Multi-Baker Map as a Model of Digital PD Control

Csernák, Gábor; Gyebrószki, Gergely; Stépàn, Gábor

doi:10.1142/s0218127416500231

Cited by 12 publications

(18 citation statements)

References 17 publications

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“…The resolution is taken into account with the Int() function, which denotes rounding towards the origin. According to the solution of the linearized, dimensionless equation of motion, the following mapping can be derived between the states at subsequent sampling instants [8]:…”

Section: Analysis Of the Micro-chaos Mapmentioning

confidence: 99%

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

Gyebrószki

Csernák²

2017

Communications in Nonlinear Science and Numerical Simulation

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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.

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Section: Analysis Of the Micro-chaos Mapmentioning

confidence: 99%

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

Gyebrószki

Csernák²

2017

Communications in Nonlinear Science and Numerical Simulation

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“…As it was shown in [7], strange sets appear in the neighbourhoods of the switching lines between any two true fixed points and even between a true and a virtual fixed point. This property is related to the fact that the control force is quantized.…”

Section: Strange Structures Between Fixed Pointsmentioning

confidence: 74%

“…It was shown in [7] that the phase-space structure can be described as a series of baker's maps. To be able to refine the concepts outlined in [7], we reiterate the notations introduced there.…”

Section: Boundary Crisis Bifurcationsmentioning

confidence: 99%

“…To be able to refine the concepts outlined in [7], we reiterate the notations introduced there. Consider the neighbourhood of the swithing line SW m (m > 0) that is between the fixed points F l on the left and Fig.…”

Section: Boundary Crisis Bifurcationsmentioning

confidence: 99%

“…These vibrations are often neglected or considered as noise in the engineering practice [6]. It was shown in [7] that several small strange attractors can coexist in the phasespace of a simple PD (proportional-differential) controlled linear system. Although the amplitude of chaotic vibrations was found to be indeed small, the large distance of the attracting structures from the desired po-G. Csernák MTA-BME Research Group on Dynamics of Machines and Vehicles Tel.…”

Section: Introductionmentioning

confidence: 99%

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Quantization-induced control error in a digitally controlled system

Csernák¹

2016

Nonlinear Dyn

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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.

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Twofold quantization in digital control: deadzone crisis and switching line collision

Gyebrószki

Csernák²

2019

Nonlinear Dyn

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Quantization, sampling and delay may cause undesired oscillations in digitally controlled systems. These vibrations are often neglected or replaced by random noise (Widrow and Kollár in Quantization noise: roundoff error in digital computation, signal processing, control, and communications, Cambridge University Press, Cambridge, 2008); however, we have shown that digital effects may lead to small amplitude deterministic chaotic solutions-the so-called microchaos (Csernák and Stépán in Int J Bifurc Chaos 5(20):1365-1378, 2010). Although the amplitude of the micro-chaotic oscillations is small, multiple chaotic attractors can appear in the state space of the digitally controlled system-situated far away from the desired state-causing significant control error (Csernák and Stépán in Proceedings of the 19th mediterranean conference on control and automation, 2011). In this paper, we are interested in the analysis of a digitally controlled inverted pendulum with both input and output quantizers along with sampling. We show that this twofold quantization creates patterns in the state space corresponding to different control effort (force or torque) values for a simple PD control. We also highlight how

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Multi-Baker Map as a Model of Digital PD Control

Cited by 12 publications

References 17 publications

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

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

Quantization-induced control error in a digitally controlled system

Twofold quantization in digital control: deadzone crisis and switching line collision

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