Cascade decompositions of linear systems in terms of realizations

The authors consider how to express systems (or equivalently, rational matrix functions) as being built up from simpler more elementary systems (or functions). A measure of the complexity of a system is the dimension of the state space or equivalently, if controllability and observability are assumed, the McMillan degree of the associated transfer function. The authors discuss two ways in which simple systems can be connected to form more complicated systems: simple cascade connections and cascade (or fractional) compositions. These are both special cases of a more general composition, which the authors call general cascade composition, which they discuss briefly. Issues of particular interest are when a given system can be decomposed as a product of systems with a one-dimensional state space, the stability of such decompositions relative to small changes in the state-space parameters of the system, and decompositions when the factors inherit some symmetry satisfied by the composite system