Digital Control as Source of Chaotic Behavior

Csernák, Gábor; Stépàn, Gábor

doi:10.1142/s0218127410026538

Cited by 20 publications

(19 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If one takes into account a processing delay that is just equal to the sampling period, the micro-chaos map becomes two-dimensional, but the attractors become one-dimensional. The proof of chaos was presented for this map and the lifetime estimation algorithm was generalized in [Csernák & Stépán, 2010] and [Csernák & Stépán, 2007]. A similar 2D map was introduced in [Garay et al, 2008] as a model of hysteresis.…”

Section: Introductionmentioning

confidence: 99%

Multi-Baker Map as a Model of Digital PD Control

Csernák¹,

Gyebrószki

Stépàn

2016

Int. J. Bifurcation Chaos

Self Cite

View full text Add to dashboard Cite

ReceivedDigital stabilization of unstable equilibria of linear systems may lead to small amplitude stochastic-like oscillations. We show that these vibrations can be related to a deterministic chaotic dynamics induced by sampling and quantization. A detailed analytical proof of chaos is presented for the case of a PD controlled oscillator: it is shown that there exists a finite attracting domain in the phase-space, the largest Lyapunov exponent is positive and the existence of a Smale horseshoe is also pointed out. The corresponding two-dimensional micro-chaos map is a multi-baker map, i.e., it consists of a finite series of baker's maps.

show abstract

Section: Introductionmentioning

confidence: 99%

Multi-Baker Map as a Model of Digital PD Control

Csernák¹,

Gyebrószki

Stépàn

2016

Int. J. Bifurcation Chaos

Self Cite

View full text Add to dashboard Cite

show abstract

“…As an example of application, we show the analysis of the so-called micro-chaos map [7], where multiple disconnected attractors -possibly consisting of distinguishable communicating repellers -are present in the state space. The behaviour of this piecewise smooth system fits into moremost of the aforementioned situations, as it exhibits a pattern of chaotic attractors and crisis phenomena with the appearance or disap-…”

mentioning

confidence: 99%

Clustered Simple Cell Mapping: An extension to the Simple Cell Mapping method

Gyebrószki

Csernák²

2017

Communications in Nonlinear Science and Numerical Simulation

Self Cite

View full text Add to dashboard Cite

When a dynamical system has a complex structure of fixed points, periodic cycles or even chaotic attractors, Cell Mapping methods are excellent tools to discover and thoroughly analyse all features in the state space. These methods discretize a region of the state space into cells and examine the dynamics in the cell state space. By determining one or more image cells for each cell, the global behaviour within the region can be quickly determined. In the simplest case -Simple Cell Mapping (SCM) method -only one image corresponds to a cell and usually a rectangular grid of cells is used. In typical applications the grid of cells is refined at specific locations. This paper, however, introduces a different approach, which is useful to expand the analysed state space region to include all features which properly characterize the global dynamics of the system. Instead of refining the initial cell state space, we start with a small initial state space region, analyse other interesting regions of the state space and incorporate them into a cluster of cell mapping solutions. By this approach, trajectories escaping the original state space region can be followed automatically and additional objects in the state space can be discovered. To illustrate the benefits of the method, we present the exploration of the phase-space of the micro-chaos map -a simple model of digitally controlled systems.Keywords: discretized state space, adaptive global analysis, algorithm, micro-chaos Highlights:• Extension of the Simple Cell Mapping method by adaptively expanding cell state space.• New state space regions can be attached to the cluster of SCM solutions.• Clustering makes parallel computation trivial.

show abstract

“…is fulfilled with any kind of norm [5]. Since the fractional numbers χ k vary during the application of the micro-chaos map, the series (37) is divergent for almost all initial conditions.…”

Section: Reformulation Of the Problemmentioning

confidence: 99%

“…Quantization and sampling may lead to small-scale chaotic vibrations in digitally controlled systems [1][2][3][4][5]. These vibrations are often neglected or considered as noise in the engineering practice [6].…”

Section: Introductionmentioning

confidence: 99%

Quantization-induced control error in a digitally controlled system

Csernák¹

2016

Nonlinear Dyn

Self Cite

View full text Add to dashboard Cite

The "noise" induced by quantization in digitally controlled systems can be modelled as small amplitude chaotic behaviour described by a simple piecewise linear map, the micro-chaos map. The chaotic nature of certain micro-chaos maps was proved rigorously in the last 20 years. It was also pointed out that several disconnected strange attractors or repellers may coexist far from the desired state of the system. The control error is influenced by the number of these strange sets and the local behaviour of trajectories in their neighbourhoods. In the present paper we focus on the exploration of the possible attractor/repeller structures in the phase space of a PD-controlled system and estimate the expected control error by several methods.

show abstract

Digital Control as Source of Chaotic Behavior

Cited by 20 publications

References 20 publications

Multi-Baker Map as a Model of Digital PD Control

Multi-Baker Map as a Model of Digital PD Control

Clustered Simple Cell Mapping: An extension to the Simple Cell Mapping method

Quantization-induced control error in a digitally controlled system

Contact Info

Product

Resources

About