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

摘要

Recent developments in quantitative feedback theory include the 'new formulation' approach in which a robust performance and robust stability problem, similar to Horowitz's traditional QFT formulation, is developed in terms of sensitivity function bounds. The motivation for this approach was to provide 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 thatthe sensitivity-based formulation exhibits some unique behavior, i.e. in terms of the open loop design bounds obtained for various choices of nominal plant. Experience has shown that these bounds will denominate (i.e. are more conservative than) thecorresponding traditional QFT bounds for the same problem; it has also been 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 certain cases, can lead to aproblem which is infeasible with respect to Bode sensitivity (i.e, requiring│S(jω)│ < 1 asω→∞), while the traditional QFT problem remains feasible. Heretofore, this behavior has not; been fully explained. In this paper, these issues arecharacterized in the simplest possible setting, focusing primarily on the behavior at zero phase angle. A 'modified' sensitivity-based QFT formulation is proposed here in which limitations on the choice of nominal plant are clearly delineated; thisformulation results in open loop design bounds which are equivalent to the traditional QFT problem at zero phase angle, while over-bounding them elsewhere. The modified formulation is also shown to meet the same necessary condition for Bode feasibility as traditional QFT. In conclusion, these issues are demonstrated by means of a basic example.