Submersive rational difference systems and their accessibility

Abstract

The paper describes an algebraic construction of the inversive difference field associated with a discrete-time rational nonlinear control system under the assumption that the system is submersive. We prove that a system is submersive iff its associated difference ideal is proper, prime and reflexive. Next, we show that Kähler differentials of the above inversive field define a module over the corresponding ring of Ore operators, and relate its torsion submodule to the vector space of autonomous one-forms, introduced elsewhere. The above results allow us to check accessibility property and simplify transfer functions with computer algebra techniques.

Transfer Equivalence and Realization of Nonlinear Input-Output Delta-Differential Equations on Homogeneous Time Scales

“…The alternative of Theorem 3.2 could be then stated in terms of torsion-freeness of this module. In [22] it is proved for discrete-time nonlinear systems whose description is based on the shift operator that the system module is torsion-free if and only if H1 is trivial. Of course, the transfer equivalence may be studied in terms of differential or difference fields [30] like it has been done for continuous-time systems in [31].…”

“…In the paper [21] the relationship between two irreducibility definitions, namely Definition 3.2 and a definition based on the notion of globally degenerate i/o space, is examined. Finally, the paper [22] is focused on discrete-time systems based on the shift operator, not covered by the time scale formalism, and proves that a rational system is submersive if and only if its associated difference ideal is proper, prime and reflexive. Additionally, it proves that the subspace H 1 (see below) for rational state equations is trivial if and only if the module of Kähler differentials associated with the submersive control system is torsion-free.…”

Ritt-Wu's characteristic set method for ordinary difference polynomial systems with arbitrary ordering

“…Furthermore, A is coherent since ∆(A i , A j ) is always zero. Equations of form (12) are often used in control theory [8] and it is important to know whether sat(A) is a reflexive prime ideal. We will use the techniques developed in this paper to prove the following result which was first given in [8].…”

Practical polynomial formulas in MIMO nonlinear realization problem

“…, s. Note that (6) is not claimed to be a realization of (4); however, the sets of solutions {u(t), y(t)} of (4) and (6) are equal. System (6) allows to construct the inversive σ-differential field K * for equations (4) in a similar manner as for state equations (3), following the lines in [8], [11] for the continuous-and discrete-time counterparts, respectively. Associate with system (4) the field K of meromorphic functions of the independent system variables {y…”

“…Here we assume K * to be given and by slight abuse of notation use the same symbol K for both. For explicit construction of K * , see [11] and [2] for the cases of shift and difference operators, respectively. Note that in the continuous-time case when σ = id K , K * = K.…”