61 research outputs found
Effective Algorithms for Parametrizing Linear Control Systems over Ore Algebras
In this paper, we study linear control systems over Ore algebras. Within this mathematical framework, we can simultaneously deal with different classes of linear control systems such as time-varying systems of ordinary differential equations (ODEs), differential time-delay systems, underdetermined systems of partial differential equations (PDEs), multidimensional discrete systems, multidimensional convolutional codes etc. We give effective algorithms which check whether or not a linear control system over some Ore algebra is controllable, parametrizable, flat or Pi-free
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Mini-Workshop: Formal Methods in Commutative Algebra: A View Toward Constructive Homological Algebra
The purpose of the mini-workshop is to bring into the same place different mathematical communities that study constructive homological algebra and are motivated by different applications (e.g., constructive algebra, symbolic computation, proof theory, algebraic topology, mathematical systems theory, D-modules, dynamical systems theory) so that they can share their results, techniques, softwares and experiences. Through the development of a unified terminology, common mathematical problems, which naturally appear when making homological algebra constructive, were discussed
Exact linear modeling using Ore algebras
Linear exact modeling is a problem coming from system identification: Given a
set of observed trajectories, the goal is find a model (usually, a system of
partial differential and/or difference equations) that explains the data as
precisely as possible. The case of operators with constant coefficients is well
studied and known in the systems theoretic literature, whereas the operators
with varying coefficients were addressed only recently. This question can be
tackled either using Gr\"obner bases for modules over Ore algebras or by
following the ideas from differential algebra and computing in commutative
rings. In this paper, we present algorithmic methods to compute "most powerful
unfalsified models" (MPUM) and their counterparts with variable coefficients
(VMPUM) for polynomial and polynomial-exponential signals. We also study the
structural properties of the resulting models, discuss computer algebraic
techniques behind algorithms and provide several examples
Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras
The present text surveys some relevant situations and results where basic
Module Theory interacts with computational aspects of operator algebras. We
tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be
published in Lecture Notes in Computer Scienc
Obstructions to Genericity in Study of Parametric Problems in Control Theory
We investigate systems of equations, involving parameters from the point of
view of both control theory and computer algebra. The equations might involve
linear operators such as partial (q-)differentiation, (q-)shift, (q-)difference
as well as more complicated ones, which act trivially on the parameters. Such a
system can be identified algebraically with a certain left module over a
non-commutative algebra, where the operators commute with the parameters. We
develop, implement and use in practice the algorithm for revealing all the
expressions in parameters, for which e.g. homological properties of a system
differ from the generic properties. We use Groebner bases and Groebner basics
in rings of solvable type as main tools. In particular, we demonstrate an
optimized algorithm for computing the left inverse of a matrix over a ring of
solvable type. We illustrate the article with interesting examples. In
particular, we provide a complete solution to the "two pendula, mounted on a
cart" problem from the classical book of Polderman and Willems, including the
case, where the friction at the joints is essential . To the best of our
knowledge, the latter example has not been solved before in a complete way.Comment: 20 page
Recent progress in an algebraic analysis approach to linear systems
This paper addresses systems of linear functional equations from an algebraic point of view. We give an introduction to and an overview of recent work by a small group of people including the author of this article on effective methods which determine structural properties of such systems. We focus on parametrizability of the behavior, i.e., the set of solutions in an appropriate signal space, which is equivalent to controllability in many control-theoretic situations. Flatness of the linear system corresponds to the existence of an injective parametrization. Using an algebraic analysis approach, we associate with a linear system a module over a ring of operators. For systems of linear partial differential equations we choose a ring of differential operators, for multidimensional discrete linear systems a ring of shift operators, for linear differential time-delay systems a combination of those, etc. Rings of these kinds are Ore algebras, which admit Janet basis or Gröbner basis computations. Module theory and homological algebra can then be applied effectively to study a linear system via its system module, the interpretation depending on the duality between equations and solutions. In particular, the problem of computing bases of finitely generated free modules (i.e., of computing flat outputs for linear systems) is addressed for different kinds of algebras of operators, e.g., the Weyl algebras. Some work on computer algebra packages, which have been developed in this context, is summarized
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Mini-Workshop: Constructive Homological Algebra with Applications to Coherent Sheaves and Control Theory
The main objective of this mini-workshop is to bring together recent developments in constructive homological algebra. There, the current state already reached a level of generality which allows simultaneous application to diverse fields of applied and theoretical mathematics. In this workshop, we want to focus on simultaneous applications to system theory on the one side and to coherent sheaves and their cohomology on the other side. Surprisingly, these apparently remote fields share a considerable amount of common constructive methods. Bringing category theory and homological algebra to the computer leads to questions in logic and type theory. One goal of this workshop is to promote and enlarge this overlap
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