Computational complexity of μ calculation

Braatz, Richard P.; Young, Peter M.; Doyle, John C.; Morari, Manfred

doi:10.1109/9.284879

Cited by 324 publications

(138 citation statements)

References 8 publications

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“…Extensive computational experience suggests that the bound is often adequate for applications in robust control [29]. This is important since the computation of µ is an NP-hard problem [5]. If µ ∆ (M ) denotes the convex upper bound of µ ∆ (M ) and M ∈ C n×n , it has been shown that the ratio g n := µ ∆ (M )/µ ∆ (M ) grows no faster than a linear function of n. In …”

Section: Theorem 21 [29]mentioning

confidence: 99%

“…Although the problem is computationally NP hard [5], [16], tight bounds have been reported in the robust control literature using exclusively convex programming techniques. In a recent work [17] tests have been developed for certifying the absence of a duality gap between the structured singular value and its convex upper bound, along with a systematic procedure for reducing the duality gap when this is non-zero.…”

Section: Introductionmentioning

confidence: 99%

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Nearest common root of polynomials, approximate greatest common divisor and the structured singular value

Halikias

Galanis

Karcanias

et al. 2012

IMA Journal of Mathematical Control and Information

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This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent AbstractIn this paper the following problem is considered: Given two co-prime polynomials, find the smallest perturbation in the magnitude of their coefficients, such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the calculation of the structured singular value of a matrix arising in robust control and a numerical solution to the problem is developed.A simple numerical example illustrates the effectiveness of the method for two polynomials of low degree. Finally, problems involving the calculation of the approximate greatest common divisor (GCD) of univariate polynomials are considered, by proposing a generalization of the definition of the structured singular value involving additional rank constraints.Keywords: Approximate common root of polynomials, approximate GCD, Sylvester resultant matrix, structured singular value, distance to singularity, structured approximations.

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Section: Theorem 21 [29]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nearest common root of polynomials, approximate greatest common divisor and the structured singular value

Halikias

Galanis

Karcanias

et al. 2012

IMA Journal of Mathematical Control and Information

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“…First, it proves undecidability of a certain robust stability problem under time-varying uncertainty. In that sense, it complements negative (NP-hardness) results on the robust stability of linear systems in the presence of time-invariant uncertainty [5,16,18,21]. Second, it leads to an undecidability result for a simple class of hybrid systems.…”

Section: Introductionmentioning

confidence: 68%

The boundedness of all products of a pair of matrices is undecidable

Blondel

Tsitsiklis

2000

Systems & Control Letters

184

139

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We show that the boundedness of the set of all products of a given pair of rational matrices is undecidable. Furthermore, we show that the joint (or generalized) spectral radius ( ) is not computable because testing whether ( )61 is an undecidable problem. As a consequence, the robust stability of linear systems under time-varying perturbations is undecidable, and the same is true for the stability of a simple class of hybrid systems. We also discuss some connections with the so-called "ÿniteness conjecture". Our results are based on a simple reduction from the emptiness problem for probabilistic ÿnite automata, which is known to be undecidable.

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“…As skew µ is a generalization of µ, the calculation of the exact value of skew µ (like µ) is NP hard [16]. It is therefore necessary to consider computationally efficient algorithms that determine upper and lower bounds on skew µ.…”

Section: Theorem 4 (Bounded Frequency Test)mentioning

confidence: 99%

State‐space µ analysis for an experimental drive‐by‐wire vehicle

Halton

Hayes

Iordanov

2008

Intl J Robust & Nonlinear

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SUMMARYThis paper considers the application of the skewed structured singular value to the robust stability of systems subject to strictly real parametric uncertainty. Three state-space formulations that counteract the discontinuous nature of this problem are detailed. It is shown that the calculation of the supremum of the structured singular value over a frequency range using these formulations transforms into a single skewed structured singular value calculation. Like the structured singular value, the exact calculation of the skewed structured singular value is an NP-hard problem. In this work, two efficient algorithms that determine upper and lower bounds on the skewed structured singular value are presented. These algorithms are critically assessed using a series of robustness analysis tests on a safety-critical experimental drive-by-wire vehicle.

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Computational complexity of μ calculation

Cited by 324 publications

References 8 publications

Nearest common root of polynomials, approximate greatest common divisor and the structured singular value

Nearest common root of polynomials, approximate greatest common divisor and the structured singular value

The boundedness of all products of a pair of matrices is undecidable

State‐space µ analysis for an experimental drive‐by‐wire vehicle

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