280,291 research outputs found
Introduction to Stability Analysis of Discrete Dynamical Systems
This manuscript analyzes the fundamental factors that govern the qualitative behavior of discrete dynamical systems. It introduces methods of analysis for stability analysis of discrete dynamical systems. The analysis focuses initially on the derivation of basic propositions about the factors that determine the local and global stability of discrete dynamical systems in the elementary context of a one dimensional, first-order, autonomous, systems. These propositions are subsequently generalized to account for stability analysis in a multi-dimensional, higher-order, non-autonomous, nonlinear, dynamical systems.Discrete Dynamical Systems, Difference Equations, Global Stability, Local Stability, Non-Linear Dynamics, Stable Manifolds
Discrete Dynamical Systems
This manuscript analyzes the fundamental factors that govern the qualitative behavior of discrete dynamical systems. It introduces methods of analysis for stability analysis of discrete dynamical systems. The analysis focuses initially on the derivation of basic propositions about the factors that determine the local and global stability of discrete dynamical systems in the elementary context of a one dimensional, first-order, autonomous, systems. These propositions are subsequently generalized to account for stability analysis in a multi-dimensional, higher-order, non-autonomous, nonlinear, dynamical systems.Discrete Dynamical Systems, Difference Equations, Global Stability, Local Stability, Non-Linear Dynamics, Stable Manifolds
Applied Koopman Operator Theory for Power Systems Technology
Koopman operator is a composition operator defined for a dynamical system
described by nonlinear differential or difference equation. Although the
original system is nonlinear and evolves on a finite-dimensional state space,
the Koopman operator itself is linear but infinite-dimensional (evolves on a
function space). This linear operator captures the full information of the
dynamics described by the original nonlinear system. In particular, spectral
properties of the Koopman operator play a crucial role in analyzing the
original system. In the first part of this paper, we review the so-called
Koopman operator theory for nonlinear dynamical systems, with emphasis on modal
decomposition and computation that are direct to wide applications. Then, in
the second part, we present a series of applications of the Koopman operator
theory to power systems technology. The applications are established as
data-centric methods, namely, how to use massive quantities of data obtained
numerically and experimentally, through spectral analysis of the Koopman
operator: coherency identification of swings in coupled synchronous generators,
precursor diagnostic of instabilities in the coupled swing dynamics, and
stability assessment of power systems without any use of mathematical models.
Future problems of this research direction are identified in the last
concluding part of this paper.Comment: 31 pages, 11 figure
On delay-dependent robust stability under model transformation of some neutral systems
summary:This paper focuses on the delay-dependent robust stability of linear neutral delay systems. The systems under consideration are described by functional differential equations, with norm bounded time varying nonlinear uncertainties in the "state" and norm bounded time varying quasi-linear uncertainties in the delayed "state" and in the difference operator. The stability analysis is performed via the Lyapunov-Krasovskii functional approach. Sufficient delay dependent conditions for robust stability are given in terms of the existence of positive definite solutions of LMIs
Stability Results for a Class of Difference Systems with Delay
Considering the linear delay difference system x(n+1)=ax(n)+Bx(n-k), where a∈(0,1), B is a p×p real matrix, and k is a positive integer, the stability domain of the null solution is completely characterized in terms of the eigenvalues of the matrix B. It is also shown that the stability domain becomes smaller as the delay increases. These results may be successfully applied in the stability analysis of a large class of nonlinear difference systems, including discrete-time Hopfield neural networks
Total stability and integral action for discrete-time nonlinear systems
Robustness guarantees are important properties to be looked for during
control design. They ensure stability of closed-loop systems in face of
uncertainties, unmodeled effects and bounded disturbances. While the theory on
robust stability is well established in the continuous-time nonlinear
framework, the same cannot be stated for its discrete-time counterpart. In this
paper, we propose the discrete-time parallel of total stability results for
continuous-time nonlinear system. This enables the analysis of robustness
properties via simple model difference in the discrete-time context. First, we
study how existence of equilibria for a nominal model transfers to sufficiently
similar ones. Then, we provide results on th
Floquet Stability Analysis of Ott-Grebogi-Yorke and Difference Control
Stabilization of instable periodic orbits of nonlinear dynamical systems has
been a widely explored field theoretically and in applications. The techniques
can be grouped in time-continuous control schemes based on Pyragas, and the two
Poincar\'e-based chaos control schemes, Ott-Gebogi-Yorke (OGY) and difference
control. Here a new stability analysis of these two Poincar\'e-based chaos
control schemes is given by means of Floquet theory. This approach allows to
calculate exactly the stability restrictions occuring for small measurement
delays and for an impulse length shorter than the length of the orbit. This is
of practical experimental relevance; to avoid a selection of the relative
impulse length by trial and error, it is advised to investigate whether the
used control scheme itself shows systematic limitations on the choice of the
impulse length. To investigate this point, a Floquet analysis is performed. For
OGY control the influence of the impulse length is marginal. As an unexpected
result, difference control fails when the impulse length is taken longer than a
maximal value that is approximately one half of the orbit length for small
Ljapunov numbers and decreases with the Ljapunov number.Comment: 13 pages. To appear in New Journal of Physic
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