This paper presents the dynamics and adaptive control of an airfoil with structural stiffness and damping uncertainties, which is subjected to atmospheric gusts. The motion of the airfoil is modeled by three degrees of freedom (DOFs), namely, pitch, plunge, and flap. A flat spot or dead-zone-type stiffness is used for modeling the flap hinge free play. The nonlinear dynamics of the system without control and parametric uncertainty, where a cubic stiffness for pitch and a linear stiffness for plunge are considered, is examined. Numerical results show that the airfoil may become unstable via a Hopf bifurcation at a flow velocity well below the linear flutter speed; if the structural damping is not sufficiently high, it may also undergo chaotic motion. It was found that a proportional–derivative controller based on the partial feedback linearized system could effectively alleviate oscillations induced by gusts at flow velocities below and above the linear flutter speed. Next, an uncertain $N$ th-order polynomial stiffness for pitch and uncertain structural damping (modeled by viscous damping) coefficients for all DOFs are assumed. Considering such uncertainties, an adaptive controller with an estimation update law is designed to stabilize the airfoil subjected to gusts. A Lyapunov function is established to prove the stability of the closed-loop system. Simulation results demonstrate the effectiveness of the designed controller.

本文介绍了具有结构刚度和阻尼不确定性的翼型的动力学和自适应控制，该翼型受到大气阵风的影响。翼型的运动由三个自由度 (DOF) 建模，即俯仰、俯仰和襟翼。平点或死区类型的刚度用于模拟襟翼铰链的自由间隙。检查没有控制和参数不确定性的系统的非线性动力学，其中考虑了俯仰的三次刚度和插入的线性刚度。数值结果表明，当流速远低于线性颤振速度时，翼型可能会通过 Hopf 分岔变得不稳定；如果结构阻尼不够高，它也可能发生混沌运动。结果表明，基于部分反馈线性化系统的比例微分控制器可以有效地缓解阵风在流速低于和高于线性颤振速度时引起的振荡。接下来，不确定$N$假定所有自由度的俯仰和不确定结构阻尼（由粘性阻尼建模）系数的三阶多项式刚度。考虑到这种不确定性，设计了具有估计更新律的自适应控制器来稳定受到阵风影响的翼型。建立Lyapunov函数来证明闭环系统的稳定性。仿真结果证明了所设计控制器的有效性。