Structure of Discrete Systems with Switched Delay (original) (raw)
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2017 25th European Signal Processing Conference (EUSIPCO), 2017
Linear time-invariant (LTI) systems are of fundamental importance in classical digital signal processing. LTI systems are linear operators commuting with the time-shift operator. For N-periodic discrete time series the time-shift operator is a circulant N × N permutation matrix. Sandryhaila and Moura developed a linear discrete signal processing framework and corresponding tools for datasets arising from social, biological, and physical networks. In their framework, the circulant permutation matrix is replaced by a network-specific N × N matrix A, called a shift matrix, and the linear shift-invariant (LSI) systems are all N × N matrices H over C commuting with the shift matrix: HA = AH. Sandryhaila and Moura described all those H for the non-degenerate case, in which all eigenspaces of A are one-dimensional. Then the authors reduced the degenerate case to the non-degenerate one. As we show in this paper this reduction does, however, not generally hold, leaving open one gap in the pr...
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Constructive Approximation, 2005
A countable collection X of functions in L 2 (IR d) is said to be a Bessel system if the associated analysis operator T * X : L 2 (IR d) → 2 (X) : f → (f, x) x∈X is well-defined and bounded. A Bessel system is a fundamental frame if T * X is injective and its range is closed. This paper considers the above two properties for a generalized shift-invariant system X. By definition, such a system has the form X = ∪ j∈J Y j , where each Y j is a shift-invariant system (i.e., is comprised of lattice translates of some function(s)) and J is a countable (or finite) index set. The definition is general enough to include wavelet systems, shift-invariant systems, Gabor systems, and many variations of wavelet systems such as quasi-affine ones and non-stationary ones. The main theme of this paper is the 'fiberization' of T * X , which allows one to study the frame and Bessel properties of X via the spectral properties of a collection of finite-order Hermitian non-negative matrices.
An Infinite-Dimensional Discrete-Time Representation for Periodic Systems
IFAC Proceedings Volumes, 1998
This note deals with the representation of continuous-time periodic systems as discrete-time invariant systems in state-space form using the lifting technique. More precisely, we derive an infinite-dimensional state equation on a Hilbert space for continuous-time periodically time-varying systems and characterize the structure of the spectrum of the discrete semigroup generator. Copyright © 1998 IFAC Resume: Cet article traite de la modelisation des sytemes periodiques a temps continu sous forme de systemes a temps discret en representation d'etat grace a la technique du lifting. On montre que la dynamique des systemes periodiques est decrite par une equation aux differences du premier ordre sur un espace Hilbertien et on caracterise la structure du spectre du g€merateur de semigroupe discret.
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In this paper we study the characterization of the asymptotical stability for discretetime switched linear systems. We first translate the system dynamics into a symbolic setting under the framework of symbolic topology. Then by using the ergodic measure theory, a lower bound estimate of Hausdorff dimension of the set of asymptotically stable sequences is obtained. We show that the Hausdorff dimension of the set of asymptotically stable switching sequences is positive if and only if the corresponding switched linear system has at least one asymptotically stable switching sequence. The obtained result reveals an underlying fundamental principle: a switched linear system either possesses uncountable numbers of asymptotically stable switching sequences or has none of them, provided that the switching is arbitrary. We also develop frequency and density indexes to identify those asymptotically stable switching sequences of the system.
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IEEE Transactions on Automatic Control, 1990
We study the behavior of discrete-time systems composed of a set of smooth transition maps coupled by a "quantized" feedback function. The feedback function partitions the state space into disjoint regions, and assigns a smooth transition function to each region. The main result is that under a constraint on the norm of the derivative of the transition maps, a bounded state trajectory with limit points in the interior of the switching regions leads to a region index sequence that is eventually periodic. Indeed, under these assumptions, we show that eventually the feedback function is determined by a finite state automaton. A similar result is proved in the case of finite state dynamic feedback.
arXiv: Dynamical Systems, 2020
The aim of this article is to find definitions for shifts of finite type and sofic shifts in a general context of symbolic dynamics. We start showing that the classical definitions of shifts of finite type and sofic shifts, as they are given in the context of finite-alphabet shift spaces on the one-dimensional monoid mathbbN\mathbb{N}mathbbN or mathbbZ\mathbb{Z}mathbbZ with the usual sum, do not fit for shift spaces over infinite alphabet or on other monoid. Therefore, by examining the core features in the classical definitions of shifts of finite type and sofic shifts, we propose general definitions that can be used in any context. The alternative definitions given for shift of finite types inspires the definition of a new class of shift spaces which intersects with the class of sofic shifts and includes shift of finite types. This new class is named finitely defined shifts, and the non-finite-type shifts in it are named shifts of variable length. For the specific case of infinite-alphabet shifts on the l...
Discrete Dynamical Systems: A Brief Survey
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This technical communique introduces a new concept of set invariance with respect to linear discrete time dynamics affected by delay. We are interested in the definition and characterization of sequences of cyclically invariant subsets in the state space. The algebraic conditions established in the late '80s for linear dynamics are generalized to invariance analysis in the presence of delays for given sequences of polyhedral sets.