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 state-ofthe-art 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 symmetric-triangular 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 symmetric-diagonalupper 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 full-state 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 or-1 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 loop-free 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 2-degree 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 open-loop 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 closed-loop 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.

A 2-DOF pitch-plunge 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 quasi-steady 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 form

The 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 ∈ R² is a vector of system output and u = [u₁, u₂]^T = [β, γ]^T ∈ R² denotes the control input vector.The drifting vector h(x, x˙ , θ₁), which is assumed to satisfy Lipschitz condition, contains uncertain nonlinearities due to the existence of k_α(α). The input gain matrix G_-s can be explicitly given as follows

where the constant matrix entries g_-ij are defined as follows

where Δ = det(Gs) = m_TI_α？m_w ² x_α²b² ≠ 0. Based on the matrix decomposition introduced by Morse (1993) and the assumptions that both the leading principal minors g11 andΔ are non-zero, the input gain matrix G_-s can be decomposed as G_s = SDU where S is a symmetric, positive-definite matrix,D is a diagonal matrix with diagonal entries +1 or -1, and U is an unknown unity upper triangular matrix. According to the SDU decomposition result previously obtained by Reddy et al. (2007), the explicit representation for S, D, and U can be given as

where 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 matrix D is known. After applying the matrix decomposition property and multiplying both sides of Eq.(5) with M = S^?1 ∈ R^2×2, Eq. (5) can be rewritten as

where M is a symmetric, positive definite matrix and f(x, x˙ ,θ₁) = M？h(x, x˙ , θ₂) ∈ R² contains parametric nonlinearities.Note that M is assumed to be bounded by

where m, m ∈ R are the minimum and maximum eigenvalues of the matrix M. The tracking error e1(t) ∈ R² for the aeroelastic system can be defined as e₁ = X_d ? X, where the desired output vector x_d(t) ∈ R² can be zero all the time in this problem. Next, to simplify the subsequent control design, the auxiliary error signals e₂(t) ∈ R² and filtered tracking error r(t) ∈ R² are introduced as follows

Then, 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 open-loop dynamics of Eq. (15) can be rewritten as follows

while above equation can be further rewritten as

where the linear parameterization term Y(x¨_d, x˙ , x, e˙ ₁, e₁, u) θ ∈ R² is defined as follows

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 proposed

where K, k_T ∈ R^2×2 are positive-definite, diagonal gain matrices while u_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 for θ(t) ∈ R^p is given as follows

where 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 that

Note 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 in U are all zero. Thus, the control law can be implemented by designing u-2 first, then using that design in the computation of u_-1. After substituting Eq. (20) into the open-loop dynamics given in Eq. (17), and then multiplying both sides by M, the following closed-loop system dynamics can be obtained

where θ？(t) ∈ R^p is a parameter estimation error defined as follows

In this section, a non-negative Lyapunov candidate function V(t, z) ∈ R is defined to analyze the stability of the full state feedback control law

which can be upper and lower bounded as

where α₁('z'') and α₂('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, z)can be obtained

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 obtained

Then, V˙ (t, z) can be further upperbounded as

According to the definition of z(t) given in Eq. (13), the upperbound for V˙ (t, z) in Eq. (31) can be expressed as

where λ₃ = min{2, λ_min(K)} while λ_min(K) represents the minimum eigenvalue of K. The constant δ is given by

where 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 positive-definite 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 ⊂ Rⁿ 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. Take r>0 such that B_r ⊂ D and suppose that

μ < α₂^?1(α₁(r))

Then, there exists a class of KL function and for every initial state x(t₀), satisfying ''x(t₀)'' < α₂^？1(α₁(r)), there is T>0(dependent on x(t0) and μ) such that the solution of (4.32)satisfies

Moreover, if D = Rⁿ and α₁ belongs to class K∞, then the above two functions hold for any initial state x(t₀), with no restriction on the magnitude of μ.

Thus, the error signal ''z'' is GUUB

where β(·, ·) is a class KL function while T depends on ''z₀''and ？. From Eq. (33), kt can be made large enough such that the upper bound for ''z'' can be made arbitrarily small.

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 ∈ R⁴ for the auxiliary error signal z(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 errors

η(t) = {η₁ ^T η₂ ^T]^T ∈ R⁴ can be defined as follows

Based on Eq. (37) as well as the definitions for z? (t) and z(t),it’s easily see that

where D_ε ∈ R^4×4 can be given as follows

After 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 follows

where g ∈ R⁴ is defined as follows

while α_i∀i = 1, 2 in Eq. (40) are chosen in a way such that

is Hurwitz. The boundary-layer system

induces a Lyapunov candidate function W(η) = η^TP₀η which has the following properties

Note that in the above equation, λ_max (P₀) denotes the maximum eigenvalue of P_-0, while P₀ ∈ R is a p.d. matrix that satisfies P₀A₀ + A₀P₀ = ？I₄. From Eq. (42), it is clear that η(t) = 0 is a globally exponentially stable equilibrium of the boundary-layer system.

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 so-called peaking phenomenon and cause instability. To suppress the peaking phenomenon on the state estimates, the full-state 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. ？ and T？ 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？ (t) is obtained through the projection algorithm in Eq. (21) with respect to ？ and r？

The saturation of input is applied outside a compact set D_c = {z(t) ∈ R⁴ 'V(t) < c} of the region of attraction domain D_z.After substituting Eq. (43) into Eq. (17), it’s easy to see that

After combining Eqs. (17, 40, 41, 45), the following closedloop error dynamics are obtained

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 of D_z × R⁴. 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 of R⁴. Let D_ε= {η(t) R⁴ W(η(t)) ≤ ρε？²} be a compact set while W(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 inside Z×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 that

where V˙ (t) and W˙ (t) can be further written as follows

In 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 follows

where 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 on h(x, ？, θ₁) as well as the results in Eqs. (18) and(43) to prove the following inequalities

where k₄, k₅ ∈ R are positive constants. Based on Eqs. (53)and (54), Eq. (52) can be rewritten as follows

From Eq. (42), it is straightforward to see that ''η(t)'' ≤ ε

Therefore, Eq. (55) can be rewritten as follows

From Eq. (26), any c > 2λ₂δ/λ₃ and z(t) ∈ ∂D_c can follow 1/2λ₃''z''² > δ. Based on above facts and Eq. (56), the following expression can be obtained

Motivated by Eq. (57), ε₁ 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 that

Next, ？(.) of Eq. (50) can be upperbounded as follows

where P₀A₀ + A₀ ^TP₀ = ？I₄. By using Eq. (41) and the fact that ε？ is strictly less than 1, the following inequality can be obtained

Note that z(t) is bounded inside D_c which proves that ''g''≤ k₁''η'' + k₂ ∀ z(t) ∈ D_c and η(t) ∈ R where k₁, k₂ > 0 are constants independent of ε？. This result further implies that

According to Eq. (42),

Then, given a choice of

a choice of ρ= 36k₂ ² ''P₀''³ ensures the following result

∀0 < ε？ < ε₂. Finally, by defining ε？1 = min{1, ε₂, ε₃}, 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 inside Z×H is bounded for all time.

By using the boundedness of z(0) and z?(0), the definition of Eq. (37) implies that

is a positive constant. From the boundedness assertion on φ(z(t), η(t)),in Eq. (45) in the set D_c × R⁴, and the closed-loop system dynamics in Eq. (46), it is also straightforward to see that z(t)meets the following linear time growth upperbound''z(t) ? z(0)'' ≤ k₃t,

where k₃ > 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 before z(t) can exit D_c. Proving this previous assertion would imply that the solution (z(t), η(t)) is in the invariant set Σ at some time T_ε which indicates that it will stay there ∀t ∈ [T_ε, ∞). Outside the invariant set,

which implies ''η'' ≥ε？k₂''P₀''. Based on Eq. (63), W˙ (η) can be upperbounded as

By solving the above differential inequality, an upperbound for W(η(t)) can be obtained as follows

where

Based on Eq. (42) and

Eq.(65) can be rewritten as follows

where σ₂ = k₆ ²''P₀''. Based on Eq. (66), 0 < ε？₁ < ε？₂ can become small enough so that W(η(t)) enters D_ε at a time

Since η(t) enters the invariant set D_ε in less than half the time it takes for z(t) to exit D_ε ,(z(t), η(t)) enters Σ during [0, T_ε]. Hence, z(t), η(t) ∈ L∞ for all times t ≤ T_ε. For t ∈ [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 in Z × H are bounded for all time.

Theorem 3: (Ultimately Boundedness Theorem) Given any solution (z(t), η(t)) that starts in Z × H and given any small

there 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 = ε₃ (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 follows

which implies that V is decreasing outside D_μ. Define D_v= {V(z) ≤ c₀(ε?) =

clearly, D_μ ⊂ D_v since c₀(ε?) is defined to be a non-decreasing scalar function. By choosing ε₄ = ε₄ (δ?) ≤ ε?₂ small enough such that the set D_μ is compact for all ε? ≤ ε₄, the set D_v is in the interior of D_c and D_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, ε₄]. Choosing ε?₃ = ε?₃ (δ?) =min{ε₃, ε₄} and T(δ?) = max{T_ε3, T_ε4}, then

Thus (z(t), η(t)) starting in Z × H are ultimately bounded.

The simulation results are presented in the following paragraphs for a nonlinear 2-DOF aeroelastic system controlled by TECS and LECS. The nonlinear wing section model was simulated using the dynamics of Eqs. (1-3). 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 and x˙ _d were simply selected as zero. The initial conditions for pitch angle α(t)and plunge displacement h(t) were selected as α(0) = 5.729 deg and h(0) = 0 m while all other variables h˙(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 as

The 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 input u_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.

The open-loop response of the system at pre-flutter speed U_∞ = 8m/s < U_F = 11.4 m/s and post-flutter 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 pre-flutter speed, the proposed method exhibited faster settling times while the limit cycle oscillations (LCOs) at the post-flutter 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.

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 two-axis 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.