Nonlinear aeroelastic phenomena, such as limit cycle oscillation [1] and free-play flutter [2], not only threaten the aircraft structure, but can even result in loss of control. This challenge attracts many researchers to develop aeroelastic control for flutter suppression and avoidance of LCO [3].Usually passive flutter control has been widely used for wing flutter suppression in practical engineering, such as mass balance, local stiffness enhancement, etc., however, this method requires extra mass and makes the airplane heavier [4]. Therefore, active flutter control has been proposed and the objectives are to suppress flutter and to make the airplane flight exceed its flutter speed.

Active flutter control is a hot topic to suppress wing flutter by using modern control theories. Many researchers proposed useful schemes for active flutter suppression. Ouyang et al. studied the influence of flap camber on the wing flutter bounds, and proposed a Linear Quadratic Guassian (LQG) based control to actively suppress the aeroelastic response by the flap deflection, and the flutter speed and frequency of the wing will be changed by the orders of the reduced order model of the morphing flap camber [5]. Na et. al. proposed a LQG control method with sliding mode control (SMC) observer to active control of composite thin-walled beam structures in compressible flow, the dynamic aeroelastic responses of the beam structures subject to sharp-edged gust, blast loads and sonic-boom are investigated by this method [6]. Compared with linear aeroelasticity, nonlinear aeroelastic issue is more attractive, which includes freeplay, time delay and cubic nonlinearities in aeroelastic systems. Neil et al. studied structure nonlinear effect, analyzed and experimented by a rigid wing supported by a nonlinear spring [7]. Lee et al. presented the effect of a cubic structure restoring force on the flutter of a two dimensional airfoil placed in an incompressible flow, and the results showed the dependence of the divergence flutter boundary on initial conditions for a soft spring, while the independence of the divergence flutter boundary on initial conditions for the hard spring [8]. To stabilize a nonlinear aeroelastic system, Singh et al. proposed an adaptive backstepping output feedback control scheme, and trajectory control of the pitch angle is achieved by this method [9]. To deal with model uncertainties of aeroelastic systems, Lin et al. suggested a decoupled fuzzy sliding mode control (DFSMC), and the DFSMC system was designed to control the plunge and pitch motions simultaneously [10].Chen et. al. proposed a terminal SMC (TSMC)for aeroelastic systems, the LCO suffered in the aeroelastic system is removed by the TSMC [11]. Mukhopadhyay introduced historical perspective on analysis and control of aeroelasticity over the century, and showed some experiments of aeroelastic control for different aircrafts [12]. Bouma et al. investigated structural and aerodynamic nonlinearities effects for three degrees of freedom (DOF) aeroelastic systems, and stall effects were introduced using quasi-steady and unsteady aerodynamic representations [13]. Vishal et al. studied the synchronization characteristics of a pitch–plunge aeroelastic system, possessing discontinuous nonlinearities in both structural and aerodynamic fronts. Meanwhile, a multi-physics fluid–structure interaction problem possessing coupled non-smooth nonlinearities can be alleviated by the use of synchronization theory [14]. Livne gave a survey on aircraft active flutter suppression (AFS) through state of the art, and AFS wind tunnel test and flight-test programs were illustrated [15].

Flutter suppression theory has also been developed by the researchers. Considering delayed control problem, a single time delay was determined according to both the critical stability condition and the stability margins, Zou et al. proposed a delayed proportional–differential (PD) controller for robust suppression the body-freedom flutter (BFF) of a flying-wing drone [16].Trapani and Guo proposed a rudderless aeroelastic fin for a flexible blended-wing-body UAV to improve divergence and flutter margin [17]. Although these methods are useful for the linear time-invariant systems, they cannot guarantee the closed-loop performances of the nonlinear and time-varying aeroelastic system. For the structural nonlinearities and unsteady flow, Zhang and Behal proposed a continuous robust controller to realize aeroelastic vibration control of a two dimensional airfoil [18]. Lee and Singh proposed an L₁ adaptive control for the aeroelastic system with unsteady aerodynamics and gust load [19].For the nonlinear aeroelastic system, Platanitis and Strganac proposed a feedback linearization combined with model reference, and it can adaptively suppress LCO of a typical wing section with LE and TE control surfaces [20]. Prime set up a linear parameter varying (LPV) model for the three DOF Nonlinear Aeroelastic Test Apparatus (NATA), and developed a H₂ robust control by linear matrix inequalities [21]. Li et al. proposed a state-dependent Riccati equation based method to control an airfoil section with control surface nonlinearities, but it depended on the accurate model [22]. To reduce dependence on the model, Tang et al. proposed a neural network based active flutter suppression under control input constraints [23], but neural networks need extensive calculations. Recently Gao introduced sliding mode control into an aeroelastic system and presented a finite-time fault-tolerant control for wing flutter suppression [24].Chen et al. proposed a free-model deep reinforcement learning algorithm for the active flutter suppression through aeroelsatic wind tunnel testing, thus the jet flow intensity can be intelligently selected according to the real-time state of the flexible wing [25].

Since the aircrafts fly in a varying environment, the influence of model uncertainties and external disturbances on aeroelastic systems have also been considered by the researchers. Pettit briefly described aerodynamic and structure uncertainties and applications of uncertainty with design optimization [26]. For the nonlinear perturbations, Hao et al. analyzed stochastic nonlinear flutter behaviors of 3-DOF wing model by the stochastic p-bifurcation analysis method [27].Lee and Singh proposed a robust higher-order super-twisting control to deal with aeroelastic system with unsteady aerodynamics [28]. To estimate unknown parameters, Prabhu and Srinivas proposed an intelligent observer and then robust control of a three degrees of freedom aeroelastic model [29], but the neural network based observer has a heavy computational load. To improve the convergence speed of the aeroelastic system, Chen et al. proposed a terminal sliding mode control to reach the target within a finite time frame [30], and Yuan et al. proposed a sliding mode observer control for a two dimensional aeroelastic system with gusts [31]. Xu et al. also applied Backstepping Sliding Mode Control (BSMC) design to suppress the LCO with uncertainties and external disturbances [32]. All these methods can suppress or attenuate external disturbances and achieve the desired robust performances, however an alternative approach is available, the so-called disturbance observer based control (DOBC). The DOBC lumps all internal uncertainties and external disturbances acting on the aeroelastic system into a single term and then identifies them by the DO. The DO can provide fast, excellent control performances and smooth control actions without using large feedback gains [33]. Liu and Whidborne have proposed a nonlinear disturbance observer to deal with bounded unknown disturbances [34], but this observer does not cope well with variable disturbances, and the output tracking responses by this DO-BSMC cannot be guaranteed to converge within finite time.

Motivated by Xu’s [32] and Chen’s work [33], this paper considers trajectory tracking control for the two-dimensional airfoil-flap aeroelastic system. The major contributions of this work are as follows:

Considering quasi-steady and unsteady aerodynamic loads, nonlinear dynamic models are set up for an aeroelastic two-dimensional airfoil with both trailing-edge and leading-edge control surfaces.

To suppress flutter, a backstepping terminal sliding-mode control is proposed for the two-dimensional airfoil-flap aeroelastic system. The benefits of this approach have TSMC merits of high robustness, fast transient response and finite time convergence, as well as backstepping control in terms of globally asymptotic stability.

To overcome the limitation of the BTSMC being dependent on prior knowledge, a nonlinear disturbance observer is designed for coping with model uncertainties and variable external bounded gust disturbances, which is different from Backstepping SMC for flutter suppression [32].

To adjust the speed where the aeroelastic response converges to zero, a learning rate is incorporated into the BTSMC design.

Simulation results show the proposed DO-BTSMC method is effective and has advantages in LCO suppression and flutter control for a two-dimensional wing with LE and TE control surfaces.

This paper is organized as follows. Section 2 establishes the nonlinear aeroelastic dynamics model of the two-dimensional airfoil-flap system. Section 3 proposes a DO-BTSMC design, and stability is analyzed for the closed-loop control system. Simulations and analysis are in Section 4. Some conclusions are given in Section 5.

The studied model is a two-DOF pitch-plunge wing section with both LE and TE control surfaces and is shown in Figure 1, both control surfaces are used as control inputs.

The physical parameters of the wing are listed as follows, a is nondimensional distance from midchord to elastic axis position, b and c denote semichord and full chord length of wing section respectively.

The aeroelastic governing models are established as follows [20] where

C_{mα, eff} , C_{mβ, eff} , and C_{mγ, eff} are the effective dynamic and control moment derivatives due to angle of attack and trailing- and leading-edge control surface deflection, respectively, about the elastic axis and are defined as follows

m_T denote total mass of pitch–plunge system, c_α and c_h denote damping of pitch and plunge respectively, k_α and k_h denote pitch and plunge stiffness respectively, s_p is wing section span, I_α denotes total pitch moment of inertia about elastic axis,ρ denotes air density, h is plunge displacement, V is freestream velocity, α is angle of attack (AOA),β and γ trailing- and leading-edge control surface deflection respectively, C_L, C_Lα, C_Lβ, C_Lγ denote wing section lift coefficients and C_m, C_mα, C_mβ, C_mγ denote wing section pitch moment coefficients at quarter-chord.

Denote the state variable vector is and uncertain functions of plunge and AOA motions are

The servo motor dynamics for the TE and LE can be modeled as a second-order system of the form where$c_{\beta s}$and$c_{\gamma s}$are the damping of the TE and LE respectively, $k_{\beta s}$ and$k_{\gamma s}$ are stiffness of the TE and LE respectively, and${\beta }_c$and${\gamma }_c$denote the desired positions of the servo motors for the TE and LE respectively.$m_{\alpha }$and $I_{\alpha 0}$denote the rotational wingbody of mass and inertia respectively, $I_{\alpha }=I_{\alpha 0}+m_{\alpha }{x}_{\alpha }^2{b}^2$,$I_{\beta }=I_{\beta 0}+m_{\beta }{r}_{\beta }^2$ and $I_{\gamma }=I_{\gamma 0}+m_{\gamma }{r}_{\gamma }^2$are the moments of inertia of the TE and LE about its pivot respectively , $m_{\beta }$,$m_{\gamma }$,$I_{\beta 0}$ and $I_{\gamma 0}$are the rotational TE and LE section of mass and inertia respectively. Let$\overline{m}=m_h+m_{\alpha }+m_{\beta }+m_{\gamma }$,$m_h$denotes a translational body of mass.

When the aerodynamic performance of the airfoil is unsteady, the equations of motion for the entire system can be set up by Lagrangian energy theory, where $\mathcal{L}$denotes Lagrangian function. The expanded form of (10) is wherea * denotes symmetric matrix element,

The aerodynamic force and moment in (8) can be calculated by Theodorsen method [35,36] where T_i (i =1,2,…,14) are the Theodorsen constants, see Appendix B [37]. $M_{\gamma }$is derived from those by Throdorsen [35,36,37].

The aerodynamic force and moments in (13)-(15) are dependent on Theodorsen function C(k), where k is the non-dimensional reduced frequency of harmonic oscillation, C(k) can be approximated by Jones method, where c₁=0.0075, c₂=0.10055, c₃=0.0455, c₄=0.3. C(s) can be presented as a state-space model as follows where

The aerodynamic force and moments can be described as the sum of non-circulatory and circulatory loads, where F_nc and F_c are non-circulatory load and circulatory load respectively. Denote${\overline{x}}_1={\left[\begin{array}{cccc}h& \alpha & \beta & \gamma \end{array}\right]}^T$, x_a = [x_a1, x_a2]^T, then we get where x_α is nondimensional distance from elastic axis to centre of mass, and meeting x_α= r_cg /b, r_cg is distance from elastic axis to centre of mass.

$T_{13}$and$T_{13d}$are the Theodorsen constants for the TE and LE respectively. Substitute (17) , (13) - (15) into (11), then yields where(3)

Denote the full state vector$\overline{x}={\left[\begin{array}{ccc}{{\overline{x}}_1}^T& {{\dot{\overline{x}}}_1}^T& x_a^T\end{array}\right]}^T$, the overall state-space model is where the A and B matrix are as follows where

Since there exists wind disturbance for the aeroelastic system, then the external disturbance D is considered and added into the system of (6) and (25) as where$D\in R^2$and $\overline{D}(t)\in R^4$denote external disturbances and bound by$‖D‖<D_U$, D_U is a real value meeting D_U≥0. Denote${\overline{f}}_2({\overline{x}}_1)=-M^{-1}\left(K(V){\overline{x}}_1+C(V){\dot{\overline{x}}}_1+V^2{E}_c{x}_a\right)$ ,

${\overline{g}}_2({\overline{x}}_1)=M^{-1}{G}_s$. The aerodynamic loads due to the wind disturbances are defined as [38] where${\omega }_G(\tau )$denotes the disturbance velocity, τ is a dimensionless time variable defined as $\tau =Vt/b$, so $D={\left[\begin{array}{cc}-L_g/m_T& M_g/I_{\alpha }\end{array}\right]}^T$.Suppose that the above disturbance can be generated by a linear exogenous system [33], where${\varsigma }_d\in R^p$,W_d, V_d are coefficient matrices with corresponding dimensions.

Airfoil aeroelastic vibration Control Problem: The control objective is to suppress the aeroelastic vibrations, so the desired smooth trajectory x_d can be chosen to be zero for all time. Alternatively the desired trajectory x_d along with the actual pitching and plunging variables is driven toward the origin.

This section gives an overview of the DO-BTSMC control for the two dimensional airfoil.

The studied aeroelastic system is the NATA model and the physical parameters of the aeroelastic system are listed in Table 1 [20,21,40].

A nonlinear backstepping nonsingular terminal SMC method is used to design a controller to suppress limit cycle oscillation (LCO) and flutter. The developed controller stabilizes the plunge and pitch motions of the two dimensional airfoil with LE and TE control surfaces within finite-time convergence. Stability analysis shows that the closed-loop aeroelastic dynamics are globally asymptotically stable. Six examples of the variable bounded disturbances of triangular gust and sine gust and model parameter uncertainties are simulated. Compared with the BSMC and PID controllers, the DO-BTSMC with learning rate achieves better flutter suppression performances in fast response (improved by 18.6%) and root mean square error( reducing 75% for α). Therefore, the effectiveness and availability of the DO- BTSMC design are demonstrated.