利用反馈控制和牛顿迭代跟踪实验中的动态不稳定现象
Using feedback control and Newton iterations to track dynamically unstable phenomena in experiments
摘要：
如果一个人想系统地探索一个动力系统的性质，他必须能够跟踪平衡点和周期轨道，而不管它们的稳定性如何。如果动力系统是一个可控的实验，那么一种方法是经典的反馈控制和牛顿迭代的组合。参数激励摆上的机械实验最近表明了这种算法的简化版本的实际可行性：时间延迟反馈控制（如Pyragas所提出的）和牛顿迭代低维方程组的组合。我们证明了算法的两个部分在鞍结分岔附近是一致稳定的：带有时滞反馈控制的实验具有一致稳定的周期轨道，而为了使控制非侵入而必须求解的二维非线性系统具有良好的条件雅可比矩阵。
英文摘要：
If one wants to explore the properties of a dynamical system systematically one has to be able to track equilibria and periodic orbits regardless of their stability. If the dynamical system is a controllable experiment then one approach is a combination of classical feedback control and Newton iterations. Mechanical experiments on a parametrically excited pendulum have recently shown the practical feasibility of a simplified version of this algorithm: a combination of time-delayed feedback control (as proposed by Pyragas) and a Newton iteration on a low-dimensional system of equations. We show that both parts of the algorithm are uniformly stable near the saddle-node bifurcation: the experiment with time-delayed feedback control has uniformly stable periodic orbits, and the two-dimensional nonlinear system which has to be solved to make the control non-invasive has a well-conditioned Jacobian.
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