TABLE OF CONTENTS Presented here is a new approach to robustness analysis. Stability properties are described in terms of system parameters, where these parameters are real. The problem addressed is the robustness of a nonlinear H ? controller against amplitude parameters. Because checking the closed-loop stability for all combinations of a large number of amplitudes is a formidable task, fast algorithms based on concepts from simplicial geometry are used.13 , 42 , 43
Horowitz39 , 60 , 62 , 63 used templates on the Nyquist diagram to demonstrate the stability attributes of a system subjected to perturbations of a given structure. These templates {det( I + L( j ?) ?): ? ? D, ? ? ?} are equivalent to the mapping:
| (4.1) |  |
| (4.2) |  |
Here ? is the set of all real frequencies. Its topology is that of a circle, generated as a section of a Riemann sphere. With D a two-dimensional manifold, the Cartesian product D x ? is a three-manifold, while a Horowitz template is a two-manifold.42 When determining stability, it is the perimeter P of the template near the origin that is examined (the inverse image of the perimeter of the template included on the walls of D x ? is checked). These boundary conditions are sufficient. Defining the boundary operator ? n as the operator that takes an n-dimensional object and determines its n 1 dimensional boundary, the previous condition...
