 =

(−mxαb2ρclα − mρb2cmα)/d, c11 =

[ Iα(ch + ρUbclα) + mxαρUb3cmα

] /d,

c21 = [ IαρUb

2 clα

( 1

2 − a

) − mxαbcα

+ mxαρUb4cmα (

2 − a

)]/ d,

c31 = (−mxαbch − mxαρUb2clα − mρUb2cmα)/d,

c41 = [ mcα − mxαρUb3clα

( 1

2 − a

)

− mρUb3cmα (

2 − a

)]/ d,

b01 = U2 ( mxαb

2 ρclβ + mρb2cmβ

) /d,

b02 = U2 (−Iαρbclβ − mxαb3ρcmβ)/d,

p0 = [ −m/d mxab/d

] .

References

[1] Y.C. Fung, An Introduction to the Theory of Aeroelasticity, Wiley, New York, 1955, pp. 207–215.

[2] E.H. Dowell (Ed.), A Modern Course in Aeroelasticity, Kluwer Academic, MA, 1995.

[3] L.C. Zhao, Z.C. Yang, Chaotic motions of an airfoil with nonlinear stiffness in incompressible flow, J. Sound Vibration 128 (2) (1990) 245–254.

[4] P.P. Friedmann, D. Guillot, E. Presente, Adaptive control of aeroelastic instabilities in transonic flow and its scaling, J. Guidance, Control, and Dynamics 20 (6) (1997) 1190–1199.

[5] M.R. Waszak, Robust multivariable flutter suppression for the bench- mark active control technology (BACT) wind-tunnel model, J. Guid- ance, Control, and Dynamics 24 (1) (1997) 143–147.

[6] R.C. Scott, S.T. Hoadley, C.D. Wieseman, M.H. Durham, Benchmark active controls technology model aerodynamic data, J. Guidance, Control, and Dynamics 23 (5) (2000) 914–921.

[7] R.M. Bennett, R.C. Scott, C.D. Wieseman, Computational test cases for the benchmark active controls model, J. Guidance, Control, and Dynamics 23 (5) (2000) 922–929.

[8] V. Mukhopadhyay, Transonic flutter suppression control law de- sign and wind-tunnel test results, J. Guidance, Control, and Dynam- ics 23 (5) (2000) 930–937.

[9] A.G. Kelkar, S.M. Joshi, Passivity-based robust control with applica- tion to benchmark controls technology wing, J. Guidance, Control, and Dynamics 23 (5) (2000) 938–947.

[10] J.M. Barker, G.J. Balas, Comparing linear parameter-varying gain- scheduled control techniques for active flutter suppression, J. Guid- ance, Control, and Dynamics 23 (5) (2000) 948–955.

[11] R.C. Scott, L.E. Pado, Active control of wind-tunnel model aeroelas- tic response using neural networks, J. Guidance, Control, and Dynam- ics 23 (6) (2000) 1100–1108.

[12] D.M. Guillot, P.P. Friedmann, Fundamental aeroservoelastic study combining unsteady computational fluid mechanics with adaptive control, J. Guidance, Control, and Dynamics 23 (6) (2000) 1117–1126.

[13] T. O’Neil, Experimental and analytical investigations of an aerolastic structure with continuous nonlinear stiffness, M.S. Thesis, Dept. of Aerospace Engineering, Texas A & M University, College Station, TX, May 1996.

[14] J. Block, T.M. Strganac, Applied active control for a nonlinear aeroelastic structure, J. Guidance, Control, and Dynamics 21 (6) (1998) 838–845.

[15] J. Ko, A.J. Kurdila, T.W. Strganac, Nonlinear control of a prototypical wing section with torsional nonlinearity, J. Guidance, Control, and Dynamics 20 (6) (1997).

S.N. Singh, W. Yim / Aerospace Science and Technology 7 (2003) 23–31 31

[16] J. Ko, T.W. Strganac, Stability and control of a structurally nonlinear aeroelastic system, J. Guidance, Control, and Dynamics 21 (5) (1998) 718–725.

[17] W. Xing, S.N. Singh, Adaptive output feedback control of a nonlinear aeroelastic structure, J. Guidance, Control, and Dynamics 23 (6) (2000) 1109–1116.

[18] J. Ko, T.W. Strganac, A.J. Kurdila, Adaptive feedback linearization for the control of a typical wing section with structutal nonlinearity, Nonlinear Dynamics 18 (1999) 289–301.

[19] R. Zhang, S.N. Singh, Adaptive output feedback control of an aeroelastic system with unstructured uncertainties, J. Guidance, Con- trol, and Dynamics 24 (3) (2001) 502–509.

[20] R.K. Prasanth, R.K. Mehra, Control of a nonlinear aeroelastic sys- tem using Euler–Lagrange theory, J. Guidance, Control, and Dynam- ics 23 (6) (2000) 1134–1139.

[21] J.-J.E. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall, New York, 1991.

[22] A. Isidori, Nonlinear Control Systems, Springer-Verlag, New York, 1989.

[23] C.P. Mracek, J.R. Cloutier, Full envelope missile longitudinal autopilot design using the state-dependent Riccati equation method, in: Proc. AIAA Guid., Navigation, and Control Conference, New Orleans, LA, 1997, pp. 1697–1705.

[24] C.P. Mracek, J.R. Cloutier, Control designs for the nonlinear bench- mark problem via the state-dependent Riccati equation method, Int. J. Robust Nonlinear Control 8 (1998) 401–433.

[25] J.R. Cloutier, C.P. Mracek, D.B. Ridgely, K.D. Hammett, State- Dependent Riccati Equation Techniques: Theory and Applications, Notes From the SDRE Workshop Conducted at the American Control Conference, Albuquerque, NM, 1998.

[26] J.R. Cloutier, D.T. Stansbery, Nonlinear hybrid bank-to-turn/skid-to- turn autopilot design, in: Proc. AIAA Guid., Navigation, and Control Conference, Montreal, Canada, 2001.