Disturbance Observer-Based Backstepping Terminal Sliding Mode Aeroelastic Control of Airfoils

Abstract

This paper studies aeroelastic control for a two-dimensional airfoil–flap system with unknown gust disturbances and model uncertainties. Open loop limit cycle oscillation (LCO) happens at the post-flutter speed. The structural stiffness and quasi-steady and unsteady aerodynamic loads of the aeroelastic system are represented by nonlinear models. To robustly suppress aeroelastic vibration within a finite time, a backstepping terminal sliding-mode control (BTSMC) is proposed. In addition, a learning rate (LR) is incorporated into the BTSMC to adjust how fast the aeroelastic response converges to zero. In order to overcome the fact that the BTSMC design is dependent on prior knowledge, a nonlinear disturbance observer (DO) is designed to estimate the variable observable disturbances. The closed-loop aeroelastic control system has proven to be globally asymptotically stable and converges within a finite time using Lyapunov theory. Simulation results of an aeroelastic two-dimensional airfoil with both trailing-edge (TE) and leading-edge (LE) control surfaces show that the proposed DO-BTSMC is effective for flutter suppression, even when subjected to gusts and parameter uncertainties.

1. Introduction

Nonlinear aeroelastic phenomena, such as limit cycle oscillation [1] and free play flutter [2], not only threaten the aircraft structures 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]. Passive flutter control has usually been widely used for wing flutter suppression in practical engineering, including 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 increase the flutter speed.

Modern control theory is often proposed to suppress wing flutter. For example, Ouyang et al. studied the influence of flap camber on the wing flutter bounds and proposed a linear quadratic Gaussian (LQG)-based control to actively suppress the aeroelastic response by the flap deflection. They found that the wing flutter speed and frequency are affected by the order of the morphing flap camber geometry model [5]. Na et al. proposed an LQG control method with a sliding mode control (SMC) observer to actively control composite thin-walled beam structures in compressible flow; the dynamic aeroelastic responses to sharp-edged gusts, blast loads, and sonic booms were investigated [6].

Aeroelastic systems are subjected to nonlinearities and other complex effects, resulting from freeplay, time delays, and cubic nonlinearities. Neil et al. explored the effects of structural nonlinearity by analyzing and experimenting with a rigid wing supported by a nonlinear spring [7]. Lee et al. examined the effect of a cubic structural restoring force on the flutter of a two-dimensional airfoil placed in incompressible flow. Their results demonstrated that the divergence flutter boundary depends on the initial conditions for a soft spring, and it remains independent of them for a 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 [9].

To address the model uncertainties in 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 an aeroelastic system was removed by the TSMC [11]. Mukhopadhyay introduced a historical perspective on the analysis and control of aeroelasticity over the century and showed some experiments of aeroelastic control for different aircraft [3].

Bouma et al. investigated structural and aerodynamic nonlinear effects for three-degree-of-freedom (DOF) aeroelastic systems; stall effects were introduced using quasi-steady and unsteady aerodynamic representations [12]. 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 [13]. Livne provided a survey on state-of-the-art aircraft active flutter suppression (AFS) through AFS wind tunnel testing and flight-test programs [14].

Flutter suppression theory has also been developed by researchers. Considering the delayed control problem, a single time delay was determined according to both the critical stability condition and the stability margins, and Zou et al. proposed a simple scheme consisting of a delayed proportional–differential (PD) controller to robustly suppress the body-freedom flutter (BBF) of a flying-wing drone [15]. Trapani and Guo proposed a rudderless aeroelastic fin for a flexible blended-wing-body UAV to improve the divergence and flutter margin [16]. Although these methods are useful for linear time-invariant systems, they cannot guarantee the closed-loop performances of nonlinear and time-varying aeroelastic systems.

To cope with structural nonlinearities and unsteady flow, Zhang and Behal proposed a continuous robust controller for aeroelastic vibration control of a two-dimensional airfoil [17]. Lee and Singh proposed an L₁ adaptive control for an aeroelastic system with unsteady aerodynamics and gust loads [18]. For nonlinear aeroelastic systems, Platanitis and Strganac proposed feedback linearization combined with a model reference that can adaptively control and suppress the LCO of a typical wing section with both leading-edge and trailing-edge control surfaces [19]. Prime proposed a linear-parameter-varying model for the three-DOF Nonlinear Aeroelastic Test Apparatus (NATA) model and developed an H₂ robust control via linear matrix inequalities [20]. Li et al. proposed a state-dependent Riccati equation-based method to control an airfoil section with control surface nonlinearities, but it depended on an accurate model [21]. To reduce dependence on the model, Tang et al. proposed neural-network-based active flutter suppression under control input constraints [22], but neural networks require extensive calculations. Recently, Gao introduced sliding-mode control into an aeroelastic system and presented a finite-time fault-tolerant control for wing flutter suppression [23]. Chen et al. proposed a model-free deep reinforcement learning algorithm for active flutter suppression through aeroelastic wind tunnel testing; thus, the jet flow intensity can be intelligently selected according to the real-time state of the flexible wing [24].

Because aircraft fly within varying environments, the influence of model uncertainties and external disturbances on aeroelastic systems have also been considered by researcher. Pettit briefly described aerodynamic and structural uncertainties and applications of uncertainty with design optimization [25]. For nonlinear perturbations, Hao et al. analyzed stochastic nonlinear flutter behaviors of a three-DOF wing model using the stochastic p-bifurcation analysis method [26]. Lee and Singh proposed a robust higher-order super-twisting control to control an aeroelastic system with unsteady aerodynamics [27].

To estimate unknown parameters, Prabhu and Srinivas proposed an intelligent observer and then robust control of a three-DOF aeroelastic model [28], 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 TSMC to reach the target within a finite time frame [11], while Yuan et al. proposed a sliding-mode observer control for a two-dimensional aeroelastic system with gusts [29]. Xu et al. also applied backstepping sliding-mode control (BSMC) design to suppress the LCO in the presence of uncertainties and external disturbances [30].

All these methods can suppress or attenuate external disturbances and achieve the desired robust performance; however, an alternative approach is available, the so-called disturbance-observer-based control (DOBC). DOBC lumps all internal uncertainties and external disturbances acting on the aeroelastic system into a single term and then identifies them using the DO. The DO can provide fast, excellent control performances and smooth control actions without requiring large feedback gains [31]. Liu and Whidborne have proposed a nonlinear disturbance observer to deal with bounded unknown disturbances [32], but this observer is less effective with variable disturbances, and the output tracking responses of this DO-BSMC are not guaranteed to converge within a finite time.

Motivated by the work of Xu [30] and Chen [31], this paper considers trajectory tracking control for a two-dimensional airfoil–flap aeroelastic system. The main contributions of this research 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 include the 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 the backstepping SMC for flutter suppression [30].

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

Simulation results show that 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.

2. Aeroelastic System Dynamic Model

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. Also shown are the wing physical parameters; a is the non-dimensional distance from the wing section mid-chord to the elastic axis position, b denotes the semi-chord length, and c is the full chord length.

The aeroelastic governing models are established as follows [19]:

$$\left[\begin{array}{cc}{m}_T& m_w{x}_{\alpha }b\\ m_w{x}_{\alpha }b& I_{\alpha }\end{array}\right]\left[\begin{array}{c}\ddot{h}\\ \ddot{\alpha }\end{array}\right]+\left[\begin{array}{cc}{c}_h& 0\\ 0& c_{\alpha }\end{array}\right]\left[\begin{array}{c}\dot{h}\\ \dot{\alpha }\end{array}\right]+\left[\begin{array}{cc}{k}_h& 0\\ 0& k_{\alpha }(\alpha )\end{array}\right]\left[\begin{array}{c}h\\ \alpha \end{array}\right]=\left[\begin{array}{c}-L\\ M\end{array}\right],$$

(1)

where

$$L=\rho V^{2}b{s}_p{C}_{L\alpha }\left(\alpha +{\displaystyle \frac{\dot{h}}{V}}+\left({\displaystyle \frac{1}{2}}-a\right)b{\displaystyle \frac{\dot{\alpha }}{V}}\right)+\rho V^{2}b{s}_p{C}_{L\beta }\beta +\rho V^{2}b{s}_p{C}_{L\gamma }\gamma ,$$

(2)

$$M=\rho V^2{b}^2{s}_p{C}_{m\alpha ,eff}\left(\alpha +{\displaystyle \frac{\dot{h}}{V}}+\left({\displaystyle \frac{1}{2}}-a\right)b{\displaystyle \frac{\dot{\alpha }}{V}}\right)+\rho V^2{b}^2{s}_p{C}_{m\beta ,eff}\beta +\rho V^2{b}^2{s}_p{C}_{m\gamma ,eff}\gamma ,$$

(3)

and C_mα,eff, C_mβ,eff, and C_mγ,eff are the effective dynamic and control moment derivatives due to the angle of attack and trailing- and leading-edge control surface deflection, respectively, about the elastic axis and are defined as follows:

$$C_{m\alpha ,eff}=\left(0.5+a\right)C_{L\alpha }+2C_{m\alpha },C_{m\beta ,eff}=\left(0.5+a\right)C_{L\beta }+2C_{m\beta },$$

$$C_{m\gamma ,eff}=\left(0.5+a\right)C_{L\gamma }+2C_{m\gamma },$$

(4)

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

The state variable vector is denoted as

$${\mathit{x}}_1={\left[\begin{array}{cc}h& \alpha \end{array}\right]}^T,{\mathit{x}}_2={\dot{\mathit{x}}}_1={\left[\begin{array}{cc}\dot{h}& \dot{\alpha }\end{array}\right]}^T,\mathit{x}={\left[\begin{array}{cc}{\mathit{x}}_1^T& {\mathit{x}}_2^T\end{array}\right]}^T,$$

(5)

$${\dot{\mathit{x}}}_2(t)={\mathit{f}}_2(\mathit{x})+{\mathit{g}}_2(\mathit{x})U(t),$$

(6)

$${\mathit{f}}_2(\mathit{x})=\left[\begin{array}{cc}-k_1\hfill & -k_2{V}^2+p(x_2)\\ -k_3\hfill & -k_4{V}^2+q(x_2)\end{array}\right]{\mathit{x}}_1+\left[\begin{array}{cc}-c_1& -c_2\\ -c_3& -c_4\end{array}\right]{\mathit{x}}_2,U=\left[\begin{array}{c}\beta \\ \gamma \end{array}\right],{\mathit{g}}_2(\mathit{x})=\left[\begin{array}{cc}{g}_{2,11}& g_{2,12}\\ g_{2,21}& g_{2,22}\end{array}\right],$$

$$\mathrm{\Xi }=m_T{I}_a-{(m_W{x}_{\alpha }b)}^2,k_1=I_a{k}_h/\mathrm{\Xi },k_3=-m_W{x}_{\alpha }b{k}_h/\mathrm{\Xi },$$

$$k_2=\left(I_a\rho s_{p}b{C}_{L\alpha }+m_W{x}_{\alpha }\rho s_p{b}^3{C}_{m\alpha }\right)/\mathrm{\Xi },k_4=-\left(m_W{x}_{\alpha }\rho s_p{b}^2{C}_{L\alpha }+m_T\rho s_p{b}^2{C}_{m\alpha ,eff}\right)/\mathrm{\Xi },$$

$$c_1=\left(I_{\alpha }\left(c_h+\rho V{s}_{p}b{C}_{L\alpha }\right)+m_W{x}_{\alpha }\rho V{s}_p{b}^3{C}_{m\alpha ,eff}\right)/\mathrm{\Xi },$$

$$c_2=\left(I_{\alpha }\rho V{s}_p{b}^2{C}_{L\alpha }\left(0.5-a\right)-m_W{x}_{\alpha }b{c}_{\alpha }\right.\left.+m_W{x}_{\alpha }\rho V{s}_p{b}^4{C}_{m\alpha ,eff}\left(0.5-a\right)\right)/\mathrm{\Xi },$$

$$c_3=\left(-m_W{x}_{\alpha }b\left(c_h+\rho Vb{s}_p{C}_{L\alpha }\right)-m_T\rho V{s}_p{b}^2{C}_{m\alpha ,eff}\right)/\mathrm{\Xi },$$

$$c_4=\left[m_T\left(c_{\alpha }-\rho V{s}_p{b}^3{C}_{m\alpha ,eff}\left(0.5-a\right)\right)\right.\left.-m_W{x}_{\alpha }\rho V{s}_p{b}^3{C}_{L\alpha }\left(0.5-a\right)\right]/\mathrm{\Xi },$$

$$g_{\mathrm{2,11}}=\left(-I_{\alpha }\rho b{s}_p{C}_{L\beta }-m_W{x}_{\alpha }\rho s_p{b}^3{C}_{m\beta ,eff}\right)V^2/\mathrm{\Xi },$$

$$g_{\mathrm{2,21}}=\left(-I_{\alpha }\rho b{s}_p{C}_{L\gamma }-m_W{x}_{\alpha }\rho s_p{b}^3{C}_{m\gamma ,eff}\right)V^2/\mathrm{\Xi },$$

$$g_{\mathrm{2,21}}=\left(m_W{x}_{\alpha }\rho s_p{b}^2{C}_{L\beta }+m_T\rho s_p{b}^2{C}_{m\beta ,eff}\right)V^2/\mathrm{\Xi },$$

$$g_{\mathrm{2,22}}=\left(m_W{x}_{\alpha }\rho s_p{b}^2{C}_{L\gamma }+m_T\rho s_p{b}^2{C}_{m\gamma ,eff}\right)V^2/\mathrm{\Xi },$$

(7)

and the uncertain functions of plunge and AOA motions are

$$q(x_2)=m_T{k}_{\alpha }/\mathrm{\Xi },p(x_2)=-m_W{x}_{a}b{k}_a/\mathrm{\Xi }.$$

(8)

The servo motor dynamics for the TE and LE can be modeled as second-order systems of the form

$$\left\{\begin{array}{l}{I}_{\beta }\ddot{\beta }+c_{\beta s}\dot{\beta }+k_{\beta s}\beta =k_{\beta s}{\beta }_c\\ I_{\gamma }\ddot{\gamma }+c_{\gamma s}\dot{\gamma }+k_{\gamma s}\gamma =k_{\gamma s}{\gamma }_c\end{array}\right.,$$

(9)

where $c_{\beta s}$ and $c_{\gamma s}$ are the dampings of the TE and LE, respectively; $k_{\beta s}$ and $k_{\gamma s}$ are the 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. The terms $m_{\alpha }$ and $I_{\alpha 0}$ denote the rotational wing body 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 their respective pivots; $m_{\beta }$ and $m_{\gamma }$ are the rotational TE and LE sections of mass, respectively; and $I_{\beta 0}$ and $I_{\gamma 0}$ are the rotational TE and LE sections of inertia, respectively. Let $\overline{m}=m_h+m_{\alpha }+m_{\beta }+m_{\gamma }$ hold, where $m_h$ denotes the translational body mass.

When the aerodynamic performance of the airfoil is unsteady, the equations of motion for the entire system can be set up using Lagrangian energy theory,

$${\displaystyle \frac{d}{dt}}\left[\begin{array}{c}\partial \mathcal{L}/\partial h\\ \partial \mathcal{L}/\partial \alpha \\ \partial \mathcal{L}/\partial \beta \\ \partial \mathcal{L}/\partial \gamma \end{array}\right]-\left[\begin{array}{c}\partial \mathcal{L}/\partial h\\ \partial \mathcal{L}/\partial \alpha \\ \partial \mathcal{L}/\partial \beta \\ \partial \mathcal{L}/\partial \gamma \end{array}\right]=-\left[\begin{array}{c}0\\ 0\\ k_{\beta s}\beta \\ k_{\gamma s}\gamma \end{array}\right]-\left[\begin{array}{c}{c}_h\dot{h}\\ c_{\alpha }\dot{\alpha }\\ c_{\beta s}\dot{\beta }\\ c_{\gamma s}\dot{\gamma }\end{array}\right]+\left[\begin{array}{c}-L\\ M_{\alpha }\\ M_{\beta }\\ M_{\gamma }\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ k_{\beta s}{\beta }_c\\ k_{\gamma s}{\gamma }_c\end{array}\right],$$

(10)

where $\mathcal{L}$ denotes a Lagrangian function. The expanded form of Equation (10) is

$$\left[\begin{array}{cccc}\overline{m}& {\xi }_1& m_{\beta }{r}_{\beta }& m_{\gamma }{r}_{\gamma }\\ \ast & {\xi }_2& I_{\beta }+x_{\beta }{m}_{\beta }{r}_{\beta }& I_{\gamma }+x_{\gamma }{m}_{\gamma }{r}_{\gamma }\\ \ast & \ast & I_{\beta }& 0\\ \ast & \ast & 0& I_{\gamma }\end{array}\right]\left[\begin{array}{c}\ddot{h}\\ \ddot{\alpha }\\ \ddot{\beta }\\ \ddot{\gamma }\end{array}\right]+\left[\begin{array}{c}{k}_{h}h+c_h\dot{h}+{\xi }_3\\ k_{\alpha }\alpha +c_{\alpha }\dot{\alpha }+{\xi }_4\\ k_{\beta s}\beta +c_{\beta \mathrm{s}}\dot{\beta }+{\alpha }^2{m}_{\beta }{r}_{\beta }{x}_{\beta }\sin \beta \\ k_{\gamma s}\gamma +c_{\gamma \mathrm{s}}\dot{\gamma }+{\alpha }^2{m}_{\gamma }{r}_{\gamma }{x}_{\gamma }\sin \gamma \end{array}\right]=\left[\begin{array}{c}-L\\ M_{\alpha }\\ M_{\beta }\\ M_{\gamma }\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ k_{\beta s}{\beta }_c\\ k_{\gamma s}{\gamma }_c\end{array}\right]$$

(11)

where * denotes an asymmetric matrix element, and

$$\begin{array}{c}{\xi }_1=(m_{\alpha }{r}_{\alpha }+m_{\beta }{r}_{\beta }+m_{\gamma }{r}_{\gamma })\cos \alpha +m_{\beta }{r}_{\beta }\cos (\alpha +\beta )+m_{\gamma }{r}_{\gamma }\cos (\alpha +\gamma ),\hfill \\ {\xi }_2=I_{\alpha }+I_{\beta }+I_{\gamma }+m_{\beta }{x}_{\beta }^2+m_{\gamma }{x}_{\gamma }^2+2x_{\beta }{m}_{\beta }{r}_{\beta }\cos \beta +2x_{\gamma }{m}_{\gamma }{r}_{\gamma }\cos \gamma ,\hfill \\ {\xi }_3=-{(\alpha +\beta )}^2{m}_{\beta }{r}_{\beta }\sin (\alpha +\beta )-{(\alpha +\gamma )}^2{m}_{\gamma }{r}_{\gamma }\sin (\alpha +\gamma )-{\alpha }^2\sin \alpha (m_{\alpha }{r}_{\alpha }+m_{\beta }{x}_{\beta }+m_{\gamma }{x}_{\gamma }),\hfill \\ {\xi }_4=-\beta (2\alpha +\beta )m_{\beta }{r}_{\beta }{x}_{\beta }\sin (\beta )-\gamma (2\alpha +\gamma )m_{\gamma }{r}_{\gamma }{x}_{\gamma }\sin (\gamma ).\hfill \end{array}$$

(12)

The aerodynamic force and moment in Equation (10) can be calculated by Theodorsen’s method [33,34]:

$$\begin{array}{c}L=\pi \rho b^2{s}_p\left(\ddot{h}+V\dot{\alpha }-ba\ddot{\alpha }-(V/\pi )T_4\dot{\beta }-(b/\pi )T_1\ddot{\beta }+(V/\pi )T_4\dot{\gamma }-(b/\pi )T_1\ddot{\gamma }\right)\hfill \\ +2\pi \rho Vb{s}_{p}C(k)\left(V\alpha +\dot{h}+b(0.5-a)\dot{\alpha }+(V/\pi )T_{10}\beta +(b/2\pi )T_{11}\dot{\beta }+(V/\pi )T_{10}\gamma +(b/2\pi )T_{11}\dot{\gamma }\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}{M}_{\alpha }=\pi \rho b^2{s}_p& \left(ba\ddot{h}-Vb(0.5-a)\dot{\alpha }-b^2(1/8+a)\ddot{\alpha }-(V^2/\pi )T_{15}(\beta \right.+\gamma )\left.-(Vb/\pi ){\mathrm{T}}_{16}(\dot{\beta }+\dot{\gamma })-2(b^2/\pi )T_{13}(\ddot{\beta }+\ddot{\gamma })\right)\\ & +2\pi \rho V{b}^2{s}_p(a+0.5)C(k)\left(V\alpha +\dot{h}+b(0.5-a)\dot{\alpha }+(V/\pi )T_{10}(\beta +\gamma )+(b/2\pi )T_{11}(\dot{\beta }+\dot{\gamma })\right),\end{array}$$

(14)

$$\begin{array}{c}{M}_{\beta }=-\rho b^2{s}_p\left(\left(-2T_9-T_1+T_4(a-0.5)\right)Vb\dot{\alpha }+2T_{13}{b}^2\ddot{\alpha }+(1/\pi )V^2(\beta +\gamma )(T_5-T_4{T}_{10})\right.\hfill \\ \left.-(1/2\pi )Vb(\dot{\beta }+\dot{\gamma })T_4{T}_{11}-(1/\pi )T_{3}b(\ddot{\beta }+\ddot{\gamma })-T_{1}b\ddot{h}\right)\hfill \\ -\rho V{b}^2{s}_p{T}_{12}C(k)\left(V\alpha +\dot{h}+b(0.5-a)\dot{\alpha }+(1/\pi )T_{10}V(\beta +\gamma )+b(1/2\pi )T_{11}(\dot{\beta }+\dot{\gamma })\right),\hfill \end{array}$$

(15)

where T_i (i = 1, 2, …, 14) are the Theodorsen constants; see Appendix B [35]. $M_{\gamma }$ is derived from these by Throdorsen [33,34,35].

The aerodynamic force and moments in Equations (13)–(15) are dependent on the Theodorsen function C(k), where k is the non-dimensional reduced frequency of harmonic oscillation; C(k) can be approximated by Jones’s method,

$$C(s)=1-{\displaystyle \frac{0.165s}{s+0.0455V/b}}-{\displaystyle \frac{0.335s}{s+0.3V/b}}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{c_{1}V/b}{s+c_{3}V/b}}+{\displaystyle \frac{c_{2}V/b}{s+c_{4}V/b}},$$

(16)

where c₁ = 0.0075, c₂ = 0.10055, c₃ = 0.0455, and c₄ = 0.3. C(s) can be presented as a state-space model as follows:

$$\left\{\begin{array}{l}\left[\begin{array}{c}{\dot{x}}_{a1}\\ {\dot{x}}_{a2}\end{array}\right]={\displaystyle \frac{V}{b}}\left[\begin{array}{cc}-c_3& 0\\ 0& -c_4\end{array}\right]\left[\begin{array}{c}{x}_{a1}\\ x_{a2}\end{array}\right]+\left[\begin{array}{c}1\\ 1\end{array}\right]r\\ y_a={\displaystyle \frac{V}{b}}\left(\begin{array}{cc}{c}_1& c_2\end{array}\right)\left[\begin{array}{c}{x}_{a1}\\ x_{a2}\end{array}\right]+0.5r\end{array}\right.,$$

(17)

where

$$r=V\alpha +\dot{h}+b(0.5-a)\dot{\alpha }+(V/\pi )T_{10}\beta +b(1/2\pi )\dot{\beta }.$$

(18)

The aerodynamic force and moments can be described as the sum of non-circulatory and circulatory loads,

$$F_{ac}=F_{nc}+F_c,$$

(19)

where F_nc and F_c are the non-circulatory and circulatory loads respectively. We denote ${\overline{x}}_1={\left[\begin{array}{cccc}h& \alpha & \beta & \gamma \end{array}\right]}^T$ and x_a = [x_a1, x_a2]^T, and then we obtain

$$F_{nc}=-M_{nc}{\ddot{\overline{x}}}_1-V{C}_{nc}{\dot{\overline{x}}}_1-V^2{K}_{nc}{\overline{x}}_1,$$

(20)

$$F_c=-\rho Vb{s}_p{R}_c\left({\displaystyle \frac{V}{b}}\left(\begin{array}{cc}{c}_1& c_2\end{array}\right)x_a+0.5\left(S_1{\dot{\overline{x}}}_1+V{S}_2{\overline{x}}_1\right)\right),$$

(21)

where x_α is the non-dimensional distance from the elastic axis to the center of mass, and x_α = r_cg/b, where r_cg is the distance from the elastic axis to the center of mass, and

$$\begin{array}{c}{M}_{nc}=\rho b^2{s}_p\left[\begin{array}{cccc}\pi & -\pi ba& -b{T}_1& -b{T}_1\\ -\pi b\overline{a}& \pi b^2(1/8+a^2)& 2b^2{T}_{13}& -b^2{T}_{13d}\\ -b{T}_1& 2b^2{T}_{13}& -T_3{b}^2/\pi & 0\\ -b{T}_1& 2b^2{T}_{13d}& 0& -T_3{b}^2/\pi \end{array}\right],\hfill \\ K_{nc}=\rho b^2{s}_p\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& T_{15}& T_{15}\\ 0& 0& T_{18}/\pi & 0\\ 0& 0& 0& T_{18}/\pi \end{array}\right],\hfill \\ R_c={\left[\begin{array}{cccc}2\pi & -2\pi b(0.5+a)& T_{12}& T_{12}\end{array}\right]}^T,S_1=\left[\begin{array}{cccc}1& b(0.5-a)& b{T}_{11}/(2\pi )& b{T}_{11}/(2\pi )\end{array}\right],\hfill \\ S_2=\left[\begin{array}{cccc}0& 1& {\mathrm{T}}_{10}/\pi & {\mathrm{T}}_{10}/\pi \end{array}\right],\hfill \end{array}$$

(22)

where $T_{13}$ and $T_{13d}$ are the Theodorsen constants for the TE and LE. respectively. Substituting Equations (17) and (13)–(15) into Equation (11) then yields

$$\left(M_s+M_{nc}\right){\ddot{\overline{\mathit{x}}}}_1+\left(C_s+V{C}_{nc}+V{C}_c\right){\dot{\overline{\mathit{x}}}}_1+\left(K_s+V^2{K}_{nc}+V^2{K}_c\right){\overline{\mathit{x}}}_1=-V^2{E}_c{x}_a+G_s{u}_c$$

(23)

where

$$\begin{array}{c}{C}_{nc}=\rho b^2{s}_p\left[\begin{array}{cccc}0& \pi & -T_4& T_4\\ 0& \pi b(0.5-\overline{a})& b{T}_{16}& b{T}_{16}\\ 0& b{T}_{17}& b{T}_{19}/\pi & 0\\ 0& b{T}_{17}& 0& b{T}_{19}/\pi \end{array}\right],C_s=\left[\begin{array}{cccc}{c}_h& 0& 0& 0\\ 0& c_{\alpha }& 0& 0\\ 0& 0& c_{\beta \mathrm{s}}& 0\\ 0& 0& 0& c_{\gamma \mathrm{s}}\end{array}\right],\hfill \\ K_s=\left[\begin{array}{cccc}{k}_h& 0& 0& 0\\ 0& k_{\alpha }& 0& 0\\ 0& 0& k_{\beta s}& 0\\ 0& 0& 0& k_{\gamma s}\end{array}\right],G_s=\left[\begin{array}{cc}0& 0\\ 0& 0\\ k_{\beta s}& 0\\ 0& k_{\gamma s}\end{array}\right],M_s=\left[\begin{array}{cccc}\overline{m}& {\xi }_1& m_{\beta }{r}_{\beta }& m_{\gamma }{r}_{\gamma }\\ \ast & {\xi }_2& I_{\beta }+x_{\beta }{m}_{\beta }{r}_{\beta }& I_{\gamma }+x_{\gamma }{m}_{\gamma }{r}_{\gamma }\\ \ast & \ast & I_{\beta }& 0\\ \ast & \ast & 0& I_{\gamma }\end{array}\right]\hfill \\ C_c={\displaystyle \frac{1}{2}}\rho b{s}_p{R}_c{S}_1,K_c={\displaystyle \frac{1}{2}}\rho b{s}_p{R}_c{S}_2,E_c=\rho s_p{R}_c\left[\begin{array}{cc}{c}_1& c_2\end{array}\right],u_{\mathrm{c}}={\left[{\mathrm{\beta }}_c,{\mathrm{\gamma }}_c\right]}^{\mathrm{T}},\hfill \end{array}$$

(24)

The full state vector is denoted as $\overline{\mathit{x}}={\left[\begin{array}{ccc}{{\overline{\mathit{x}}}_1}^T& {{\dot{\overline{\mathit{x}}}}_1}^T& x_a^T\end{array}\right]}^T$, and the overall state-space model is

$$\dot{\overline{\mathit{x}}}=\mathit{A}\overline{\mathit{x}}+\mathit{B}u,$$

(25)

where the A and B matrices are as follows:

$$\mathit{A}=\left[\begin{array}{ccc}0& I& 0\\ -M^{-1}K(V)& -M^{-1}C(V)& -V^2{M}^{-1}{E}_c\\ V{K}_a& C_a& V{A}_a\end{array}\right],\mathit{B}=\left[\begin{array}{c}0\\ M^{-1}{G}_s\\ 0\end{array}\right],$$

(26)

where

$$\begin{array}{c}M=M_{nc}+M_s,K(V)=K_s+V^2{K}_{nc}+V^2{K}_c,C(V)=C_s+V{C}_{nc}+V{C}_c,\hfill \\ K_a=\left[\begin{array}{c}1\\ 1\end{array}\right]S_2,C_a=\left[\begin{array}{c}1\\ 1\end{array}\right]S_1,A_a={\displaystyle \frac{1}{b}}\left[\begin{array}{cc}-c_3& 0\\ 0& -c_4\end{array}\right].\hfill \end{array}$$

(27)

Since there exists wind disturbance for the aeroelastic system, the external disturbance D is considered and added into the system of Equations (6) and (25) as

$$\left\{\begin{array}{cc}{\dot{\mathit{x}}}_2(t)={\mathit{f}}_2(\mathit{x})+{\mathit{g}}_2(\mathit{x})U(t)+\mathit{D}(t),\hfill & \hfill \left(\mathrm{Quasi}\text{-}\mathrm{steady} \mathrm{load} \mathrm{case}\right)\\ {\ddot{\overline{\mathit{x}}}}_1(t)=-M^{-1}\left(K(V){\overline{\mathit{x}}}_1+C(V){\dot{\overline{\mathit{x}}}}_1+V^2{E}_c{x}_a\right)+M^{-1}{G}_{s}u+\overline{\mathit{D}}(t),\hfill & \hfill \left(\mathrm{Unsteady} \mathrm{load} \mathrm{case}\right)\end{array}\right.$$

(28)

where $\mathit{D}\in R^2$ and $\overline{\mathit{D}}(t)\in R^4$ denote external disturbances and are bounded by $‖D‖<D_U$, where D_U is a real value meeting D_U ≥ 0, ${\overline{\mathit{f}}}_2({\overline{\mathit{x}}}_1)=-M^{-1}\left(K(V){\overline{\mathit{x}}}_1+C(V){\dot{\overline{\mathit{x}}}}_1+V^2{E}_c{x}_a\right)$, and ${\overline{\mathit{g}}}_2({\overline{\mathit{x}}}_1)=M^{-1}{G}_s$. The aerodynamic loads due to the wind disturbances are defined as [36]

$$L_g=\rho V^{2}b{s}_p{C}_{L\alpha }{\omega }_G(\tau )/V=\rho Vb{s}_p{C}_{L\alpha }{\omega }_G(\tau ),$$

(29)

$$M_g=(0.5+a)b{L}_g,$$

(30)

where ${\omega }_G(\tau )$ denotes the disturbance velocity and τ is a dimensionless time variable defined as $\tau =Vt/b$, so $\mathit{D}={\left[\begin{array}{cc}-L_g/m_T& M_g/I_{\alpha }\end{array}\right]}^T$. Suppose that the above disturbance can be generated by a linear exogenous system [31],

$$\left\{\begin{array}{l}{\dot{\varsigma }}_d(t)=W_d{\varsigma }_d(t)\\ D(t)=V_d{\varsigma }_d(t)\end{array}\right.,$$

(31)

where ${\varsigma }_d\in R^p$, W_d, and 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 towards the origin.

3. DO-BTSMC Aeroelastic Control Design

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

3.1. Nonlinear Disturbance Observer Design

Considering unknown disturbances in the aeroelastic system (28), a nonlinear disturbance observer (NDO) is proposed to improve the tracking precision [31],

$$\left\{\begin{array}{l}{\dot{z}}_d(t)=(W_d-l(x)V_d)z_d(t)+W_d{p}_d(x)-l(x)\left(V_d{p}_d(x)+f_2(x)+g_{2}U(t)\right)\\ {\widehat{\varsigma }}_d(t)=z_d(t)+p_d(x)\\ \widehat{D}(t)=V_d{\widehat{\varsigma }}_d(t)\end{array}\right.,$$

(32)

where $\widehat{D}$ is the estimated disturbance, l (x) is the vector of observer gains to be tuned for performance, z_d is the internal state vector, and p_d(x) is a nonlinear function to be designed. The NDO gain l(x) is determined by

$$l(x)={\displaystyle \frac{\partial p_d(x)}{\partial x}},$$

(33)

Integrating l(x₂) with respect to the aeroelastic system state x yields

$$p_d(x)=L_{d}Li{e}_f^{r-1}\hslash (x)$$

(34)

where L_d is a gain vector, Lie_f denotes a Lie derivative of the function f(x), and r is the relative degree. We select $\hslash (x_2)=x_2$. The nonlinear disturbance estimation error is $d=\mathit{D}-\widehat{D}$.

It has been shown that the estimation $\widehat{D}$ of the NDO approaches the disturbance D(t) exponentially if the observer gain l(x) is chosen such that (32) is globally exponentially stable for all $x\in R^n$ [37]. The estimation of harmonic disturbance, $\widehat{D}$, yielded by the DO of Equation (32) approaches the disturbance, D, and is globally exponentially stable if the following inequalities are satisfied [31]:

$$V={\displaystyle \frac{1}{2{\gamma }_d}}{d}^{T}d>0,\dot{V}<0$$

(35)

Hence, the disturbance compensator can be designed as

$$u_d(t)=-g_2^{-1}\widehat{D}(t)$$

(36)

When the nonlinear disturbance observer is used, $\widehat{D}$ will approach D, so the disturbance can be attenuated through the disturbance compensator (36), and the system (28) can be transformed as follows:

$$\begin{gathered} {\dot{x}}_2(t)=f_2(x_1,x_2)+g_{2}U(t)+\mathit{D} \\ =f_2(x_1,x_2)+g_2\left(U_{BTSMC}(t)+u_d(t)\right)+\mathit{D}(t) \\ =f_2(x)+g_2(x)U_{BTSMC}(t)+d, \end{gathered}$$

(37)

The observable disturbances, such as 1-cosine winds, usually can be measured and estimated by the observer. It has been shown that the estimation $\widehat{\mathit{D}}(t)$ of the NDO approaches the disturbance D(t) exponentially if the error dynamics for d(t) is globally exponentially stable [31]. By a similar method, the disturbances $\overline{\mathit{D}}(t)$ for unsteady loads can also be estimated.

3.2. DO-BTSMC-Based Aeroelastic Control Design

To drive the aeroelastic output x to converge to the desired value vector x_d, the aeroelastic controller is designed as follows:

The tracking error vector of the plunge displacement and pitch angle is defined as

$${\mathit{z}}_1={\mathit{x}}_1-{\mathit{x}}_{1d},$$

(38)

and its derivative is

$${\dot{\mathit{z}}}_1={\dot{\mathit{x}}}_1-{\dot{\mathit{x}}}_{1d}={\mathit{x}}_2-{\dot{\mathit{x}}}_{1d}$$

(39)

The stabilizing function ${\varphi }_1(x)$ is defined as the virtual speed vector of ${\dot{\mathit{x}}}_{1d}$:

$${\varphi }_1(x_1)={\dot{\mathit{x}}}_{1d}-K_1{\mathit{z}}_1,$$

(40)

where ${\dot{\mathit{x}}}_{1d}$ represents the reference or virtual speed vector, and K₁ > 0 is often chosen as a diagonal matrix to simplify the design; i.e., K₁ =diag(k₁₁, k₁₂), where k_1i (i = 1, 2) is a constant value.

The Lyapunov function V₁ is constucted as follows:

$$V_1={\displaystyle \frac{1}{2}}{\mathit{z}}_1^T{\mathit{z}}_1.$$

(41)

The speed tracking error vector for the plunge and pitch dynamics is defined as

$${\mathit{z}}_2={\dot{\mathit{x}}}_1-{\varphi }_1(x_1)={\mathit{x}}_2-{\varphi }_1(x_1).$$

(42)

and its derivative is

$${\dot{\mathit{z}}}_2={\ddot{\mathit{x}}}_1-{\ddot{\mathit{x}}}_{1d}+K_1{\dot{\mathit{z}}}_1.$$

(43)

The detailed derivation is given in the Appendix A.

Remark 1. 

If there are uncertainties for the system (28), then f₂ and g₂ will include uncertain terms; the uncertain terms can be regarded as disturbances and added into D, and then the observer (32) can be used with nominal f₂ and g_2.

Therefore, the structure of the DO-BTSMC controller for aeroelastic control is designed as shown in Figure 2, which includes the aeroelastic dynamics system of the two-dimensional airfoil, an adaptive BTSMC used for aeroelastic flutter suppression, a nonlinear DO used to estimate observable disturbances, and an adaptive disturbance estimator used to observe the unobservable disturbances. A learning rate is applied to speed up the convergence of the proposed sliding-mode control.

4. Discussion

The studied aeroelastic system is the NATA model, and the physical parameters of the aeroelastic system are listed in

Table 1. Parameters of the two-dimensional wing.

Quasi-Steady Aerodynamic Model		Unsteady Aerodynamic Model
Parameter	Value	Parameter	Value
mwing	4.34 kg	mh	6.815 kg
mW(total)	5.23 kg	mα	5.715 kg
mT	15.57 kg	mβ	0.537 kg
sp	0.5945 m	mγ	0.5 kg
b	0.1905 m	Iα	0.119 kg·m2
a	−0.6719	Iβ	1 × 10−5 kg·m2
c	0.381 m	Iγ	1 × 10−5 kg·m2
ICG.wing	0.04342 kg·m2	rα	0.040 m
Icam	0.04697 kg·m2	rβ	0
ch	27.43 kg/s	rγ	0
cα	0.0360 kg·m2/s	rac	−0.033 m
ρ	1.225 kg/m3	r3/4c	0.233 m
rcg	−b(0.0998+a)	Lβ	0.233 m
CLα	6.757	Lγ	-0.01 m
CLβ	3.774	kβs	766.08 Iβ
Cmα	0	cβs	41.82 Iβ
Cmβ	−0.6719	kγs	530.24 Iγ
CLγ	−0.1566	cγs	44.27 Iγ
Cmγ	−0.1005	cα	0.205 Nm·s/rad
kh	2844 N/m	ch	27.43 Nm·s/rad
xα	rcg/b m	ωnβs	27.68 rad/s
fLE	0.15	ζβs	0.7555
fTE	0.2	ωnγs	23.03 rad/s
Iα	Icam + ICG.wing + mwing(rcg)2 kg⸱m2	ζγs kh	0.961 2844 N/m
kα	12.77 + 53.47α + 1003α2Nm/rad	kα	27.96 − 167.63α + 552.55α2 + 1589.3α3 − 3247.2α4

[19,20,38].

4.1. Flutter Suppression

In this section, the flutter suppression of the airfoil is studied. For the two-dimensional airfoil, the polynomial model of the stiffness is found by static measurements on the nonlinear pitch cam, k_α(α) = 12.77 + 53.47α + 1003α² Nm/rad [19]. When the airspeed is increased to 11.4m/s, the LCO phenomenon appears. The output responses are shown in Figure 3. To suppress this LCO and flutter vibrations, different scenarios are studied. First, the proposed BTSMC is used to suppress this flutter. The initial conditions of the two-dimensional aeroelastic airfoil are as follows: h(0)= −0.01 m, α(0) = 0.1 rad, $\dot{h}(0)=-0.1$ m/s, $\dot{\alpha }(0)=-2$ rad/s, and the nominal airspeed V = 11.4m/s. The parameters of the BTSMC are designed after several design iterations as follows: K₁ = diag(0.2,0.1), h_s = diag(0.5,0.5), ς = diag(30,45), φ_s = 0.4, β_s = 1, p = 5, q = 3, β = diag(1,1), α₀ = 10, β₁ = 3, and λ = 1.8diag(1,1). The simulation results refer to Figure 4.

It can be seen from Figure 4 that the dynamic responses of the proposed nonlinear aeroelastic system flutter under the initial input during the first 2 s, and the resulting dynamic responses under the BTSMC are stable after 2 s. The FFT results of plunge displacements and AOA with and without BTSMC control are shown in Figure 3 and Figure 4, the vibration frequency is 2.7 Hz (see Figure 3), and the amplitude of h_FF changes from 1.5 × 10⁻³ to 1.25 × 10⁻⁴, while the amplitude of α_FFT changes from 0.1 to 7.4 × 10⁻³, which shows that the flutter is suppressed by the proposed BTSMC.

For high stiffness nonlinearity in pitch, k_α(α) can be modeled as the following polynomial type [20]:

k_α = k_α0 + k_α1α + k_α2α² + k_α3α³ + k_α4α⁴     

      = 27.96 − 167.63α + 552.55α² + 1589.3α³ − 3247.2α⁴

The flutter speed is related to its stiffness; the flutter equation for the p-k method can be written as follows using Equation (23) [39]:

$$\left[\left({\displaystyle \frac{V}{L_{ref}}}\right)\mathrm{M}{p}^2+\mathrm{K}-{\displaystyle \frac{1}{2}}\rho V^2\left({\mathrm{Q}}^R(M,k)+{\displaystyle \frac{p}{k}}{\mathrm{Q}}^I(M,k)\right)\right]\cdot \{\mathrm{q}\}=0.$$

(44)

where M and K are the generalized mass and stiffness matrices, respectively; Q^R and Q^I are the real and imaginary parts of the generalized aerodynamic force matrix that depends on Mach number M and reduced frequency k. V is the velocity of the undisturbed flow, and L_ref is the reference length. q denotes the eigenvector of the flutter equation; p = g_s + iω_s is the non-dimensional Laplace complex parameter and is also referred to as an eigenvalue of Equation (44). The imaginary part of p must meet the following consistency condition: $k=\left({\displaystyle \frac{L_{ref}}{V}}\right)\cdot \mathrm{Im}(p)$, i.e., $k=\left({\displaystyle \frac{L_{ref}}{V}}\right)\cdot \omega$, and $\omega =\sqrt{{\displaystyle \frac{\mathrm{K}}{\mathrm{M}}}}$. When the damping curve {(V_s, g_s)} for the vibration mode intersects with g_s = 0, the associated speed is the flutter speed. Therefore, for a given k, when the stiffness K is increased, the frequency is increased, and the flutter speed V_s is increased. Here, the high stiffness is applied, and the flutter speed is increased from V = 11.4 m/s to V = 13.2 m/s. With this stiffness model, the simulation results are shown in Figure 5 and Figure 6.

It can be seen from Figure 5 and Figure 6 that the flutter speed is increased from V = 11.4 m/s to V = 13.2 m/s by the high stiffness nonlinearity, and the output responses of the plunge and pitch motions rapidly converge with the proposed BTSMC control.

Considering a high angle of attack (AOA) in the post-flutter behavior, a square wave command of 10deg is tracked, and the output responses are shown in Figure 7.

It can be seen from Figure 7 that the 10° AOA command has been accurately tracked by the proposed BTSMC, but the control surfaces saturate, resulting in small oscillations of the output responses.

4.2. Flutter Suppression Under Unknown Wind Disturbances

Case I: for quasi-steady aerodynamic loads

There are two kinds of gusts to be studied here, that is, a triangular-type gust and a sine gust, as follows [30,36]:

$$w_G(\tau )=\left\{\begin{array}{ll}2w_0{\displaystyle \frac{\tau }{{\tau }_G}}\left(H(\tau )-H(\tau -{\displaystyle \frac{{\tau }_G}{2}})\right)+2w_0\left({\displaystyle \frac{\tau }{{\tau }_G}}-1\right)\left(H(\tau -{\tau }_G)-H(\tau -{\displaystyle \frac{{\tau }_G}{2}})\right),\hfill & \hfill \left(\mathrm{triangular} \mathrm{gust}\right)\\ H(\tau )w_0\sin \left(0.5\tau \right),\hfill & \left(\sin \mathrm{e} \mathrm{gust}\right)\hfill \end{array}\right..$$

(45)

where H(·) denotes the Heaviside step function used to describe the typical velocity distribution time-history corresponding to the gust loads; this triangular gust lasts 0.5 s from t = 0 to t = 0.5 s, i.e., t_G = 0.05 s, w₀ = 0.3 m/s, ${\tau }_G=V{t}_G/b$.

Where w₀ = 0.3 m/s, and this sine gust lasts 120 s from t = 0 to t = 120 s.

Note that the triangular gust is very similar to the traditional 1-cosine gust-type function, both of which can be classified as ephemeral disturbances. Since gusts will result in a lift force and pitch moment change, which then drive the vertical speed and pitch rate change, for simplification, we regard them as external disturbances by

$$\mathit{D}(t)={\mathrm{\Gamma }}_w{w}_G(t)$$

(46)

where ${\mathrm{\Gamma }}_w$ is the gust effectiveness matrix, which can be approximated by Equations (29) and (30),

$${\mathrm{\Gamma }}_w={\left[\begin{array}{cc}-\rho {(V+D_w)}^{2}b{s}_p{C}_{L\alpha }/(V{m}_T)& \rho {(V+D_w)}^2{b}^2{s}_p{C}_{m\alpha }/(V{I}_{\alpha })\end{array}\right]}^T,$$

(47)

The disturbance parameters of (31) are

$$W_d=\left[\begin{array}{cc}0& 0.5\\ -0.5& 0\end{array}\right],V_d=\left[\begin{array}{cc}2& 0\end{array}\right],$$

(48)

The associated DO gains of (33) are designed as follows:

$$l(h,\dot{h})={\left[\begin{array}{cc}20& 6\end{array}\right]}^T,l(\alpha ,\dot{\alpha })={\left[\begin{array}{cc}5& 0.4\end{array}\right]}^T,\left(\mathrm{for}\mathrm{sine}\mathrm{gust}\right)$$

(49)

$$l(h,\dot{h})={\left[\begin{array}{cc}2.5& 0.17\end{array}\right]}^T,l(\alpha ,\dot{\alpha })={\left[\begin{array}{cc}5& 0.4\end{array}\right]}^T,\left(\mathrm{for}\mathrm{triangular}\mathrm{gust}\right)$$

(50)

Remark 2. 

In this paper, single (1-cos) gusts (i.e., triangular gusts) and continuous sinusoidal (harmonic) gusts are considered. It can be seen from Equations (49) and (50) that the observer gains of l (h,$\dot{h}$) are different for different kinds of gusts, which shows that the DO is adaptive for the gusts. Different kinds of gusts can be detected by short-range light detection and ranging (Lidar) wind measurement.

The parameter of the adaptive observer (Equation (A19)) is set as γ_d = 0.2. To illustrate our method in trajectory tracking control, a BSMC design from [40] is used for comparison. The sliding-mode surface is designed as

$$\mathit{s}={\mathit{\lambda }}_1{\mathit{z}}_1+{\mathit{z}}_2$$

(51)

and the BSMC controller is designed as follows:

$${\mathit{U}}_{BSMC}={(g_2)}^{-1}\left(-(K_1+{\lambda }_1){\dot{\mathit{z}}}_1-f_2+{\ddot{\mathit{x}}}_{1d}-\widehat{\mathit{d}}-\left(h\mathit{s}+\varsigma \mathrm{sgn}(\mathit{s})\right)\right)$$

(52)

A PID controller is applied as follows:

$${\mathit{U}}_{\mathrm{PID}}={(g_2)}^{-1}\left(-(K_{p,z_1}{\mathit{z}}_1+K_{I,z_1}{\displaystyle {\int }_0^t{\mathit{z}}_{1}dt})\right.\left.-(K_{p,z_2}{\mathit{z}}_2+K_{I,z_2}{\displaystyle {\int }_0^t{\mathit{z}}_{2}dt})-f_2+{\ddot{\mathit{x}}}_{1d}-\widehat{\mathit{d}}\right)$$

(53)

The parameters of the studied controllers are designed as shown in

Table 2. Selected parameters of the controller (quasi-steady case).

Controller	Parameters	Value
PID	kp,z1, ki,z1, kp,z2, ki,z2	diag(0.1,0.1), diag(0.1,0.1), diag(1,1), diag(0.4,0.4)
BTSMC		The same as Section 4.1
BSMC	K1, λ1	diag(0.4,0.2), diag(0.5,0.5)
	hs, ς	diag(0.5,0.5), diag(50, 30)
	φs, βs	0.4, 1

after several design iterations.

The simulation results using the proposed DO-BTSMC, BSMC, and PID design are shown in Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 and

Table 3. Performance data comparison under triangular and sine gusts.

Gust Input	Controller	Data	Rising Time (s)	Over- Shoots	RMSE
Triangular gust (Case I, Example 1)	No LR (with DO)	h	0.1770	0.0134	0.0023
		α	–	–	0.0129
	LR (with DO)	h	0.1187	0.0136	0.0022
		α	–	–	0.0120
Sine gust (Case I, Example 2)	No DO (No LR)	h	0.1084	0.0104	0.0027
		α	–	–	0.0160
	DO (No LR)	h	0.1205	0.0124	0.0018
		α	–	–	0.0110

‘–’ denotes a non-statistical variable in Table 3.

and

Table 4. Performance data comparison under triangular gust (quasi-steady aerodynamic loads).

Controller	Variant	Rising Time (s)	Over- Shoots	RMSE
DO-PID	h	0.1467	0.0067	0.0026
	α	-	-	0.0345
DO- BTSMC	h	0.1187	0.0136	0.0021
	α	-	-	0.0120
DO- BSMC	h	0.1215	0.0118	0.0022
	α	-	-	0.0153

.

Example 1. 

Firstly, we consider different cases of the BTSMC design with and without the learning rate α_r under the triangular gust of (45), and the DO is applied; the output responses are shown in Figure 8 and Figure 9 and Table 3, where the reference inputs of root mean square error (RMSE) are h_cmd = 0 and α_cmd = 0.

Example 2. 

Secondly, we consider the case of the BTSMC design with and without a DO under the sine gust of (45), and there is no learning rate; the output responses are shown in Figure 10 and Figure 11 and Table 3.

Considering a triangular gust input, it can be seen from Figure 8 and Figure 9 and Table 3 that the resulting dynamic responses of the proposed nonlinear aeroelastic system are stable, and the rise time of plunge responses for BTSMC with a learning rate and DO is shorter than that of the BTSMC without a learning rate but with a DO, which shows the tracking speed of the BTSMC with a learning rate for the plunge motions is faster than that of the BTSMC without a learning rate, but the LR can produce small overshoots; see Figure 8. The root mean square errors of the plunge and pitch motion for the BTSMC with an LR and a DO are smaller than those for the BTSMC without an LR. This shows the learning rate of the BTSMC can improve the control performance.

Now considering a sine gust input, Figure 10 and Figure 11 and Table 3 show that the resulting dynamic responses of the proposed nonlinear aeroelastic system are stable, and the steady-state errors of the BTSMC with a DO for the plunge and pitch motions are less than those of the BTSMC without a DO. The associated control inputs for the LE control surface are smaller than for the BTSMC without a DO. This shows the disturbance observer of (32) and (A19) can effectively estimate an unknown time-varying gust, and the disturbance compensator reduces the disturbance effect for the aeroelastic system.

Finally, we consider the different flutter suppression approaches of the BTSMC, BSMC, and PID under the triangular gust of Equation (45). The output responses obtained by using the proposed design with the DO of (32) and an adaptive observer (Equation (A19)) are shown in Table 4 and Figure 12, Figure 13 and Figure 14. The sign function of the SMC design is replaced by the tangent hyperbolic function with a boundary layer (where φ_s = 0.4).

It can be seen from Figure 12 and Table 4 that the resulting dynamic responses of the proposed nonlinear aeroelastic system are convergent to the desired values, and the rise time of plunge responses for DO-BTSMC is shorter than that of the DO-BSMC and DO-PID control, which shows that the BTSMC response is the fastest among them. The output responses of the PID control easily oscillate under triangular gusts while the responses of the BTSMC and BSMC design are steady after the transition process, which shows the robust performances of the BTSMC and BSMC. The root mean square errors of the plunge and pitch motion for the DO-BTSMC are the smallest, while those for the PID control are the highest, and RMSEs for the DO-BSMC are between them, showing that the tracking precision is the highest for the DO-BTSMC, second highest for the DO-BSMC, and worst for the PID control. Figure 10 shows that the triangular gust of (45) can be effectively estimated and that the unobservable disturbances including uncertainties are estimated by the adaptive observer (Equation (A19)) (see Figure 14), which reduces the effect of a triangular gust on the aeroelastic system.

Case II: for unsteady aerodynamic load.

As the post-flutter behavior of the airfoil is intrinsically unsteady, the unsteady aerodynamic loads are calculated by Equations (10)–(27), and the parameters of the studied controllers are designed as shown in

Table 5. Selected parameters of the controller (unsteady case).

Controller	Parameters	Value
PID	kp,z1, ki,z1, kp,z2, ki,z2	diag(0.1,0.1,0.1,0.1), diag(0.1,0.1,0.1,0.1), diag(1,1,1,1), diag(0.4,0.4, 0.4, 0.4)
BTSMC	K1, λ1	K1 = diag(0.2,0.1,0.1,0.1), λ = 1.8 × diag(1,1,1,1).
	p, q	5, 3
	β, α0, β1	β = diag(1,1,1,1), α0 = 10, β1 = 3,
	hs, ς	hs = diag(0.5,0.5,0.5,0.5), ς = diag(30,50, 30, 30),
	φs, βs	φs = 0.4, βs = 1
BSMC	K1, λ1	diag(0.2,0.1,0.1,0.1), diag(0.5,0.5,0.5,0.5)
	hs, ς	diag(0.5,0.5,0.5,0.5), diag(30,50, 30, 30)
	φs, βs	0.4, 1

after several design iterations.

The associated DO gains of Equation (33) are designed as follows:

$$\begin{array}{c}l(h,\dot{h})={\left[\begin{array}{cc}25& 0.25\end{array}\right]}^T,l(\alpha ,\dot{\alpha })={\left[\begin{array}{cc}50& 0.4\end{array}\right]}^T,l(\beta ,\dot{\beta })={\left[\begin{array}{cc}0.5& 0.04\end{array}\right]}^T,l(\gamma ,\dot{\gamma })={\left[\begin{array}{cc}0.5& 0.04\end{array}\right]}^T,\hfill \\ \left(\mathrm{for}\mathrm{sine}\mathrm{gust}\right)\hfill \end{array}$$

(54)

$$\begin{array}{c}l(h,\dot{h})={\left[\begin{array}{cc}25& 0.25\end{array}\right]}^T,l(\alpha ,\dot{\alpha })={\left[\begin{array}{cc}50& 0.4\end{array}\right]}^T,l(\beta ,\dot{\beta })={\left[\begin{array}{cc}5& 0.4\end{array}\right]}^T,l(\gamma ,\dot{\gamma })={\left[\begin{array}{cc}5& 0.4\end{array}\right]}^T,\hfill \\ \left(\mathrm{for}\mathrm{triangular}\mathrm{gust}\right)\hfill \end{array}$$

(55)

Example 3. 

Firstly, we consider the different cases for the BTSMC design with and without the learning rate α_r under the triangular gust of (45), and the external disturbances given by Equation (46), where  ${\mathrm{\Gamma }}_w$ is the gust effectiveness matrix, which can be approximated as

$${\mathrm{\Gamma }}_w={\left[\begin{array}{cccc}-\rho {(V+D_w)}^{2}b{C}_{L\alpha }/(V{m}_T)& M_D& M_D/2& M_D/2\end{array}\right]}^T$$

(56)

 where  $M_D=\rho {(V+D_w)}^2{b}^2{s}_p{C}_{m\alpha }/(V{I}_{\alpha })$, the disturbance parameters are the same as those in Example 1 with w₀ = 3 m/s, and the DO is applied; the output responses are shown in Figure 15, Figure 16 and Figure 17.

The control inputs are shown in Figure 18.

The wind disturbances are estimated by the proposed DO (32) and are shown in Figure 19.

Example 4. 

Secondly, we consider the case of the BTSMC design with and without a DO under the sine gust of (45); the disturbance parameters are the same as those for Example 2, and there is no learning rate. The output responses are shown in Figure 20, Figure 21, Figure 22 and Figure 23 and in

Table 6. Performance data comparison under triangular and sine gusts with unsteady aerodynamic loads.

Gust Input	Controller	Data	Rising Time (s)	Over- Shoots	RMSE
Triangular gust (Case II, Example 3)	No LR (with DO)	h	0.2438	0.0010	0.0022
		α	-	0.1758	0.0285
	LR (with DO)	h	0.2278	0.0015	0.0020
		α	-	0.1782	0.0281
Sine gust (Case II, Example 4)	No DO (No LR)	h	0.1704	0.0045	0.0019
		α	-	0.2312	0.0267
	DO (No LR)	h	0.1447	0.0025	0.0013
		α	-	0.2144	0.0218

.

The control inputs refer to Figure 23.

Considering the triangular gust input, it can be seen from Figure 15, Figure 16 and Figure 17 and Table 6 that the resulting dynamic responses of the proposed nonlinear aeroelastic system are stable, and the rise time of plunge responses for BTSMC with LR and DO is shorter than that of for the BTSMC without an LR but with a DO, which shows the tracking speed of the BTSMC with a learning rate and DO for the plunge motions is faster than that of the BTSMC without a learning rate, but the LR can produce a small overshoot; see Figure 15, Figure 16 and Figure 17. The RMSE of the plunge and pitch motion for the BTSMC with a learning rate and DO is smaller than that of the BTSMC without an LR. This shows the learning rate of the BTSMC can improve the control performances.

Now consider the sine gust input. Figure 20, Figure 21, Figure 22 and Figure 23 and Table 3 show that the resulting dynamic responses of the proposed nonlinear aeroelastic system are stable, and the steady-state errors of the BTSMC with a DO for the plunge and pitch motions are less than those of the BTSMC without a DO, and the associated control inputs are smaller than those of the BTSMC without a DO. This shows the disturbance observer of (32) and (A19) can effectively estimate an unknown time-varying gust and that the disturbance compensator reduces the disturbance effect for the aeroelastic system. Figure 19 shows that the external disturbances can be estimated quickly and accurately.

4.3. Robust Flutter Suppression with Parameter Uncertainties and Disturbances

Case I: for quasi-steady aerodynamic loads

Example 5. 

Consider the effect of parameter uncertainties on the aeroelastic system. The parameter uncertainties of k_h, k_α, and c_h are set to be ±1%, and the uncertainties of c_α and C_Lα are set to be ±30% and ±7%, respectively. The external disturbance is set as the sine gust of (45). The controller parameters are designed and listed in Table 2, and the sign function of the SMC design is replaced by a continuous approximation of the sign function$sgn\left(s\right)\approx s/\sqrt{s^{T}s+{\beta }_s}$ with β_s = 1. Since the DO-BTSMC can deal with model uncertainties through adaptive disturbance estimation and gain regulation, we use the proposed DO-BTSMC design, and the simulation results are shown in

Table 7. Performance data comparison under parameter uncertainty and sine gust (quasi-steady case).

Controller	Variant	Rising Time (s)	Over- Shoots	RMSE
DO-PID	h	0.1481	0.0077	0.0025
	α	-	-	0.0447
DO- BTSMC	h	0.1205	0.0124	0.0018
	α	-	-	0.0110
DO- BSMC	h	0.1275	0.0045	0.0014
	α	-	-	0.0173

and Figure 24, Figure 25, Figure 26 and Figure 27.

It can be seen from Figure 24 and Table 7 that the resulting dynamic responses of the proposed nonlinear aeroelastic system are convergent, and the rise time of plunge responses for DO-BTSMC is the shortest. This shows that the BTSMC response is the fastest among them and demonstrates the DO-BTSMC has a fast transient response and finite time-convergence. But the BTSMC produces larger overshoots than the BSMC and PID designs. This is because larger inputs are required for finite-time convergence during the transition process for the DO-BTSMC. The RMSEs of the plunge and pitch motion show that the tracking precision of the DO-BTSMC is the best, followed by DO-BSMC and then the PID control. Meanwhile, the output responses of the PID control are vibrating under the sine gust, which shows that the robust performance of the PID controller is poorer than that of the BTSMC and BSMC. Figure 25 shows that the sine gust of Equation (45) can be effectively estimated, and the unobservable disturbances are also estimated by the adaptive observer of (Equation (A19)) as shown in Figure 26, which reduces the effect of the sine gust on the aeroelastic system. Figure 27 shows that the control surface deflections of the DO-BTSMC are the smallest and fastest, but the control inputs of the DO-PID control are the largest and easily cause control surface saturation; this also demonstrates the robust performances of the BTSMC.

Case II: for unsteady aerodynamic load

Example 6. 

For unsteady aerodynamic loads with disturbances and model parameter uncertainties, the parameters of disturbances and uncertainties are the same as those for Example 5, and we use the proposed DO-BTSMC design. The simulation results are shown in

Table 8. Performance data comparison under parameter uncertainty and sine gust (unsteady case).

Controller	Variant	Rising Time (s)	Over- Shoots	RMSE
DO-PID	h	0.1916	0.0024	0.0017
	α	-	0.2137	0.0199
DO- BTSMC	h	0.1415	0.0025	0.0010
	α	-	0.2147	0.0183
DO- BSMC	h	1.3885	0.0004	0.0010
	α	-	0.1645	0.0151

and Figure 28, Figure 29, Figure 30, Figure 31, Figure 32 and Figure 33.

It can be seen from Figure 28, Figure 29 and Figure 30 and Table 8 that the resulting dynamic responses of the proposed nonlinear aeroelastic system are convergent, and the rise time of plunge responses for DO-BTSMC is the shortest, which shows the BTSMC response is the fastest and shows that the DO-BTSMC has a fast transient response and finite-time convergence. But the BTSMC produces larger overshoots than the BSMC and the PID design. This is because a larger input is required for finite-time convergence during the transition process for the DO-BTSMC. The root mean square errors of the plunge and pitch motion for the DO-BTSMC are 0.0010 m and 0.0183 rad, respectively, and those for the DO-BSMC are 0.0010 m and 0.0151 rad respectively, while those for the PID control are 0.0017 m and 0.0199 rad, respectively, which shows that the tracking precision of the DO-BSMC is the best, followed by the DO-BTSMC and then the PID control. Although the tracking precision of the DO-BSMC is better than that of the DO-BTSMC, the rise time of the plunge motion response for DO-BSMC is very slow, which shows its vulnerability to damping.

The disturbance can be estimated by the proposed DO Equations (32) and (A19); the outputs are shown in Figure 31 and Figure 32.

The control inputs refer to Figure 33.

Figure 31 shows that the sine gust of (45) can be effectively estimated and the unobservable disturbances are also estimated by the adaptive observer of (A19) as shown in Figure 32, which reduces the effect of the sine gust on the aeroelastic system. Figure 33 shows that the control surface deflections of DO-BTSMC are the smallest and fastest. In addition, the control inputs of the DO-BSMC control are the largest and easily cause control surface saturation; this also demonstrates the DO-BSMC has small damping and reduced tracking performance.

5. Conclusions

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 with finite-time convergence. Stability analysis shows that the closed-loop aeroelastic dynamics are globally asymptotically stable. Six examples with 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 a learning rate achieves better flutter suppression performances with a fast response (improved by 18.6%) and reduced root mean square error (by 75% for α). Therefore, the effectiveness and availability of the DO-BTSMC design are demonstrated.