利用特征根延拓法快速生成时滞系统稳定图
Fast generation of stability charts for time-delay systems using continuation of characteristic roots
摘要：
许多动态过程都含有时滞，因此它们的动力学行为可以用时滞微分方程（DDE）来描述。研究动态系统的稳定性是至关重要的，但分析时滞系统的稳定性是具有挑战性的，因为DDE是无限维的。我们提出了一种新的方法来快速生成稳定性图表的DDE使用连续的特征根（CCR）。在我们的CCR方法中，一个DDE的特征方程的根被写为感兴趣的参数的隐函数，并推导出的连续方程的形式常微分方程（ODEs）。数值延拓，然后确定在参数空间中的所有点的特征根，原来的DDE的稳定性，然后可以很容易地确定。所提出的方法的一个关键的优点是，系统的线性无关常微分方程的解决，而不是解决一个大的本征值问题在域中的每个网格点的典型策略。因此，CCR方法显着减少了确定DDE稳定性所需的计算工作量。正如我们用几个例子证明的那样，CCR方法生成高度精确的稳定性图表，并且比Galerkin近似方法快10倍。
英文摘要：
Many dynamic processes involve time delays, thus their dynamics are governed by delay differential equations (DDEs). Studying the stability of dynamic systems is critical, but analyzing the stability of time-delay systems is challenging because DDEs are infinite-dimensional. We propose a new approach to quickly generate stability charts for DDEs using continuation of characteristic roots (CCR). In our CCR method, the roots of the characteristic equation of a DDE are written as implicit functions of the parameters of interest, and the continuation equations are derived in the form of ordinary differential equations (ODEs). Numerical continuation is then employed to determine the characteristic roots at all points in a parametric space; the stability of the original DDE can then be easily determined. A key advantage of the proposed method is that a system of linearly independent ODEs is solved rather than the typical strategy of solving a large eigenvalue problem at each grid point in the domain. Thus, the CCR method significantly reduces the computational effort required to determine the stability of DDEs. As we demonstrate with several examples, the CCR method generates highly accurate stability charts, and does so up to 10 times faster than the Galerkin approximation method.
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