Rings of skew polynomials in algebraical approach to control theory. (English) Zbl 0874.16022

Ježek, Jan

Kybernetika 32, No. 1, 63-80 (1996).

The paper contains the background mathematics for systems and control theory. The author generalizes the polynomial approach in the control theory from time invariant systems to time varying ones. For that purpose, the algebraical rings are equipped with some additional operations as: shift, difference or derivation. Based on that, the introduced skew polynomials are non-commutative but satisfy a commutation equation. The theory of skew polynomials originated from O. Ore in 1933. Rings with derivation were introduced by H. W. Raudenbush, jun. in the same year. The unified approach to the continuous-time systems and the discrete-time ones is a contribution of the author.

Reviewer: J.Duda (Brno)

MSC:

16S36	Ordinary and skew polynomial rings and semigroup rings
16W25	Derivations, actions of Lie algebras
93A99	General systems theory
93B25	Algebraic methods
16W20	Automorphisms and endomorphisms

Keywords:

discrete-time systems; rings with derivation; control theory; time invariant systems; skew polynomials

References:

[1]	W. Greub: Linear Algebra. Springer Verlag, New York 1975. · Zbl 0317.15002
[2]	N. Jacobson: Structure of Rings. American Mathematical Society, Providence, R.I. 1956. · Zbl 0073.02002
[3]	V. Kučera: Discrete Linear Control - The Polynomial Approach. Wiley, Chichester 1979.
[4]	O. Øre: Theory of non-commutative polynomials. Ann. of Math. 34 (1933), 480-508. · Zbl 0007.15101 · doi:10.2307/1968173
[5]	H. W. Raudenbush, Jr.: Differential fields and ideals of differential forms. Ann. of Math. 34 (1933), 509-517. · Zbl 0007.15103 · doi:10.2307/1968174