On characterizing sensitivity-based and traditional formulations for quantitative feedback theory

Recent developments in quantitative feedback theory include the 'new formul ation' approach in which a robust performance and robust stability problem, similar to Horowitz's traditional QFT formulation, is developed in terms o f sensitivity function bounds. The motivation for this approach was to prov ide the basis for a more rigorous treatment of nonminimum phase systems and /or plants characterized by mixed parametric and non-parametric uncertainty models. However, it has been found in practice that the sensitivity-based formulation exhibits some unique behaviour, i.e. in terms of the open loop design bounds obtained for various choices of nominal plant. Experience has shown that these bounds will dominate (i.e. are more conservative than) th e corresponding traditional QFT bounds for the same problem; it has also be en observed that the degree to which this occurs varies with choice of the nominal plant. Further, it has been found that the choice of nominal, in ce rtain cases, can lead to a problem which is infeasible with respect to Bode sensitivity (i.e. requiring \S(j omega)\ < 1 as omega --> infinity), while the traditional QFT problem remains feasible. Heretofore, this behaviour h as not been fully explained. In this paper, these issues are characterized in the simplest possible setting, focusing primarily on the behaviour at ze ro phase angle. A 'modified' sensitivity-based QFT formulation is proposed here in which limitations on the choice of nominal plant are clearly deline ated; this formulation results in open loop design bounds which are equival ent to the traditional QFT problem at zero phase angle, while over-bounding them elsewhere. The modified formulation is also shown to meet the same ne cessary condition for Bode feasibility as traditional QFT. In conclusion, t hese issues are demonstrated by means of a basic example.