Simultaneous stabilization of linear systems and interpolation with rational functions

Blondel, Vincent D.

doi:10.1007/978-1-4471-0807-8_12

Cited by 13 publications

(9 citation statements)

References 10 publications

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“…The quantity β(p) can be computed by standard software. 3 It is the reciprocal of the H ∞ norm for the state space realization (A, B, C, D), where A, B, C and D are respectively the companion matrix for p (with its negated coefficients in the first column), the identity matrix, the first row of the identity matrix, and a zero row, since then C(sI − A)…”

Section: Experimental Analysismentioning

confidence: 99%

“…But we also believe that it is very useful to focus on a specific, challenging example, to demonstrate the potential of the approach. More than a decade ago, Blondel [2, p.150], [3] Blondel also offered a kilo of Belgian chocolate for a solution of a special case: Problem 1.2: Show whether or not 0.9 is in the range of values for δ for which stabilization is possible.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, Problem 1.1 reduces to determining δ * . Results on the range of analytic functions proved in [6] imply that δ * < 0.99998 [3], [4], and it is known from the experiments reported in [13] that δ * > 0.937.…”

Section: Introductionmentioning

confidence: 99%

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Stabilization via Nonsmooth, Nonconvex Optimization

Burke

Henrion

Lewis

et al. 2006

IEEE Trans. Automat. Contr.

138

101

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Nonsmooth variational analysis and related computational methods are powerful tools that can be effectively applied to identify local minimizers of nonconvex optimization problems arising in fixedorder controller design. We support this claim by applying nonsmooth analysis and methods to a challenging "Belgian chocolate" stabilization problem posed in 1994: find a stable, minimum phase, rational controller that stabilizes a specified second-order plant. Although easily stated, this particular problem remained unsolved until 2002, when a solution was found using an 11th order controller.Our computational methods find a stabilizing 3rd order controller without difficulty, suggesting explicit formulas for the controller and for the closed loop system, which has only one pole with multiplicity 5. Furthermore, our analytical techniques prove that this controller is locally optimal in the sense that there is no nearby controller with the same order for which the closed loop system has all its poles further left in the complex plane. Although the focus of the paper is stabilization, once a stabilizing controller is obtained, the same computational techniques can be used to optimize various measures of the closed loop system, including its complex stability radius or H ∞ performance.

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Section: Experimental Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stabilization via Nonsmooth, Nonconvex Optimization

Burke

Henrion

Lewis

et al. 2006

IEEE Trans. Automat. Contr.

138

101

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show abstract

“…Since this issue is deeply concerned with an interesting and meaningful nature on the notion of the stabilization from the system theoretic points of view (cf. [4] [5]), the simultaneous stabilization of three systems has attracted many attentions of researchers in the control and system theory from not only practical but also theoretical points of view. In the standard system theory, there are many studies related to the simultaneous stabilization problem.…”

Section: Introductionmentioning

confidence: 99%

“…In the standard system theory, there are many studies related to the simultaneous stabilization problem. As representative works on this issue, we can cite the references [1], [2], [3], [4], [5], [7], [8], [9], [17], [19]. Particularly, as stated in [2], it was shown that the simultaneous stabilizability of three linear systems is rationally undecidable, that is This work is not supported by any organizations All of the authors are with Graduate School of Engineering Science, Osaka University, Machikaneyama 1-3, Toyonaka, Osaka, 560-8531, Japan kaneko@ft-lab.sys.es.osaka-u.ac.jp to say, it is not possible to find necessary and sufficient conditions for the simultaneous stabilization of three linear systems in the sense that there exist no ways for checking the simultaneous stabilizability by algebraic, logical, and sign test operations.…”

Section: Introductionmentioning

confidence: 99%

A sufficient condition for a triple of linear systems to be simultaneously stabilizable in a behavioral framework

Kaneko

Fujii²

Proceedings of the 44th IEEE Conference on Decision and Control

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In this paper, we address simultaneous stabilization problem for a triple of linear systems within a behavioral framework. First, we provide a necessary and sufficient condition for a given pair of linear systems to be simultaneously stabilizable in a behavioral framework. Next, we show that a simultaneous stabilizer has a symmetric structure, which is also a self-standing interesting result from the theoretical points of view. By using these results, we give a new necessary and sufficient condition for a given pair of linear systems to be simultaneously stabilizable. This new condition is described in terms of the solvability of polynomial matrix equations consisting of polynomial matrices inducing kernel or image representations of the behaviors of given two behaviors. Finally, by using this new condition, we give a sufficient condition for a triple of linear systems to be simultaneously stabilizable in terms of the behaviors.

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Simultaneous Stabilization Problem in a Behavioral Framework

Kaneko

2015

Mathematical Control Theory II

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Simultaneous stabilization of linear systems and interpolation with rational functions

Cited by 13 publications

References 10 publications

Stabilization via Nonsmooth, Nonconvex Optimization

Stabilization via Nonsmooth, Nonconvex Optimization

A sufficient condition for a triple of linear systems to be simultaneously stabilizable in a behavioral framework

Simultaneous Stabilization Problem in a Behavioral Framework

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