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

Stable linear fractional transformations (SLET's) 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. The problem of representing a plant as a SLFT of another plant such that the order of the original plant is exactly equal to the sums of the orders of the SLET 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 SLFT's, it is shown that if two plants are related by a 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.