Stable linear fractional transformations with applications to stabilization and multistage H∞ control design

Abstract

Stable linear fractional transformations (SLFTs) resulting from a 2 × 2-block unit Z in the ring of stable real rational proper matrices are considered in this paper. Several general properties are obtained, including properties with respect to possible pole-zero cancellations and a generic McMillan degree relationship between a transfer matrix and its image under an SLFT. The problem of representing a plant as an SLFT of another plant such that the order of the original plant is exactly equal to the sums of the orders of the SLFT and of the new plant is solved. All such representations can be found by searching for all matching pairs of stable invariant subspaces associated with the plant. In relation to applications of SLFTs, it is shown that if two plants are related by an SLFT, then a one-to-one correspondence between their two respective sets of all stabilizing controllers can be established via a different SLFT. Also, it is shown how to decompose a standard H∞ control problem by means of SLFT into two individual H∞ subproblems, the first involving a nominal plant model and the second involving a certain frequency-shaped approximation error. An example is presented to illustrate the idea of decomposing the complexity of an H∞ control problem.