Canonical State Space Realizations of IO Systems via Gröbner Bases

In Theorem 6.3.11, we presented Fliess’ unconstructive proof that every IO behavior over a univariate polynomial ring admits an observable state-space realization, which is unique up to similarity. In this chapter, we use the Grobner basis theory to construct the canonical observability and observer realizations of a behavior. If the behavior is controllable, we also construct the canonical controllability and controller realizations. The canonical realizations depend only on the 30 given IO behavior and on a chosen term order in the Grobner basis theory, but not on the particular matrices that define the behavior. If the transfer matrix of the IO behavior is proper, these realizations can be built with elementary building blocks. In particular, the proper compensators and observers from Chaps. 10 and 11 can be implemented as state-space behaviors in this fashion.