Computational Topology in Robust Control

9510656在不确定反馈系统的分析和设计中，传统的鲁棒性问题可以通过考虑不确定参数的空间来解决，不确定参数被交叉超曲面分割成两部分，其中满足规范的部分和不满足规范的部分被分隔开来。一个重要的观察是，这种分离的超曲面可能会变得非常复杂，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函数的可能性外，超曲面的结构不稳定性似乎是解释真正的u函数相对于“某些”参数缺乏连续性这一令人尴尬的问题的唯一途径。用于构建和显示超曲面的简单算法的原型代码-使用新兴的计算几何技术-已经处于可操作的边缘。该建议的第一个目标是使用最先进的计算几何、各向异性网格等将该代码进一步开发成一个功能更强大、用户友好的代码。该建议的第二个目标是开发用于Nyguist地图奇异性的代数代码(Grobner基)，该代码可能揭示超曲面的结构不稳定性或性能相对于舍入误差缺乏连续性的可能性。我们将特别关注分层不确定空间上的奇点。***

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. ***