Representations and structural properties of periodic systems

Aleixo, José; Polderman, Jan Willem; Rocha, Paula

doi:10.1016/j.automatica.2007.03.013

Cited by 14 publications

(15 citation statements)

References 8 publications

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“…On the other hand, ℓ can clearly be reconstructed from w 1 =2ℓ and w 2 =3ℓ. Similar phenomena can also be observed with periodic systems 16, and have already been reported for systems over rings in 17.…”

Section: Controllabilitysupporting

confidence: 83%

On multidimensional convolutional codes and controllability properties of multidimensional systems over finite rings

Zerz

2010

Asian Journal of Control

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Section: Controllabilitysupporting

confidence: 83%

On multidimensional convolutional codes and controllability properties of multidimensional systems over finite rings

Zerz

2010

Asian Journal of Control

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“…If F is a field, N = Z ⊃ L = Zn and B ⊂ W N × is a closed subspace the invariance means q n • B = B. This is the case treated in Kuijper and Willems (1997) and Aleixo et al (2007a).…”

Section: Relation To Previous Workmentioning

confidence: 96%

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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“…The main results of this paper described above are contained in Sections 5 and 6. Section 3 describes the module-behavior duality for periodic behaviors on the time-axis N. For the time-axis Z the theory is simpler and was treated with similar methods in [5] where we also explained the relation with previous work [11], [1], [2]. In Section 4 we apply Morita theory to derive essential notions for and properties of periodic behaviors and their dual f.g. B-left modules, for instance autonomy, controllability, the existence and characterization of input/output (IO) structures and left and right coprime factorizations.…”

Section: ]mentioning

confidence: 99%

“…They solve open problems that were raised in [1, §7]. In contrast to [11], [1] and [5] and in accordance with [9], [2] and [16] (in the LTI case) we consider N -periodic systems on the time-axis N ∋ t of natural numbers and not on Z. A periodic system is a linear time-varying (LTV) system whose coefficient functions a are N -periodic, i.e., satisfy a(t + N ) = a(t) for t ∈ N. If F denotes any field or, in Sections 5 and 6, the field R or C of real or complex numbers, the coefficient functions form the commutative algebra F Z/ZN of functions from Z/ZN to F where we pose a(t) := a(t + ZN ) for t ∈ N. The monoid N acts on a ∈ F Z/ZN via algebra isomorphisms by (j • a)(t + ZN ) := a(j + t + ZN ).…”

Section: Introductionmentioning

confidence: 99%

Robust stabilization of discrete-time periodic linear systems for tracking and disturbance rejection

Bourlès

Oberst

2016

Math. Control Signals Syst.

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In analogy to the Kučera-Youla parametrization we construct and parametrize all stabilizing controllers of a stabilizable linear periodic discrete-time input/output system, the plant. We establish a necessary and sufficient algebraic condition for the existence of controllers among these for which the output of the plant tracks a given reference signal in spite of disturbance signals on the input and the output of the plant. With a minor additional assumption the tracking stabilizing controllers are robust. As in the linear time-invariant (LTI) case the reference and disturbance signals are assumed to be generated by an autonomous system. Our results are the analogues for periodic behaviors of the corresponding LTI results of Vidyasagar. A completely different approach to stabilization and control of discrete periodic systems was developed by Bittanti and Colaneri. We derive a categorical duality between periodic behaviors over the time-axis of natural numbers and finitely generated modules over a suitable noncommutative ring of difference operators and use this for the proof of the main stabilization and control results. Morita's theory of equivalences between module categories is employed as an essential algebraic tool. All results of the paper are constructive.

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Representations and structural properties of periodic systems

Cited by 14 publications

References 8 publications

On multidimensional convolutional codes and controllability properties of multidimensional systems over finite rings

On multidimensional convolutional codes and controllability properties of multidimensional systems over finite rings

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

Robust stabilization of discrete-time periodic linear systems for tracking and disturbance rejection

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