406 research outputs found
Counting and computing regions of -decomposition: algebro-geometric approach
New methods for -decomposition analysis are presented. They are based on
topology of real algebraic varieties and computational real algebraic geometry.
The estimate of number of root invariant regions for polynomial parametric
families of polynomial and matrices is given. For the case of two parametric
family more sharp estimate is proven. Theoretic results are supported by
various numerical simulations that show higher precision of presented methods
with respect to traditional ones. The presented methods are inherently global
and could be applied for studying -decomposition for the space of parameters
as a whole instead of some prescribed regions. For symbolic computations the
Maple v.14 software and its package RegularChains are used.Comment: 16 pages, 8 figure
Transverse Contraction Criteria for Existence, Stability, and Robustness of a Limit Cycle
This paper derives a differential contraction condition for the existence of
an orbitally-stable limit cycle in an autonomous system. This transverse
contraction condition can be represented as a pointwise linear matrix
inequality (LMI), thus allowing convex optimization tools such as
sum-of-squares programming to be used to search for certificates of the
existence of a stable limit cycle. Many desirable properties of contracting
dynamics are extended to this context, including preservation of contraction
under a broad class of interconnections. In addition, by introducing the
concepts of differential dissipativity and transverse differential
dissipativity, contraction and transverse contraction can be established for
large scale systems via LMI conditions on component subsystems.Comment: 6 pages, 1 figure. Conference submissio
Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design
We describe an elementary algorithm to build convex inner approximations of
nonconvex sets. Both input and output sets are basic semialgebraic sets given
as lists of defining multivariate polynomials. Even though no optimality
guarantees can be given (e.g. in terms of volume maximization for bounded
sets), the algorithm is designed to preserve convex boundaries as much as
possible, while removing regions with concave boundaries. In particular, the
algorithm leaves invariant a given convex set. The algorithm is based on
Gloptipoly 3, a public-domain Matlab package solving nonconvex polynomial
optimization problems with the help of convex semidefinite programming
(optimization over linear matrix inequalities, or LMIs). We illustrate how the
algorithm can be used to design fixed-order controllers for linear systems,
following a polynomial approach
On Stability of Parametrized Families of Polynomials and Matrices
The Schur and Hurwitz stability problems for a parametric polynomial
family as well as the Schur stability problem for a compact set of real
matrix family are considered. It is established that the Schur stability
of a family of real matrices is equivalent to the nonsingularity
of the family {2â2+â¶â,â[â1,1]} if has at
least one stable member. Based on the Bernstein expansion of a
multivariable polynomial and extremal properties of a multilinear
function, fast algorithms are suggested
Quantum Spectrum Testing
In this work, we study the problem of testing properties of the spectrum of a
mixed quantum state. Here one is given copies of a mixed state
and the goal is to distinguish whether 's
spectrum satisfies some property or is at least -far in
-distance from satisfying . This problem was promoted in
the survey of Montanaro and de Wolf under the name of testing unitarily
invariant properties of mixed states. It is the natural quantum analogue of the
classical problem of testing symmetric properties of probability distributions.
Here, the hope is for algorithms with subquadratic copy complexity in the
dimension . This is because the "empirical Young diagram (EYD) algorithm"
can estimate the spectrum of a mixed state up to -accuracy using only
copies. In this work, we show that given a
mixed state : (i) copies
are necessary and sufficient to test whether is the maximally mixed
state, i.e., has spectrum ; (ii)
copies are necessary and sufficient to test with
one-sided error whether has rank , i.e., has at most nonzero
eigenvalues; (iii) copies are necessary and
sufficient to distinguish whether is maximally mixed on an
-dimensional or an -dimensional subspace; and (iv) The EYD
algorithm requires copies to estimate the spectrum of
up to -accuracy, nearly matching the known upper bound. In
addition, we simplify part of the proof of the upper bound. Our techniques
involve the asymptotic representation theory of the symmetric group; in
particular Kerov's algebra of polynomial functions on Young diagrams.Comment: 70 pages, 6 figure
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Big data clustering: Data preprocessing, variable selection, and dimension reduction
[no abstract available
Parametric Robust Control and System Identification: Unified Approach
During the period of this support, a new control system design and analysis method has been studied. This approach deals with control systems containing uncertainties that are represented in terms of its transfer function parameters. Such a representation of the control system is common and many physical parameter variations fall into this type of uncertainty. Techniques developed here are capable of providing nonconservative analysis of such control systems with parameter variations. We have also developed techniques to deal with control systems when their state space representations are given rather than transfer functions. In this case, the plant parameters will appear as entries of state space matrices. Finally, a system modeling technique to construct such systems from the raw input - output frequency domain data has been developed
Spectrum analysis of LTI continuous-time systems with constant delays: A literature overview of some recent results
In recent decades, increasingly intensive research attention has been given to dynamical systems containing delays and those affected by the after-effect phenomenon. Such research covers a wide range of human activities and the solutions of related engineering problems often require interdisciplinary cooperation. The knowledge of the spectrum of these so-called time-delay systems (TDSs) is very crucial for the analysis of their dynamical properties, especially stability, periodicity, and dumping effect. A great volume of mathematical methods and techniques to analyze the spectrum of the TDSs have been developed and further applied in the most recent times. Although a broad family of nonlinear, stochastic, sampled-data, time-variant or time-varying-delay systems has been considered, the study of the most fundamental continuous linear time-invariant (LTI) TDSs with fixed delays is still the dominant research direction with ever-increasing new results and novel applications. This paper is primarily aimed at a (systematic) literature overview of recent (mostly published between 2013 to 2017) advances regarding the spectrum analysis of the LTI-TDSs. Specifically, a total of 137 collected articles-which are most closely related to the research area-are eventually reviewed. There are two main objectives of this review paper: First, to provide the reader with a detailed literature survey on the selected recent results on the topic and Second, to suggest possible future research directions to be tackled by scientists and engineers in the field. © 2013 IEEE.MSMT-7778/2014, FEDER, European Regional Development Fund; LO1303, FEDER, European Regional Development Fund; CZ.1.05/2.1.00/19.0376, FEDER, European Regional Development FundEuropean Regional Development Fund through the Project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19.0376]; National Sustainability Program Project [LO1303 (MSMT-7778/2014)
Parallel Matrix-free polynomial preconditioners with application to flow simulations in discrete fracture networks
We develop a robust matrix-free, communication avoiding parallel, high-degree
polynomial preconditioner for the Conjugate Gradient method for large and
sparse symmetric positive definite linear systems. We discuss the selection of
a scaling parameter aimed at avoiding unwanted clustering of eigenvalues of the
preconditioned matrices at the extrema of the spectrum. We use this
preconditioned framework to solve a block system arising in the
simulation of fluid flow in large-size discrete fractured networks. We apply
our polynomial preconditioner to a suitable Schur complement related with this
system, which can not be explicitly computed because of its size and density.
Numerical results confirm the excellent properties of the proposed
preconditioner up to very high polynomial degrees. The parallel implementation
achieves satisfactory scalability by taking advantage from the reduced number
of scalar products and hence of global communications
Estimating the rank of the spectral density matrix
The rank of the spectral density matrix conveys relevant information in a variety of statistical modelling scenarios. This note shows how to estimate the rank of the spectral density matrix at any given frequency. The method presented is valid for any hermitian positive de?nite matrix estimate that has a normal asymptotic distribution with a covariance matrix whose rank is known. JEL Classification: C12, C32, C52Spectral Density Matrix, Tests of Rank
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