polynomial systems

We study the realization problem for discrete-time input/output polynomial systems. These are formalized using tools from commutative algebra and algebraic geometry as systems whose state spaces are algebraic varieties, or more abstractly the set of k-points of an affine k-scheme, where k is an arbitrary infinite field. The input/output behaviors of such systems are described by "polynomial response maps" in which outputs are polynomial functions of past inputs. The main results show that every polynomial response map admits a canonical (quasi-reachable and algebraically observable) realization, which is unique up to isomorphism. The key to the approach is to linearize dynamics by considering a "dual" system in which states are functions defined on states. Finite dimensionality of the canonical realization, and its polynomiality, are characterized in terms of the space, algebra, and field of observables of the map, as well as in terms of algebraic input/output difference equations, and by a Jacobian rank criterion. A particular subclass consists of the maps that we call "bounded," defined by the property that their degree in the past inputs is uniformly bounded. Bounded maps are shown to be finitely realizable if and only if they are realizable by finite-dimensional state-affine systems, whose theory in turn reduces to that of rational formal power series. We also study the lattice of quasi-reachable realizations of a given map, including normal realizations. This work is an update of the PhD thesis written by the author in 1976; connections to recent work, including "Koopman-like" linearizations, are briefly discussed as well.

Different notions of observability are compared for systems defined by polynomial difference equations. The main result states that, for systems having the standard property of (multiple-experiment initial-state) observability, the response to a generic input sequence is sufficient for final-state determination. Some remarks are made on results for nonpolynomial and/or continuous-time systems. An identifiability result is derived from the above.

Considered here are a type of discrete-time systems which have algebraic constraints on their state set and for which the state transitions are given by (arbitrary) polynomial functions of the inputs and state variables. The paper studies reachability in bounded time, the problem of deciding whether two systems have the same external behavior by applying finitely many inputs, the fact that finitely many inputs (which can be chosen quite arbitrarily) are sufficient to separate those states of a system which are distinguishable, and introduces the subject of realization theory for this class of systems.