The feedback control of Hopf bifurcation of nonlinear aeroelastic systems with asymmetric aerodynamic lift force and nonlinear elastic forces of the airfoil is discussed. For the Hopf bifurcation analysis, the eigenvalue problems of the state matrix and its adjoint matrix are defined. The Puiseux expansion is used to discuss the variations of the non-semi-simple eigenvalues, as the control parameter passes through the critical value to avoid the difficulty for computing the derivatives of the non-semi-simple eigenvalues with respect to the control parameter. The method of multiple scales and center-manifold reduction are used to deal with the feedback control design of a nonlinear system with non-semi-simple eigenvalues at the critical point of the Hopf bifurcation. The first order approximate solutions are developed, which include gain vector and input. The presented methods are based on the Jordan form which is the simplest one. Finally, an example of an airfoil model is given to show the feasibility and for verification of the present method.

中文翻译：

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Hopf分岔临界点非半单特征值非线性气动弹性系统的反馈控制

讨论了具有非对称气动升力和翼型非线性弹性力的非线性气动弹性系统Hopf分岔的反馈控制。对于Hopf分岔分析，定义了状态矩阵及其伴随矩阵的特征值问题。Puiseux展开用于讨论非半简单特征值的变化，因为控制参数通过临界值以避免计算非半简单特征值相对于控制参数的导数的困难。采用多尺度和中心流形约简方法处理Hopf分岔临界点处具有非半单特征值的非线性系统的反馈控制设计。开发了一阶近似解，其中包括增益向量和输入。所提出的方法基于最简单的 Jordan 形式。最后，给出了一个翼型模型的例子，以表明本方法的可行性和验证。