Abstract
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 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 these patterns lead to chaotic attractors or periodic cycles with superimposed chaotic oscillations.
Highlights
IntroductionDigitally controlled devices are becoming more and more popular, as the field of automation, smart devices, and the Internet of Things continuously grows
Nowadays, digitally controlled devices are becoming more and more popular, as the field of automation, smart devices, and the Internet of Things continuously grows.The three main digital effects: sampling, quantization and processing delay are usually present in all kinds of digitally controlled devices [13,18]
We have shown in our previous works [2,4] that in case of rounding and sampling, digitally controlled systems can exhibit small amplitude chaotic oscillations— the so-called micro-chaos
Summary
Digitally controlled devices are becoming more and more popular, as the field of automation, smart devices, and the Internet of Things continuously grows. The three main digital effects: sampling, quantization and processing delay are usually present in all kinds of digitally controlled devices [13,18]. Because of high-performance applications are featuring fast CPUs, high resolution analog-to-digital (adc) and digital-to-analog converters (dac), these effects were often negligible—in the last years—thanks to the small sampling time, fast computation and high resolution of quantizers. The corresponding digital effects: sampling, quantization and delay are becoming more significant. We have shown in our previous works [2,4] that in case of rounding and sampling, digitally controlled systems can exhibit small amplitude chaotic oscillations— the so-called micro-chaos. Rounding partitions the state space into bands corresponding to different control effort values, and sampling adds irregularity to switching events. Two new types of bifurcations are introduced: deadzone crisis (Sect. 3.3) and collision of switching lines (Sect. 4.1)
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