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

Computing diagonal form and Jacobson normal form of a matrix using Gr\"obner bases

In this paper we present two algorithms for the computation of a diagonal form of a matrix over non-commutative Euclidean domain over a field with the help of Gr\"obner bases. This can be viewed as the pre-processing for the computation of Jacobson normal form and also used for the computation of Smith normal form in the commutative case. We propose a general framework for handling, among other, operator algebras with rational coefficients. We employ special "polynomial" strategy in Ore localizations of non-commutative $G$-algebras and show its merits. In particular, for a given matrix $M$ we provide an algorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$ holds. The polynomial approach allows one to obtain more precise information, than the rational one e. g. about singularities of the system. Our implementation of polynomial strategy shows very impressive performance, compared with methods, which directly use fractions. In particular, we experience quite moderate swell of coefficients and obtain uncomplicated transformation matrices. This shows that this method is well suitable for solving nontrivial practical problems. We present an implementation of algorithms in SINGULAR:PLURAL and compare it with other available systems. We leave questions on the algorithmic complexity of this algorithm open, but we stress the practical applicability of the proposed method to a bigger class of non-commutative algebras

Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{\"o}bner bases

This paper is a sequel to "Computing diagonal form and Jacobson normal form of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We present a new fraction-free algorithm for the computation of a diagonal form of a matrix over a certain non-commutative Euclidean domain over a computable field with the help of Gr\"obner bases. This algorithm is formulated in a general constructive framework of non-commutative Ore localizations of $G$-algebras (OLGAs). We split the computation of a normal form of a matrix into the diagonalization and the normalization processes. Both of them can be made fraction-free. For a matrix $M$ over an OLGA we provide a diagonalization algorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$ holds and $D$ is diagonal. The fraction-free approach gives us more information on the system of linear functional equations and its solutions, than the classical setup of an operator algebra with rational functions coefficients. In particular, one can handle distributional solutions together with, say, meromorphic ones. We investigate Ore localizations of common operator algebras over $K[x]$ and use them in the unimodularity analysis of transformation matrices $U,V$. In turn, this allows to lift the isomorphism of modules over an OLGA Euclidean domain to a polynomial subring of it. We discuss the relation of this lifting with the solutions of the original system of equations. Moreover, we prove some new results concerning normal forms of matrices over non-simple domains. Our implementation in the computer algebra system {\sc Singular:Plural} follows the fraction-free strategy and shows impressive performance, compared with methods which directly use fractions. Since we experience moderate swell of coefficients and obtain simple transformation matrices, the method we propose is well suited for solving nontrivial practical problems.Comment: 25 pages, to appear in Journal of Symbolic Computatio

Algorithmic aspects of algebraic system theory

The mathematical roots of system and control theory date back to the paper On Governors by J. C. Maxwell published 1868 in Proceedings of the Royal Society of London. The seminal work of R.E. Kalman established system theory as a mathematical discipline in the 1950s. The contribution of U. Oberst, which appeared in 1990, gives fundamental insight for algebraic system theory. A very important algebraic property of the signal space is realized to be highly copious for signals and systems there, namely the property of the signal space to be an injective cogenerator over the underlying operator ring. The goal of algebraic system theory is the structural analysis of dynamical systems using algebraic tools. These systems may arise from various practical problems settled for instance in a scientific, technical or economical area. Classically linear time-invariant systems with field coefficients are studied. In the recent past variations of these systems have proved to be worthy for extended studies. From the applied point of view, there is obviously the interest to consider corresponding generalizations. From the algebraic point of view, some particular settings are very interesting for further investigations since ring theory and homological algebra provide a deep insight. Beyond theoretical studies, the computer algebra machinery allows the enormous benefit of constructive analyses. This thesis elaborates both aspects,the theoretical and the computational, in parallel. It is organized as follows. Chapter 1 and Chapter 2 serve for an extended introduction. System theoretical aspects are provided in Chapter 1. Basic concepts and definitions are presented and furthermore the following chapters are motivated from the system theoretical point of view. Chapter 3 studies systems with coefficients in a finite ring, in contrast to the classical case. The general motivation for this framework stems mainly from communication theory. However, the extension leads to problems like zero-divisors and the principal ideal domain property is lost. Therefore concepts useful for coding fail to generalize straightforwardly. In the field case the so-called predictable degree property is useful for many areas of system theory, ranging from controller parameterization to minimal realizations of linear systems over fields. This property does not carry over directly to the ring case. The paper The predictable degree property and row reducedness for systems over a finite ring by M. Kuijper, R. Pinto, J. W. Polderman and P. Rocha establishes a new framework which allows the adoption of that classical result in a novel setting. By the tool of Gröbner bases these results are extended to a more general framework which additionally allows concrete calculations. For this purpose the notion of the so-called minimal Gröbner p-basis is established and the connection to known results is pointed out. Chapter 4 is focused on one-dimensional systems with time-varying rational coefficients. This leads to the non-commutative operator ring called rational Weyl algebra which is a principal ideal domain. Therefore the non-commutative analogon to the Smith form, the so-called Jacobson form, exists. This normal form can be used to obtain a decomposition into a controllable and an autonomous subsystem of the corresponding linear abstract system. Furthermore the order of the underlying ordinary differential equation system is obtained directly. But computational problems known from the commutative counterpart even increase due to the non-commutative structure, namely the explosive growth of the coefficients. A novel approach which can be applied in a completely fraction free framework is presented in this chapter. This approach shows first how to obtain a decoupled form. It should be stressed that this decoupled form may even be interesting by itself. Further we show how to obtain a normal form from the decoupled form. The implementation is realized as a library called jacobson.lib for the computer algebra system Singular::Plural. This implementation is compared with all implementations which are available to the best of our knowledge. A behavioral approach to linear exact modeling is formulated for one-dimensional systems with constant coefficients by J.C. Willems. This problem of system identification is extended to polynomial-exponential signals in a multi-dimensional time-varying model class in Chapter 5. These model classes are summarized in the so-called Ore algebras. The idea of this approach is to derive a model describing the observed data and containing as much information as possible. It turns out that the particular model classes yield a very precise description, as pointed out in the case of continuous systems. Two alternative possibilities to calculate the models will be presented, one of them working in a purely commutative framework

Algorithmic aspects of algebraic system theory

The mathematical roots of system and control theory date back to the paper On Governors by J. C. Maxwell published 1868 in Proceedings of the Royal Society of London. The seminal work of R.E. Kalman established system theory as a mathematical discipline in the 1950s. The contribution of U. Oberst, which appeared in 1990, gives fundamental insight for algebraic system theory. A very important algebraic property of the signal space is realized to be highly copious for signals and systems there, namely the property of the signal space to be an injective cogenerator over the underlying operator ring. The goal of algebraic system theory is the structural analysis of dynamical systems using algebraic tools. These systems may arise from various practical problems settled for instance in a scientific, technical or economical area. Classically linear time-invariant systems with field coefficients are studied. In the recent past variations of these systems have proved to be worthy for extended studies. From the applied point of view, there is obviously the interest to consider corresponding generalizations. From the algebraic point of view, some particular settings are very interesting for further investigations since ring theory and homological algebra provide a deep insight. Beyond theoretical studies, the computer algebra machinery allows the enormous benefit of constructive analyses. This thesis elaborates both aspects,the theoretical and the computational, in parallel. It is organized as follows. Chapter 1 and Chapter 2 serve for an extended introduction. System theoretical aspects are provided in Chapter 1. Basic concepts and definitions are presented and furthermore the following chapters are motivated from the system theoretical point of view. Chapter 3 studies systems with coefficients in a finite ring, in contrast to the classical case. The general motivation for this framework stems mainly from communication theory. However, the extension leads to problems like zero-divisors and the principal ideal domain property is lost. Therefore concepts useful for coding fail to generalize straightforwardly. In the field case the so-called predictable degree property is useful for many areas of system theory, ranging from controller parameterization to minimal realizations of linear systems over fields. This property does not carry over directly to the ring case. The paper The predictable degree property and row reducedness for systems over a finite ring by M. Kuijper, R. Pinto, J. W. Polderman and P. Rocha establishes a new framework which allows the adoption of that classical result in a novel setting. By the tool of Gröbner bases these results are extended to a more general framework which additionally allows concrete calculations. For this purpose the notion of the so-called minimal Gröbner p-basis is established and the connection to known results is pointed out. Chapter 4 is focused on one-dimensional systems with time-varying rational coefficients. This leads to the non-commutative operator ring called rational Weyl algebra which is a principal ideal domain. Therefore the non-commutative analogon to the Smith form, the so-called Jacobson form, exists. This normal form can be used to obtain a decomposition into a controllable and an autonomous subsystem of the corresponding linear abstract system. Furthermore the order of the underlying ordinary differential equation system is obtained directly. But computational problems known from the commutative counterpart even increase due to the non-commutative structure, namely the explosive growth of the coefficients. A novel approach which can be applied in a completely fraction free framework is presented in this chapter. This approach shows first how to obtain a decoupled form. It should be stressed that this decoupled form may even be interesting by itself. Further we show how to obtain a normal form from the decoupled form. The implementation is realized as a library called jacobson.lib for the computer algebra system Singular::Plural. This implementation is compared with all implementations which are available to the best of our knowledge. A behavioral approach to linear exact modeling is formulated for one-dimensional systems with constant coefficients by J.C. Willems. This problem of system identification is extended to polynomial-exponential signals in a multi-dimensional time-varying model class in Chapter 5. These model classes are summarized in the so-called Ore algebras. The idea of this approach is to derive a model describing the observed data and containing as much information as possible. It turns out that the particular model classes yield a very precise description, as pointed out in the case of continuous systems. Two alternative possibilities to calculate the models will be presented, one of them working in a purely commutative framework

Journal de Théorie des Nombres de Bordeaux 19 (2007), 683–701

S-extremal strongly modular lattice