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Adaptive backstepping control of a nonlinear aeroelastic system
LIU Songdan, LI Daochun, XIANG Jinwu
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:For a two-dimensional airfoil with leading-edge and trailing-edge control surfaces, the nonlinear aeroelastic equations under the supposition of quasi-steady aerodynamic forces were established and were described in state space form. The control variables of the leading and trailing edges were coupled resulting that the backstepping control method could not be used directly. To solve the problem, two equivalent control laws were newly defined. Supposing that the system has parametric uncertainty in the cubic nonlinearity in pitch, an adaptive control law was designed based on Lyapunov stability theory. In order to verify the validation of the control law, the dynamic equations were solved numerically by using Runge-Kutta method. The simulation results show that the open-loop aeroelastic system is unstable with limit cycle oscillation, while the close-loop system reaches to stable as a result of the adaptive control law. With double control surfaces, the flutter critical velocity is improved after the control design. Taking the limits of the control surface deflection in reality into account, the invalidation problems of the single control surface are discussed. Just considering the effectiveness of the single control surface, the system using the trailing edge control surface is better than that of using the leading edge control surface.
Key words: two-dimensional airfoil     nonlinear aeroelasticity     flutter     Lyapunov stability theory     adaptive backstepping control

 图 1 前/后缘控制面二元机翼Fig. 1 Two-dimensional airfoil with leading and trailing edge control surfaces

c1>0,c4>0,c4+0.036mT/d>0,则负定.

 图 2 系统仿真结果Fig. 2 System simulation results
3.1 双控制面控制

U=12 m/s,气动弹性系统的自适应控制结果如图3所示.当t＜10 s时,控制输入u1=0,u2=0,即此时为开环响应,此时系统形成了振幅稳定的极限环振荡.10 s后,开始施加控制,从数值仿真结果可以看出,施加控制后不久系统稳定到零点.

 图 3 模型的自适应控制结果(U=12 m/s)Fig. 3 Model adaptive control results (U=12 m/s)

 图 4 限制控制面的模型自适应控制(U=12 m/s)Fig. 4 Model adaptive control with limited control surface (U=12 m/s)

 图 5 两种情况的模型自适应控制(U=12 m/s)Fig. 5 Two different model adaptive controls (U=12 m/s)

 图 6 限制控制面的单前缘控制系统响应(U=12 m/s)Fig. 6 Response of control system only by leading edge with limited control surface (U=12 m/s)

1) 双控制面在反演自适应控制律的作用下，可以使开环不稳定的系统稳定.

2) 仅对比单前/后缘控制面的作用,后缘控制面比前缘控制面对系统控制更有效.

3) 双控制面的共同作用比单后缘控制面的作用更快地使系统稳定.其中，单后缘控制的闭环颤振临界速度较开环颤振临界速度提高172.38%.

4) 考虑到实际情况中存在控制面偏转限制，当来流速度达到一定值时，闭环系统仍然会出现颤振现象.

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#### 文章信息

LIU Songdan, LI Daochun, XIANG Jinwu

Adaptive backstepping control of a nonlinear aeroelastic system

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(6): 1128-1134.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0423