Robust Control Theory in Hilbert Space

Feintuch, Avraham

doi:10.1007/978-1-4612-0591-3

Cited by 92 publications

(122 citation statements)

References 3 publications

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“…This result also applies to solve the robustness problem of feedback systems in the gap metric (38) in the TV case as outlined in (11; 21; 33), since the latter was shown in (11) to be equivalent to a special version of the mixed sensitivity problem (20).…”

Section: Theorem 3 Introduce the Orthogonal Projection π As Followsmentioning

confidence: 84%

Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach

Djouadi¹

2011

Recent Advances in Robust Control - Theory and Applications in Robotics and Electromechanics

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Section: Theorem 3 Introduce the Orthogonal Projection π As Followsmentioning

confidence: 84%

Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach

Djouadi¹

2011

Recent Advances in Robust Control - Theory and Applications in Robotics and Electromechanics

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“…It is possible to extend the definition of positive operator [10] on Hilbert spaces to positive operators on the corresponding space of truncated functions.…”

Section: Some Preliminariesmentioning

confidence: 99%

“…∀ > 0, with ≥ 0 since is an accretive operator on 2 (R 0+ ; R ), such that the real constant > 0 if the operator is strongly accretive, where ( ) > 0 is the minimum modulus of since it is one-to-one and of closed range, [10]. Now, (51) implies that : 2 (R 0+ ; R ) → 2 (R 0+ ; R ) is asymptotically accretive, incrementally asymptotically positive, incrementally asymptotically passive (since the operator is self-adjoint), asymptotically positive, and asymptotically passive, since, in addition, the operator maps "0" into "0" and lim →∞ inf ( ) ≥ 0.…”

Section: Remark 18 Note That Theorem 14 and Corollaries 16 And 17mentioning

confidence: 99%

“…It is wellknown that the asymptotic hyperstability formalism covers particular cases, the so-called Lure's and Popov's absolute stability problems. See, for instance, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. On the other hand, the so-called passivity property of dynamic systems can be described in the time-domain in terms of evolution of the so-called storage functions [6] and translates in the global Lypunov stability of all the feedback systems integrated by a feed-forward hyperstable controlled system and any controller belonging to the class of controllers satisfying a Popov's type inequality.…”

Section: Introductionmentioning

confidence: 99%

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On Some Relations between Accretive, Positive, and Pseudocontractive Operators and Passivity Results in Hilbert Spaces and Nonlinear Dynamic Systems

Sen

2017

Discrete Dynamics in Nature and Society

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This paper investigates some parallel relations between the operators ( − ) and in Hilbert spaces in such a way that the pseudocontractivity, asymptotic pseudocontractivity, and asymptotic pseudocontractivity in the intermediate sense of one of them are equivalent to the accretivity, asymptotic accretivity, and asymptotic accretivity in the intermediate sense of the other operator. If the operators are self-adjoint then the obtained accretivity-type properties are also passivity-type properties. Such properties are very relevant in stability theory since they refer to global stability properties of passive feed-forward, in general, nonlinear, and time-varying controlled systems controlled via feedback by elements in a very general class of passive, in general, nonlinear, and time-varying controllers. These results allow the direct generalization of passivity results in controlled dynamic systems to wide classes of tandems of controlled systems and their controllers, described by -operators, and their parallel interpretations as pseudocontractive properties of their counterpart ( − )-operators. Some of the obtained results are also directly related to input-passivity, output-passivity, and hyperstability properties in controlled dynamic systems. Some illustrative examples are also given in the framework of dynamic systems described by extended square-integrable input and output signals.

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“…We would like to thank both Prof. V. Blondel of Louvainla-Neuve University (Belgium) and Prof. A. Feintuch of the Ben-Gurion University (Israel) for interesting discussions on stable range and strong stabilization. Both of them already knew that the concept of stable range was interesting for the study of strong stabilization [3,9,10] (see also the bibliography of [2] and Chapter 6 of [8]). Independently and quite recently, we were led to the concept of stable range while we were developing a K-theoretical approach to stabilization problems (see the forthcoming [29] for more information).…”

mentioning

confidence: 99%

On a General Structure of the Stabilizing Controllers Based on Stable Range

Quadrat¹

2004

SIAM J. Control Optim.

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Abstract. In this paper, we prove that some stabilizing controllers of a plant, which admits a left/right-coprime factorization, have a special form where their stable and unstable parts are separated. The dimension of the unstable part depends on the algebraic concept of stable range of the ring A of SISO stable plants. Moreover, we prove that, if the stable range of A is equal to 1, then every plant-defined by a transfer matrix with entries in the quotient field of A and admitting a left/right-coprime factorization-can be stabilized by a stable controller (strong stabilization). In particular, using a result of 1. Introduction. The fractional representation approach to analysis and synthesis problems was developed in the eighties in order to express in a unique mathematical framework several questions on stabilization problems. In that framework, we can study internal stabilization (existence of an internally stabilizing controller), parametrization of all stabilizing controllers, strong stabilization (possibility of stabilizing a plant by means of a stable controller), simultaneous stabilization (possibility of stabilizing a set of plants by means of a single controller), metrics of robustness (gap or graph topologies), H ∞ or H 2 -optimal controllers, etc. See [2, 6, 42] for more details.Recently, the reformulation of the fractional representation approach to analysis and synthesis problems within an algebraic analysis approach has allowed us to obtain new necessary and sufficient conditions for internal stabilizability and for the existence of (weakly) left/right/doubly coprime factorizations in the general setting [25,26,24]. Moreover, all the rings of SISO stable plants (used in this framework) over which one of the previous properties is satisfied were completely characterized [25,26,24]. In [27,28], a new parametrization of all stabilizing controllers of a stabilizable plant was developed. It generalizes the Youla-Kučera parametrization [42] for stabilizable plants which do not necessarily admit doubly coprime factorizations. All these results show that a natural mathematical framework for the study of stabilization problems is the so-called K-theory [22,32]. See [29] for more details.

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Robust Control Theory in Hilbert Space

Abstract: Library of Congress Cataloging-in-Publication Data Feintuch, Avraham. Robust control theory in Hilbert space / Avraham Feintuch. p. cm. -(Applied mathematical sciences ; 130) IncJudes bibliographical referenees (p.

Cited by 92 publications

References 3 publications

Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach

Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach

On Some Relations between Accretive, Positive, and Pseudocontractive Operators and Passivity Results in Hilbert Spaces and Nonlinear Dynamic Systems

On a General Structure of the Stabilizing Controllers Based on Stable Range

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