Micro-chaotic dynamics due to digital sampling in hybrid systems of Filippov type (original) (raw)

Micro-Chaos in Relay Feedback Systems with Bang-Bang Control and Digital Sampling

Proceedings of the 18th IFAC World Congress, 2011

We investigate a class of linear relay feedback systems with bang-bang control and with the control input applied at discrete time instances. Using a third order system as a representative example we show that stable oscillations with so-called sliding motion, with sliding present in continuous time system, loose the sliding segment of evolution, but do not loose their stability if the open loop system is stable. We then carry on our investigations and consider a situation when stable self-sustained oscillations are generated with the unstable open loop system. In the latter case a transition from a stable limit cycle to micro-chaotic oscillations occurs. The presence of micro-chaotic oscillations is shown by considering a linearised map that maps a small neighbourhood of initial conditions back to itself. Using this map the presence of the positive Lyapunov exponent is shown. The largest Lyapunov exponent is then calculated numerically for an open set of sampling times, and it is shown that it is positive. The boundedness of the attractor is ensured for sufficiently small sampling times; with the sampling time tending to zero these switchings become faster and they turn into sliding motion. It is the presence of the underlying sliding evolution that ensures the boundedness of the chaotic attractor. Our finding implies that what may be considered as noise in systems with digital control should actually be termed as micro-chaotic behaviour. This information may be helpful in designing digital control systems where any element contributing to what appears as noise should be suppressed.

Chaotic Behavior in a Switched Dynamical System

Modelling and Simulation in Engineering, 2008

We present a numerical study of an example of piecewise linear systems that constitute a class of hybrid systems. Precisely, we study the chaotic dynamics of the voltage-mode controlled buck converter circuit in an open loop. By considering the voltage input as a bifurcation parameter, we observe that the obtained simulations show that the buck converter is prone to have subharmonic behavior and chaos. We also present the corresponding bifurcation diagram. Our modeling techniques are based on the new French native modeler and simulator for hybrid systems called Scicos (Scilab connected object simulator) which is a Scilab (scientific laboratory) package. The followed approach takes into account the hybrid nature of the circuit.

The State-space Model of Micro-chaos

International Journal of Mathematical Models and Methods in Applied Sciences, 2021

Micro-chaos is the phenomenon when the sampling, the delay and the round-off lead to small amplitude chaotic oscillations in a digitally controlled system. It has been proved mathematically during the last few years in a couple of simple cases that the evolving vibrations are indeed chaotic. In this study, we partially generalize these results to the case when an originally unstable state of a system is stabilized by digital feedback control. It is pointed out that this type of systems are sensitive to initial conditions and there exists a finite attracting domain in their phase-space. We also show that the oscillations, related to micro-chaos may have a considerable influence on the accuracy and settling time of the control system. The application of numerical techniques is unavoidable in the case of chaotic systems. Several possibilities are highlighted in the paper for the numerical determination of important characteristics of microchaotic oscillations.

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

2011

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 un expected 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.

Digital Control as Source of Chaotic Behavior

International Journal of Bifurcation and Chaos, 2010

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.

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

Nonlinear Dynamics

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 micro-chaos (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 q...

Chaotic Signal Generation from a Switched Dynamical System

2007

In this paper, we explore some modeling capabilities of Scicos. Our aim is to generate chaos from a simple hybrid dynamical system. We give the chaotic dynamics of the voltage-mode controlled buck converter circuit in open loop as a case study. By considering the voltage input as a bifurcation parameter, we observe that the obtained Scicos simulations show that the buck converter is prone to subharmonic behavior and chaos. We also present the corresponding bifurcation diagram.

ICSV 14 Cairns • Australia 9-12 July , 2007 SOMETIMES DIGITAL CONTROL LEADS TO CHAOS

2007

In the present paper, we introduce and analyse a mechanical s ystem in which the digital implementation of a linear control loop may lead to chaotic behavi our. The amplitude of the evolving oscillations is usually very small, this is why these are cal led micro-chaotic vibrations. As a consequence of the digital effects, i.e., the sampling and t he round-off error, the behaviour of the system can be described by a three dimensional piecewise linear map, the micro-chaos map. We examine a 2D version of the micro-chaos map and prove that t he map is chaotic.

Structures within the Quantization Noise: Micro-Chaos in Digitally Controlled Systems

IFAC-PapersOnLine

Quantization, sampling and delay in digitally controlled systems can cause undesired oscillations (Csernák and Stépán, 2011), which-depending on the nature of the uncontrolled system-may cause issues of various importance. In many cases, these oscillations can be treated as quantization noise (Widrow and Kollár, 2008), and can be handled elegantly with the corresponding quantization theory. However, we are interested in the structure and patters of quantization in case of a digitally controlled inverted pendulum with input and output quantizers and sampling. We show the patterns of control effort in case of a simple PD control and highlight how these patternsalong with the dynamics of the controlled system-lead to attractors or periodic cycles with superimposed chaotic oscillations.