The uniform distribution: A rigorous justification for its use in robustness analysis

Barmish, B. Ross; Lagoa, Constantino

doi:10.1007/bf01211503

Cited by 189 publications

(113 citation statements)

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“…A justiÿcation of the choice of the uniform density may be found for instance in Bai, Tempo, and Fu (1998) and Barmish and Lagoa (1997). However, the results in this paper hold not only for the uniform distribution, but for the more general class of unitarily invariant distributions over B , see (Calaÿore et al, 2000a).…”

Section: Introductionmentioning

confidence: 82%

A probabilistic framework for problems with real structured uncertainty in systems and control

Calafiore

Dabbene

2002

Automatica

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The objective of this paper is twofold. First, the problem of generation of real random matrix samples with uniform distribution in structured (spectral) norm bounded sets is studied. This includes an analysis of the distribution of the singular values of uniformly distributed real matrices, and an e cient (i.e. polynomial-time) algorithm for their generation. Second, it is shown how the developed techniques may be used to solve in a probabilistic setting several hard problems involving systems subject to real structured uncertainty. ?

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Section: Introductionmentioning

confidence: 82%

A probabilistic framework for problems with real structured uncertainty in systems and control

Calafiore

Dabbene

2002

Automatica

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“…Similar distributional variations on the robustness radius and its various special cases have been studied before, in [5,6]. In this article, we provide a general definition for the concept, and afterwards show how to reduce the radius computation for some specific families of distributions to computing it for significantly smaller subfamilies.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic Robustness Radii with n:th Order Convex Distributions

Kangasniemi¹,

Nikkarila²

2016

IJMO

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Abstract-We analyze a probabilistic variation on the classic robustness radius. Instead of measuring the distance to the closest destabilizing state, we look at a set of probability distributions around our initial state. The probabilistic radius is the radius of the largest sphere, for which none of the distributions supported within it takes the expected values of system parameters outside a prescribed set. We show that when the set of acceptable parameter values is closed and convex, the radius for the family of distributions which are nondecreasing with respect to distance reduces to the same radius for uniform distributions. We generalize this result for distributions of higher order convexity or concavity with respect to distance, obtaining an equal radius using a simple family of polynomials.Index Terms-Robustness radii, distributional robustness, decision analysis, control theory, aircraft control.

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“…In order to overcome these difficulties, a different paradigm has recently emerged. This new paradigm studies uncertain feedback systems from a probabilistic point of view; see, e.g., [3], [13], [25], [31], [41], and [43]; additional references can be found in the survey papers [38] and [39]. This framework is not alternative to worst case robust control, but it provides useful and complementary information to the control engineer.…”

Section: Introductionmentioning

confidence: 99%

Randomized algorithms for probabilistic robustness with real and complex structured uncertainty

Calafiore

Dabbene

Tempo

2000

IEEE Trans. Automat. Contr.

146

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Abstract-In recent years, there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems. Unlike classical worst case methods, these algorithms provide probabilistic estimates assessing, for instance, if a certain design specification is met with a given probability. One of the advantages of this approach is that the robustness margins can be often increased by a considerable amount, at the expense of a small risk. In this sense, randomized algorithms may be used by the control engineer together with standard worst case methods to obtain additional useful information.The applicability of these probabilistic methods to robust control is presently limited by the fact that the sample generation is feasible only in very special cases which include systems affected by real parametric uncertainty bounded in rectangles or spheres. Sampling in more general uncertainty sets is generally performed through overbounding, at the expense of an exponential rejection rate.In this paper, randomized algorithms for stability and performance of linear time invariant uncertain systems described by a general -1 configuration are studied. In particular, efficient polynomial-time algorithms for uncertainty structures 1 consisting of an arbitrary number of full complex blocks and uncertain parameters are developed.

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The uniform distribution: A rigorous justification for its use in robustness analysis

Cited by 189 publications

References 6 publications

A probabilistic framework for problems with real structured uncertainty in systems and control

A probabilistic framework for problems with real structured uncertainty in systems and control

Probabilistic Robustness Radii with n:th Order Convex Distributions

Randomized algorithms for probabilistic robustness with real and complex structured uncertainty

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