An algebraic analysis approach to linear time-varying systems

Zerz, Eva

doi:10.1093/imamci/dni047

Cited by 45 publications

(69 citation statements)

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“…In the continuous case the signal space has to be enlarged to become injective over the larger ring of differential operators (Fröhler and Oberst 1998;Zerz 2006;Bourlès and Marinescu 2011).…”

Section: 4)mentioning

confidence: 99%

The injectivity of the canonical signal module for multidimensional linear systems of difference equations with variable coefficients

Bourlès

Marinescu

Oberst

2015

Multidim Syst Sign Process

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We consider discrete behaviors with varying coefficients. Our results are new also for one-dimensional systems over the time-axis of natural numbers and for varying coefficients in a field, we derive the results, however, in much greater generality: Instead of the natural numbers we use an arbitrary submonoid N of an abelian group, for instance the standard multidimensional lattice of r-dimensional vectors of natural numbers or integers. We replace the base field by any commutative self-injective ring F, for instance a direct product of fields or a quasi-Frobenius ring or a finite factor ring of the integers. The F-module W of functions from N to F is the canonical discrete signal module and is a left module over the natural associated noncommutative ring A of difference operators with variable coefficients. Our main result states that this module is injective and therefore satisfies the fundamental principle: An inhomogeneous system of linear difference equations with variable coefficients has a solution if and only if the right side satisfies the canonical compatibility conditions. We also show that for the typical cases of partial difference equations and in contrast to the case of constant coefficients the A-module W is not a cogenerator. We also generalize the standard one-dimensional theory for periodic coefficients to the multidimensional situation by invoking Morita equivalence.Dedicated to Professor Ettore Fornasini on the occasion of his seventieth birthday. B U. Oberst

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Section: 4)mentioning

confidence: 99%

The injectivity of the canonical signal module for multidimensional linear systems of difference equations with variable coefficients

Bourlès

Marinescu

Oberst

2015

Multidim Syst Sign Process

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“…Then V (x, x,u ) is a non-commutative left and right principal ideal domain [4], [21]. A matrix U E Va ( x" ) is called x,x,u unimodular if there exists a matrix U-1 E Va ( x" ) with…”

Section: Algebraic Observabilitymentioning

confidence: 99%

“…Remark 1: The above normal form is called Smith form or Jacobson form [4], [21]. Since V (x, x,u ) is even Euclidean [4], the matrices U (x, x,u ) and V (x, x,u ) can be obtained by repeating elementary row and column operations for the…”

Section: Algebraic Observabilitymentioning

confidence: 99%

Flatness-based tracking control of nonlinear differential algebraic systems with geometric index one

Satō

2013

52nd IEEE Conference on Decision and Control

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Flatness-based tracking control design with ob server is studied for nonlinear differential algebraic systems with geometric index one. It is shown that a concept of algebraic observability is useful for observer design of a flatness-based tracking control. Through a simple example of an algebraically observable system, it is demonstrated that a proposal flatness based tracking control design with observer is effective.

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“…Several authors have studied systems of this type (Bourlès, 2005;Ilchmann and Mehrmann, 2005;Pommaret and Quadrat, 1998;Zerz, 2006). The main tool is a non-commutative analogue of the Smith form which is known as the Jacobson form.…”

Section: Continuous Time-varying Systemsmentioning

confidence: 99%

Behavioral Systems Theory: A Survey

Zerz¹

2008

International Journal of Applied Mathematics and Computer Science

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We survey the so-called behavioral approach to systems and control theory, which was founded by J. C. Willems and his school. The central idea of behavioral systems theory is to put the focus on the set of trajectories of a dynamical system rather than on a specific set of equations modelling the underlying phenomenon. Moreover, all signal components are treated on an equal footing at first, and their partition into inputs and outputs is derived from the system law, in a way that admits several valid cause-effect interpretations, in general.

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An algebraic analysis approach to linear time-varying systems

Cited by 45 publications

References 0 publications

The injectivity of the canonical signal module for multidimensional linear systems of difference equations with variable coefficients

The injectivity of the canonical signal module for multidimensional linear systems of difference equations with variable coefficients

Flatness-based tracking control of nonlinear differential algebraic systems with geometric index one

Behavioral Systems Theory: A Survey

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