Stability of quantitative feedback designs and the existence of robust QFT controllers

Abstract

AbstractQuantitative feedback theory (QFT) has received much criticism for a lack of clearly stated mathematical results to support its claims. Considered in this paper are two important fundamental questions: (i) whether or not a QFT design is robustly stable, and (ii) does a robust stabilizer exist. Both these are precursors for synthesizing controllers for performance robustness. Necessary and sufficient conditions are given to resolve unambiguously the question of robust stability in SISO systems, which in fact confirms that a properly executed QFT design is automatically robustly stable. This Nyquist‐type stability result is based on the so‐called zero exclusion condition and is applicable to a large class of problems under some simple continuity assumptions. In particular, the class of uncertain plants include those in which there are no right‐half plane pole‐zero cancellations over all plant uncertainties. A sufficiency condition for a robust stabilizer to exist is derived from the well‐known Nevanlinna‐Pick theory in classical analysis. Essentially the same condition may be used to answer the question of existence of a QFT controller for the general robust performance problem. These existence results are based on an upper bound on the nominal sensitivity function. Also considered is QFT design for a special class of interval plants in which only the poles and the DC gain are assumed uncertain. The latter problem lends itself to certain explicit computations that considerably simplify the QFT design problem.