Robust stability, Morse theory and singularity

This paper develops a new approach to robust stability problems. Following the basic theme of the Morse theory, this approach emphasizes the close relationships between such robustness properties as extremal points of the Nyquist map and the topology of the uncertainty space, assumed to be a compact differentiable manifold. Plots of Morse critical points on the uncertainty manifold reveal a complicated topology even for low dimensional problems. Structural stability of Nyquist map relative to “certain” parameters is also investigated. Finally, from the critical points/values plots, a decomposition of the Nyquist map is obtained. This decomposition has the property that the Nyquist map has its extremal points on the boundary of the cells