Real and complex stability radii of polynomial matrices

Genin, Y.; Stefan, R.; Dooren, Paul Van

doi:10.1016/s0024-3795(02)00278-1

Cited by 27 publications

(28 citation statements)

References 18 publications

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“…Proof. The proof is similar to the proof of Proposition 22 in [11]. For all ω ∈ R, we have the relation This is equivalent to the assertion of the theorem.…”

Section: Correcting the Pseudospectral Abscissamentioning

confidence: 63%

A predictor–corrector type algorithm for the pseudospectral abscissa computation of time-delay systems

Gümüşsoy

Michiels

2010

Automatica

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The pseudospectrum of a linear time-invariant system is the set in the complex plane consisting of all the roots of the characteristic equation when the system matrices are subjected to all possible perturbations with a given upper bound. The pseudospectral abscissa is defined as the maximum real part of the characteristic roots in the pseudospectrum and, therefore, it is for instance important from a robust stability point of view. In this paper we present an accurate method for the computation of the pseudospectral abscissa of retarded delay differential equations with discrete pointwise delays. Our approach is based on the connections between the pseudospectrum and the level sets of an appropriately defined complex function. The computation is done in two steps. In the prediction step, an approximation of the pseudospectral abscissa is obtained based on a rational approximation of the characteristic matrix and the application of a bisection algorithm. Each step in this bisection algorithm relies on checking the presence of the imaginary axis eigenvalues of a complex matrix, similar to the delay free case. In the corrector step, the approximate pseudospectral abscissa is corrected to any given accuracy, by solving a set of nonlinear equations that characterize extreme points in the pseudospectrum contours.

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“…Proof. The proof is similar to the proof of Proposition 22 in [11]. For all ω ∈ R, we have the relation This is equivalent to the assertion of the theorem.…”

Section: Correcting the Pseudospectral Abscissamentioning

confidence: 63%

A predictor–corrector type algorithm for the pseudospectral abscissa computation of time-delay systems

Gümüşsoy

Michiels

2010

Automatica

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“…The real stability radius can be computed from (10), or by algorithms proposed in [24]. However, the computation of r R (M ) involves the minimization of unimodal functions [23], which is a challenging problem [25].…”

Section: A Robustness Index: Stability Radiusmentioning

confidence: 99%

How retroactivity impacts the robustness of genetic networks

Mou

Vecchio

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Abstract-This paper studies how retroactivity impacts the robustness of gene transcription networks against parameter perturbations. By employing the linearization technique and the real stability radius, we provide comparisons of the robustness between gene transcription networks with retroactivity and ones without retroactivity. Both numerical and analytical results show that retroactivity tends to decrease such robustness. This finding in turn implies that modular genetic networks tend to be more robust against parameter perturbations.

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“…They needs only conversion of the system operator into state-space description. Although many methods for computing norms for linear time-invariant systems [4], [7], [8] which are essential in a computer aided control system design [9] there are very difficult to find methods applicable for linear time-varying systems.…”

Section: Computation the Norm Of The Time-varying Systemmentioning

confidence: 99%

Discrete-Time, Linear Periodic Time-Varying System Norm Estimation Using Finite Time Horizon Transfer Operators

Orłowski

2010

Automatika

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Original scientific paperThe norm is one of the fundamental concepts of linear algebra and functional analysis. The notion of the norm is often employed in engineering, e.g. in control engineering, where main application is calculating the norm of the transfer function. Unfortunately existing methods are applicable for systems that can be described using Laplace transform, i.e. linear time-invariant (LTI) systems. An operational equivalent of the transfer function for linear timevarying systems is transfer operator. The transfer operator defined for finite time horizon can be described by finite dimensional matrix. Although for infinite time horizon the operator is infinite dimensional. In the paper a method for norm estimation of transfer operator defined on infinite time horizon is proposed. The method is applicable for linear time-varying, discrete-time systems given in general state-space form. The method takes advantage of the properties of the transfer operator norm on a finite time horizon. Theoretical considerations are complemented by numerical examples.Key words: Norm estimation, Discrete-time systems, Time-varying systems, Non-stationary systems Estimacija norme diskretnih, periodički vremenski promjenljivih, linearnih sustava primjenom prijenosnog operatora s konačnim vremenskih horizontom. Norma je jedan od osnovnih koncepata linearne algebre i funkcionalne analize. Pojam norme sečesto koristi kod inžinjera, npr. kod upravljanja, gdje je jedna od glavnih aplikacija računanje norma prijenosne funkcije. Nažalost postojeće metode su primjenjive samo na sustave koji se mogu opisati koristeći Laplaceovu transformaciju, tj. linearne vremenski nepromjenjive sustave. Ekvivalent prijenosnoj funkciji za linearne vremenski promjenjive sustave je prijenosni operator. Prijenosni operator definiran za konačni vremenski horizont može se opisati konačno dimenzionalnom matricom. Iako je za beskonačni vremenski horizont operator beskonačno dimenzionalan. U radu je predložena metoda za estimaciju norme prijenosnog operatora definiranog na beskonačnom vremenskom horizontu. Metoda je primjenjiva na linearne vremenski promjenjive diskretne sustave zadane u obliku prostora stanja. Metoda koristi svojstva norme prijenosnog operatora za konačni vremenski horizont. Teoretska promišljanja nadopunjena su numeričkim primjerima.Ključne riječi: estimacija norme, diskretni sustavi, vremenski promjenjivi sustavi, nestacionarni sustavi

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Real and complex stability radii of polynomial matrices

Cited by 27 publications

References 18 publications

A predictor–corrector type algorithm for the pseudospectral abscissa computation of time-delay systems

A predictor–corrector type algorithm for the pseudospectral abscissa computation of time-delay systems

How retroactivity impacts the robustness of genetic networks

Discrete-Time, Linear Periodic Time-Varying System Norm Estimation Using Finite Time Horizon Transfer Operators

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