ABSTRACT
Such robust stability issues as Kharitonov theorem, structured singular values, multivariable phase margin, etc., are given a unifying interpretation in the context of algebraic topology. The central issue is whether the mapping from the manifold of structured uncertainties to the Nyquist plot commutes with the boundary operator. This operator is central in algebraic topology and formalizes the intuitive notion of “boundary” that one can capture from elementary geometry. In the simple Kharitonov case, the Nyquist map and the boundary commute, leading to the result that to check stability it suffices to check stability on the boundary. However, in the overwhelming majority of other cases, the Nyquist map and the boundary do not commute. Our central result is an application of the simplicial approximation theorem: The Nyquist map always has an arbitrarily accurate simplicial approximation from a triangulation of the uncertain manifold to a triangulation of the Nyquist plot, with the property that inside each simplex of the triangulation the simplicial approximation and the boundary do commute. The amount of refinement of the triangulation needed to establish a simplicial map is a measure of the complexity of the problem. With the simplicial approximation, the crossover region is determined, up 340to the resolution afforted by the triangulation of the uncertainty manifold, by a fast combinatorial algorithm that does not require checking all points on the grid. Finally, the topology of the separating boundary between the stability and instability regions, that is sometimes quite complicated, is computed by an algebraic procedure without the need for intensive computation.
