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Computation of the minimum destabilizing volume for interval and affine families of polynomials

机译：多项式的区间和仿射族的最小失稳体积的计算

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The authors study the computation of the minimum destabilizing volume for interval and polytopic families of polynomials. Roughly speaking, this is equivalent to determining the smallest box in parameter space which contains unstable polynomials. This new concept is an alternative to the robustness margin for the case when the radii of the box are unknown but only a lower bound for each of them is given. As stated, this problem requires the solution of a nonlinear optimization problem. In this paper, they show that via a proper reformulation, it can be recast as a one-dimensional optimization problem which requires checking a vertex condition at each step. It is interesting to observe that the vertices involved are artificially constructed, and they do not correspond to the vertices of the box in parameter space. Finally, they show that in the case of interval polynomials the number of vertices required is linear in the number of uncertain parameters, while in the polytopic case this number may not be polynomial in the worst case. Two examples, showing the efficacy of this new concept for interval and affine families, conclude the paper.
机译：作者研究了多项式的区间和多项式族的最小去稳定体积的计算。粗略地说，这等效于确定参数空间中包含不稳定多项式的最小框。当盒子的半径未知但只给出每个盒子的下限值时，这个新概念可以替代鲁棒性余量。如上所述，这个问题需要解决非线性优化问题。在本文中，他们表明，通过适当的重构，可以将其重铸为一维优化问题，该问题需要在每个步骤中检查顶点条件。有趣的是，所涉及的顶点是人为构造的，并且它们不对应于参数空间中盒子的顶点。最后，他们表明，在区间多项式的情况下，所需的顶点数在不确定参数的数量中是线性的，而在多义题的情况下，该数目在最坏的情况下可能不是多项式。本文总结了两个例子，证明了这一新概念对间隔和仿射族的有效性。

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