A parameterization of stabilizing controllers over commutative rings

Mori, Kazuyoshi

doi:10.1109/cdc.1999.832919

Cited by 3 publications

(19 citation statements)

References 20 publications

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“…Since the plant p is of the single-input single-output, the set I defined in Definition 2.4 of [3] is equal to {{1}, {2}}, say I 1 = {1} and I 2 = {2} as in the proof of Proposition 4.1. Two generalized elementary factors Λ pI1 and Λ pI2 are given as (2).…”

Section: Construction Of Stabilizing Controllersmentioning

confidence: 99%

“…Two generalized elementary factors Λ pI1 and Λ pI2 are given as (2). Since in the cases A = Z[ √ 5i] and A = R[x 2 , x 3 ] any causal transfer functions are stabilizable, Λ pI1 + Λ pI2 = A holds by Theorem 2.1 in [3]. We should find λ I1 ∈ Λ pI1 and λ I2 ∈ Λ pI2 such that λ I1 + λ I2 = 1.…”

Section: Construction Of Stabilizing Controllersmentioning

confidence: 99%

“…From the proof of Theorem 2.1 in [3] (Theorem 3.2 of [2]), a stabilizing controllers of the plant p is given as the following form…”

Section: Construction Of Stabilizing Controllersmentioning

confidence: 99%

“…In the set H(p; A) in (6), q ij is corresponding to the (i, j)-entry of the matrix Q used in § 4 of [3]. One can observe that (1 + √ 5i)h 11 = −2h 12 , so that p = −h 12 h −1 11 .…”

Section: Construction Of Stabilizing Controllersmentioning

confidence: 99%

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

Mori

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

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Section: Construction Of Stabilizing Controllersmentioning

confidence: 99%

Section: Construction Of Stabilizing Controllersmentioning

confidence: 99%

“…From the proof of Theorem 2.1 in [3] (Theorem 3.2 of [2]), a stabilizing controllers of the plant p is given as the following form…”

Section: Construction Of Stabilizing Controllersmentioning

confidence: 99%

Section: Construction Of Stabilizing Controllersmentioning

confidence: 99%

See 2 more Smart Citations

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

Mori

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

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“…Recently the first author [7] has developed a parameterization of stabilizing controllers, which is based on the results of this paper and which does not require coprime…”

Section: Proof Of Theorem 33 We Prove the Following Relations In Ormentioning

confidence: 99%

Feedback stabilization over commutative rings: further study of the coordinate-free approach

Mori

Abe²

Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228)

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Abstract. This paper is concerned with the coordinate-free approach to control systems. The coordinatefree approach is a factorization approach but does not require the coprime factorizations of the plant. We present two criteria for feedback stabilizability for MIMO systems in which transfer functions belong to the total rings of fractions of commutative rings. Both of them are generalizations of Sule's results in [SIAM J. Control Optim., 32-6, 1675-1695(1994)]. The first criterion is expressed in terms of modules generated from a causal plant and does not require the plant to be strictly causal. It shows that if the plant is stabilizable, the modules are projective. The other criterion is expressed in terms of ideals called generalized elementary factors. This gives the stabilizability of a causal plant in terms of the coprimeness of the generalized elementary factors. As an example, a discrete finite-time delay system is considered.

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Feedback Stabilization over Commutative Rings: Further Study of the Coordinate-Free Approach

Mori

Abe²

2001

SIAM J. Control Optim.

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This paper is concerned with the coordinate-free approach to control systems. The coordinatefree approach is a factorization approach but does not require the coprime factorizations of the plant. We present two criteria for feedback stabilizability for MIMO systems in which transfer functions belong to the total rings of fractions of commutative rings. Both of them are generalizations of Sule's results in [SIAM J. Control Optim., 32-6, 1675-1695(1994]. The first criterion is expressed in terms of modules generated from a causal plant and does not require the plant to be strictly causal. It shows that if the plant is stabilizable, the modules are projective. The other criterion is expressed in terms of ideals called generalized elementary factors. This gives the stabilizability of a causal plant in terms of the coprimeness of the generalized elementary factors. As an example, a discrete finite-time delay system is considered. ). 1 rings. Further, we will not use the theory of algebraic geometry.The paper is organized as follows. In § 2, we give mathematical preliminaries, set up the feedback stabilization problem, present the previous results, and define the causality of the transfer functions. In § 3, we state our main results. As a preface to our main results, we also introduce there the notion of the generalized elementary factor of a plant. In § 4, wegive intermediate results which we will utilize in the proof of the main theorem. In § 5, we prove our main theorem. In § 6, we discuss the causality of the stabilizing controllers. Also, in order to make the contents clear, we present examples concerning a discrete finite-time delay system in § 3, 4, 5 in series. Preliminaries.In the following we begin by introducing the notations of commutative rings, matrices, and modules, commonly used in this paper. Then we give in order the formulation of the feedback stabilization problem, the previous results, and the causality of transfer functions. Notations.Commutative Rings. In this paper, we consider that any commutative ring has the identity 1 different from zero. Let R denote a commutative ring. A zerodivisor in R is an element x for which there exists a nonzero y such that xy = 0. In particular, a zerodivisor x is said to be nilpotent if x n = 0 for some positive integer n. The set of all nilpotent elements in R, which is an ideal, is called the nilradical of R. A nonzerodivisor in R is an element which is not a zerodivisor. The total ring of fractions of R is denoted by F (R).The set of all prime ideals of R is called the prime spectrum of R and is denoted by Spec R. The prime spectrum of R is said to be irreducible as a topological space if every non-empty open set is dense in Spec R.We will consider that the set of stable causal transfer functions is a commutative ring, denoted by A. Further, we will use the following three kinds of rings of fractions:

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A parameterization of stabilizing controllers over commutative rings

Cited by 3 publications

References 20 publications

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

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

Feedback stabilization over commutative rings: further study of the coordinate-free approach

Feedback Stabilization over Commutative Rings: Further Study of the Coordinate-Free Approach

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