利用新的接近度指标避免分叉

Giorgos Altanis, N. Maratos
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摘要

本文研究了在不解决任何困难的优化问题(如最近分岔问题)的情况下,监测和避免平衡点的折叠(鞍节点)分岔问题。提出了新的标量指标来监测分叉的接近度。这些指标是基于雅可比矩阵的QR分解,它们在可行性边界内部是方向可微的。这允许我们在不可控参数空间中计算最陡下降方向和在可控参数空间中计算最陡上升方向。前者用于估计到可行性边界的距离,后者通过一种简单的算法来设计可控制参数,使接近度指标保持在一定阈值以上,从而避开可行性边界。其中一个建议的指标是渐近线性相对于距离的分岔值被接近,导致准确的估计距离的分岔。
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Avoiding fold bifurcations with the help of new proximity indices
This work studies the problem of monitoring and avoiding fold (saddle-node) bifurcations of equilibria without solving any difficult optimization problems, such as the closest bifurcation problem. New scalar indices are proposed to monitor the proximity to fold bifurcations. These indices are based on the QR factorization of the Jacobian matrix, and they are directionally differentiable in the interior of the feasibility boundary. This allows us to compute steepest descent directions in the uncontrollable parameter space and steepest ascent directions in the controllable parameter space. The former are used to estimate the distance to the feasibility boundary, while the latter are used by a simple algorithm which designs the controllable parameters in a way that the proximity indices maintain values above a certain threshold, thus avoiding the feasibility boundary. One of the proposed indices is asymptotically linear with respect to the distance to the bifurcation value being approached, resulting in accurate estimates of the distance to the bifurcation.
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