SKEWED /х PROBLEMS IN ROBUSTNESS ANALYSIS

The aim of this chapter is twofold. The first one is to detail some of the physical problems, which are solved in the rest of the book. The second one is to illustrate the usefulness of the skewed /л (i. e. v ) ap­proach for engineering problems. To this aim, a large class of important practical problems is considered, which requires the skewed ц tool rather than the classical ц tool. See also chapter 11 for the presentation of a specific problem, which uses an extension of the д and skewed ^ tools, namely the problem of computing a robust delay margin in the presence of model uncertainties.

Sections 1. and 2. give two examples, in which the problem of check­ing the robustness properties of a closed loop reduces to the problem of checking a small gain condition despite model uncertainties: as proved in chapter 8 (section 2.), this is a skewed ц problem involving an augmen­ted model uncertainty. See also chapter 11 (subsection 3.3) for an other example of problem which reduces to checking a small gain condition despite model uncertainties.

The difficult QFT problem of translating closed loop frequency do­main specifications into open loop specifications on the MIMO controller frequency response can also be solved in an approximate way with the skewed n tool. Here again, the problem reduces to the issue of checking a small gain condition despite a model uncertainty in the controller fre­quency response: see (Ferreres and LeGorrec, 1999) for further details (in a controller reduction context).

Three other examples of skewed /і problems are given in sections 3. to 5., namely the direct computation of the maximal s. s.v. over a frequency interval, the maximization of the domain of the allowable model uncer­tainties and the analysis of gain-scheduled or robust adaptive controllers.