Continuous robust control for aeroelastic vibration control of a 2-D airfoil under unsteady flow

Abstract

In this paper, a particular class of 2-D aeroelastic systems accounting for structural nonlinearities in pitching displacement and operating in an unsteady aerodynamic incompressible flowfield description is considered. By using the flap hinge torque of a trailing-edge flap surface as the control input, a continuous controller is proposed to suppress the aeroelastic vibrations of the wing section model. The control design based on the choice of the pitching angle as the output variable yields a semi-global asymptotic stability result. Furthermore, the system is theoretically shown to be robust to external disturbances. Numerical simulation results verify the efficacy of the proposed control strategy toward suppressing aeroelastic vibration in both pre- and post-flutter flight speed regimes under a multitude of external disturbances.

Keywords

Aeroservoelasticity unsteady aerodynamics disturbance mitigation aeroelasticity robust control nonlinear control

1. Introduction

The development of active controllers for aeroelastic stability and flutter suppression is an important topic that has been investigated by a number of researchers. To model the system dynamics more accurately, structural nonlinearities are often considered (Marzocca et al., 2001; Dowell et al., 2003): the presence of these nonlinearities often leads to a variety of phenomena, such as limit cycle oscillations (LCOs) and chaotic vibrations. These topics have been extensively addressed in Lee et al. (1999). The uncertainty in the 2-D wing section structure is modeled as parametric uncertainty in the third- and fifth-order stiffness coefficients of the springs in the pitching and plunging degrees of freedom (DOFs). To increase the flight envelope, numerous researchers have designed controllers to stabilize the aeroelastic oscillations with a single control surface, namely, a trailing-edge flap.

Early researchers designed and implemented linear controllers. Mukhopadhyay (1992) developed a methodology for synthesizing optimal feedback control laws and applied them to active aeroelastic vibration suppression problems. Gangsaas (1981) provided practical gust load alleviation and flutter suppression control laws based on the linear quadratic Gaussian (LQG) methodology. A partial-state feedback control law using pole placement technique was developed by Karpel (1982) while Horikawa and Dowell (1979) adopted proportional gain feedback methods, developed from root locus plots, to perform flutter suppression and load alleviation.

Starting in the late nineties, researchers began to apply nonlinear control design techniques for control of flutter. Ko et al. (1997) designed a nonlinear controller for a prototypical wing section with torsional nonlinearity. In Xing and Singh (2000), an output feedback adaptive control law was proposed by Xing and Singh for a 2-D wing section with only pitching angle and plunging displacement available, whereas in Ko et al. (1999), a partial-state feedback linearizing control scheme accounting for structural nonlinearity was proposed utilizing a single trailing-edge actuator for flutter suppression. Neural-network-based control designs were provided to suppress the aeroelastic vibrations of a nonlinear wing section in Zhang et al. (2012). Adaptive control strategies derived for flutter suppression with unstructured uncertainties have also been applied to aeroelastic systems with only trailing-edge flap in Zeng and Singh (1998). Based on the immersion and invariance approach, the structural nonlinearity problem of the aeroelastic wing sections was solved by immersion- and invariance-based adaptive control design in Lee and Singh (2009). They proposed a robust control law for the global regulation of a 2-D aeroelastic system with only measurements of the pitch angle. The system parameters were assumed to be unknown, while the bounds on the uncertainties were assumed to be known in the control design. Another robust control strategy with the purpose of active flutter suppression of a nonlinear 2-D wing-flap system was presented in Zhang and Soffker (2009). A proportional-integral observer (PI-observer) was used to estimate not only the system states but also the bounds of the nonlinearities. Based on the estimated system states, an optimized state feedback robust controller was provided and demonstrated to be efficient. In Lee et al. (2012) $L$ ₁ adaptive control law was developed for the suppression of aeroelastic oscillations using the pitch angle and pitch rate feedback in the presence of large parameter uncertainties and triangular, sinusoidal, and exponential gust loads. In some recent research works (e.g. see Lazarus and Crawley, 1992; Platanitis and Strganac, 2004, 2005; Behal et al., 2006a,b; Reddy et al., 2007; Wang et al., 2010; Li et al., 2011; Wang et al., 2011, 2012), both leading-edge and trailing-edge flaps have been utilized as control surfaces to improve the system performance via design of multiple-input, multiple-output (MIMO) control strategies. Several recent contributions related to the active control of aircraft wings exhibiting nonlinear behavior have been discussed at length in Mukhopadhyay et al. (2000a,b,c).

In most of the aforementioned research works, quasi-steady aerodynamics have been considered as an approximation of the flow-induced loads on the airfoil that cause aeroelastic oscillations. However, more accurate modeling requires the ability to consider the memory derived from the unsteady aerodynamics. Block and Strganac (1998) designed a full state feedback control law in which the unsteady dynamics were included in the aerodynamic model and approximated by Theodorsen's function while the nonlinear pitching stiffness was derived from experimental data from the Texas A&M experiment testbed. A backstepping-based output feedback nonlinear control strategy for flutter suppression was proposed in Bhoir and Singh (2004) based on the aerodynamic model utilized in Block and Strganac (1998). In their following work (Bhoir and Singh, 2005), the authors developed a suboptimal nonlinear control law based on the state-dependent Riccati equation method. A similar approach has been utilized more recently in Li et al. (2010) where the effect of control surface nonlinearities and actuation time delay has been studied. Singh and Wang (2002) proposed a backstepping-based output feedback adaptive control to stabilize a nonlinear aeroelastic system under the assumption that the parameters of the system are completely unknown. Most of the aforementioned works incorporating the unsteady aerodynamics have represented the control surface as a second-order system whose dynamics do not couple with the pitch and plunge dynamics. This coupling was considered in our recent work in Zhang et al. (2013) where an adaptive controller was developed based on a 2-D wing section model accounting for unsteady aerodynamics; the model utilized in Zhang et al. (2013) constituted a corrected version (Mozaffari-Jovin et al., 2013) of the model that was first introduced in York (1980) and adopted later by other authors (Librescu et al., 2005).

In this paper, a continuous robust controller is proposed to asymptotically stabilize the 2-D airfoil system under unsteady flow with unstructured nonlinear uncertainties and bounded unknown external disturbances. The goal of this paper is to design a single-input/single-output (SISO) continuous robust controller to suppress the vibrations for the aeroelastic system using a trailing-edge flap as the actuator. The novelty of this work is that we theoretically guarantee robustness to external disturbances which (to the best of our knowledge) has not been achieved before using an unsteady formulation. The proposed controller requires only minimal knowledge of the system model, namely, the sign of the control gain. The design of this control law is motivated by a continuous robust SISO result presented in Xian et al. (2004) which was applicable to flat systems. The challenge in extending the controller of Xian et al. (2004) to the 2-D system model utilized in this paper lies in proving the stability of the internal dynamics induced by the combination of the unsteady aerodynamics and the dynamics of the plunging DOF which is not directly controlled. These internal dynamics appear as a disturbance term in the dynamics of the pitching displacement. We address these internal dynamics in a two-step fashion by utilizing input-to-state stability (ISS) concepts (Khalil, 1996). First, we prove that the system internal dynamics can be bounded as the summation of a nondecreasing function of the tracking error of the chosen output variable (specifically, the pitching DOF) and the bounds of the external disturbance. Then, given this boundedness of the internal states of the system as a function of the output error, a Lyapunov-based analysis is applied to show that semiglobal asymptotic stability can be obtained for the pitching angle tracking error under application of the proposed continuous robust controller. The advantages of the proposed robust controller compared with the adaptive control designed in our previous work in Zhang et al. (2013) are three-fold: firstly, the proposed control design is low-order since there are no parameters that need to be estimated; secondly, this is a partial-state feedback controller in which only the output variable and its derivative are utilized, and finally, the control design theoretically guarantees robustness to a class of bounded disturbances.

The remainder of the paper is organized in the following manner. In Section 2, we introduce the system dynamics followed by the open-loop error system development in Section 3. In Section 4, we analyze the system internal dynamics and show its boundedness functions. Controller design and Lyapunov-based stability analysis are proposed in Section 5. Simulation results are presented in Section 6 to validate the effectiveness of the proposed controller. Conclusions are provided in Section 7.

2. Mathematical model

A wing section model with two DOF in plunging and pitching is presented in Figure 1, where the flap hinge torque of the trailing-edge flap is taken as control input. The aeroelastic governing equations are developed based on the model proposed in York (1980), Librescu et al. (2005) and Mozaffari-Jovin et al. (2013):

M \overset{··}{Y} (t) + K Y (t) = \underset{︸}{L c (t)} Control Load - \underset{︸}{L (t)} Aero Load - \underset{︸}{L g (t)} Gust Load

(1)

where L (t) and L _g(t) represent the unsteady aerodynamic load and gust loads, respectively,

Y = Δ [h b α β] T

, and M and K denote the mass and stiffness matrices which are defined as

M = [bm S α S β bS α I α I β + S β b (c - a) bS β I β + S β b (c - a) I β], K = [K h 000 k α (α) 000 K β]

(2)

while L _c is the control load which is defined as

L c = [001] T T c (t)

where T_c(t) is the control input, which is the flap hinge torque used to control the flap deflection β. The aerodynamical load in (1) is expressed as

L (t) = [L (t) M (t) T (t)] T

, where the lift L(t), the aerodynamic pitching moment M(t), and the flap torque T(t) are given by

L (t) = π ρ b 2 [b \overset{··}{h} b - ba \overset{··}{α} + b 2 π Φ 4 \overset{··}{β} + 2 V \overset{\cdot}{h} b + 2 V (1 - a) \overset{\cdot}{α} + V π (Φ 3 + Φ 2) \overset{\cdot}{β} + 2 V 2 b α + 2 V 2 π b Φ 1 β - 2 V α 1 b B 1 - 2 V α 2 b B 2] M (t) = π ρ b 2 [- ab 2 \overset{··}{h} b + b 2 (18 + a 2) \overset{··}{α} + b 24 π Φ 7 \overset{··}{β} - 2 bV f (12 + a) \overset{\cdot}{h} b - 2 bVa (12 - a) \overset{\cdot}{α} + Vb π {12 Φ 6 - (12 + a) Φ 2} \overset{\cdot}{β} - 2 V 2 (12 + a) α + V 2 π {Φ 5 - 2 (12 + a) Φ 1} β + 2 V (12 + a) α 1 B 1 + 2 V (12 + a) α 2 B 2] T (t) = π ρ b 2 [b 22 π Φ 4 \overset{··}{h} b + b 24 π Φ 7 \overset{··}{α} + b 24 π Φ 12 \overset{··}{β} + Vb π Φ 8 \overset{\cdot}{h} b + Vb π (Φ 92 + Φ 8) \overset{\cdot}{α} + Vb 2 π 2 (Φ 11 + Φ 2 Φ 8) \overset{\cdot}{β} + V 2 π Φ 8 α + V 2 π 2 (Φ 10 + Φ 1 Φ 8) β - V π Φ 8 α 1 B 1 - V π Φ 8 α 2 B 2]

(3)

In (3) above, Φ_i = Φ_i(φ) are Theodorsen's constants (Zhang et al., 2013) which have been listed in Appendix A, φ = arccos(−c), while all other symbols have been defined in Nomenclature. Furthermore, B₁(t) and B₂(t) are the state variables associated with the unsteady aerodynamics that have the following dynamics:

\overset{\cdot}{B} 1 (t) + w 1 B 1 (t) = \overset{\cdot}{Q} (t) \overset{\cdot}{B} 2 (t) + w 2 B 2 (t) = \overset{\cdot}{Q} (t)

(4)

where

w i = β i (V b), i = 1, 2,

while Q(t) is a measure of circulation that is expressed as follows:

Q (t) = \overset{\cdot}{h} + b (12 - a) \overset{\cdot}{α} + b 2 π Φ 2 \overset{\cdot}{β} + V π Φ 1 β + V α

(5)

In (1), the gust loading L _g(τ) is defined as follows (Marzocca et al., 2001; Librescu et al., 2005):

L g (τ) = [L G (τ) M yG (τ) T T (τ)] T = ρ Vb \int_{0}^{τ} [I LG (τ - τ 0) I MG (τ - τ 0) I fG (τ - τ 0)] T w G (τ 0) d τ

(6)

where τ = Vt/b is the dimensionless time and w_G(τ) is the gust variable velocity which is assumed to be C⁰, while I_LG, I_MG and I_fG are the related aerodynamic indicial functions. For incompressible flow, they are expressed as

[I LG (τ - τ 0) I MG (τ - τ 0) I fG (τ - τ 0)] T = [4 π \overset{\cdot}{ψ} bI LG (12 - a) 0] T

(7)

where ψ denotes the Kussner function which is approximated by

ψ = 1 - 0.5 e - 0.13 τ - 0.5 e - τ

(8)

From equations (1) to (5) the state space representation of the overall system dynamics can be obtained as follows:

\overset{\cdot}{x} = A x + f (α) + Bu + G w g y = C T x

(9)

where the state variable vector x is defined as follows:

x = Δ [\overset{\cdot}{ξ} \overset{\cdot}{α} \overset{\cdot}{β} ξ α β B 1 B 2] T

(10)

where the dimensionless plunging distance h/b is denoted by ξ, while A, B, C, G, and f(α) are, respectively, parameter matrices and a nonlinear vector, whose explicit definitions have been given in Appendix A. The vector w_g

= Δ [w L (t) w M (t) w T (t)] T

denotes the external gust disturbance which is C¹ since w_G(τ) ∈ C⁰.

3. Open-loop error system development

Given only measurements of the pitching displacement α and its time derivative $\overset{\cdot}{α}$ , the objective of this paper is to design a control strategy to drive the pitching displacement α to zero. From (9), the pitching displacement dynamics are expressed as follows:

\overset{··}{α} = λ (x) + b 2 u + G 2 w g

(11)

where the expression for λ (x) is given in Appendix A, while b₂ and G₂, respectively, denote the second component of the control input vector B and the second row of the disturbance matrix G. Since we are interested in designing a continuous controller using an integral of the standard signum function, we will need to analyze a higher-order system whose state vector comprises x and

\overset{\cdot}{x}

. To this end, we begin by taking the derivative of (11) and utilizing (9) to obtain the dynamics in the following form amenable to further analysis:

α (3) = \overset{\cdot}{λ} (x) + b 2 \overset{\cdot}{u} + d (t)

(12)

where d(t) is a time-varying bounded disturbance term for which an explicit expression has been provided in Appendix A. To facilitate the following controller design, the tracking error e₁ is defined as

e 1 = α d - α

(13)

where α_d is the desired bounded output variable that is designed to be C³ smooth in deference to the requirements of the subsequent control design. To suppress the pitching displacement, α_d can be chosen as identically zero or one can design a desirable smooth trajectory along which α_d will be driven to zero. Next, the auxiliary error signals e₂ and r are defined as follows:

e 2 = \overset{\cdot}{e} 1 + e 1

(14)

r = \overset{\cdot}{e} 2 + e 2

(15)

Based on above definitions, a composite error signal e(t) can be defined as follows:

e = [e 1 e 2 r] T

(16)

By taking the derivative of r, substituting from the derivative of (14) and applying the definitions given in (12), (14), and ( 15), we obtain

1 | b 2 | \overset{\cdot}{r} = 1 | b 2 | [(α . . . d + 2 \overset{\cdot}{e} 2 - \overset{\cdot}{e} 1) - \overset{\cdot}{λ} (x)] - sgn (b 2) \overset{\cdot}{u} - 1 | b 2 | d (t)

(17)

We now arrange (17) into the following open-loop dynamics form:

1 | b 2 | \overset{\cdot}{r} = N (x, \overset{\cdot}{x}, t) - sgn (b 2) \overset{\cdot}{u} - e 2

(18)

where the auxiliary function

N (x, \overset{\cdot}{x}, t)

is defined as follows:

N = 1 | b 2 | [(α . . . d + 2 \overset{\cdot}{e} 2 - \overset{\cdot}{e} 1) - \overset{\cdot}{λ} (x)] - 1 | b 2 | d (t) + e 2

(19)

Note here that the function

N (x, \overset{\cdot}{x}, t)

will be split into different parts based on their boundedness to facilitate the subsequent control design.

4. Analysis of internal dynamics

Before we delve into the control design, we introduce a lemma which can provide the boundedness of system states in terms of the input, output, and initial state values. We start by transforming (9) into a normal form by applying a linear diffeomorphism (Zhang et al., 2013) Ϝ:R⁸ → R⁸ (see Appendix A for an explicit definition of Ϝ) to the original system states x(t) as follows:

Ϝ x = [α \overset{\cdot}{α} ζ T] T

(20)

such that

d dt [α \overset{\cdot}{α} ζ] = FAx + F f (α) + Ϝ B u + Ϝ Gw g (t)

(21)

where (9) has been utilized; here, Ϝ has been designed s.t.

Ϝ B = [0 b 2 0 6 \times 1] T

, which implies that the ζ≜Ϝ_subx ∈ R^6×1 dynamics are independent of the control variable u as follows:

\overset{\cdot}{ζ} = Ϝ sub A x + Ϝ sub f (α) + Ϝ sub Gw g (t)

(22)

where

Ϝ sub

is the appropriate submatrix of Ϝ derived from (20). To facilitate further analysis, we define a higher-order state vector

Z = [μ T η T] T

as follows where

μ = Δ [α \overset{\cdot}{α} \overset{··}{α}] T

and

η = Δ [ζ T \overset{\cdot}{ζ} T] T .

By utilizing (12) and the time derivative of (22), the dynamics of Z(t) can be succinctly written as follows:

\overset{\cdot}{Z} = F (Z) + T \overset{\cdot}{u} + D (t)

(23)

while F, T, and D are defined as follows:

F (Z) = Δ [\overset{\cdot}{α} \overset{··}{α} \overset{\cdot}{x} T \partial λ (x) \partial x f η (μ, η)], T = Δ [00 b 2 0 12 \times 1], D (t) = Δ [00 d (t) d ζ (t)]

(24)

where the auxiliary vector f_η(μ,η) ∈ R^12×1 and the disturbance vector function d_ζ(t) ∈ R^12×1 are defined as follows:

f η (μ, η) = Δ [\overset{\cdot}{ζ} T Ϝ sub A \overset{\cdot}{x} + Ϝ sub \overset{\cdot}{f} (α)] T, d ζ (t) = Δ [0 1 \times 6 (Ϝ sub G \overset{\cdot}{w} g (t)) T] T

(25)

It is clear to see from the definitions above that f_η(μ,η) can be written out as follows:

f η (μ, η) = f η (0, η) + f η (μ, 0)

(26)

Since f_η(0,η) denotes the nominal zero dynamics and the system is assumed to be minimum phase, a converse theorem (Khalil, 1996) can be employed to obtain the following upper bound:

η T f η (0, η) \leq - K η ‖ η ‖ 2

(27)

where K_η is a system constant. With all of the above definitions, an ISS-like lemma is introduced as follows.

Lemma 1

Consider the system (24) with a bounded initial state Z₀; given a positive constant C₀ such that ‖‖ η₀≜η(t₀)‖‖ < C₀, there exists a corresponding positive constant C₁ such that ∀t > t₀, the internal dynamics η satisfy

‖ η ‖ \leq ρ 1 (‖ e ‖) ‖ e ‖ + C 1

(28)

where ρ₁(·) is a positive invertible nondecreasing function.

Proof

We define a positive definite function V₁(t) as follows:

V 1 = 12 η T η

(29)

Taking the time derivative of V₁ along the trajectory of (23) yields

\overset{\cdot}{V} 1 = η T f η (0, η) + η T (f η (μ, 0) + d ζ (t))

(30)

where we have utilized (26). By adding and subtracting η^Tf_η(μ_d,0) to the right-hand side of the above expression and using (27),

\overset{\cdot}{V} 1

can be bounded as

\overset{\cdot}{V} 1 \leq - K η ‖ η ‖ 2 + ‖ η ‖ (‖ f η (μ, 0) - f η (μ d, 0) ‖ + ‖ f η (μ d, 0) + d ζ (t) ‖)

An application of the mean value theorem (Xian et al., 2004) allows us to obtain the inequality

‖ f η (μ, 0) - f η (μ d, 0) ‖ \leq ρ 0 (‖ e ‖) ‖ e ‖

where ρ₀(·) is a positive nondecreasing function and e(t) is a composite error signal previously defined in (16). The desired higher-order output trajectory

μ d = Δ [α d \overset{\cdot}{α} d \overset{··}{α} d] \in L \infty

is bounded by design while the disturbance d_ζ(t) is bounded by assumption. Thus, one can obtain a further upper bound on

\overset{\cdot}{V} 1

as follows:

\overset{\cdot}{V} 1 \leq - K η ‖ η ‖ (‖ η ‖ - ρ 1 (‖ e ‖) ‖ e ‖ - C)

(31)

where

C = Δ 1 K η ‖ f η (μ d, 0) + d ζ (t) ‖ \infty

and

ρ 1 (\cdot) = Δ 1 K η ρ 1 (‖ e ‖) ‖ e ‖

is an invertible nondecreasing function. We define a set D₁ as follows:

D 1 = {η | ‖ η ‖ \leq ρ 1 (‖ e ‖) ‖ e ‖ + C}

which is invariant because

\overset{\cdot}{V} 1

is negative on its boundary. Based on this definition, we define another set D_η s.t. D₁ ⊂ D_η as follows:

D η = {η | ‖ η ‖ \leq ρ 1 (‖ e ‖) ‖ e ‖ + C 1}

where C₁ is a positive constant defined as

C 1 = Δ C 0 + C

(32)

where C₀ is the least upper bound of ‖‖η₀‖‖ . It is clear to see that D_η is invariant for ∀t > t₀ which implies that η(t) ∈ D_η ∀t ≥ t₀ since ‖‖η₀‖‖ ∈ D_η. This completes the proof.▪

Remark 2

We note here that the plunging displacement h was used as the output variable for the adaptive design on a similar model published in Zhang et al. (2013) since we could not obtain stable zero dynamics for the choice of α as output given the model parameters used therein. For the proposed work, we were able to find parameter settings that led to stable zero dynamics when using α as the output. In typical aeroelastic literature, α is indeed the preferred variable that is directly controlled. It is important to note that this control methodology could work with any choice of output variable so long as that choice results in stable zero dynamics.

5. Control design and stability analysis

Given that both the pitch degree and its derivative are measurable and the sign of control gain is known, we are motivated by the structure of open-loop error dynamics in (18) to propose a continuous controller as follows:

u = sgn (b 2) [(k s + 1) e 2 - (k s + 1) e 2 (t 0) + \int_{t 0}^{t} [(k s + 1) e 2 (τ) + k d sgn (e 2 (τ))] d τ]

(33)

where k_s and k_d are positive gain constants. To facilitate the subsequent analysis, we rewrite (33) in dynamic form as follows:

\overset{\cdot}{u} = sgn (b 2) [(k s + 1) r + k d sgn (e 2 (t))]

(34)

After substituting (34) into (18), the closed-loop dynamics can be expressed in the following manner:

1 | b 2 | \overset{\cdot}{r} = N - (k s + 1) r - k d sgn (e 2 (t)) - e 2

(35)

It is clear to see from the definitions of μ and η as well as (20) and (53) that

N (x, \overset{\cdot}{x}, t)

defined in (19) can be split into two parts as follows:

N (x, \overset{\cdot}{x}, t) = N μ (μ, t) + N η (η)

(36)

where

N η (η) = Δ K N η η

and

K N η \in R 1 \times 12

is a constant vector comprised of system constants. After defining N_μd(t)≜N_μ(μ_d,t) and Ñ_μ≜N_μ(μ,t) − N_μd(t), one can rewrite the closed-loop dynamics of (35) as follows:

1 | b 2 | \overset{\cdot}{r} = N μ d + \tilde{N} μ + N η - (k s + 1) r - k d sgn (e 2 (t)) - e 2

(37)

One can utilize (28) to upper-bound N_η as follows:

‖ N η ‖ \leq ρ 2 (‖ e ‖) ‖ e ‖ + C 2

(38)

while the mean value theorem (Xian et al., 2004) can be utilized to upper-bound Ñ_μ in the following manner:

‖ \tilde{N} μ ‖ \leq ρ 3 (‖ e ‖) ‖ e ‖

(39)

where ρ₂(·) and ρ₃(·) are positive nondecreasing functions while C₂ is a system constant. To facilitate the subsequent stability analysis, the following lemmas are needed.

Lemma 3

For any differentiable function e(t) :R⁺ → R with e(t), ė(t) ∈ ℒ_∞, there exist positive constants ɛ₁ and ɛ₂ such that

\int_{t 0}^{t} | \overset{\cdot}{e} (t) | d τ \leq ɛ 1 + ɛ 2 \int_{t 0}^{t} | e (t) | d τ + | e (t) | \forall t > t 0

Proof

The proof of Lemma 3 can be found in Stepanyan and Kurdila (2009).▪

Lemma 4

An auxiliary function L(t) ∈ R is defined as follows:

L (t) = r (\bar{N} μ d - k d sgn (e 2))

(40)

where

\bar{N} μ d = N μ d + C 2 sgn (r)

If the control gain k_d is chosen to be sufficiently large as follows:

k d > (1 + ɛ 2) | | \bar{N} μ d | | \infty

(41)

then the integral of L(t) can be upper-bounded in the following manner:

\int_{t 0}^{t} L (τ) d τ \leq ε ≜ ɛ 1 | | \bar{N} μ d | | \infty + k d | e 2 (t 0) |

(42)

Proof.

The proof of Lemma 4 is given in Appendix B.▪

The stability analysis is carried out in two steps. First, the error signals e₁,e₂, and r are proven to be ultimately bounded. Then, we utilize Lemmas 3 and 4 to prove the semiglobal asymptotic stability of the error signals. We begin by defining a nonnegative Lyapunov candidate function V₂ as follows:

V 2 = 12 e_{1}^{2} + 12 e_{2}^{2} + 121 | b 2 | r 2

(43)

Since the Lyapunov function is continuously differentiable, V₂ can be bounded as

min {12, 12 | b 2 |} ‖ e ‖ 2 \leq V 2 (e) \leq max {12, 12 | b 2 |} ‖ e ‖ 2

(44)

The time derivative of (43) along (37) can be obtained as follows:

\overset{\cdot}{V} 2 = e 1 e 2 - e_{1}^{2} - e_{2}^{2} - (k s + 1) r 2 + r [N μ d + \tilde{N} μ + N η - k d sgn (e 2 (t))]

By utilizing the bounds of (38) and (39) in the above expression, one can obtain the following upper bound for

\overset{\cdot}{V} 2

\overset{\cdot}{V} 2 \leq - 12 (e_{1}^{2} + e_{2}^{2}) - (k s + 1) r 2 + | r | (ρ 3 (‖ e ‖) ‖ e ‖ + ρ 2 (‖ e ‖) ‖ e ‖ + C 2) + r [N μ d - k d sgn (e 2 (t))] \leq - 12 (e_{1}^{2} + e_{2}^{2}) - (k s + 1) r 2 + | r | ρ (‖ e ‖) ‖ e ‖ + C 4 | r |

where ρ(‖‖e‖‖)≜ρ₃(‖‖e‖‖) + ρ₂(‖‖e‖‖) while C₄≜‖‖N_μd + C₂sgn(r) − k_dsgn(e₂(t))‖‖_∞. After completing squares to obtain the inequality

- k s r 2 + | r | ρ (‖ e ‖) ‖ e ‖ \leq 14 k s ρ 2 (‖ e ‖) ‖ e ‖ 2,

one can further upper-bound

\overset{\cdot}{V} 2

as follows:

\overset{\cdot}{V} 2 \leq - 12 (e_{1}^{2} + e_{2}^{2}) - r 2 + 14 k s ρ 2 (‖ e ‖) ‖ e ‖ 2 + C 4 | r |

where

\overset{\cdot}{V} 2

can be further bounded as

\overset{\cdot}{V} 2 \leq - 12 ‖ e ‖ 2 - (12 - 14 σ 1) r 2 + 14 k s ρ 2 (‖ e ‖) ‖ e ‖ 2 + σ 1 C_{4}^{2}

by utilizing the identity

C 4 | r | \leq 14 σ 1 | r | 2 + σ 1 C_{4}^{2}

where σ₁ is an arbitrary positive constant. By choosing σ₁ such that

12 - 14 σ 1 \geq 0,

\overset{\cdot}{V} 2

can be bounded as

\overset{\cdot}{V} 2 \leq - (12 - 14 k s ρ 2 (‖ e ‖)) ‖ e ‖ 2 + σ 1 C_{4}^{2}

Finally, one can utilize (44) to obtain the following upper bound rendering the following inequality:

\overset{\cdot}{V} 2 \leq - C 5 V 2 + σ 1 C_{4}^{2}

(45)

for all ‖‖e‖‖ ≤

ρ - 1 (\sqrt{2 k s})

, where C₅ is a positive constant. It is now easy to conclude that V₂ is semi-global uniformly ultimately bounded which implies that e₁, e₂, r ∈ ℒ_∞. Therefore, we can apply Lemma 3 and subsequently Lemma 4 to define a nonnegative function P(t) as follows:

P = Δ ε - \int_{t 0}^{t} L (τ) d τ \geq 0

(46)

To show asymptotic stability, we begin by considering the following Lyapunov function candidate:

V 3 = V 2 + P

(47)

By utilizing the definition of (46), the time derivative of V₃(t) along (37) can be obtained as follows:

\overset{\cdot}{V} 3 = e 1 e 2 - e_{1}^{2} - e_{2}^{2} - (k s + 1) r 2 + r [N μ d + N η + \tilde{N} μ - k d sgn (e 2 (t))] - L (t)

Following similar procedures as utilized to bound

\overset{\cdot}{V} 2,

we can find an upper bound for

\overset{\cdot}{V} 3

as follows:

\overset{\cdot}{V} 3 \leq - 12 ‖ e ‖ 2 - k s r 2 + r ρ (‖ e ‖) ‖ e ‖ + {r [N μ d + C 2 sgn (r) - k d sgn (e 2 (t))] - L (t)}

The term in braces nets out to zero by virtue of (40) so that the upper bound above can be simplified as follows:

\overset{\cdot}{V} 3 \leq - (12 - 14 k s ρ 2 (‖ e ‖)) ‖ e ‖ 2

(48)

Then one can state from (48) that

\overset{\cdot}{V} 3 (t)

is negative semi-definite on the region

S = {e \in ℜ 3 | | | e | | < ρ - 1 (\sqrt{2 k s})}

(49)

We note here that the region of attraction S can be made arbitrarily large to include any initial value of e₁(t), e₂(t), r(t) by increasing the control gain k_s. From (47) and (48), one can infer that e₁(t), e₂(t), r(t) ∈ ℒ_∞ ∩ ℒ₂ which implies that

\overset{\cdot}{e} 1 (t), \overset{\cdot}{e} 2 (t) \in L \infty

from (14) and (15 ). From the boundedness and smoothness of α_d(t) and the previous boundedness assertions, once can infer that

α (t), \overset{\cdot}{α} (t), \overset{··}{α} (t) \in L \infty

from the definitions of (13), (14), and (15). From (20) and (28), it is easy to see that

x (t), \overset{\cdot}{x} (t) \in L \infty

which implies from (35) that

\overset{\cdot}{r} (t) \in L \infty

. Thus the right-hand side of the inequality of (48) is uniformly continuous which allows us to utilize Barbalat's lemma to show the asymptotic stability of e₁(t), e₂(t), r(t) to the origin. This result is valid under any bounded C⁰ external disturbance ω_G(τ). It is also clear to see by virtue of (11) that the control input u(t) remains bounded for all time as well. Thus, all system states and the control input remain bounded for all time during closed-loop operation. While the nominal internal dynamics are asymptotically stable, we do not expect the internal states to converge to the origin under application of finite external disturbances; however, we can guarantee boundedness of all system states under bounded disturbance.

6. Simulation results

6.1. Model and controller parameters

In this section, numerical simulation results are presented for a nonlinear 2-D wing flag structural model under unsteady flow controlled by the flap hinge torque of the trailing-edge flap. External disturbances are also taken into consideration to validate the controller effectiveness. The wing section model is simulated using the dynamics of equations (1) and (3). The nonlinear polynomial representing pitching spring stiffness is chosen as k_α(α) = K_α(1 − 22.1α + 1315.5α² + 8580α³ − 17289.7α⁴) Nm/rad. The parameter settings are chosen as shown in Table 1 leading to a flutter speed of V_f = 83.56 ms⁻¹.

Table 1.

Model parameters.

Parameter	Value	Parameter	Value
b	0.305 m	K _h	500m
a	−0.5	K _α	1200I_α
c	0.5	K _β	15000I_β
m	90 kg/m	ρ	1.225 kg/m³
I _β	0.552 kgm²/m	I _α	22.08 kgm²/m
S _β	0.441 kg	S _α	7.062 kg

The desired trajectory variable α_d and its derivatives are simply selected to be zero. The initial conditions for pitch angle α (t) and plunge displacement h(t) are set to be α (0) = 5.729° and h(0) = 0.1 m while all other state variables are initially selected to be zero. In this paper, the following three external disturbances are considered according to Marzocca et al. (2001). The first type of external disturbance is modeled as a triangular gust, whose velocity distribution function is given as

w G (τ) = 2 w 0 τ τ G (H (τ) - H (τ - τ G 2)) - 2 w 0 (τ τ G - 1) (H (τ - τ G) - H (τ - τ G 2))

(50)

where H(·) denotes a unit step function and τ_G = V_ft_G/b given t_G = 0.25 s. This triangular gust lasts for 0.5 s from t = 0 to 0.5 s. The second disturbance is modeled as a sustained disturbance beyond the transient response time of the closed-loop aeroelastic system. It is given in the form of a graded gust with its velocity distribution shown as

w G (τ) = H (τ) w 0 (1 - e - 0.75 τ)

(51)

The third type of external disturbance is modeled as a sinusoidal gust with the following velocity distribution function w_G(τ):

w G (τ) = H (τ) w 0 \sin ω τ

(52)

where ω = 0.5 rad/s. For all three disturbances, we set w₀ to be 0.5 m/s for both pre- and post-flutter speed simulations. These external disturbances cover three main kinds of disturbance: ephemeral disturbance, steady sustained disturbance, and time-varying sustained disturbance. The contribution of these disturbances to the system response will be presented in the form of gust loading as shown in (6). Both the disturbance velocity distribution and the corresponding gust load will be presented along with the system response. For all simulations, we implement the controller of (33) with gains selected as k_s = 50 and k_d = 40.

6.2. Numerical simulation results

In this section, we run various simulations to test the nominal system response as well as that under the external disturbances given in equations (50) to (52). Figure 2 shows the open-loop and closed-loop response of the wing section model at pre-flutter speed V = 0.9V_f without disturbance. The controller in the closed-loop simulation was turned on at t = 1.5 s. It can be seen from Figure 2 that the pitching displacement converged in 2 s. The plunging displacement converged according to the internal dynamics of the system: it is driven to zero in 2 s as shown in (2).

Figure 3 shows the open-loop and closed-loop responses of the model section at post-flutter speed V = 1.1V_f without disturbance. As shown in Figure 3, the controller is turned on at t = 1.5 s. The LCOs of the pitching and plunging displacements are suppressed within 2 s. The triangular gust disturbance is the first type of external disturbance considered in the simulation. Since an external gust disturbance impacts the aerodynamics in terms of gust loading L_g(t), it would be intuitive to analyze the response of the wing section model with the corresponding plot of gust load. Since from (6) and (7) we can see that I_MG and I_fG are linear functions of I_LG, we only show the plot of I_LG. As shown in Figure 4, the gust starts at 1 s and lasts for 0.5 s. However, after filtering by Kussner's function (8), the loading produced by the gust is seen to be smoother and longer-lasting than the gust itself. Figure 5 presents the open- and closed-loop response of the wing section model under triangular gust at pre-flutter speed with V = 0.9V_f. As shown in Figure 5, the triangular gust has an obvious impact on the plunge and pitch displacement between t = 1 s and t = 4 s. Figure 6 shows the open- and closed-loop response of the wing section model under triangular gust at post-flutter speed with V = 1.1V_f.

In Figures 5 and 6, the controller is turned on at t = 2 s and in both of these two simulations, plunge and pitch displacement are driven to zero in 2 s. While the pitching and plunging disturbances converge to the origin relatively quickly, the flap torque converges to zero only at the rate of convergence of the external disturbance. The third set of simulations was run under the graded external gust disturbance; as shown in Figure 7, the graded gust load starts at t = 1 s and exponentially goes to a constant loading value as time goes on.

At both pre- and post-flutter speeds as shown in Figures 8 and 9, the proposed controller is turned on at t = 2 s and is seen to successfully drive the output to zero. As expected, the persistent disturbance causes the control signal to not converge to zero along with the output. However, it is clear to see that the control signal is able to compensate for the unknown graded disturbance injected into the wing section model.

Sinusoid gust disturbance was considered in the fourth set of simulations. The disturbance and its gust load is presented in Figure 10. The open- and closed-loop responses of the wing section model under sinusoid gust disturbance at pre- and post-flutter speed are displayed in Figures 11 and 12, respectively. The disturbance starts at t = 0 and the controller is turned on at t = 2 s at both pre- and post- flutter speed. As shown in Figures 11 and 12, after the pitching degree (which is the directly controlled DOF) converges, there are still small residual oscillations in the responses of h and β. Although the system (9) is minimum-phase and the output α converges under the proposed control, due to the sustained bounded external disturbance, the zero dynamics just stay bounded but are not able to converge as predicted by the bound shown in (28).

To further investigate the impact of sinusoidal disturbance on the system response, we set all initial states to be zero. In this case, the impact of sinusoid disturbance on the system would be more distinctly visible. Figure 13 shows the plots of open- and closed-loop responses under sinusoidal gust at pre-flutter speed. Figure 13 shows that the pitching DOF converges very rapidly after the control is turned on at t = 2 s. As discussed before, the other states of the system stayed bounded. Due to the sustained disturbance, the plunging displacement and flap angle are not able to converge but the control torque is able to compensate for the sinusoidal disturbance in the controlled DOF. Figure 14 shows the open-loop and closed-loop response under sinusoidal gust at post-flutter speed with zero initial states. As shown in Figure 14, the controller is turned on at t = 2 s and pitching displacement converges rapidly. However, due to the sustained disturbance, the plunging displacement does not converge but its amplitude is reduced to an extent.

7. Conclusion

The active aeroelastic control of 2-D wing-flap systems which are operating in an incompressible flowfield and exposed to three different kinds of external gust loads is modeled. By utilizing the property of ISS, the system internal states are proved to be bounded as a function of the output tracking error and its derivatives. Then, a continuous robust controller is proposed to suppress the aeroelastic vibration subject to external disturbance on a nonlinear plunging and pitching wing section subject to unsteady aerodynamics. The control strategy requires minimal system and disturbance knowledge and is implemented by a trailing-edge flap torque. Lyapunov-based stability analysis is provided to obtain a semiglobal asymptotic stability result on the pitching DOF tracking error. The robustness and efficacy of the controller is validated by simulation results under different operating conditions and disturbances. The investigated gust disturbances in this paper are the most typical ones encountered. While it is impossible to present the system response under all possible kinds of disturbances, one can expect satisfactory simulation results under other disturbances based on the rigorous stability analysis presented in this paper. Our future work will focus on addressing aeroelastic vibration control using a combination of leading- and trailing-edge flaps such that both pitching and plunging displacements can be shown to be completely suppressed both theoretically and in numerical simulation under arbitrary disturbance profiles.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Nomenclature

System matrix of the system state space equation

Nondimensional distance from mid-chord to elastic axis

Control input matrix of the system state space equation

B _i

Aerodynamic lag state variables

Semi-chord of the wing [m]

Output matrix of the system state space equation

C(k)

Theodorsen's function

Nondimensional distance from mid-chord to flap hinge line

D(t)

Duhamel integration

e₁,e₂,r,e

Tracking error, filtered tracking error, and composite error

f (·), f _c(·)

Nonlinear function vectors

Ϝ,Ϝ_sub

State transformation matrix

Plunging displacements [m]

Identity matrix

I _α

Inertia of wing section per unit span about elastic axis [kgm²/m]

I _β

Inertia of flap per unit span about hinge root [kgm²/m]

I_LG,I_MG,I_fG

Aerodynamic indicial functions

Stiffness matrix

k_h,k_α

Structural stiffnesses in plunging [N/m] and pitching [Nm]

k_s,k_d

Control gain

K _β

Torsional stiffness of the flap [Nm]

The aerodynamic lift per unit span [N/m]

Unsteady aerodynamic load vector

L _c

Control load vector

L ₀

Lipschitz constant

Aerodynamic pitching moment per unit span [Nm/m]

Airfoil mass per unit length [kg/m]

N,N_d,Ñ

Unstructured nonlinear uncertainties

Filtered plunging displacement error

S_α,S_β

Static moment per unit span of pitch angle and flap angle [kgm]

Flap torque per unit span [N/m]

T _c

Control torque at flap hinge per unit span [N/m]

Time variables [s]

V,V_f

Flight and flutter speeds [m/s]

w_g(t)

External gust disturbance

State vector

Column vector of plunge, pitch, and flap displacement

Pitching displacement [rad]

Trailing edge flap angle [rad]

Zero dynamics vector

μ,ζ

States of system normal form

Air density [kgm³]

Dimensionless time

Φ_i

Theodorsen's constants

sgn(·)

Sign function

‖‖·‖‖

Euclidean norm

f(t) ∈ C^k

f(t) has k continuous derivatives

Appendix A: aeroelastic model constants

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