Abstract
Digital effects (quantization, sampling and delay) can lead to small amplitude chaotic oscillations, called micro-chaos [1, 2]. Often, these vibrations are neglected due to their small amplitude or replaced by random noise, but doing so one might be unable to capture some important behavior of the digitally controlled system.One notable example is the PD-controlled inverted pendulum with quantization at the calculated control effort. Taking digital effects into account leads to separated chaotic attractors in the state space. While the amplitude of the chaotic oscillations is indeed small, these attractors are situated rather far from the desired position, introducing considerable control error. Micro-chaos is undoubtedly present in ideal models of computer-controlled mechanical systems, however an important question is still open: does it persist if a more complex model of reality is used? For instance, does it survive in the presence of dry friction?This paper answers the latter question analyzing the micro-chaos in a system with Coulomb friction. We introduce the so-called hybrid micro-chaos map that describes the behavior of a digitally controlled system with dry friction. Then, the theoretical analysis of this map is presented and numerical results are provided that were acquired using a new mathematical tool, the Clustered Simple Cell Mapping method.Lastly, we conclude, that the phenomena of micro-chaos can withstand the presence of Coulomb friction and chaotic attractors can coexist with sticking zones in the state space.
Highlights
In our past works, we have analyzed various digitally controlled systems exhibiting micro-chaos: small amplitude chaotic oscillations
We conclude, that the phenomena of micro-chaos can withstand the presence of Coulomb friction and chaotic attractors can coexist with sticking zones in the state space
Even though we have shown that properly taking quantization into account results in different behavior compared to the neglection or replacement by random-noise approach, we have not addressed an important question yet: Does micro-chaos persist if friction is present?
Summary
We have analyzed various digitally controlled systems exhibiting micro-chaos: small amplitude chaotic oscillations. The model of inverted pendulum has a great significance in this area, as well, during the analysis of stick-balancing [6] which can provide important results to understand the control strategy implemented by the human brain when compared with measurements As it was presented in [7], quantization can contribute to the stabilization of unstable dynamical systems in the presence of feedback delay in neural control. The present paper is mainly based on the results of [8], where the dynamics of a digitally controlled inverted pendulum was analyzed without dry friction As it was proved, sampling and quantization leads to chaotic behavior and the appearance of coexisting chaotic attractors and repellors in the phase-space of the considered system. Before formulating the equations for the quantized case, we add a new term to Eq (2) that corresponds to Coulomb friction (see Fig. 2):
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