Modeling of symbolic systems: Part II - Hilbert space construction for model identification and order reduction

This paper, which is the second of two parts, is built upon the vector space of symbolic systems represented by probabilistic finite State automata (PFSA) reported in the first part. This second part addresses the Hilbert space construction for model identification, where order reduction is achieved via orthogonal projection. To this end, a family of inner products is constructed and the norm induced by an inner product is interpreted as a measure of information contained in the PFSA, which also quantifies the error due to model order reduction. A numerical example elucidates the process of model order reduction by orthogonal projection from the space of PFSA onto a subspace that belongs to the class of shifts of finite type.