Stabilization of linear systems by rotation
Stabilization of linear systems by rotation
复制标题
通过旋转稳定线性系统
DOI：
10.1016/j.jde.2006.12.005
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发表时间：
2007
影响因子：
2.4
通讯作者：
A. Ilchmann
中科院分区：
文献类型：
--
作者：
H. Crauel;T. Damm;A. Ilchmann
本文对具有负迹的确定性线性系统引入了“旋转镇定”的概念。该概念包含Meerkov在20世纪70年代提出的众所周知的“振动稳定化”概念，并且是Arnold及其同事在20世纪80年代提出的随机系统“噪声稳定化”的确定性版本。它表明，线性系统的负迹可以通过添加一个反对称矩阵，乘以一个适当的标量所谓的“增益函数”（可能是一个常数），这是足够大的稳定。为了克服什么是“足够大”的问题，我们还提出了一个伺服机制，它通过从轨迹学习来调整增益函数，直到最终轨迹趋于零。这种方法可以表明，一个Meerkov的假设振动稳定是多余的。此外，虽然Meerkov和Arnold及其同事假设稳定的周期函数或噪声具有足够大的频率和幅度，但我们还提供了一种伺服机制来在确定性设置中动态地确定该函数。
We introduce the concept of “stabilization by rotation” for deterministic linear systems with negative trace. This concept encompasses the well-known concept of “vibrational stabilization” introduced by Meerkov in the 1970s and is a deterministic version of ‘stabilization by noise’ for stochastic systems as introduced by Arnold and coworkers in the 1980s. It is shown that a linear system with negative trace can be stabilized by adding a skew-symmetric matrix, multiplied by a suitable scalar so-called “gain function” (possibly a constant) which is sufficiently large. To overcome the problem of what is “sufficiently large”, we also present a servo mechanism which tunes the gain function by learning from the trajectory until finally the trajectory tends to zero. This approach allows to show that one of Meerkov's assumptions for vibrational stabilization is superfluous. Moreover, while Meerkov as well as Arnold and coworkers assume that a stabilizing periodic function or the noise has sufficiently large frequency and amplitude, we also provide a servo mechanism to determine this function dynamically in a deterministic setup.