1 Introduction

Today's technology refers to the deep relation between the physical and computing worlds, the two worlds integrated and connected through sensors and actuators. This integration has appeared in several forms, such as the Internet of things (IoT), industrial 4.0, the Industrial Internet, machine-to-machine (M2M), Internet of Everything, the Smarter Planet, TSensors (Trillion Sensors), and The Fog computing environment. All previous systems are Cyber-Physical Systems (CPS) which are an integration of computation with physical processes [1, 2]. CPS systems face challenges like complex structure, poorly defined safety properties, interaction with complex and stochastic environments that are difficult to model, and the interactions between the system and environment exposed to failure [3, 4]. The interaction between physical processes and devices (embedded processors) in a Fog computing environment is influenced by the sampling rate, delays, jitters, packet losses, and resource contention. The sampling rate, delays, and time requirements are the main constraints in CPS systems, and they are traded off with other parameters like power saving and performance. The time requirements in CPS systems are determined by the following constraints: frequency, chronological order, simultaneity, latency, and temporal assurance or time predictive, which are related to safety and reliability [5–7].

The stochastic physical environments behave in a non-known manner, which is difficult to predict. In a Fog computing environment, the execution within the worst case time (WCET) or achieving the Nyquist condition in the sampling process ($f_s\ge 2f$: $f$ signal frequency, $f_s$ sampling frequency) are not the main goals. The more important issue is the ability to predict the temporal behavior of sensors at the system edge, and this is a basic requirement for Precision Timed (PRET) Machines or embedded processors in CPS Systems. The timing constraints in embedded processors are related to the hardware and software architecture. The hardware includes instruction cycle count, memory access delay, bandwidth, and cache substitution technology, and on the other hand, the software architecture includes operating system and scheduling algorithms, in addition to the inter-event arrival time and worst-case execution time, and the minimum and maximum sampling rate [8–10].

This paper focuses on the sensor signal, which passes through two states in general, the transient state (1) which is the sensor output that goes from the initial value (zero) to the system's working value, and the steady state (2), where the system works at its systematic values. Figure 1 shows the supposed states of the signal $x(t)$ during system operation. The sensor signal in a steady state is almost constant or has small fluctuations, except for some sudden changes that occur sometimes; these sudden changes in the signal will not be noticed by the processor if it works at a fixed low sampling frequency, and it can capture the changes if it works at the maximum sampling frequency.

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The supposed states of the sensor's signal during its lifetime in a system

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00667

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As mentioned above, the sampling rate must be able to capture and predict the temporal behavior of sensors at the system edge, on the other hand, processor utilization, performance, and energy saving are important design issues, especially when the processor implements more than one function. To achieve these issues, an adaptive sampling rate is taken into account in this paper. A higher sampling rate gives higher accuracy and better response time, and a lower sampling rate only saves processor time without reducing the noise and jitter.

No fixed sampling frequency for all applications, a good frequency is one that combines experiment and mathematical analysis of the sensor signal. There are several systematic approaches to determining the sampling frequency, the characteristics of the sensor used in the application, the noise, and the highest and lowest sampling rate of the processor [11].

On the other hand, the sampled time varies depending on some parameters, including processor performance and utilization, processor energy, signal behavior, Nyquist condition, WCET and other issues depending on the application.

To optimally use the embedded processor and improve its performance, we proposed an adaptive sampling algorithm based on the first derivative of the sensor signal, to capture the sudden changes of the signal in a steady state.

The main contributions of the paper are listed as follows:

• The paper proposes a lightweight adaptive sampling algorithm for IoT devices (embedded processors) in CPS systems.

• The proposed algorithm is based on the first derivative of the sensor signal, which can capture the sudden changes of the signal in early time.

• The proposed algorithm enhances processor utilization and performance in the CPS system.

• The paper presents a mathematical model of the sampling frequency, which can be enhanced in future works depending on the application.

The rest of this paper is organized as follows. Section 2 illustrates some important related works. In Section 3, the proposed adaptive sampling algorithm is presented. The simulation results and discussion are illustrated in Section 4. The conclusion is presented in Section 5.

2 Related works

In the following, we summarize some studies that presented different ideas for obtaining an adaptive sampling rate.

Rieger and Taylor propose a low-power analog system, which adjusts the converter clock rate to perform a peak-picking algorithm on the second derivative of the input signal, and establish Adaptive sampling as a practical method to reduce the sample data volume [12]. Feng et al. exploit a strong correlation of radar echo pulses to introduce a compressive sampling (CS) method to implement analog-to-information conversion (AIC) for sub-Nyquist radar target detection [13]. Qaisar et al. use an adaptive rate ADC which is based on the Cross-Level Sampling Scheme (LCSS), which can adapt its conversion activity according to the input signal local variations [14]. Mishali and Eldar propose a modulated wideband converter, which multiplies the analog signal by a bank of periodic waveforms, and the product is low-pass filtered and sampled uniformly at a low rate [15]. Jaraczewski et al. discuss the methods of measuring electrical quantities by devices with low computational efficiency and a low sampling frequency up to 1 kHz. The main advantage of this new method is that it achieves a balance between processing power and accuracy in calculating the most important electrical signal indicators (such as power, RMS, and THD) [16]. Koutsoubelias et al. propose a protocol based on the Kalman filter following the dynamic changes in the sensor data generation rate [17]. Homjakovs proposed an adaptive sampling approach in which analog signal samples are taken depending on their activity [18]. Alexandru and Dragotti investigate the problem of timing-based sampling of non-bandlimited signals within the Finite Rate of Innovation (FRI) setting and show how it can nonuniformly sample these signals using a compact-support kernel that satisfies the generalized Strang-Fix conditions [19]. Bhandari et al. introduce the concept of “Unlimited Sampling”, and in addition to [15] they use Self-Reset ADCs (SR-ADCs), which allow for sensing modulo samples [20]. Liu and Feng introduced an adaptive dead band-triggered communication scheme for sampled-data systems [21].

Tadokoro et al. propose two adaptive frequency estimation methods using variable sampling processing. One of them is a synchronous addition and subtraction method and the other is a notch filter method [22]. Wang and Wan present a group of local time domain theorems for sampling differentiable continuous analog signals through Lagrange interpolation and show how to use them for adaptive control of the sampling rate [23]. Petkovski et al. explore Chaikin's algorithm for the generation of arbitrary curves to reconstruct non-uniformly sampled signals. The sampling is adapted to the signal shape [24]. Li et al. present a technique for adaptive ocean sampling based on a maximum differential algorithm (MDA) using ocean sampling platforms equipped with multiple sensors [25]. Czaczkowska et al. propose adaptive nonuniform sampling algorithms [26]. Fox et al. introduces and investigates signal reconstruction algorithms that are given MSVR sampling conditions for vehicular CPS applications [27].

Corso et al. in [3] summered importance sampling algorithms for estimating the probability of failure in the CPS system, including the cross-entropy method, multilevel splitting, classification-based importance sampling, and state-dependent importance sampling. Termehchi and Rasti in [5] used self-triggered methods to determine the next maximum acceptable sampling moment and to allocate resources in the network subsystem, aiming to minimize power consumption in the industrial CPS. Jazayeri et al. in [7] used Deep Reinforcement Learning (DRL) algorithm to find the fast response and improve the network delay as well as reducing the energy consumption in the mobile, Fog Devices and Cloud. Ghobaei-Arani et al. in [28] presented a task scheduling algorithm based on moth-flame optimization algorithm to assign an optimal set of tasks to fog nodes to meet the satisfaction of quality of service requirements of CPS applications in such a way that the total execution time of tasks is minimized. Shahidinejad et al. in [29] proposed a lightweight authentication protocol for IoT devices using a three-layer scheme, Including IoT device layer, trust center at the edge layer, and cloud service providers.

In [30], the authors proposed an Adaptive Stochastic Gradient Descent Algorithm to evaluate the risk of fetal abnormality, the findings of this work suggest that proposed innovative method can successfully classify the anomalies linked with nuchal translucency thickening. In [2], the authors proposed an AI-enabled IoT-CPS Algorithm which doctors can utilise to discover diseases in patients based on AI. The experimental results demonstrate that the proposed algorithm is more efficiently compared with existing algorithms. In [31], a decentralized OFSD control strategy was proposed, to handle the global output feedback sampled-data (OFSD) control problem for cyber-physical systems (CPSs) described by nonstrict-feedback large-scale nonlinear systems with denial-of-service attacks. The authors in [32] presented an effective micro-genetic algorithm in order to choose suitable destinations between physical hosts for VMs. The researcher in [4] proposed a new routing protocol with the cluster structure for IoT networks using blockchain-based architecture for software-defined networking (SDN) controller.

3 Proposed adaptive sampling algorithm

We design a proposed algorithm in several stages, starting with analyzing the sensor signal, then calculating the sampling frequency corresponding to the signal changes, and finally the algorithm formulation.

3.1 Analytic study of a sensor signal

Let $x(t)$ be an analog output signal of a sensor. The signal $x(t)$ changes in two ways, either a first-order change or a second-order change, and mathematically formulated as first derivative $\frac{dx}{dt}$ or second derivative $\frac{d^{2}x}{{dt}^2}$ as shown in Fig. 2a. And maybe the change from a higher order, and mathematically is written as n^th derivative $\frac{d^{n}x}{{dt}^n}$, where $n$ is a finite integer.

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Sensor signal, showing the first and second derivative in (a), and the signal's slow and fast rate of change in (b)

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00667

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As we show in Fig. 2a, the second-order change is a sequence of two first-order changes, so any order of change is a combination of a set of first-order changes. The changing rate might be slow (slow rate) and might be fast rate. Also, it is associated with the change of horizon angle as shown in Fig. 2b. The value of the angle is associated with the speed of change, as the changing rate is faster as the angle is larger, and vice versa. Therefore, determining the angle will help to determine the sampling frequency. To capture all signal changes, the sampling rate must be increasing with signal frequency.

Based on the above, we propose a ${\theta }_{\mathit{min}}$ is the angle where the sampling frequency is minimum, and ${\theta }_{\mathit{max}}$ is the angle where the sampling frequency is maximum.

The first derivative of the curve at any point is the tangent of the curve at this point as shown in Fig. 3a, we define the parameter $\alpha$ as follows:

$\alpha =\tan \theta$(2)

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The two states of change in signal, slow rate state and fast rate state depending on angle $\theta$ with the horizon

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00667

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The value of $\alpha$ changes in range $[0,\infty [$ depending on the angle $\theta$ with the horizon, as follows:

$\alpha =\{\begin{array}{c}\left[0,1\right]\hspace{0.17em}0°\le \theta \le 45°\\ [1,\infty ]\hspace{0.17em}45°<\theta \le 90°\end{array}$(3)

The curve of signal changes slowly in the region where $0°\le \theta \le 45°$, we called it area (1) or slow rate area, and the second area (2), where $45°<\theta \le 90°$ is fast rate area as shown in Fig. 3b. On the other hand, from relation (2), we obtain the curve in Fig. 4. As we see, we can divide the curve into two regions, the first one where $0°\le \theta \le 45°$ is a linear region and the second one where $45°<\theta \le 90°$ is a nonlinear region.

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The relation between $\alpha$ and $\theta$ (where $0°<\theta \le 90°$)

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00667

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3.2 Calculation of the parameter $\mathit{\alpha }$

When the signal changes suddenly in a steady state, this change takes the form of a sinusoidal signal as shown in Fig. 5, we can specify this instant change as follows:

$x(t)=x_{\mathit{max}}\sin \omega t$(4)

Where $\omega =2\pi f$, $f$ is the signal frequency at the time of change, and $x_{\mathit{max}}$ is the maximum amplitude.

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The sudden change in the sensor signal

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00667

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We illustrate in Fig. 6a the rising edge of change in signal, which we defined as first derivative of the signal. If we assume $T_s$ is the sampling period of the embedded processor, or $f_s=1/T_s$ is the sampling frequency. The difference between actual signal $x(t)$ and sample signal $x_s(t)$ at time $t$ is the error $\epsilon (t)$, which we can formulate as an algebraic value as follows:

$\epsilon (t)=\left|x(t)-x_s(t)\right|$(5)

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The relation between sampling error and angle $\theta$ (1st derivative), (a): the sampling period $T_s$ and the error $\epsilon$ at the middle of the period $T_s$, (b): the error $\epsilon$ at various angles (${\theta }_1,{\theta }_2,{\theta }_3$) as a function of time $\tau$ measured at an arbitrary point in the period $T_s$

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00667

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As shown in Fig. 6b, the value of $\epsilon$ increases with increasing the slope of the curve or with increasing the angle $\theta$. If we assume $t_k$ is the origin of the time coordinate, we can write the error as a function of time as $\epsilon (\tau )$. Then, we have the following:

${\theta }_1>{\theta }_2>{\theta }_3\hspace{0.17em}\Rightarrow \hspace{0.17em}{\epsilon }_1(\tau )>{\epsilon }_2(\tau )>{\epsilon }_3(\tau ),or\hspace{0.17em}\epsilon (\tau )\propto \theta$

Typically, we use one of three basic methods to obtain dc signal from ac signal [33]: rms -to-dc conversion, peak detection, and ac-to-MAV conversion. In the case of MAV (mean absolute value), the error follows a linear behavior. On the other hand, Ac/MAV converters rely on the particular relation between the rms (root mean square) voltage of a sine wave and its mean absolute value after rectification. For a full-wave rectified sine wave, we have the mean absolute value of signal $x(t)$ as follows:

$x_s\left(MAV\right)=\frac{\omega }{\pi }{\int }_0^{\pi /\omega }{x}_{\mathit{max}}\sin \omega t\hspace{0.17em}dt=\frac{2\hspace{0.17em}{x}_{\mathit{max}}}{\pi }$(6)

For a small sinusoidal change, we suppose these two approximates. In the first one, we consider the error $\epsilon (\tau )$ is measured in the middle of the sampling period, i.e. between $t_k\hspace{0.17em}\text{and}\hspace{0.17em}{t}_{k+1}$ as shown in Fig. 6a, and the second one, the mean absolute value MAV for a signal at the time of change is equal to sampling signal $x_s(t)$. As a result of our proposals and the relations (5) and (6) we calculate the error as follows:

$\epsilon (\tau )=\left|x(\tau )-\frac{2\hspace{0.17em}{x}_{\mathit{max}}}{\pi }\right|$(7)

For a small period time $T_s$, the sine is approximated by its Taylor expansion, truncated after the second term

$x(\tau )=x_{\mathit{max}}\sin \omega \tau =x_{\mathit{max}}\hspace{0.17em}\left(\omega \tau -\frac{{\omega }^3{\tau }^3}{3!}+O\left({\tau }^5\right)\right)$(8)

From the previous, the error is written as

$\epsilon (\tau )=\frac{2\hspace{0.17em}{x}_{\mathit{max}}}{\pi }-x_{\mathit{max}}\hspace{0.17em}\left(\omega \tau -\frac{{\omega }^3{\tau }^3}{3!}\right)$(9)

From Fig. 6b and the relation (2), and using an acceptable approximation for small changes, we calculate the parameter $\alpha$ as follows:

$\alpha =\tan \theta =\frac{\epsilon }{\tau },0\le \tau \le T_s$(10)

We measure the error in the middle of range $\left[t_k,t_{k+1}\right]$, i.e. $\tau =T_s/2$ then

$\epsilon \left(T_s/2\right)=\frac{2\hspace{0.17em}{x}_{\mathit{max}}}{\pi }-x_{\mathit{max}}\omega \hspace{0.17em}\frac{T_s}{2}\left(1-{\omega }^2\frac{{T_s}^2}{24}\right)$(11)

Finally, $\alpha$ is written as follows:

$\alpha =\tan \theta =\frac{\epsilon }{T_s/2}=\frac{4\hspace{0.17em}{x}_{\mathit{max}}}{\pi T_s}-x_{\mathit{max}}\omega \hspace{0.17em}\left(1-{\omega }^2\frac{{T_s}^2}{24}\right)$(12)

We reformulate the previous relation, where $T_s=1/f_s$ and $\omega =2\pi f$, we obtain

$\alpha =\frac{4\hspace{0.17em}{x}_{\mathit{max}}}{\pi }{f}_s-x_{\mathit{max}}2\pi f\hspace{0.17em}\left(1-\frac{{\left(2\pi f\right)}^2}{24{f_s}^2}\right)=x_{\mathit{max}}\left[1.3\hspace{0.17em}{f}_s-6.3\hspace{0.17em}f\hspace{0.17em}\left(1-1.6\frac{f^2}{{f_s}^2}\right)\right]$(13)

3.3 Calculation of the sampling frequency ${\mathit{f}}_{\mathit{s}}$

As we see in relation (4), the signal frequency during change is $f$. Now, we assume the sampling frequency $f_s$ of the embedded processor varies between Maximum frequency $f_{smax}$ and Minimum frequency $f_{smin}$. In comparison with the angle of curve slope, the sampling frequency should be

$f_s=\{\begin{array}{c}{f}_{smin}\hspace{0.17em}\theta ={\theta }_{\mathit{min}}\\ f_{smax}\hspace{0.17em}\theta ={\theta }_{\mathit{max}}\end{array}$(14)

The angle $\theta$ changes in the range $\theta \in \left[0,\frac{\pi }{2}\right]$, we divide this range into slow rate area and fast rate area, as shown in Fig. 3b. Then the relation (3) becomes

$f_s=\{\begin{array}{c}{f}_{smin}\hspace{0.17em}\theta =0°\\ f_{smax}\hspace{0.17em}\theta =90°\end{array}$

For $f_{smax}$ sufficiently big compared with $f_{smin}$, we divide the sampling frequency range into two ranges $\left[f_{smin},f_{smax}/2\right],\text{and}\hspace{0.17em}\left[f_{smax}/2,f_{smax}\right]$, the previous relation becomes as follows:

$f_s=\{\begin{array}{c}\in \left[f_{smin},f_{smax}/2\right]\hspace{0.17em}0\le \alpha <1\\ \in \left[f_{smax}/2,f_{smax}\right]\hspace{0.17em}\alpha \ge 1\end{array}$(15)

Then the sampling system has two states as following:

$\{\begin{array}{c}state\hspace{0.17em}1\hspace{0.17em}{f}_{smin}\le f_s<\hspace{0.17em}{f}_{smax}/2,0\le \alpha <1\\ state\hspace{0.17em}2\hspace{0.17em}{f}_{smax}/2\le f_s<f_{smax},\alpha \ge 1\end{array}$

In state1, the sampling frequency $f_s$ changes linearly between two values $\left\{f_{smin},f_{smax}/2\right\}$ (see Fig. 4), then we can formulate the sampling frequency in state1 as follows:

$f_s=\alpha .\hspace{0.17em}{f}_{smax}/2+\left(1-\alpha \right).f_{smin},0\le \alpha \le 1$(16)

Where $\theta =0°\iff \alpha =\tan 0°=0,\Rightarrow f_s=f_{smin}$ $\theta =45°\iff \alpha =\tan 45°=1,\Rightarrow f_s=f_{smax}/2$

In state2, the sampling frequency increases in a nonlinear manner from $f_{smax}/2$ to $f_{smax}$. The nonlinear part in Fig. 4 behaves as same as the function $\left\{f(\alpha )=\left(1-2^{-\alpha }\right),\alpha \ge 1\right\}$ as shown in Fig. 7.

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The output characteristic of function $f(\alpha )=\left(1-2^{-\alpha }\right)$ drawing by Ptolemy II

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00667

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Then the sampling frequency in state2 is formulated as follows:

$f_s=f_{smax}\left(1-2^{-\alpha }\right),\alpha \ge 1$(17)

Where

$\theta =45°\iff \alpha =\tan 45°=1,\Rightarrow f_s=f_{smax}/2$

$\theta =90°\iff \alpha \to \infty ,\Rightarrow f_s=f_{smax}$

Finally, the 1^st derivative adaptive sampling frequency is given by the following relation:

$f_s=\{\begin{array}{c}\alpha .\hspace{0.17em}\frac{f_{smax}}{2}+\left(1-\alpha \right).f_{smin},\alpha \in \left[0,1\right]\\ f_{smax}\left(1-2^{-\alpha }\right),\alpha \in [1,\infty [\end{array}$(18)

where $\alpha =x_{\mathit{max}}\left[1.3\hspace{0.17em}{f}_s-6.3\hspace{0.17em}f\hspace{0.17em}\left(1-1.6\frac{f^2}{{f_s}^2}\right)\right]$

3.4 Algorithm realization

The proposed algorithm consist of four steps, step 1 initializing the data, step 2 reading the frequency $\mathrm{f}$ of input signal, then calculating the condition parameter $\alpha$ (see relation 18) in step 3, and at last calculating the sampling frequency ${\mathrm{f}}_{\mathrm{s}}$ (see relation 18) depending on the value of $\alpha$ in step 4. The flowchart and the pseudo code of proposed algorithm are shown in Fig. 8.

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The flowchart and the pseudo code of the proposed algorithm

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00667

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The time complexity of the proposed algorithm on general constant O(1) for one calculation cycle, because the used statement is “If-then-else". The time complexity for the life cycle of the sensor signal is linear O(n) depending on the sampled frequency ${\mathrm{f}}_{\mathrm{s}}$, where $n\approx {\mathrm{f}}_{\mathrm{s}}$. In other words, the time complexity of the algorithm is polynomial and it is effective.

4 Simulation and results

The algorithm was implemented and tested as a hybrid model using Ptolemy II (Version 11.0.1_20180619) [34], which is a Java-based software that studies modeling, simulation, and design of concurrent, real-time, embedded systems. It focuses on assembly of concurrent components and uses well-defined models of computation that govern the interactions between components. A major problem area being addressed is the use of heterogeneous mixtures of models of computation. The Ptolemy project includes a number of support packages, such as graph, providing graph-theoretic manipulations, math, providing matrix and vector math and signal processing functions, plot, and providing visual display of data. Also, it has C Code generator and can generate code for some models. It is suitable for designing and testing various types of models and algorithms.

The computer that runs the Ptolemy program has the following resources (CPU: Intel(R) Core(TM) i5-4300U CPU @ 1.90 GHz 2.50 GHz, RAM: 4.00 GB, GPU: GeForce GT 720M). The proposed embedded processor is ATmega328, which works at clock frequency 16 MHz. ATmega328 is a microcontroller with multi-uses, it is used in small and medium applications and the results it gives can be applied to other embedded processors, it provides a ${\mathbf{f}}_{\text{smax}}=\mathbf{1}\mathbf{M}\mathbf{H}\mathbf{z}$ as the maximum sampling frequency, and ${\mathbf{f}}_{\text{smin}}=\mathbf{10}\mathbf{k}\mathbf{H}\mathbf{z}$ as the minimum sampling frequency.

Figure 9 shows the components used to execute the algorithm. The first component is used to generate sampling frequency ${\mathbf{f}}_{\mathbf{s}}$ is formulated in relation (18), the first component presents the system states, state1 (slow rate) and state2 (fast rate). The second component is used to generate $\mathit{\alpha }$ based on 1st derivative of the sensor signal.

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Modeling of the proposed algorithm using Ptolemy II

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00667

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4.1 Simulation

The algorithm has been tested and the result is shown in Figs 10 and 11. In Fig. 10, the signal we applied is still stable until $t\approx 5.5\hspace{0.17em}\mathit{sec}$, then it starts to change with a low frequency between ($t=5.5\hspace{0.17em}\mathit{sec}\hspace{0.17em}and\hspace{0.17em}t=8\hspace{0.17em}\mathit{sec}$), then the frequency of the signal increases in the interval [8 s, 10.02 s] but it still in state1 (slow rate), at $t=10.1\hspace{0.17em}\mathit{sec}$ the frequency increased and the signal moved to state2(fast rate), in the other hand sampling frequency increased in the same manner, which it is $f_{\mathrm{s}}=f_{\text{smin}}=10\text{kHz}$ in the begin till $t\approx 5.5\hspace{0.17em}\mathit{sec}$, then increased to $f_{\mathrm{s}}\approx 21\text{kHz}$ at (1), $f_{\mathrm{s}}\approx 233\text{kHz}$ at (2), and $f_{\mathrm{s}}\approx 564\text{kHz}$ at (3).

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The results of testing the algorithm, where $\alpha =0$ in (1) no change in signal, $0\le \alpha \le 1$ in (2) slow rate state, and $\alpha >1$ in (3) fast rate state, and the sampling frequency $564\text{kHz}\ge {\mathrm{f}}_{\mathrm{s}}\ge 10\text{kHz}$

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00667

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Increase the frequency of the signal and $\alpha >2$ in (3) and the sampling frequency ${\mathrm{f}}_{\mathrm{s}}\ge 0.8\text{MHz}$

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00667

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Figure 10D shows the sampling density, it is clear that the density of samples increases with sampling frequency. Another test is shown in Fig. 11, where the sampling frequency increases and crosses $f_{\mathrm{s}}\approx 0.8\text{MHz}$ when becomes $\alpha >2$ in state2.

From the results, it is clear that the increasing sampling frequency when the signal begins in change, is the same as the sampling rate. Also, the linear relation between ${\mathrm{f}}_{\mathrm{s}}$ and $\alpha$ in state1, and the sampling frequency ${\mathrm{f}}_{\mathrm{s}}$ increases in a nonlinear manner with $\alpha$ in state2 (see Fig. 11C). As mentioned in the introduction, achieving the Nyquist condition in the sampling algorithm is not the main goal, because for all signal frequencies ($f$) the minimum sampling frequency ${\mathrm{f}}_{\text{smin}}\ge 2f$ and the Shanon-Nquist sampling theorem is satisfied, and in all results the condition ${\mathrm{f}}_{\text{smin}}\ge 2f$ is achieved.

Finally, and from the previous results, we can conclude the relationship between sampling frequency and the slope of the signal (the angle $\theta$ with horizon). Figure 12 shows how frequency changes with angle $\theta$, and we observe from the curve a Zeno behavior [35] in simulation at $\left(\theta >73°\hspace{0.17em}\text{or}\hspace{0.17em}\alpha >3\right)$, which reflects the rapid change in signal.

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Sampling frequency as function to angle $\theta$ with the horizon

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00667

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5 Conclusions

In Cyber-Physical Systems, neither the worst-case time nor achieving the Nyquist condition in the sampling process are the main goals. The most important thing is the ability to predict the time and the temporal behavior of the system variables, which is the basic requirement for embedded processors or Precision Timed (PRET) Machines in CPS. The sensors are the main source of non-estimated temporal behavior, and with these Trillion Sensors, the matter becomes more complicated.

To improve the CPS's performance, the usability of computing platforms and a good estimate of time are required. An adaptive sampling algorithm for cyber-physical system (embedded processors) has been proposed in this paper, where the sampling frequency is based on the 1st derivative of the sensor signal. The results showed how sampling frequency $f_s$ increased during signal change and proved the validity of this algorithm. The density of the samples increased linearly in the first state (slow rate state) and nonlinear manner in the second state (fast rate state). The proposed algorithm is simple, it contributes to improving the performance, and the usability of the embedded processor, and reduces power consumption in the system. This algorithm helps in realizing the time conditions in Cyber-Physical Systems.

The proposed algorithm has many applications in the real world, like monitoring human health, especially in those who suffer from diseases such as heart, blood pressure, and diabetes, and need permanent monitoring. Also in electrical networks, monitoring cooling water in nuclear plants, remote childcaree, mentoring oxygen pressure in hospitals, monitoring the temperature and pressure during chemical reactions, and other precision industrial applications. In all previous applications, the physical quantities are constant during normal operation, except in some cases a sudden change happens. These kinds of application need a lightweight adaptive algorithm with a little overhead, like the proposed algorithm that can capture these sudden changes.