Pseudospectra and stability radii for analytic matrix functions with application to time-delay systems

Michiels, Wim; Green, Kirk; Wagenknecht, Thomas; Niculescu, Silviu–Iulian

doi:10.1016/j.laa.2006.02.036

Cited by 42 publications

(66 citation statements)

References 15 publications

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“…The -pseudospectrum σ of a matrix A is defined as σ := {λ ∈ C : λ ∈ σ(A + δA), where δA < }, (6) where σ denotes the spectrum, · is an arbitrary matrix norm and δA is a perturbation matrix [19]. It is known, that Eq.…”

Section: Pseudospectra and Stability Radiusmentioning

confidence: 99%

“…If w M → ∞, then no perturbation on the mass matrix M is allowed [19]. This formalism allows the perturbations satisfying…”

Section: Weighted Complex Stability Radiusmentioning

confidence: 99%

“…Real stability radius for time-delay systems was introduced in [21] and in [13]. The formulas for unstructured and weighted structured complex stability radii for time-delay systems were given in [19] and [35], respectively.…”

mentioning

confidence: 99%

“…First, the effect of the uncertainties of the modal parameters are analyzed for time-domain techniques using the stability radii concept based on [24,12,21,19]. Then, inspired by the time-domain concept, a new frequency-domain estimation of the robust stability boundaries is proposed using the measured FRFs directly.…”

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confidence: 99%

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Robust stability analysis of machining operations

Hajdu

Insperger

Stépàn

2016

Int J Adv Manuf Technol

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Prediction of machine tool chatter requires the characterization of dynamic of the machine-toolworkpiece system by means of frequency response functions (FRFs). Uncertainties of the measured FRFs result in uncertainties of the calculated stability diagrams, therefore robustness of stability prediction against parameter perturbations is of high importance. Although there exist methods to determine robust stability in terms of stability radii, these methods either give a conservative estimate of the real uncertainties or are limited to perturbations of a few modal parameters, only. In this paper, a frequency-domain approach is presented to determine robust stability boundaries using the measured FRFs directly without any modal parameter identification. The method is based on an envelope fitting around the measured FRFs combined with some considerations of the single-frequency method. The application of the method is demonstrated in case of a turning operation, where the machine tool structure is characterized by a series of FRF measurements.

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Section: Pseudospectra and Stability Radiusmentioning

confidence: 99%

“…If w M → ∞, then no perturbation on the mass matrix M is allowed [19]. This formalism allows the perturbations satisfying…”

Section: Weighted Complex Stability Radiusmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Robust stability analysis of machining operations

Hajdu

Insperger

Stépàn

2016

Int J Adv Manuf Technol

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“…In this work we consider the particular case arising again from DDEs, i.e. we compute the ε−pseudospectrum of the linear unbounded operator A which is the infinitesimal generator associated to systems of DDEs such as (3) [9]. Since this operator is infinite dimensional, its pseudospectrum is approximated by discretizing A into a suitable matrix A n via pseudospectral differencing methods as reported in the previous section, for details see [5].…”

Section: ε−Pseudospectramentioning

confidence: 99%

An adaptive algorithm for efficient computation of level curves of surfaces

2009

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A new efficient algorithm for the computation of z = constant level curves of surfaces z = f (x, y) is proposed and tested on several examples. The set of z-level curves in a given rectangle of the (x, y)-plane is obtained by evaluating f on a first coarse square grid which is then adaptively refined by triangulation to eventually match a desired tolerance. Adaptivity leads to a considerable reduction in terms of evaluations of f with respect to uniform grid computation as in Matlab®'s contour. Furthermore, especially when the evaluation of f is computationally expensive, this reduction notably decreases the computational time. A comparison of performances is shown for two real-life applications such as the determination of stability charts and of ε−pseudospectra for linear time delay systems. The corresponding Matlab code is also discussed.

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Pseudospectra, stability radii and their relationship with backward error for structured nonlinear eigenvlaue problems

Ahmad,

Nag

2024

Math Methods in App Sciences

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This paper discusses pseudospectra and stability radii for structured nonlinear matrix functions, such as Hermitian, skew‐Hermitian, H‐even, H‐odd, complex symmetric, and complex skew‐symmetric. To compute pseudospectra and stability radii, eigenvalue backward error is required. Hence, we initially present the structured eigenvalue backward error. Subsequently, we compute the structured pseudospectra using the obtained results for the eigenvalue backward error of a class of structured nonlinear matrix functions. Finally, we discuss the stability radii of the above‐structured problems arising in different applications. The paper also generalizes the results on the eigenvalue backward error of matrix polynomials in the literature for the above structures.

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Pseudospectra and stability radii for analytic matrix functions with application to time-delay systems

Cited by 42 publications

References 15 publications

Robust stability analysis of machining operations

Robust stability analysis of machining operations

An adaptive algorithm for efficient computation of level curves of surfaces

Pseudospectra, stability radii and their relationship with backward error for structured nonlinear eigenvlaue problems

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