Effective algorithms for parametrizing linear control systems over Ore algebras

Chyzak, Frédéric; Quadrat, Alban; Robertz, Daniel

doi:10.1007/s00200-005-0188-6

Cited by 100 publications

(215 citation statements)

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“…In Fliess et al (2010), there is a remark about the possibility to extend this framework to ODEs with polynomial coefficients (see Remark 1 in page 133). Being not aware of this remark but inspired by works developed in the symbolic computation community (Chyzak et al (2005)), in Ushirobira et al (2016), we show that the algebraic approach can be extended to signal x which is defined by an OD system with polynomial coefficients. Indeed, within the algebraic approach to parameter estimation, the signal is studied in the frequency domain by means of the operational calculus.…”

Section: Introductionmentioning

confidence: 67%

“…Classes of functional equations can be studied by means of the so-called Ore algebras. For more details, see Chyzak et al (2005) and the references therein.…”

Section: Classes Of Signals X Under Studymentioning

confidence: 99%

“…Let us suppose that K is a field and let D = K(t) ∂ be the noncommutative ring of OD operators in ∂ t with coefficients in the field K(t) of rational functions in t with coefficients in K. An element of D is of the form (Chyzak et al (2005)). If x is a D-finite function, then the set ∂ i t x = x (i) i=0,...,n of cardinality n+1 admits at least one relation over K(t), i.e., there exist a i ∈ K(t) for i = 0, .…”

Section: Estimation Of An Expansion Of X Into a Basismentioning

confidence: 99%

“…Indeed, within the algebraic approach to parameter estimation, the signal is studied in the frequency domain by means of the operational calculus. The Laplace transform can still be used for OD systems with polynomial coefficients since the time variable t is then transformed into −∂ s , where ∂ s = d ds is the derivation with respect to the Laplace variable s. Within the algebraic analysis approach (Kashiwara et al (1986); Chyzak et al (2005)), the Laplace transform is an automorphism of the Weyl algebra A 1 (k) = k t, ∂ t | ∂ t t = t ∂ t + 1 of OD operators with polynomial coefficients in t over a field k of characteristic zero (e.g., k = Q, R), i.e., the k-linear map L : A 1 (k) −→ A 1 (k) defined by L (t) = −∂ s and L (∂ t ) = s is an automorphism of k-algebras, where the target Weyl algebra is in the complex variable s, i.e., A 1 (k) = k s, ∂ s | ∂ s s = s ∂ s + 1 . The algebraic approach to parameter estimation can then be developed similarly to the linear time-invariant OD systems.…”

Section: Introductionmentioning

confidence: 99%

“…. , polynomials (Abramowitz et al (1964); Chyzak et al (2005)). Note that the algebraic parameter estimation problem was studied for Taylor expansion series (Mboup et al (2009)) and initiated for Fourier expansions, i.e., expansions into the orthogonal basis {e i n θ } n∈Z of L 2 (T), where T = {z ∈ C | |z| = 1} is the unit torus of C (Ushirobira et al (2012(Ushirobira et al ( , 2013).…”

Section: Introductionmentioning

confidence: 99%

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Towards an effective study of the algebraic parameter estimation problem

Quadrat

2017

IFAC-PapersOnLine

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The paper aims at developing the first steps toward a symbolic computation approach to the algebraic parameter estimation problem defined by Fliess and Sira-Ramirez and their coauthors. In this paper, within the algebraic analysis approach, we first give a general formulation of the algebraic parameter estimation for signals which are defined by ordinary differential equations with polynomial coefficients such as the standard orthogonal polynomials (Chebyshev, Jacobi, Legendre, Laguerre, Hermite, . . . polynomials). Based on a result on holonomic functions, we show that the algebraic parameter estimation problem for a truncated expansion of a function into an orthogonal basis of L 2 defined by orthogonal polynomials can be studied similarly. Then, using symbolic computation methods such as Gröbner basis techniques for (noncommutative) polynomial rings, we first show how to compute ordinary differential operators which annihilate a given polynomial and which contain only certain parameters in their coefficients. Then, we explain how to compute the intersection of the annihilator ideals of two polynomials and characterize the ordinary differential operators which annihilate a first polynomial but not a second one. These results, which are at the core of the algebraic parameter estimation, are implemented in the Non-A package built upon the OreModules software.

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Section: Introductionmentioning

confidence: 67%

“…Classes of functional equations can be studied by means of the so-called Ore algebras. For more details, see Chyzak et al (2005) and the references therein.…”

Section: Classes Of Signals X Under Studymentioning

confidence: 99%

Section: Estimation Of An Expansion Of X Into a Basismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Towards an effective study of the algebraic parameter estimation problem

Quadrat

2017

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The inverse syzygy problem in algebraic systems theory

Seiler

Zerz

2010

Proc Appl Math and Mech

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We present a constructive solution of the inverse syzygy problem over arbitrary coherent rings and show how it can be used to compute certain extension groups. The inverse syzygy problemIn algebraic systems theory, linear systems are mathematically modelled by modules over a ring D (typically a ring of linear differential or difference operators) and control theoretic properties of a system are related to homological properties of the corresponding module. In this note, we are mainly concerned with controllability and show that for a very large class of rings it is equivalent to torsionlessness of the system module. The latter property in turn can be effectively verified via an inverse syzygy problem provided that over D it is possible to solve effectively the direct syzygy problem.Due to lack of space, we can only present some results; for proofs and more details we refer to [1]. From a very theoretical point of view, the inverse syzygy problem was already solved by Auslander and Bridger [2]. An effective solution was first presented by Oberst [3] who also noticed the connection to controllability. Like all subsequent works (see e. g. [4]), our results are based on Oberst's algorithm, but significantly enlarge the class of admissible rings D. This algorithm can be extended to a construction of certain extension groups of the system module. In [5][6][7], one can find discussions of various notions of controllability and their relation to the vanishing of extension groups.In the sequel, we will only assume that D is a coherent ring, i. e. that any finitely generated left or right ideal of D can also be finitely presented (in other words, its syzygy module is also finitely generated). In particular, we do not assume that D is some sort of (possibly non-commutative) polynomial ring and we explicitly allow that D may contain zero divisors: in contrast to previous works, we do not need the existence of a quotient field of D. For a D-module M, we denote by M * = Hom D (M, D) its dual module (note that M * is a right D-module for a left D-module M). All considered modules are assumed to be finitely generated.Let M, N be two free left D-modules and β : M → N a module homomorphism. The direct syzygy problem consists of finding a free left D-module P and a homomorphism α : P → M such that im (α) = ker (β), whereas in the inverse syzygy problem we search for a free left D-module Q and homomorphism γ : N → Q such that im (β) = ker (γ).Since we are dealing with homomorphisms between free modules, these can be represented by matrices with entries in D: we write β(P) = PB with a matrix B of appropriate dimensions and β * (Q * ) = BQ * for the dual map β * : N * → M * . Solving the direct syzygy problem for β corresponds to computing the left syzygies of the rows of B. For many classes of rings, effective algorithms are known for this task; in particular, Gröbner bases can be used for polynomial rings. Oberst's algorithmObviously, the inverse syzygy problem corresponds to the direct problem with the arrows reverted and a dualisa...

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Effective Algebraic Analysis Approach to Linear Systems over Ore Algebras

Cluzeau¹,

Koutschan²,

Quadrat³

et al. 2020

Algebraic and Symbolic Computation Methods in Dynamical Systems

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The purpose of this paper is to present a survey on the effective algebraic analysis approach to linear systems theory with applications to control theory and mathematical physics. In particular, we show how the combination of effective methods of computer algebra − based on Gröbner basis techniques over a class of noncommutative polynomial rings of functional operators called Ore algebras − and constructive aspects of module theory and homological algebra enables the characterization of structural properties of linear functional systems. Algorithms are given and a dedicated implementation, called OREALGEBRAIC-ANALYSIS, based on the Mathematica package HOLONOMICFUNCTIONS, is demonstrated.

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Effective algorithms for parametrizing linear control systems over Ore algebras

Cited by 100 publications

References 25 publications

Towards an effective study of the algebraic parameter estimation problem

Towards an effective study of the algebraic parameter estimation problem

The inverse syzygy problem in algebraic systems theory

Effective Algebraic Analysis Approach to Linear Systems over Ore Algebras

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