Publications about 'systems over rings' |
Articles in journal or book chapters |
We prove that for any family of n-dimensional controllable linear systems, continuously parameterized by up to three parameters, and for any continuous selection of n eigenvalues (in complex conjugate pairs), there is some dynamic controller of dimension 3n which is itself continuously parameterized and for which the closed-loop eigenvalues are these same eigenvalues, each counted 4 times. An analogous result holds also for smooth parameterizations. |
New results are given on the pole-shifting problem for commutative rings, and these are then applied to conclude that rings of continuous, smooth, or real-analytic functions on a manifold X are PA rings if and only if X is one-dimensional. |
We present various comments on a question about systems over rings posed in a recent note by Sharma, proving that a ring R is pole-assignable if and only if, for every reachable system (F,G), G contains a rank-one summand of the state space. We also provide a generalization to deal with dynamic feedback. |
A controller is shown to exist, universal for the family of all systems of fixed dimension n, and m controls, which stabilizes those systems that are stabilizable, if certain gains are large enough. The controller parameters are continuous, in fact polynomial, functions of the entries of the plant. As a consequence, a result is proved on polynomial stabilization of families of systems. |
This paper provides an introduction to definitions and known facts relating to the stabilization of parametrized families of linear systems using static and dynamic controllers. New results are given in the rational and polynomial cases. |
A polynomially parametrized family of continuous-time controllable linear systems is always stabilizable by polynomially parametrized feedback. (Note: appendix had a MACSYMA computation. I cannot find the source file for that. Please look at journal if interested, but this is not very important. Also, two figures involving root loci are not in the web version.) |
Given a continuous-time family of finite dimensional single input linear systems, parametrized polynomially, such that each of the systems in the family is controllable, there exists a polynomially parametrized control law making each of the systems in the family stable. |
Various types of transfer matrix factorizations are of interest when designing regulators for generalized types of linear systems (delay differential. 2-D, and families of systems). This paper studies the existence of stable and of stable proper factorizations, in the context of the thery of systems over rings. Factorability is related to stabilizability and detectability properties of realizations of the transfer matrix. The original formulas for coprime factorizations (which are valid, in particular, over the field of reals) were given in this paper. |
Problems that appear in trying to extend linear control results to systems over rings R have attracted considerable attention lately. This interest has been due mainly to applications-oriented motivations (in particular, dealing with delay-differential equations), and partly to a purely algebraic interest. Given a square n-matrix F and an n-row matrix G. pole-shifting problems consist in obtaining more or less arbitrary characteristic polynomials for F+GK, for suitable ("feedback") matrices K. A review of known facts is given, various partial results are proved, and the case n=2 is studied in some detail. |
This paper studies the problem of obtaining minimal realizations of linear input/output maps defined over rings. In particular, it is shown that, contrary to the case of systems over fields, it is in general impossible to obtain realizations whose dimiension equals the rank of the Hankel matrix. A characterization is given of those (Noetherian) rings over which realizations of such dimensions can he always obtained, and the result is applied to delay-differential systems. |
An abstract operator approach is introduced, permitting a unified study of discrete- and continuous-time linear control systems. As an application, an algorithm is given for deciding if a linear system can be built from any fixed set of linear components. Finally, a criterion is given for reachability of the abstract systems introduced, giving thus a unified proof of known reachability results for discrete-time, continuous-time, and delay-differential systems. |
This paper deals with observability properties of realizations of linear response maps defined over commutative rings. A characterization is given for those maps which admit realizations which are simultaneously reachable and observable in a strong sense. Applications are given to delay-differential systems. |
A lattice characterization is given for the class of minimal-rank realizations of a linear response map defined over a (commutative) Noetherian integral domain. As a corollary, it is proved that there are only finitely many nonisomorphic minimal-rank realizations of a response map over the integers, while for delay -differential systems these are classified by a lattice of subspaces of a finite-dimensional real vector space. |
It is well known that principal rings are strong Fatou rings. We construct here a more general type of strong Fatou rings. We also prove that the monoid of divisor classes of a noetherian strong Fatou ring contains only the zero element, and that the dimension of such a ring is at most two. |
An elementary presentation is given of some of the main motivations and known results on linear systems over rings, including questions of realization and control. The analogies and differences with the more standard case of systems over fields are emphasized throughout. |
Two classes of rings which occur in linear system theory are introduced and compared. Characterizations of one of them are given in terms, of integral extensions (every finite extension of R is integral) and Cayley--Hamilton type matrix condition. A comparison is made in the case of no zero-divisors with Ore domains. |
This paper studies some problems appearing in the extension of the theory of linear dynamical systems to the case in which parameters are taken from noncommutative rings. Purely algebraic statements of some of the problems are also obtained. Through systems defined by operator rings, the theory of linear systems over rings may be applied to other areas of automata and control theory; several such applications are outlined. |
Conference articles |
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