T. Sánchez-Giralda

Brunovsky's canonical form for linear dynamical systems over commutative rings

Abstract This paper is devoted to studying the action of the feedback group on linear dynamical systems over a commutative ring R with unit. We characterize the class of m-input n-dimensional reachable linear dynamical systems ∑ = (F, G) over R that are feedback equivalent to a system ∑c = (Fc, Gc) with Brunovskys canonical form. This characterization is obtained in terms of the minors of the matrices G ∑ i = (G, FG, …, F i − 1 G) for 1 ⩽i ⩽ n .

Feedback invariants for linear dynamical systems over a principal ideal domain

Abstract This paper studies the action of the feedback group F n,m on m -input, n -dimensional reachable linear dynamical systems over a principal ideal domain R . For such a 2-dimensional system ∑ a complete set of invariants is given which characterizes the feedback class of ∑. In particular it is characterized, in terms of these invariants, when ∑ has the feedback cyclization property. The particular cases R = Z and R = R[ X ] are studied in some detail. Finally, when n is arbitrary, the feedback classification is given for the class of reachable systems ∑ = ( F , G ) such that G is a matrix with at least n − 1 invariant factors equal to one.

On the duality principle for linear dynamical systems over commutative rings

Abstract The main result in this paper characterizes those commutative rings R having the property that every linear dynamical system over R verifies the duality principle [i.e., the system Σ is observable (reachable) if and only if the dual system Σ t is reachable (observable)]. This characterization is given in terms of the finitely generated faithful ideals of R , and it generalizes a result due to Ching and Wyman for the noetherian case. In case R satisfies the additional condition of being a reduced ring, we prove that the duality principle holds in R if and only if the height of every finitely generated ideal of R is zero.

Pole-shifting for linear systems over commutative rings☆

Abstract This paper deals with the pole-shifting problem for non-necessarily reachable linear systems. The notion of PS ring is introduced in the same way as the notion of pole assignable ring is given for reachable systems. We prove that a bcs ring is a PS ring and that over a Prufer domain these properties are equivalent. Finally, we study when the converse of the pole-shifting theorem is verified.

CANONICAL FORMS FOR 2-DIMENSIONAL LINEAR SYSTEMS OVER COMMUTAT IVE RINGS

ABSTRACT In this paper we study the action of the feedback group on m-input, 2-dimensional linear dynamical systems over a commutative ring R, in order to calculate canonical forms. We define a set for each element g of R . For a class of systems, a complete set of canonical forms can be constructed associated with pairs , where f belongs to . If the ring is an elementary divisor domain, in particular a P.I.D., this method applies to all reachable systems. When R is a Dedekind domain, a formula is obtained for calculating using the factorization of gR in powers of prime ideals. We also establish two conditions on the ring which imply that each set , is finite. For the rings and , we calculate explicitely the canonical forms, improving by this way the known results about the number of feedback classes over these rings. Finally, effective calculations are made when R is the ring of integers of an algebraic field, using methods of Computational Algebra.