Twofold quantization in digital control: deadzone crisis and switching line collision
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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 socalled microchaos (Csernák and Stépán in Int J Bifurc Chaos 5(20):1365–1378, 2010). Although the amplitude of the microchaotic 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.
Keywords
Microchaos Digital effects Rounding Border collision1 Introduction
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]. Because of highperformance applications are featuring fast CPUs, high resolution analogtodigital (adc) and digitaltoanalog converters (dac), these effects were often negligible—in the last years—thanks to the small sampling time, fast computation and high resolution of quantizers.
Currently, however, low and mediumcost controllers (from Atmel\(^{\copyright }\) AVRs in Arduinos to ARM Cortex\(^{\copyright }\) ST microcontrollers) are becoming widespread in several applications, which usually use 8–12 bit adcs/dacs and communicate in larger and wider networks which often introduce noticeable lags. That is, 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 socalled microchaos. Rounding partitions the state space into bands corresponding to different control effort values, and sampling adds irregularity to switching events. Trajectories are allowed to cross switching lines unnoticed for a random amount of time—until the next sampling occurs. It can happen that multiple chaotic attractors appear in the state space of these systems and—depending on the initial conditions—the system may arrive to an attractor far away from the desired state. Thus, the control errors becomes large. Usually, the size of the chaotic attractor is so small that the solution is practically stable [14]. Depending on the nature of the instability of the uncontrolled system, periodic orbits with superimposed chaotic oscillations can also appear [6, 7, 12].
Note that the explicit control of chaos itself is not our goal with the PDcontrol. However, an elegant feedback control approach was introduced in [17] and applied to a simple system in [1].
The single quantization cases—where either only the measured state or only the outgoing control effort is quantized—are well known [3, 5]. In some cases, when both quantizations are present, the less significant can be neglected, and one can return to a single quantization case.
Our current aim is to create a model for twofold quantization from which the single quantization cases can be inherited, and to discover the range of quantization resolutions for which the effect of the less important quantizer is negligible.
In this paper, the effects of twofold quantization are presented on an inverted pendulum with a simple PDcontrol. Two new types of bifurcations are also introduced: deadzone crisis (Sect. 3.3) and collision of switching lines (Sect. 4.1).
2 Digitally controlled inverted pendulum
Consider a single degreeoffreedom (DoF) inverted pendulum with digital control, i.e. the measured states and the output control torque are sampled and quantized. The processing delay is neglected and the controller realizes zeroorderhold, see Fig. 1. The measured angle \(\varphi \) and angular velocity \({\dot{\varphi }}\) are quantized according to input resolution \(r_{{\mathrm {I}}}\), and the calculated control effort M is quantized with output resolution \(r_{{\mathrm {O}}}\).
Note, that the resolution of the angular velocity \({\dot{\varphi }}_i\) is \(r_{{\mathrm {I}}}/\tau \). Thus, according to the definition of the dimensionless time T, one can write \({\dot{\varphi }}_i\,\tau /r_{{\mathrm {I}}} = \varphi '_i/r_{{\mathrm {I}}}\). This results in the same dimension in displacement and velocity with the same quantization resolutions, \(r_{{\mathrm {I}}}\) at the input and \(r_{{\mathrm {O}}}\) at the output.
In some cases, one of the quantizations is dominant over the other, and therefore, the quantization with higher resolution can be neglected, and one of the single quantization models can be used (where either the input or the output is quantized) [3]. However, our goal is to analyse the joint effect of twofold quantization and examine the transition between the twofold and single quantization cases. Doing so, we can also highlight those ranges, where neglecting the less influential quantizer is valid.
2.1 Characteristic displacement for unit resolution output quantization
2.2 Characteristic displacement for unit resolution input quantization
In Equations (\(4_{{\mathrm {I}}}\)–\(4_{{\mathrm {O}}}\)), a single quantization ratio (\(\rho \)) characterizes the ratio of input and output quantization resolutions. For large \(\rho _{{\mathrm {I}}}\) or small \(\rho _{{\mathrm {O}}}\) values, the input quantization dominates, and the outer quantization can be practically neglected. Similarly, for large \(\rho _{{\mathrm {O}}}\) or small \(\rho _{{\mathrm {I}}}\) values, the output quantization is more significant. Lastly, when the characteristic displacements \(X_{{\mathrm {I}}}\) and \(X_{{\mathrm {O}}}\) are equal, \(\rho _{{\mathrm {O}}} = \rho _{{\mathrm {I}}} = 1\), therefore both quantizations have the same unit resolution.
Consequently, it can be firmly stated that none of the singleparameter twofold quantization equations (\(4_{{\mathrm {I}}}\)) or (\(4_{{\mathrm {O}}}\)) can be solely used to analyse the transition to both single quantization cases.
3 Numerical analysis of the microchaos map
3.1 Microchaos map
For the input quantization case, however, these domains are rectangular areas since \(F_i = \hat{P}\,{\mathrm {Int}}(x_{i})+\hat{D}\,{\mathrm {Int}}(x'_{i})\). Consequently, the quantization results in a grid of horizontal and vertical switching lines (see Figs. 5 and 6).
In case of input quantization, our general observation is that a periodic orbit (with superimposed chaotic oscillations) appears around the internal structure of repellers. Depending on the parameters, one or more chaotic attractors spanning over multiple control effort bands can be found, see Fig. 5.
3.2 Cell mapping results
Utilizing Clustered Simple Cell Mapping [9], it is possible to automatically extend the analysed state space region and also execute cell mapping in a parallel computing environment.
We have generated a series of SCM solutions by sweeping the parameter \(\rho _{{\mathrm {I}}}\) for some fixed \(\alpha , \beta , \hat{P}\) and \(\hat{D}\) values. Figure 8 shows the transition from \(\rho _{{\mathrm {I}}}=0\) to \(\rho _{{\mathrm {I}}}=16\) at \(\hat{P}=0.007, \hat{D}=0.02, \alpha = 0.074\) and \(\delta = 0\), which correspond to the parameters of a realistic experimental device. Here the output quantization case has eight separated chaotic attractors (fourfour on both sides, see Fig. 9top) and as the quantization ratio increases, these attractors eventually become repellers. At \(\rho _{{\mathrm {I}}}=1.28\) (see Fig. 9bottom), the outermost attractors disappear resulting in a more favourable state space configuration in terms of control error.
We trace back the aforementioned results to two phenomena: as it can be seen in Fig. 7, the switching lines become jagged, and consequently regions appear in the state space corresponding to onlyP and onlyD control (socalled input deadzones, see Fig. 6), due to the quantization of measured values. In the next section, we examine the effect of these new deadzones.
3.3 Deadzone crisis
In the case of twofold quantization, during the variation of the quantization ratio, the borders of the output deadzone (uncontrolled region between the \({\textsc {sw}}_{\pm 1}\) switching lines, where the control effort is \(F=0\)) and input deadzones (deadzones around x and \(x'\) axes, where either part of the PDcontrol is offline) move, thus state space objects (e.g. attractors or periodic orbits) can disappear or qualitatively change. This is called deadzone crisis.
To illustrate a possible scenario, consider Fig. 10. As the quantization ratio \(\rho _{{\mathrm {I}}}\) increases, the steps on the switching lines grow. At the intersection of the xaxis and the switching line, the switching line becomes locally vertical in the range of the input quantizer’s deadzone and the attractor adapts to this by expanding proportionally. At a certain point—as the switching line gets close to the stable manifold of the nearby saddle point—a deadzone crisis happens, and the solution will be able to escape from the chaotic attractor, leaving a transient chaotic repeller behind.
During the transition from the output quantization to twofold quantization, a series of deadzone crises occur and eventually all chaotic attractors turn to repellers. The interactions of the repellers lead to a newly formed recurring orbit with superimposed chaotic oscillations (see Sect. 3.2 and Fig. 8).
Based on these results, it is obvious that the nonsmooth, stairlike shape of the switching lines play an important role in manipulating state space objects by opening up escape possibilities from the previously closed domains of chaotic attractors.
To gain a deeper insight in this phenomenon, the following section examines the topology of switching lines.
4 Analysis of switching lines
4.1 Switching line collision
When switching line collision occurs, trajectories gain the ability to bypass certain control bands by passing through a switching line intersection point. In the case of PDcontrol—if there is no switching line collision—bands corresponding to the same control effort are connected domains. However, if the switching lines \({\textsc {sw}}_{{\mathrm {m}}}\) and \({\textsc {sw}}_{\mathrm {m+1}}\) collide, the band \(F_i = m\) becomes disconnected (see Fig. 12).
Since the quantization \({\mathrm {Int}}(i)\) has a discontinuity (see Fig. 14), we analyse a small neighbourhood \(\varepsilon \) around \(x=i\,\rho _{{\mathrm {I}}}\) and express the collision between the upper corner of the lower switching line (\({\textsc {sw}}_m\)) and the lower corner of the upper switching line (\({\textsc {sw}}_{m+1}\)).
These kind of slcs will be referred to as firstorder switching line collisions (while in general, the \(k{{\mathrm {th}}}\) order slc means the collision of \({\textsc {sw}}_m\) and \({\textsc {sw}}_{m+k}\)).
Here we considered only the case of positive \(\hat{P}\) and \(\hat{D}\), but a similar analysis can be carried out for negative control parameters, as well.
In the following sections, we show how the transition between the twofold and single quantization cases affects the switching lines.
4.2 Transition from twofold quantization to output quantization
4.3 Transition from twofold quantization to input quantization
Switching lines corresponding to input quantizations form a regular grid of horizontal (Int(y)) and vertical (Int(x)) lines (see Fig. 6right). The square shaped domains (or rectangle shaped domains around the axes) between switching lines correspond to integer value linear combination of the control parameters, e.g. \(F=i\,\hat{P}+j\,\hat{D}\) control effort at \({\mathrm {Int}}(x)=i,\,{\mathrm {Int}}(x')=j\).

Condition 1 switching lines must partition the state space into square shaped domains, i.e. for every \(x'\) value, crossing \(x=i,\,(i\in {\mathbb {Z}})\) values must result in a switch in the control effort value. Similarly, for every x value, crossing \(x'=j,\,(j\in {\mathbb {Z}})\) must also result in a switch.

Condition 2 for each domain between the switching lines, the control effort value should be the same as in the case of input quantization.
It can be seen, that once Condition 1 is satisfied (and the structure of the state space matches the input quantization case), the control effort value of twofold quantization will be within an error of \(\rho _{{\mathrm {O}}}\) to the control effort value of the input quantization case [see Eq. (8)]. It follows therefore, that \(\rho _{{\mathrm {O}}}\rightarrow 0\) will satisfy Condition 2.
5 Conclusion
We have shown that twofold quantization in digital control can be characterized by the quantization ratio, corresponding to the ratio of input and output quantizers’ resolution. We have presented that twofold quantization can be reduced to a single quantization case (input or output quantization) if an appropriate quantization ratio \(\rho \) is used and its limit \(\rho \rightarrow 0\) is analysed.
The microchaos map corresponding to a digitally controlled inverted pendulum was presented and the Clustered Simple Cell Mapping method was used to analyse the effect of varying the quantization ratio. Numerical results revealed, that the chaotic attractors disappear or merge due to the change of the switching lines and deadzones corresponding to input quantization.
We have presented, that deadzone crisis occurs, when the innermost stair on the switching line—governed by the input quantization deadzone—grows large enough to collide with other state space objects. Analysing an example transition from output to twofold and finally input quantization, we have highlighted that a series of deadzone crises happen and separated chaotic attractors merge into a single recurring orbit.
Another interesting effect, the switching line collision was also introduced, which can induce qualitative changes in the state space of continuous flows. Since the solutions of maps are allowed to “jump” in the phasespace, the effects of slc are less pronounced in the case of maps. This is the reason why no slcrelated sudden bifurcations were detected during the analysis of the microchaos map.
From practical point of view, it is possible to improve the properties of the control for a given application, by carrying out an analysis of the quantization ratio and selecting a favourable range as illustrated in Sect. 3.2. Doing so, one can also find out how to improve a certain controlled system, i.e. which quantizer should be replaced by a higherresolution one. In some cases one can even arrive to an unnatural conclusion, that using lowerresolution output quantizer or larger sampling time will actually result in lower control error. Similar results were found in [15, 16], where the quantization improved the stability properties of the controlled system.
Notes
Acknowledgements
Open access funding provided by Budapest University of Technology and Economics (BME). This research was supported by the Hungarian National Science Foundation under Grant No. NKFI128422. The research reported in this paper was supported by the Higher Education Excellence Program of the Ministry of Human Capacities in the frame of Artificial intelligence research area of Budapest University of Technology and Economics (BME FIKPMI).
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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