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Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things

Pubblicato online: 15 Jul 2022
Volume & Edizione: AHEAD OF PRINT
Pagine: -
Ricevuto: 09 Feb 2022
Accettato: 31 Mar 2022
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License
Formato
Rivista
eISSN
2444-8656
Prima pubblicazione
01 Jan 2016
Frequenza di pubblicazione
2 volte all'anno
Lingue
Inglese
Abstract

This article proposes a software fusion processing method for multi-sensor monitoring the Internet of Things based on the fractional differential filtering algorithm. The purpose is to solve the difference between the detection data caused by the difference in each production information sampling node's equipment performance and working environment. The article applies the above algorithm to the remote detection data fusion processing experiment of the discrete manufacturing system of the Internet of Things. The research found that the algorithm realizes the remote high-precision measurement of the Internet of Things sensor information. The fractional integral operator has high fusion accuracy in remote detection data fusion processing.

Keywords

MSC 2010

Introduction

The Internet of Things connects all objects through perception technology and identification technology and the Internet for information exchange and communication [1]. This provides universal data analysis and services based on heterogeneous Internet infrastructure and ubiquitous sensors and networks. It finally realizes the one-to-one mapping relationship between data and objects and provides query interfaces [2]. This paper proposes a discrete system differential data fusion guided by integral discrete in the Internet of Things. We first adopt the Internet of Things technology to uniformly collect and integrate the terminal data of each discrete manufacturing system under the distributed system. In addition, after summarizing all the information, we use the integral discrete guidance method to process all the differentiated data obtained to achieve the effective fusion of all data [3]. Finally, we use a set of 6 types of data of 100 nodes to test the system's performance.

Proposal of Integral Discrete Guided Algorithm

An integral discrete guidance algorithm is an algorithm that uses the calculus method to fuse and uniformly process the distributed and differentiated data in the system network. We use the integration idea under the distributed system to organically combine various data in the network system [4]. Finally, the efficient integration of differentiated data is realized. The system model of the integral discrete guidance algorithm is defined as: {E(I)=Ω|DυIdxdy+2¯I0IL22IBVυ(Ω)xΩyΩ \left\{{\matrix{{E\left(I \right) = \int_\Omega {\left| {{D^\upsilon}I\left\| {dxdy + \bar 2\left\| {{I_0} - I} \right\|_{{L_2}}^2} \right.} \right.}} \hfill \cr {I \in B{V^\upsilon}\left(\Omega \right)} \hfill \cr {x \in \Omega} \hfill \cr {y \in \Omega} \hfill \cr}} \right.

|DυI|=(DxυI)2+(DyυI)2 \left| {{D^\upsilon}I} \right| = \sqrt {{{\left({D_x^\upsilon I} \right)}^2} + {{\left({D_y^\upsilon I} \right)}^2}} is the system model of the integral discrete guided algorithm. DxυI D_x^\upsilon I is the integral derivative of the integral discrete guidance algorithm concerning point x. DyυI D_y^\upsilon I is the integral derivative of the integral discrete guidance algorithm concerning point y.

We need to adopt the minimization thought processing method in the integral discrete guidance algorithm [5]. Therefore, we use the variational nature of the integral discrete guided algorithm to construct the corresponding differential data guiding function, which we define as: g(ε)=E(I+εφ)=Ω|Dυ(I+εφ)I||dxdy+λ2||I0Iεφ||L22 g\left(\varepsilon \right) = E\left({I + \varepsilon \varphi} \right) = \int_\Omega {\left| {{D^\upsilon}\left({I + \varepsilon \varphi} \right)I} \right|\left| {dxdy + {\lambda \over 2}} \right|\left| {{I_0} - I - \varepsilon \varphi} \right||_{{L_2}}^2}

ε is any real number. φ is an arbitrary function. We use different data to guide the basic properties of functional extremes [6]. When φC0(Ω) \varphi \in C_0^\infty \left(\Omega \right) and gε'(0)=0 g_\varepsilon^{'}\left(0 \right) = 0 , the corresponding solution guiding equation can be obtained: Ω(DxυI|DυI|Dxυφ+DyυI|DυI|Dyυφ)dxdyλΩ(I0I)φL2dxdy \int_\Omega {({{D_x^\upsilon I} \over {\left| {{D^\upsilon}I} \right|}}D_x^\upsilon \varphi + {{D_y^\upsilon I} \over {\left| {{D^\upsilon}I} \right|}}D_y^\upsilon \varphi)dxdy - \lambda \int_\Omega {({I_0} - I){\varphi _{{L_2}}}dxdy}}

We need to eliminate arbitrary functions to get the final difference data guide model. Therefore, ϕ adopts the identity formula in the differential data guidance model of discrete integral guidance: Ωfgdxdy=λ2πΩf^g^¯dω1dω2=0 \int_\Omega {fgdxdy = {\lambda \over {2\pi}}\int_\Omega {\hat f\,{\bar {\hat g}}d{\omega _1}d{\omega _2} = 0}}

Processing the difference data fusion algorithm guided by integral discrete can get: λ2πΩ(D^xυφD^xυI¯|DυI|+D^yυφD^yυI¯|DυI|)dω1dω2λ22πΩφ^(I^0¯I^¯)dω1dω2=0 {\lambda \over {2\pi}}\int_\Omega {({\hat D}_x^\upsilon \varphi {{\overline {{\hat D}_x^\upsilon I}} \over {\left| {{D^\upsilon}I} \right|}} + {\hat D}_y^\upsilon \varphi {{\overline {{\hat D}_y^\upsilon I}} \over {\left| {{D^\upsilon}I} \right|}})} d{\omega _1}d{\omega _2} - {{{\lambda ^2}} \over {2\pi}}\int_\Omega {{\hat \varphi} (\overline {{{\hat I}_0}} - {\bar {\hat I}})d{\omega _1}d{\omega _2} = 0}

The transformation properties of the difference data fusion algorithm guided by integral discrete are: D^xυφ=(j2πω1)υφ^,D^yυφ=(j2πω2)υφ^ {\hat D}_x^\upsilon \varphi = {\left({j2\pi {\omega _1}} \right)^\upsilon}{\hat \varphi},{\hat D}_y^\upsilon \varphi = {\left({j2\pi {\omega _2}} \right)^\upsilon}{\hat \varphi}

We substitute formula (6) into formula (5) and eliminate the coefficients to obtain: Ω((j2πω1)υφ^D^xυI¯|DυI|+(j2πω2)υφ^D^yυI¯|DυI|)dω1dω2λΩφ^(I^0¯I^¯)dω1dω2=0 \int_\Omega {({{\left({j2\pi {\omega _1}} \right)}^\upsilon}{\hat \varphi} {{\overline {{\hat D}_x^\upsilon I}} \over {\left| {{D^\upsilon}I} \right|}} + {{\left({j2\pi {\omega _2}} \right)}^\upsilon}{\hat \varphi} {{\overline {{\hat D}_y^\upsilon I}} \over {\left| {{D^\upsilon}I} \right|}})d{\omega _1}d{\omega _2} - \lambda \int_\Omega {{\hat \varphi} (\overline {{{\hat I}_0}} - {\bar {\hat I}})d{\omega _1}d{\omega _2} = 0}}

The arbitrariness of the function φ in the differential data fusion algorithm is guided by integral discrete. We can use the basic lemma of the variational method of the differential data fusion algorithm to get: (j2πω1)υD^xυI¯|DυI|+(j2πω2)υD^yυI¯|DυI|λ(I^0¯I^¯)=0 {\left({j2\pi {\omega _1}} \right)^\upsilon}{{\overline {{\hat D}_x^\upsilon I}} \over {\left| {{D^\upsilon}I} \right|}} + {\left({j2\pi {\omega _2}} \right)^\upsilon}{{\overline {{\hat D}_y^\upsilon I}} \over {\left| {{D^\upsilon}I} \right|}} - \lambda \left({\overline {{{\hat I}_0}} - {\bar {\hat I}}} \right) = 0

We use the inverse transformation of the differential data fusion algorithm guided by integral discreteness to get the partial differential equation of the corresponding differential data fusion algorithm as: Re{[Dxυ¯(DxυI|DυI|+Dyυ¯(DyυI|DυI|)]}λ(I^0¯I^¯)=0 {\mathop{\rm Re}\nolimits} \{[\overline {D_x^\upsilon} ({{D_x^\upsilon I} \over {\left| {{D^\upsilon}I} \right|}} + \overline {D_y^\upsilon} ({{D_y^\upsilon I} \over {\left| {{D^\upsilon}I} \right|}})]\} - \lambda (\overline {{{\hat I}_0}} - {\bar {\hat I}}) = 0

Finally, the gradient descent method of the differential data fusion algorithm can be used to obtain the model of the differential data fusion algorithm based on discrete integral guidance: It=Re{[Dxυ¯(DxυI|DυI|+Dyυ¯(DyυI|DυI|)]}+λ(I^0¯I^¯)=0 {{\partial I} \over {\partial t}} = - {\mathop{\rm Re}\nolimits} \{[\overline {D_x^\upsilon} ({{D_x^\upsilon I} \over {\left| {{D^\upsilon}I} \right|}} + \overline {D_y^\upsilon} ({{D_y^\upsilon I} \over {\left| {{D^\upsilon}I} \right|}})]\} + \lambda (\overline {{{\hat I}_0}} - {\bar {\hat I}}) = 0

Then the difference fusion of the difference data fusion algorithm guided by discrete integral guidance in each direction is defined as: DxI(x,y)=I(x,y)I(x1,y)=1NI(ω1,ω2)(1ej2πω1/N)ej2π(xω1+yω2)/N {D_x}I\left({x,y} \right) = I\left({x,y} \right) - I\left({x - 1,y} \right) = {1 \over {\sqrt N}}I(\mathop {{\omega _1},\,{\omega _2}}\limits^ \wedge)(1 - {e^{j2\pi {\omega _1}/N}}){e^{j2\pi \left({x{\omega _1} + y{\omega _2}} \right)/N}} DyI(x,y)=I(x,y)I(x,y1)=1NI(ω1,ω2)(1ej2πω1/N)ej2π(xω1+yω2)/N {D_y}I\left({x,y} \right) = I\left({x,y} \right) - I\left({x,y - 1} \right) = {1 \over {\sqrt N}}I(\mathop {{\omega _1},\,{\omega _2}}\limits^ \wedge)(1 - {e^{j2\pi {\omega _1}/N}}){e^{j2\pi \left({x{\omega _1} + y{\omega _2}} \right)/N}}

Therefore, according to the nature of the differential data fusion algorithm guided by integral discrete, the corresponding relationship between the spatial domain and the frequency domain of the first-order difference is defined as: DxI(x,y)(1ej2πω1/N)I(ω1,ω2)DyI(x,y)(1ej2πω2/N)I(ω1,ω2) \matrix{{{D_x}I\left({x,y} \right) \leftrightarrow (1 - {e^{- j2\pi {\omega _1}/N}})I(\mathop {{\omega _1},\,{\omega _2}}\limits^ \wedge)} \cr {{D_y}I\left({x,y} \right) \leftrightarrow (1 - {e^{- j2\pi {\omega _2}/N}})I(\mathop {{\omega _1},\,{\omega _2}}\limits^ \wedge)} \cr}

In the same way, it can be seen that the corresponding relationship between the spatial domain and the frequency domain of the differential data fusion algorithm guided by integral discrete is: DxnI(x,y)(1ej2πω1/N)nI(ω1,ω2)DynI(x,y)(1ej2πω2/N)I(ω1,ω2) \matrix{{D_x^nI(x,y) \leftrightarrow {{(1 - {e^{- j2\pi {\omega _1}/N}})}^n}I(\mathop {{\omega _1},\,{\omega _2}}\limits^ \wedge)} \cr {D_y^nI(x,y) \leftrightarrow (1 - {e^{- j2\pi {\omega _2}/N}})I(\mathop {{\omega _1},\,{\omega _2}}\limits^ \wedge)} \cr}

We extend the integer-order n to the fractional-order υ to obtain the numerical calculation formula of the differential data fusion algorithm based on discrete integral guidance: DxυI(x,y)(1ej2πω1/N)υI(ω1,ω2)DyυI(x,y)(1ej2πω2/N)υI(ω1,ω2) \matrix{{D_x^\upsilon I\left({x,y} \right) \leftrightarrow {{(1 - {e^{- j2\pi {\omega _1}/N}})}^\upsilon}I(\mathop {{\omega _1},\,{\omega _2}}\limits^ \wedge)} \cr {D_y^\upsilon I\left({x,y} \right) \leftrightarrow {{(1 - {e^{- j2\pi {\omega _2}/N}})}^\upsilon}I(\mathop {{\omega _1},\,{\omega _2}}\limits^ \wedge)} \cr}

In this way, the difference data fusion algorithm guided by the integral discrete can obtain the conjugate operator of the differential operator: DxυI(x,y)¯(1ej2πω1/N)υ¯I(ω1,ω2)DyυI(x,y)¯(1ej2πω2/N)υ¯I(ω1,ω2) \matrix{{\overline {D_x^\upsilon I(x,y)} \leftrightarrow \overline {{{(1 - {e^{- j2\pi {\omega _1}/N}})}^\upsilon}} I(\mathop {{\omega _1},\,{\omega _2}}\limits^ \wedge)} \cr {\overline {D_y^\upsilon I(x,y)} \leftrightarrow \overline {{{(1 - {e^{- j2\pi {\omega _2}/N}})}^\upsilon}} I(\mathop {{\omega _1},\,{\omega _2}}\limits^ \wedge)} \cr}

There is known difference data I(x, y), (x, y) ∈ Ω. Assume that the different data I(x, y) is independent of each other in the X and I(x, ·) axis directions. Therefore, only the fractional calculus numerical calculation of the difference data Y in the direction of the X axis is considered. From the G — L definition of fractional calculus, it can be seen that the fractional differential of the difference data I(x, ·) is: aGDtυI(x,)=(1h)j=1N1(1)jΓ(υ+1)Γ(j+1)Γ(υj+1) _a^GD_t^\upsilon I\left({x, \cdot} \right) = \left({{1 \over h}} \right)\sum\limits_{j = 1}^{N - 1} {{{\left({- 1} \right)}^j}{{\Gamma \left({\upsilon + 1} \right)} \over {\Gamma \left({j + 1} \right)\Gamma \left({\upsilon - j + 1} \right)}}}

We use the different data fusion algorithm guided by integral discreteness to fuse the original data in different directions to form new fused output data. This can greatly improve the difference between different data of different sensor nodes in the discrete manufacturing system [7]. The flow chart of the differential data fusion algorithm guided by integral discrete is shown in Figure 1. It can be seen from Figure 1 that the differential data fusion algorithm using discrete integral guidance first needs to integrate all the data. Then judge the difference of all the data. The integration method is used to process the different data and fuse the different data. Finally, it is judged whether the system algorithm is over according to the quality of the data fusion effect.

Figure 1

Flow chart of the differential data fusion algorithm guided by integral discrete

Differentiated data fusion under the Internet of Things based on discrete integral guidance
Internet of Things Technology

The basic characteristics of the Internet of Things technology are mainly reflected in the following aspects [8]:

(1) Comprehensive perception. Signal detection and perception are carried out through sensors, detectors, two-dimensional codes, and even transducers. This provides preconditions for intelligent analysis and network resource sharing.

(2) Reliable delivery. Through the establishment of the Internet of a Things technology platform and the integration with the Internet, the transmission of parameter information, the sharing of information resources, and real-time processing are realized.

(3) Intelligent processing. Through modern signal and information processing technology (data mining technology, cloud processing technology, neural network system technology, pattern recognition technology), information processing and analysis of massive data and extraction of useful information [9]. In this way, the object-oriented operation and control are implemented, and the user's purpose is finally achieved.

Differentiated data fusion under the Internet of Things based on discrete integral guidance

The objective function of the differentiated data fusion system under the Internet of Things based on discrete integral guidance is defined as: aGDtυI(x)=j=1N1(1)j(υj)I(xj) _a^GD_t^\upsilon I\left(x \right) = \sum\limits_{j = 1}^{N - 1} {{{\left({- 1} \right)}^j}\left({\matrix{\upsilon \cr j \cr}} \right)I\left({x - j} \right)}

Assume that the differentiated data fusion function Cυ(x) satisfies: Cυ(x)={(1)x(υx),x00,x<0 {C^{\upsilon \left(x \right)}} = \left\{{\matrix{{{{\left({- 1} \right)}^x}\left({\matrix{\upsilon \cr x \cr}} \right),} & {x \ge 0} \cr {0,} & {x < 0} \cr}} \right.

Therefore, the system of differentiated data fusion can be expressed as: aGDtυI(x)=Cυ(x)*I(x) _a^GD_t^\upsilon I\left(x \right) = {C^{\upsilon \left(x \right)}}*I\left(x \right)

So we use the function of the BVυ(Ω) space of differentiated data fusion to obtain the differentiated data fusion under the Internet of Things based on discrete integral guidance as: E(I)=Ω|cυ*I|dxdy+λ2|I0I|L22 E\left(I \right) = \int_\Omega {\left| {{c^\upsilon}*I} \right|dxdy + {\lambda \over 2}\left| {{I_0} - I} \right|_{{L_2}}^2}

In: {|cυ*I|=(cυ*I)x2+(cυ*I)y2(cυ*I)x=Ωcυ(z)I(xz,y)dz(cυ*I)y=Ωcυ(z)I(x,yz)dz \left\{{\matrix{{\left| {{c^\upsilon}*I} \right| = \sqrt {\left({{c^\upsilon}*I} \right)_x^2 + \left({{c^\upsilon}*I} \right)_y^2}} \hfill \cr {{{\left({{c^\upsilon}*I} \right)}_x} = \int_\Omega {{c^{\upsilon \left(z \right)}}} I\left({x - z,y} \right)dz} \hfill \cr {{{\left({{c^\upsilon}*I} \right)}_y} = \int_\Omega {{c^{\upsilon \left(z \right)}}} I\left({x,y - z} \right)dz} \hfill \cr}} \right.

We get the expression of the differentiated data. It satisfies the following formula: g(ε)=E(I+εφ)=Ω|cυ(I+εφ)|dxdy+λ2|I0I+εφ|L22 g\left(\varepsilon \right) = E\left({I + \varepsilon \varphi} \right) = \int_\Omega {\left| {{c^\upsilon}\left({I + \varepsilon \varphi} \right)} \right|dxdy + {\lambda \over 2}\left| {{I_0} - I + \varepsilon \varphi} \right|_{{L_2}}^2}

We still use the basic properties of the functional extremum. Let g′(0) = 0 obtain: Ω(cυ*I)x(cυ*φ)x+(cυ*I)y(cυ*φ)y|cυ*I|dxdyΩ(I0I)φdxdy=0 \int_\Omega {{{{{\left({{c^\upsilon}*I} \right)}_x}{{\left({{c^\upsilon}*\varphi} \right)}_x} + {{\left({{c^\upsilon}*I} \right)}_y}{{\left({{c^\upsilon}*\varphi} \right)}_y}} \over {\left| {{c^\upsilon}*I} \right|}}} dxdy - \int_\Omega {\left({{I_0} - I} \right)\varphi dxdy = 0}

Because when the differentiated data in the system (x, y) ∉ R is I(x, y) =0. So when Ω → R2 is: R2fgdxdy=R2f^g^¯dω1dω2 \int_{{R^2}} {fgdxdy = \int_{{R^2}} {\overline {{\hat f}{\hat g}} d{\omega _1}d{\omega _2}}}

According to the differential data equation, the final transformation form of the objective function of the differential data fusion system under the Internet of Things guided by integral discreteness can be obtained as: 12πΩ((cυ*^I)x¯|cu*I|)(cυ*^φ)xdω1dω2+12πΩ((cυ*^I)y¯|cu*I|)(cυ*^φ)ydω1dω2λ12πΩ(I^¯0I^¯)φ^dω1dω2=0 \matrix{{{1 \over {2\pi}}\int_\Omega {\left({{{\overline {{{\left({{c^\upsilon}{\hat *I}} \right)}_x}}} \over {\left| {{c^u}*I} \right|}}} \right)} {{\left({{c^\upsilon}{\hat *\varphi}} \right)}_x}d{\omega _1}d{\omega _2} +} \hfill \cr {{1 \over {2\pi}}\int_\Omega {\left({{{\overline {{{\left({{c^\upsilon}{\hat *I}} \right)}_y}}} \over {\left| {{c^u}*I} \right|}}} \right)} {{\left({{c^\upsilon}{\hat *\varphi}} \right)}_y}d{\omega _1}d{\omega _2} -} \hfill \cr {\lambda {1 \over {2\pi}}\int_\Omega {\left({{{\bar {\hat I}}_0} - \bar {\hat I}} \right)} {{\hat \varphi}} d{\omega _1}d{\omega _2} = 0} \hfill \cr}

The differentiated data fusion algorithm under the Internet of Things guided by integral discreteness can continuously collect data information in the Internet of Things system. It merges the differences of the differentiated data [10].

System experiment and result analysis
Description of the experimental environment

The hardware equipment used in the experiment is brand Dell, and the software environment is IntelPentiumE2180 @ 4GHzCPU / 2.00GB / Windows7. The article tests differentiated data fusion algorithms’ performance pros and cons based on discrete integral guidance under the Internet of Things. We use Matlab simulation software to build an experimental environment [11]. The article uses a set of 100 random difference data to conduct a system experiment. The detailed experimental parameters are shown in Table 1.

Description of experimental parameters.

Project Parameter Description
Number of difference data nodes 100
Number of different data types 6
Differential data distribution type Random distribution difference
Difference tolerance Within the set range, a standard deviation of 1
Difference processing method Equal treatment
Result analysis

The system experiment is carried out under the system experiment environment constructed in Table 1. The distribution of the original data is shown in Figure 2.

Figure 2

Original differentiated data

It can be seen from Figure 2 that the data of each discrete manufacturing system in the original state presents the characteristics of the individual combination. It shows strong clustering nder the same rule [12]. The article uses a differentiated data fusion algorithm based on discrete integral guidance to process the above data under the Internet of Things. The result is shown in Figure 3.

Figure 3

Differential data fusion results under the Internet of Things based on discrete integral guidance

Through the comparison of Figure 2 and Figure 3, it can be seen that the original data has obvious partition characteristics due to the differences in equipment and sources [13]. We use a differentiated data fusion algorithm under the Internet of Things based on discrete integral guidance to process the data, and the data distribution characteristics are more random. The data spectrum before and after the differential data fusion is shown in Figure 4.

Figure 4

Comparison of data spectrum before and after differentiated data fusion

It can be seen from Figure 4 that the spectral characteristics of the original data show a situation where six peaks are merged. The original data distribution is relatively regular, so the data peak characteristic is very strong. After the differential fusion algorithm processes the data, the data spectrum becomes a uniformly distributed random spectrum. This shows that the differences between the data have been merged.

Conclusion

A discrete system differential data fusion in the Internet of Things guided by integral discrete is studied. The data of each node needs to be processed separately. This article proposes a discrete system differential data fusion in the Internet of Things guided by integral discrete. Then treat all data as research objects for unified processing. Then, the method of discrete integral guidance is used to process all the obtained differential data. We integrate the data we have to judge the differences of all the data. We use the integration method to process the different data and fuse the different data to combine effectively. Finally, a set of 6 types of data of 100 nodes is used for experiments. The results show that the discrete system differential data fusion in the Internet of Things is guided by integral discreteness and the average distribution of the data spectrum. So the algorithm has a very good application value.

Figure 1

Flow chart of the differential data fusion algorithm guided by integral discrete
Flow chart of the differential data fusion algorithm guided by integral discrete

Figure 2

Original differentiated data
Original differentiated data

Figure 3

Differential data fusion results under the Internet of Things based on discrete integral guidance
Differential data fusion results under the Internet of Things based on discrete integral guidance

Figure 4

Comparison of data spectrum before and after differentiated data fusion
Comparison of data spectrum before and after differentiated data fusion

Description of experimental parameters.

Project Parameter Description
Number of difference data nodes 100
Number of different data types 6
Differential data distribution type Random distribution difference
Difference tolerance Within the set range, a standard deviation of 1
Difference processing method Equal treatment

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