Feedback stabilization over commutative rings with no right-/left-coprime factorizations

Mori, Kazuyoshi

doi:10.1109/cdc.1999.832920

Cited by 5 publications

(12 citation statements)

References 9 publications

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“…Moreover, 12) and thus, by Proposition 4.4 of [17], M is a projective A-module of rank 2. Thus, by Theorem 3.2, P is internally stabilizable.…”

Section: Internal Stabilizationmentioning

confidence: 94%

“…Therefore, it is not possible to parametrize all their stabilizing controllers by means of the Youla-Kučera parametrization [4,28]. These results allow us to explain the counterexamples exhibited in [1,12]. We prove that, over a projective-free domain A (e.g., H ∞ (C + ), RH ∞ ), every stabilizable system admits doubly coprime factorizations.…”

mentioning

confidence: 94%

“…the other hand, the ones developed in [12,13,24,26,27,28,29]. We refer to [17] for the development of the algebraic analysis approach used in this second part, as well as for some results that will be continually used in what follows.…”

mentioning

confidence: 99%

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The Fractional Representation Approach to Synthesis Problems: An Algebraic Analysis Viewpoint Part I: (Weakly) Doubly Coprime Factorizations

Quadrat¹

2003

SIAM J. Control Optim.

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Abstract. In this second part of the paper [A. Quadrat, SIAM J. Control Optim., 40 (2003), pp. 266-299], we show how to reformulate the fractional representation approach to synthesis problems within an algebraic analysis framework. In terms of modules, we give necessary and sufficient conditions for internal stabilizability. Moreover, we characterize all the integral domains A of SISO stable plants such that every MIMO plant-defined by means of a transfer matrix whose entries belong to the quotient field K = Q(A) of A-is internally stabilizable. Finally, we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in [M. C. Smith, IEEE Trans. Automat. Control, 34 (1989) Introduction. Using the algebraic analysis viewpoint of the fractional representation approach to analysis and synthesis problems [5,28,29], developed in the first part of the paper [17], we give necessary and sufficient conditions for internal stabilizability. Moreover, using these results, we prove that every multi-input multi-output (MIMO) plant-defined by means of a transfer matrixT are matrices whose entries belong to an integral domain A of single input single output (SISO) stable plants-is internally stabilizable iff A is a Prüfer domain [6,23]. From the fact that the intersection between coherent Sylvester domains (see [17] for more details) and Prüfer domains are just Bézout domains, we also recover the result of Vidyasagar [29]: every MIMO plant admits doubly coprime factorizations iff A is a Bézout domain. Hence, if the algebra A is a Prüfer domain but not a Bézout domain, there exist plants which are internally stabilizable but fail to admit doubly coprime factorizations. Therefore, it is not possible to parametrize all their stabilizing controllers by means of the Youla-Kučera parametrization [4,28]. These results allow us to explain the counterexamples exhibited in [1,12]. We prove that, over a projective-free domain A (e.g., H ∞ (C + ), RH ∞ ), every stabilizable system admits doubly coprime factorizations. Finally, we show that the previous results allow us to recover, on the one hand, the results of [25] and, on

show abstract

“…Moreover, 12) and thus, by Proposition 4.4 of [17], M is a projective A-module of rank 2. Thus, by Theorem 3.2, P is internally stabilizable.…”

Section: Internal Stabilizationmentioning

confidence: 94%

mentioning

confidence: 94%

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The Fractional Representation Approach to Synthesis Problems: An Algebraic Analysis Viewpoint Part I: (Weakly) Doubly Coprime Factorizations

Quadrat¹

2003

SIAM J. Control Optim.

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“…The common feature of the results is that they require the existence of coprime factorizations. This is problematic since all plants do not possess coprime factorizations [1,14], or their existence is not known [10,15].…”

Section: Introductionmentioning

confidence: 99%

A fractional representation approach to the robust regulation problem for SISO systems

Laakkonen

Quadrat

2017

Systems & Control Letters

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The purpose of this article is to develop a new approach to the robust regulation problem for plants which do not necessarily admit coprime factorizations. The approach is purely algebraic and allows us dealing with a very general class of systems in a unique simple framework. We formulate the famous internal model principle in a form suitable for plants defined by fractional representations which are not necessarily coprime factorizations. By using the internal model principle, we are able to give necessary and sufficient solvability conditions for the robust regulation problem and to parameterize all robustly regulating controllers.

show abstract

“…In the following we show that (15) in Theorem 4.2 still holds even in the case where there do not exist right-/left-coprime factorizations over A.…”

Section: The General Casementioning

confidence: 75%

A parameterization of stabilizing controllers over commutative rings

Mori

Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304)

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Feedback stabilization over commutative rings with no right-/left-coprime factorizations

Cited by 5 publications

References 9 publications

The Fractional Representation Approach to Synthesis Problems: An Algebraic Analysis Viewpoint Part I: (Weakly) Doubly Coprime Factorizations

The Fractional Representation Approach to Synthesis Problems: An Algebraic Analysis Viewpoint Part I: (Weakly) Doubly Coprime Factorizations

A fractional representation approach to the robust regulation problem for SISO systems

A parameterization of stabilizing controllers over commutative rings

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