Stability and performance robustness analysis with highly structured uncertainties: necessary and sufficient test

The author presents an extension of a technique of analyzing the stability of linear time-invariant interconnected systems with uncertainties: each element of the frequency response matrix is assumed to lie inside a polygon at each frequency. It is shown that the exterior boundary of the image of the Cartesian product of these polygons under the map det[I+MΔ] is contained in the boundary of the image of the Cartesian product of the boundaries of these polygons under the same map. Thus, the image can be constructed from the images of the edges. The computational technique of R.R.E. de Gaston and M.G. Safonov (1988) can then be applied to perform a nonconservative robustness analysis of the feedback system