# Abscissa

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### Zhendong Sun – One of the best experts on this subject based on the ideXlab platform.

• ##### Performance analysis of switched linear systems under arbitrary switching via generalized coordinate transformations
Science China Information Sciences, 2018
Co-Authors: Meili Lin, Zhendong Sun

Abstract:

For a continuous-time switched linear system, the spectral Abscissa is defined as the worst-case divergence rate under arbitrary switching, which is critical for characterizing the asymptotic performance of the switched system. In this study, based on the generalized coordinate transformations approach, we develop a computational scheme that iteratively produces sequences of minimums of matrix set $\mu_1$ measures, where the limits of the sequences are upper bound estimates of the spectral Abscissa. A simulation example is presented to illustrate the effectiveness of the proposed scheme.

• ##### Upper bound estimation of the spectral Abscissa for switched linear systems via coordinate transformations
Kybernetika, 2018
Co-Authors: Meili Lin, Zhendong Sun

Abstract:

In this paper, we develop computational procedures to approximate the spectral Abscissa of the switched linear system via square coordinate transformations. First, we design iterative algorithms to obtain a sequence of the least $\mu_1$ measure. Second, it is shown that this sequence is convergent and its limit can be used to estimate the spectral Abscissa. Moreover, the stopping condition of Algorithm 1 is also presented. Finally, an example is carried out to illustrate the effectiveness of the proposed method.

• ##### Approximating the Spectral Abscissa for Switched Linear Systems via Coordinate Transformations
Journal of Systems Science and Complexity, 2016
Co-Authors: Meili Lin, Zhendong Sun

Abstract:

In this paper, an approach of square coordinate transformation is proposed to approximate the spectral Abscissa for continuous-time switched linear systems. By applying elementary transformations iteratively, a series of minimums of least μ 1 matrix set measures are obtained, which are utilized to approximate the spectral Abscissa of the switched system. The approach is developed into tractable numerical algorithms that provide upper bound estimates of the spectral Abscissa. Numerical simulations show the effectiveness of the proposed method.

### Wim Michiels – One of the best experts on this subject based on the ideXlab platform.

• ##### Polynomial (chaos) approximation of maximum eigenvalue functions
Numerical Algorithms, 2019
Co-Authors: Luca Fenzi, Wim Michiels

Abstract:

This paper is concerned with polynomial approximations of the spectral Abscissa function (defined by the supremum of the real parts of the eigenvalues) of a parameterized eigenvalue problem, which are closely related to polynomial chaos approximations if the parameters correspond to realizations of random variables. Unlike previous work, we highlight the major role of this function smoothness properties. Even if the eigenvalue problem matrices are analytic functions of the parameters, the spectral Abscissa function may not be differentiable, and even non-Lipschitz continuous, due to multiple rightmost eigenvalues counted with multiplicity. This analysis demonstrates smoothness properties not only heavily affect the approximation errors of the Galerkin and collocation based polynomial approximations, but also the numerical errors in the evaluation of coefficients in the Galerkin approach with integration methods. A documentation of the experiments, conducted on the benchmark problems through the software Chebfun, is publicly available.

• ##### A predictor-corrector type algorithm for the pseudospectral Abscissa computation of time-delay systems
Automatica, 2010
Co-Authors: Suat Gumussoy, Wim Michiels

Abstract:

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 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 characterizes the extreme points in the pseudospectrum contours.

• ##### Computing the pseudospectral Abscissa of time-delay systems
IFAC Proceedings Volumes, 2009
Co-Authors: Suat Gumussoy, Wim Michiels

Abstract:

Abstract The pseudospectra of a linear time-invariant system are the sets 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 are defined as the maximum real part of the characteristic roots in the pseudospectra and, therefore, they are for instance important from a robust stability point of view. In this paper we present a numerical 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 pseudospectra and the level sets of an appropriately defined complex function. These connections lead us to a bisection algorithm for the computation of the pseudospectral Abscissa, where each step relies on checking the presence of imaginary axis eigenvalues of an appropriately defined operator. Because this operator is infinite-dimensional a predictor-corrector approach is taken. In the predictor step the bisection algorithm is applied where the operator is discretized into a matrix, yielding approximations for the pseudospectral Abscissa. The effect of the discretization is fully characterized in the paper. In the corrector step, the approximate pseudospectral Abscissa are corrected to any given accuracy, by solving a set of nonlinear equations that characterize extreme points in the pseudospectra contours.

### Jokke Hasa – One of the best experts on this subject based on the ideXlab platform.

• ##### representation growth of compact linear groups
Transactions of the American Mathematical Society, 2019
Co-Authors: Jokke Hasa, Alexander Stasinski

Abstract:

We study the representation growth of simple compact Lie groups
and of SLn(O), where O is a compact discrete valuation ring, as well as the
twist representation growth of GLn(O). This amounts to a study of the Abscissae of convergence of the corresponding (twist) representation zeta functions.
We determine the Abscissae for a class of Mellin zeta functions which include
the Witten zeta functions. As a special case, we obtain a new proof of the
theorem of Larsen and Lubotzky that the Abscissa of Witten zeta functions is
r/κ, where r is the rank and κ the number of positive roots.
We then show that the twist zeta function of GLn(O) exists and has the
same Abscissa of convergence as the zeta function of SLn(O), provided n does
not divide char O. We compute the twist zeta function of GL2(O) when the
residue characteristic p of O is odd and approximate the zeta function when
p = 2 to deduce that the Abscissa is 1. Finally, we construct a large part of
the representations of SL2(Fq[[t]]), q even, and deduce that its Abscissa lies in
the interval [1, 5/2].

• ##### representation growth of compact linear groups
arXiv: Representation Theory, 2017
Co-Authors: Jokke Hasa, Alexander Stasinski

Abstract:

We study the representation growth of simple compact Lie groups and of $\mathrm{SL}_n(\mathcal{O})$, where $\mathcal{O}$ is a compact discrete valuation ring, as well as the twist representation growth of $\mathrm{GL}_n(\mathcal{O})$. This amounts to a study of the Abscissae of convergence of the corresponding (twist) representation zeta functions.
We determine the Abscissae for a class of Mellin zeta functions which include the Witten zeta functions. As a special case, we obtain a new proof of the theorem of Larsen and Lubotzky that the Abscissa of Witten zeta functions is $r/\kappa$, where $r$ is the rank and $\kappa$ the number of positive roots.
We then show that the twist zeta function of $\mathrm{GL}_n(\mathcal{O})$ exists and has the same Abscissa of convergence as the zeta function of $\mathrm{SL}_n(\mathcal{O})$, provided $n$ does not divide $\text{char}\,{\mathcal{O}}$. We compute the twist zeta function of $\mathrm{GL}_2(\mathcal{O})$ when the residue characteristic $p$ of $\mathcal{O}$ is odd, and approximate the zeta function when $p=2$ to deduce that the Abscissa is $1$. Finally, we construct a large part of the representations of $\mathrm{SL}_2(\mathbb{F}_q[[t]])$, $q$ even, and deduce that its Abscissa lies in the interval $[1,\,5/2]$.