Gábor Csernák

DIGITAL CONTROL AS SOURCE OF CHAOTIC BEHAVIOR

In the present paper, we introduce and analyze a mechanical system, in which the digital implementation of a linear control loop may lead to chaotic behavior. The amplitude of such oscillations is usually very small, this is why these are called micro-chaotic vibrations. As a consequence of the digital effects, i.e. the sampling, the processing delay and the round-off error, the behavior of the system can be described by a piecewise linear map, the micro-chaos map. We examine a 2D version of the micro-chaos map and prove that the map is chaotic.

Life Expectancy of Transient Microchaotic Behaviour

Abstract The application of digital control may lead to so-called transient chaotic behaviour. In the present paper, we analyse a simple model of a digitally controlled mechanical system, which may create such vibrations. As a consequence of the digital effects, i.e., the sampling and the round-off error, the behaviour of this system can be described by a one-dimensional piecewise linear map. The lifetime of chaotic transients is usually characterized by the so-called escape rate. In the literature, the reciprocal of the escape rate is considered to be the expected duration of the transient chaotic phenomenon. We claim that this approach is not always fruitful, and present a different way of calculating the mean lifetime in the case of one-dimensional piecewise linear maps. Our method might also be used to solve diffusion problems in one-dimensional models of periodic arrays.

Sampling and round-off, as sources of chaos in PD-controlled systems

It is well-known that nonlinear terms in the governing equations of dynamical systems may lead to chaotic behaviour. With this fact in mind, a well-trained engineer must be able to decide which system of equations can be linearized without a significant change in the solution. However, if the linearized dynamical system in question is part of a digital control loop, the interaction between the original mechanical or electrical system and the control system may still lead to unexpected behaviour due to the so-called digital effects. Our goal is to analyze the problem of computer-controlled stabilization of unstable equilibria, with the application of the PD control scheme. We consider the problem of the inverted pendulum, with linearized equations of motion. As a consequence of the digital effects, i.e., the sampling and the round-off error, the solutions of the system can be described by a two dimensional piecewise linear map. We show that this system may perform chaotic behaviour. Although the amplitude of the evolving oscillations is usually very small, several disconnected strange attractors may coexist in certain parameter domains, rather far from the desired equilibrium position. We claim that - since the amplitude is small - the nonlinearity of the digital control system is the primary source of the stochastic-like vibrations of the inverted pendulum, instead of the nonlinearity of the mechanical system.

Multi-Baker Map as a Model of Digital PD Control

Digital 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.

Micro-chaotic Behaviour in PD-controlled Systems

Abstract The nonlinearity caused by the application of digital control may lead to chaotic behaviour. Our goal is to analyse the problem of computer-controlled stabilization of unstable equilibria, with the application of the PD control law. We consider a linear equation of motion. However, as a consequence of the digital effects, i.e., the sampling and the round-off error, the solutions of the system can be described by a two dimensional piecewise linear map. We show that this system may perform chaotic behaviour. The amplitude of the evolving oscillations is usually very small, this is why these are called micro-chaotic vibrations.

Methods for the Quick Analysis of Micro-chaos

Micro-chaos is a phenomenon when sampling, round-off and processing delay (shortly, digital effects) lead to chaotic oscillations with small amplitude. In previous works [1], the so-called micro-chaos maps of various digitally controlled unstable linear mechanical systems were derived and the possibility of the coexistence of several disconnected attractors was highlighted. The typical size of these attractors is usually negligible from the practical point of view, but the distance of the farthest attractor from the desired state can be rather large. This is why the phenomenon of micro-chaos can be the source of significant control error. In this paper, a set of numerical methods (e.g. cell mapping techniques for the exploration of the phase-space structure) is assembled in order to create a toolkit for the quick analysis of micro-chaotic behaviour. The elaborated methods are tested on models of PD-controlled unstable systems and the practically important characteristics of chaotic behaviour are determined.

Life Expectancy Calculations of Transient Chaotic Behaviour Using Approximate 1D Maps

In engineering practice, chaotic oscillations are often observed which disappear suddenly. This phenomenon is often referred to as transient chaos. The duration of these oscillations varies stochastically. In this work two methods are presented for the simple estimation of the expected length of the chaotic behaviour. As an example, the Lorenz system is considered at some specific parameter values.