The Cascade Decompositions of a given System Vs the Linear Fractional Decompositions of Its Transfer Function*

The external description of a system [A,B,C,D] is given by its transfer function W(h). Classically in designing a system with given transfer function the key step is to express W(h) as a linear functional combination of simpler functions. Naturally such a decomposition of a transfer function must correspond to some type of decomposition of the system [A,B,C,D] into smaller subsystems. In this article we describe the correspondence precisely. It leads us to define a reducing pair of subspaces S1 (9 S2 for a system with state space X to be a pair which gives a direct sum decomposition X $ 1 (9 $2 for X such that $1 is (A,B) invariant and $2 is (C,A) invariant. It turns out that each reducing pair for a system gives rise to a small family of linear fractional decompositions of the transfer function. Conversely, any non-trivial "minimal" decomposition of the transfer function of a minimal system corresponds to a reducing pair for that system.