Computational Topology in Robust Control

小行星9510656 传统的鲁棒性问题在不确定反馈系统的分析和设计，可以通过考虑不确定参数的空间被分割成两部分的交叉超曲面分离的不确定性的部分，其中规格满足，不满足规格。 一个重要的观察是，这种分离的超曲面可以变得非常复杂，并且可以找到其同调的复杂性界限。除了从这种几何情况计算u函数的可能性之外，超曲面的结构不稳定性确实似乎是解释真实的u函数相对于“某些”参数缺乏连续性这一尴尬问题的唯一途径。 一个原型代码的单纯算法建设和显示的超曲面-使用新兴的计算几何技术-已经在成为操作的边缘。 该提案的第一个目标是进一步开发该代码，使其成为一个更强大，用户友好的代码，使用最先进的计算几何，各向异性网格等。 该提案的第二个目标是为Nyguist映射的奇异性开发代数代码（Grobner基），该代码可以揭示超曲面的结构不稳定性或相对于舍入误差缺乏连续性的可能性。 特别注意将致力于奇异性分层不确定性空间。 ***

9510656 Jonckheere Traditional robustness issues in uncertain feedback systems analysis and design can be approached by considering the space of uncertain parameters cut into two pieces by the crossover hypersurface separating that part of the uncertainty where specifications are met from that part where specifications are not met. A crucial observation is that this separating hypersurface can get very complicated and complexity bounds on its homology can be found. In addition to the possibility of computing the u function from this geometric situation, structural instability of the hypersurface has indeed appeared to be the only way to explain the embarrassing problem of lack of continuity of the real u-function relative the "certain" parameters. A prototype code for simplicial algorithm construction and display of the hypersurface - using the emerging computational geometry technology - is already on the verge of becoming operational. The first objective of this proposal is to further develop this code into a more powerful, user-friendly one using state-of- the-art computational geometry, anisotropic gridding, etc. The second objective of this proposal is to develop algebraic code (Grobner basis) for singularity of the Nyguist map that could reveal the potential for structural instability of the hypersurface or lack of continuity of performance relative to rounding errors. Special attention will be devoted to singularity over a stratified uncertainty space. ***