Robust Adaptive Output Feedback Control Design for a MultiInput MultiOutput Aeroelastic System
 Author: Wang Z., Behal A., Marzocca P.
 Organization: Wang Z.; Behal A.; Marzocca P.
 Publish: International Journal Aeronautical and Space Sciences Volume 12, Issue2, p179~189, 30 June 2011

ABSTRACT
In this paper, robust adaptive control design problem is addressed for a class of parametrically uncertain aeroelastic systems.A fullstate robust adaptive controller was designed to suppress aeroelastic vibrations of a nonlinear wing section. The design used leading and trailing edge control actuations. The full state feedback (FSFB) control yielded a global uniformly ultimately bounded result for twoaxis vibration suppression. The pitching and plunging displacements were measurable; however, the pitching and plunging rates were not measurable. Thus, a high gain observer was used to modify the FSFB control design to become an output feedback (OFB) design while the stability analysis for the OFB control law was presented. Simulation results demonstrate the efficacy of the multiinput multioutput control toward suppressing aeroelastic vibrations and limit cycle oscillations occurring in pre and postflutter velocity regimes.

KEYWORD
Aeroelastic systems , Multiinput multioutput control , Robust adaptive control , Output feedback control

1. Introduction
Reduced weight, increased structural flexibility and operating speed certainly increase the likelihood of flutter occurring within the aircraft operational envelope. The mission profiles of next generation of flying vehicles will require adaptable airframes to best meet varying flight conditions. However, geometry changes could possibly incur aeroelastic instabilities, such as flutter, at transition points during the mission. New requirements imposed on the design next generation vehicles have called for increasing structural flexibility, and high maneuverability while maintaining the ability to operate safely in severe environmental conditions. Consequently, developing and implementing active control technology has become very important. In the last two decades, the advances in active control technology have rendered the applications of active flutter suppression and active vibrations control systems feasible. Great research efforts are currently devoted to the aeroelastic active control and flutter suppression of flight vehicles. Librescu and Marzocca (2005) presented stateoftheart advances in these areas. Recent contributions related to the active control of an aircraft wing are discussed in length in a publication by Mukhopadhyay (2000a, b, 2011).
Behal et al. (2006b), Guijjula et al. (2005), Plantantis and Srganac (2004) presented novel research that improved the performance of adaptive schemes through wing extensions containing two control surfaces. Platanitis and Strganac(2004) presented an adaptive scheme that utilized fullstate feedback. However, the uncertainty in the coupling between the control inputs was not taken into account.Rather, an inversion of the nominal input gain matrix was utilized to decouple the control inputs. Gujjula et al. (2005)designed adaptive and radial basis functions neural network controllers in order to compensate for system nonlinearity.Furthermore, a projection operator was utilized to assure that the input gain matrix estimate remained invertible.To sidestep the need for projection, Behal et al. (2006b)utilized a symmetrictriangular decomposition of the input gain matrix in order to design singularity free controllers for the leading edge control surfaces (LECS) and trailing edge control surfaces (TECS). This control design required fullstate feedback as well as a filtered tracking error. Reddy et al.(2007) design an output feedback (OFB) adaptive controller that used backstepping coupled with a symmetricdiagonalupper triangular factorization. Wang et al. (2011) proposed a modular OFB controller to suppress aeroelastic vibrations on an unmodeled nonlinear wing section subject to a variety of external disturbances. Wang et al. (2010a) presented a detailed report of the adaptive and robust control of nonlinear aeroelastic models.
As previously discussed, Behal et al. (2006a) and Reddy et al. (2007) proposed adaptive control algorithms that utilized a backstepping approach that led to significant overparameterization and a very complicated control design.The adaptive control design work proposed in Behal et al.(2006b) does not utilize backstepping but the control design is fullstate feedback which requires a measurement not only of the output variables but also their time derivatives.
The work in this paper removes the restrictions exhibited in the works by Behal et al. (2006a, b), Reddy et al. (2007).Consequently, the tracking error may converge to the origin.The work presented here exploited the robust adaptive OFB control scheme presented by Wang et al. (2010b).
The input gain matrix was assumed to be real with nonzero leading principal minors. By employing the matrix decomposition approach introduced by Morse (1993), the input gain matrix can be decomposed into the product of a symmetric p.d. matrix, a known diagonal matrix with +1 or1 on the diagonal, and a unity upper triangular matrix. This triangular structure was exploited in the following adaptive control design and stability analysis since it allowed us to design algebraic loopfree control signals
u_{i} (t )∀i = 1, 2 sequentially. Next, the design of a full state feedback (FSFB)adaptive control law that yielded a global uniformly ultimately bounded (GUUB) result for the tracking error through a Lyapunov analysis was conducted. Finally, motivated by the high gain observer (HGO) presented by Atassi and Khalil(1999), an OFB control law was designed when only the system output vector was available for measurement. The simulation results on the 2degree of freedom (DOF) wing section model show practical convergence under both FSFB and OFB control laws.This paper is organized as follows. Section 2 introduces the system dynamics. Then, the control objective is defined and the openloop error system is developed to facilitate the subsequent control design. Section 3 presents the robust adaptive feedback control design followed by a Lyapunov based analysis of stability of the closedloop system. In Section 4, a solution based on a HGO is proposed to design the OFB controller. Simulation results to confirm the performance and robustness of the controller are presented in Section 5. Concluding remarks are provided in Section 6.
2. Aeroelastic Model Configuration and Error System Development
A 2DOF pitchplunge wing section based on previous models with both LECS and TECS is presented in Fig. 1. The system dynamics is given as follows
In Eq. (1), the quasisteady lift L(h˙, α？, α, β, γ) and aerodynamic moment M(h˙, α？, α, β, γ) are defined as
where
C_{mα} _{?eff} , Cmβ?eff, and Cmγ?eff are given in the following formThe governing equation can be written into the inputoutput representation using Eqs. (2) and (3). Motivated by Chen et al. (2008) and Zhang et al. (2005), this representation will facilitate the ensuing robust adaptive feedback control design
where x = [α,
h ]^{T} ∈R ^{2} is a vector of system output and u = [u _{1},u _{2}]^{T} = [β, γ]^{T} ∈R ^{2} denotes the control input vector.The drifting vectorh(x, x˙ , θ_{1}) , which is assumed to satisfy Lipschitz condition, contains uncertain nonlinearities due to the existence ofk _{α}(α). The input gain matrix G_{s} can be explicitly given as followswhere the constant matrix entries g_{ij} are defined as follows
where Δ = det(Gs) =
m _{T}I _{α}？m _{w} ^{2}x _{α}^{2}b ^{2} ≠ 0. Based on the matrix decomposition introduced by Morse (1993) and the assumptions that both the leading principal minors g11 andΔ are nonzero, the input gain matrixG _{s} can be decomposed asG _{s} =SDU whereS is a symmetric, positivedefinite matrix,D is a diagonal matrix with diagonal entries +1 or 1, andU is an unknown unity upper triangular matrix. According to theSDU decomposition result previously obtained by Reddy et al. (2007), the explicit representation forS ,D , and U can be given aswhere the notation sign(*) represents the standard signum function. In this paper, the signs of the leading principal minors of the input gain matrix
G s are assumed to be known for purposes of control design, which implies that the diagonal matrixD is known. After applying the matrix decomposition property and multiplying both sides of Eq.(5) withM =S ^{?1} ∈ R^{2×2}, Eq. (5) can be rewritten aswhere M is a symmetric, positive definite matrix and
f(x, x˙ ,θ_{1}) =M？h(x, x˙ , θ_{2}) ∈ R^{2} contains parametric nonlinearities.Note that M is assumed to be bounded bywhere m, m ∈ R are the minimum and maximum eigenvalues of the matrix
M . The tracking error e1(t) ∈ R^{2} for the aeroelastic system can be defined ase_{1} =X_{d} ?X , where the desired output vectorx _{d}(t) ∈ R^{2} can be zero all the time in this problem. Next, to simplify the subsequent control design, the auxiliary error signalse _{2}(t) ∈ R^{2} and filtered tracking errorr (t ) ∈ R^{2} are introduced as followsThen, based on the above error system definitions, a composite error signal z(t) can be defined as follows
After taking the time derivative of r and substituting the derivative of e2, the following equation can be obtained as follows
After multiplying both sides of Eq. (14) by M and applying the definitions given in Eqs. (9) and (12), Eq. (14) can be rewritten as
Furthermore, given a strictly upper triangular matrix
U =DU ?D , the openloop dynamics of Eq. (15) can be rewritten as followswhile above equation can be further rewritten as
where the linear parameterization term Y(x¨_{d}, x˙ , x, e˙ _{1}, e_{1}, u) θ ∈ R^{2} is defined as follows
3. State Feedback Control Development
3.1 FSFB control design
In this section, all state variables in Eq. (1) are assumed to be available for measurement. To facilitate the ensuing control design, the following auxiliary control matrix is given as
where Y
_{i} denotes the ith row of the measurable regression matrix Y ∈R ^{2×p}. Based on previous definition as well as the subsequent stability analysis, the following state feedback adaptive control law is proposedwhere K,
k _{T} ∈ R^{2×2} are positivedefinite, diagonal gain matrices whileu _{i} ∈ R ∀i = 1, 2 is the ith element of the control input vector u. Note that the robustifying term kTTr in Eq. (20) is used to ensure uniformity of the stability. The parameter adaptation law forθ (t ) ∈R^{p} is given as followswhere the constant adaptation gain matrix Γ ∈
R^{p×p} is diagonal and positive definite while p denotes the dimensions of the unknown parameter vector. Based on the results presented by Pomet and Praly (1992), the parameter projection operator Proj{·} is designed to bound parameter estimates θ？ (t ) in a known compact set Ω_{ε} such thatNote that the strict triangular structure of
U in Eq. (18)implies that u1 only depends on u2 but u2 can be determined independently of other control inputs since the diagonal elements inU are all zero. Thus, the control law can be implemented by designing u2 first, then using that design in the computation of u_{1}. After substituting Eq. (20) into the openloop dynamics given in Eq. (17), and then multiplying both sides by M, the following closedloop system dynamics can be obtainedwhere θ？(
t ) ∈R^{p} is a parameter estimation error defined as follows3.2 Stability analysis
In this section, a nonnegative Lyapunov candidate function V(t,
z ) ∈R is defined to analyze the stability of the full state feedback control lawwhich can be upper and lower bounded as
where α_{1}('z'') and α_{2}('z'') are class K_{∞}^{4} functions given as
where the assumption stated in Eq. (10) has been utilized. After taking the time derivative of Eq. (25), and then substituting Eqs. (12) and (23), the following result for V(
t, )can be obtainedz By applying the expressions given in Eq. (19), the expression in Eq. (28) can be rewritten as follows
where
k_{T} , i represents the ith diagonal element for the gain matrix kT. By adding and subtracting ''θ ?''^{2}/4k_{T} , _{1} and ''θ ?''^{2}/4k_{T} , _{2} to the right hand side of Eq. (29), the following result can be obtainedThen, V˙ (
t , z) can be further upperbounded asAccording to the definition of z(t) given in Eq. (13), the upperbound for V˙ (
t ,z ) in Eq. (31) can be expressed aswhere λ_{3} = min{2, λ_{min}
(K) } while λ_{min}(K) represents the minimum eigenvalue ofK . The constant δ is given bywhere the supremum exists since parameter projection operator defined in Eq. (21) ensures the boundedness of the parameter estimates, which implies that the parameter estimates error is also bounded. From Eq. (32), it is also easy to show that
where
while γ('z'') is a positivedefinite function.From the results in Eqs. (27) and (34), all conditions for the following theorem (Khalil, 1996) are satisfied.
Theorem 4.18 (Khalil, 1996): Let D ⊂
R^{n} be a domain that contains the origin and V:[0, ∞)×D →R be a continuously differentiable function such that∀
t ≥ 0 and ∀x ≥ D, where α1 and α2 are class K functions and W3(x) is a continuous positive definite function. Taker >0 such thatB_{r} ⊂ D and suppose thatμ < α_{2}^{?1}(α_{1}(r))
Then, there exists a class of KL function and for every initial state
x (t _{0}), satisfying ''x (t _{0})'' < α_{2}^{？1}(α_{1}(r)), there isT >0(dependent on x(t0) and μ) such that the solution of (4.32)satisfiesMoreover, if
D =R^{n} and α_{1} belongs to classK ∞, then the above two functions hold for any initial state x(t _{0}), with no restriction on the magnitude of μ.Thus, the error signal ''
z '' is GUUBwhere β(·, ·) is a class KL function while T depends on ''
z _{0}''and ？. From Eq. (33), kt can be made large enough such that the upper bound for ''z'' can be made arbitrarily small.4. OFB Control Development
4.1 High gain observer
In this section, it is assumed that the only measurements available are the pitching and plunging displacements, while the remaining states can be estimated by using of HGO. An estimated composite error signal z？(
) = [t e？ _{1} ^{T},r ^{T}]^{T} ∈R ^{4} for the auxiliary error signalz(t) can be obtained via the following HGO (Atassi and Khalil, 1999)where α
_{i} ∈ R ∀i = 1, 2 are gain constants and ε？ is a small positive constant. In order to facilitate the analysis in the singularly perturbed form, the scaled observer errorsη(
t ) = {η_{1} ^{T} η_{2} ^{T}]^{T} ∈R^{4} can be defined as followsBased on Eq. (37) as well as the definitions for z? (
) andt z (t ),it’s easily see thatwhere D_{ε} ∈
R ^{4×4} can be given as followsAfter differentiating Eq. (37) as well as taking advantage of the definition in Eq. (11), Eq. (12) and the design of Eq. (36),the following observer error dynamics can be obtained
where
i = 1, 2. Thus, a more compact form of Eq. (39) is represented as followswhere g ∈
R ^{4} is defined as followswhile α
_{i} ∀i = 1, 2 in Eq. (40) are chosen in a way such thatis Hurwitz. The boundarylayer system
induces a Lyapunov candidate function W(η) =
η ^{T} P _{0}η which has the following propertiesNote that in the above equation, λ_{max} (
P _{0}) denotes the maximum eigenvalue ofP _{0}, whileP _{0} ∈R is a p.d. matrix that satisfiesP _{0}A _{0} +A _{0}P _{0} =？I_{4} . From Eq. (42), it is clear thatη (t ) = 0 is a globally exponentially stable equilibrium of the boundarylayer system.4.2 OFB control law
From the observer error dynamic in Eq. (39), the solution of
η (t ) contains terms like 1ε？/ 1e for some ω > 0, which may introduce the socalled peaking phenomenon and cause instability. To suppress the peaking phenomenon on the state estimates, the fullstate control design in Eq. (20) is modified to an OFB saturated control as follows (Atassi and Khalil, 1999)where sat{} is the standard saturation function used to limit the magnitude of the control signal to avoid the peak phenomenon.
？ andT？ are the regression and auxiliary matrix defined in Eqs. (18) and (19) with respect to sat{z？} instead of z(t).K ,k _{T} are defined in Eq. (20),θ？ ( is obtained through the projection algorithm in Eq. (21) with respect tot )？ andr？ The saturation of input is applied outside a compact set
D_{c} = {z (t ) ∈R ^{4} 'V(t) < c} of the region of attraction domain D_{z}.After substituting Eq. (43) into Eq. (17), it’s easy to see thatAfter combining Eqs. (17, 40, 41, 45), the following closedloop error dynamics are obtained
4.3 OFB stability results
The OFB stability proof in this paper can be split into three steps in order to reduce the overall complexity. First, the existence of a positively invariant set for the solutions of Eq.(46) will be verified. Then, the boundedness of solutions of Eq. (46) is regained in the second step provided the trajectory(
z (t ),η (t )) starts inside a compact subset ofD_{z} ×R^{4} . Finally,global ultimate boundedness for Eq. (46) is recovered.In the ensuing stability analysis, Z is defined to be any compact set in the interior of Dz such that
Z ⊂Dc ⊂Dz ,H is defined to be any compact set in the interior ofR^{4} . Let D_{ε}= {η(t )R^{4} W(η(t )) ≤ ρε？^{2}} be a compact set whileW(t) was defined in Eq. (42), ρ is a positive constant that is yet to be selected while ε？ is the HGO constant. In the following stability analysis, (z (t),η (t)) is considered to start insideZ×H .Theorem 1: (Invariant Set Theorem) Given Σ =
D_{c} ×D_{ε} ,there exists an ε？_{1} > 0 such that ∀ε？ ∈ (0, ε？_{1}], Σ is a positively invariant set for the trajectory (z (t), η(t )).First of all, given the following composite Lyapunov function V_{c}(
z , η)where V(
z ) and W(η ) have been previously defined in Eqs. (25) and (42). Then, after taking the time derivative of Eq. (47) along the trajectory of Eq. (46), it’s straightforward to see thatwhere V˙ (
t ) and W˙ (t ) can be further written as followsIn this theorem, our goal is to prove that V˙ _{c}'_{∂Σ} ≤ 0 while the notation ∂Σ denotes the boundary of the compact set Σ.Inside the set Σ =
D _{c} ×D _{ε}, saturation does not apply on the control law. Thus, the term V˙ (t ) in Eq. (49) can be obtained as followswhere Eqs. (45) and (46) were utilized in above equation.
By using the results in Eqs. (28) and (32), Eq. (51) can be rewritten as follows
where
？ (t ) =Y ？？ . When the closed loop solution is inside the invariant set Σ, the locally Lipschitz condition can be used onh(x, ？, θ_{1}) as well as the results in Eqs. (18) and(43) to prove the following inequalitieswhere k_{4}, k_{5} ∈
R are positive constants. Based on Eqs. (53)and (54), Eq. (52) can be rewritten as followsFrom Eq. (42), it is straightforward to see that ''η(
t )'' ≤ εTherefore, Eq. (55) can be rewritten as follows
From Eq. (26), any c > 2λ_{2}δ/λ_{3} and
z (t ) ∈ ∂D _{c} can follow 1/2λ_{3}''z''^{2} > δ. Based on above facts and Eq. (56), the following expression can be obtainedMotivated by Eq. (57), ε_{1} can be defined as
, where β ∈ R is defined as follows
Then ∀0 < ε？ < ε1 and
η (t ) ∈ D_{ε}, based on the result in Eq.(51), it is easy to see thatNext, ？(.) of Eq. (50) can be upperbounded as follows
where
P _{0}A _{0} +A _{0} ^{T}P _{0} =？I_{4} . By using Eq. (41) and the fact that ε？ is strictly less than 1, the following inequality can be obtainedNote that
z (t ) is bounded insideD_{c} which proves that ''g ''≤ k_{1}''η'' + k_{2} ∀z (t ) ∈D_{c} and η(t ) ∈ R where k_{1}, k_{2} > 0 are constants independent of ε？. This result further implies thatAccording to Eq. (42),
Then, given a choice of
a choice of ρ= 36k_{2} ^{2} ''
P _{0}''^{3} ensures the following result∀0 < ε？ < ε_{2}. Finally, by defining ε？1 = min{1, ε_{2}, ε_{3}}, Eq. (63)implies that Σ =
D_{c} ×D_{ε} is an invariant set ∀ε？ ∈ (0, ε？_{1}].Theorem 2: (Boundedness Theorem) There exists an ε？_{2} ≤ε？_{1} such that ∀ε？ ∈ (0, ε？_{2}], any trajectory (
z (t),η (t )) that starts insideZ×H is bounded for all time.By using the boundedness of
z (0) andz? (0), the definition of Eq. (37) implies thatis a positive constant. From the boundedness assertion on
φ (z (t),η (t)),in Eq. (45) in the setD_{c} ×R^{4} , and the closedloop system dynamics in Eq. (46), it is also straightforward to see thatz (t )meets the following linear time growth upperbound''z (t ) ?z (0)'' ≤ k_{3}t ,where k_{3} > 0 is a positive constant. Thus, the existence of a time T_{c} is shown to be independent of ε？ such that
z (t ) ∈D_{c} ∀t ∈ [0,T_{c} ]. Our aim in this theorem is to show that ε？ can be picked in such a way that if η_{i} (t ) starts outside the invariant set Σ, it can be made to enter the invariant set beforez (t ) can exitD_{c} . Proving this previous assertion would imply that the solution (z (t ),η (t )) is in the invariant set Σ at some timeT _{ε} which indicates that it will stay there ∀t ∈ [T_{ε} , ∞). Outside the invariant set,which implies ''η'' ≥ε？k_{2}''
P _{0}''. Based on Eq. (63), W˙ (η) can be upperbounded asBy solving the above differential inequality, an upperbound for W(
η (t )) can be obtained as followswhere
Based on Eq. (42) and
Eq.(65) can be rewritten as follows
where σ_{2} = k_{6} ^{2}''
P _{0}''. Based on Eq. (66), 0 < ε？_{1} < ε？_{2} can become small enough so that W(η(t )) entersD _{ε} at a timeSince η(
t ) enters the invariant setD _{ε} in less than half the time it takes forz (t ) to exitD _{ε} ,(z (t ),η (t )) enters Σ during [0,T _{ε}]. Hence,z (t ),η (t ) ∈ L∞ for all timest ≤T _{ε}. Fort ∈ [0,T _{ε}], the trajectory (z (t), η(t )) is bounded by virtue of Eq. (64) and Eq. (66). Thus, all closedloop trajectories (z (t), η(t )) starting inZ × H are bounded for all time.Theorem 3: (Ultimately Boundedness Theorem) Given any solution (
z (t),η (t )) that starts inZ × H and given any smallthere exists an 0 < ε？3(δ？) < ε？2 and T(δ？) > 0 such that''
z (t )'' ≤ δ？/2 and ''η(t )'' ≤ δ？/2∀t ≥ T(δ？) and ∀ε？ ∈ (0, ε？_{3}].Based on Eq. (66),
Thus, for any given small value δ？, it’s clearly to see that ε3 = ε_{3} (δ？) ≤ ε？_{2} such that ∀ε？ ∈ (0, ε？_{3}], the following upperbound can be defined
Inside the invariant set Σ, Eq. (56) can be utilized to obtain the following conclusion
where μ ∈ R is defined as follows
Given a compact set
implies that V˙ (
t ) can be upperbounded as followswhich implies that V is decreasing outside D_{μ}. Define D_{v}= {V(
z ) ≤ c_{0}(ε?) =clearly, D_{μ} ⊂ D_{v} since c_{0}(ε?) is defined to be a nondecreasing scalar function. By choosing ε_{4} = ε_{4} (δ?) ≤ ε?_{2} small enough such that the set
D _{μ} is compact for all ε? ≤ ε_{4}, the setD _{v} is in the interior ofD _{c} andD _{v} ⊂ {z :''z (t )'' ≤δ?/2}. From above results and the upperbound in Eq. (70), it is straightforward to show that the set Σ_{ub} =D _{v} ×D _{ε} is positively invariant. Moreover, any trajectory in Σ will enter Σ_{ub} in a finite time T_{ε4} = T_{ε4} (δ?)∀ε ∈ (0, ε_{4}]. Choosing ε?_{3} = ε?_{3} (δ?) =min{ε_{3}, ε_{4}} and T(δ?) = max{T _{ε3},T _{ε4}}, thenThus (
z (t ), η(t)) starting inZ × H are ultimately bounded.5. Simulations and Results
5.1 Model and control parameters
The simulation results are presented in the following paragraphs for a nonlinear 2DOF aeroelastic system controlled by TECS and LECS. The nonlinear wing section model was simulated using the dynamics of Eqs. (13). The model parameters utilized in the simulation were the same as used by Platanitis and Strganac (2004) and are listed in Table 1.
The desired trajectory variables
x _{d}x _{d} andx˙ _{d} were simply selected as zero. The initial conditions for pitch angle α(t)and plunge displacementh(t) were selected as α(0) = 5.729 deg andh (0) = 0 m while all other variablesh˙(t) , α˙(t) , h¨(t) ,α¨(t) , and the parameter estimates were also set to zero. In the simulation, the signs of the leading principal minors of the input gain matrix Gs were embedded in the diagonal matrix D which can be given asThe magnitude of both the leading edge β(t) and trailing edge
r(t) flaps was limited to 15 deg. Since the control design contained an adaptation scheme that involved integration of the filtered error signal r in Eq. (21), the control input saturation led to the windup problem. The following approach to limit the error signal r？ was proposed according to the magnitude of original control inputu_{i} (Astrom and Rundqwist, 1986)where rb denotes the limited filtered error and is used in the parameter update law Eq. (21) while u, designed in Eq. (43), denotes the actual control signal for the actuator with saturation bound
u_{b} = 15 deg. The OFB control was implemented via the HGO defined in Eq. (36) and control law in Eq. (43). The parameters for the controller and observer in these simulations are listed in Table 2.5.2 Results
The openloop response of the system at preflutter speed U_{∞} = 8m/s < U_{F} = 11.4 m/s and postflutter speed U_{∞} = 13.28m/s> U_{F} = 11.4 m/s is given in Fig. .2 From Fig. 3,the convergence of the error to the origin under the proposed robust adaptive method is shown. During the preflutter speed, the proposed method exhibited faster settling times while the limit cycle oscillations (LCOs) at the postflutter speed regimes were totally suppressed. In Fig.3 b, the system was allowed to evolve uncontrolled to produce LCOs due to the nonlinear pitch stiffness. The control was turned on at t = 5 s.
6. Conclusions
A robust adaptive OFB controller was proposed to suppress parametrically uncertain aeroelastic vibrations on the wing section model. The control strategy was implemented via leading (γ) and trailing (β)edge control surfaces. The system structure and parameters, with the exception of the signs of the principal minors of the input matrix, were assumed to be unknown in the control design. By using a Lyapunov based method for design and analysis, GUUB results were obtained on the twoaxis vibration errors. HGO was used to design OFB control when only the output displacements were measurable. Future work will include the experimental evaluation of the robust adaptive controller in the windtunnel laboratory at Clarkson University.

[Fig. 1.] Two degree of freedom aeroelastic system with both leading and trailingedge control surfaces.

[Table 1.] Wing section parameters

[Table 2.] Simulation parameters

[Fig. 2.] penloop response (a) at preflutter speed U∞ = 8 m/s; (b) atpostflutter speed U∞ = 13.28 m/s. LECS: leading edge controlsurface TECS: trailing edge control surface.

[Fig. 3.] Closedloop response (a) at preflutter speed U∞ = 8 m/s; (b) at postflutter speed U∞ = 13.28 m/s. LECS: leading edge control surface TECS: trailing edge control surface.