Computational Topology in Robust Control

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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. ***