The “purification” and modeling of multi-agent monitoring system for industrial textiles

Abstract

To improve the quality of industrial textile products and ensure the accuracy of the real-time collected textile production data, we first proposed a new data fitting method, which uses the integral and differential quotient operator to obtain the basic spline function and B-spline curve by operator deduction, and then used high-order approximation method for differential operator to get the precise spline fitting function. Second, a multi-agent network-monitoring system model was built, and its work principle, main function, collaborative process, and scheduling algorithm were introduced. The application results show that the proposed data fitting method is optimum and the results are accurate. Furthermore, the system structure model based on multi-agent meets the requirements of quality control of industrial textiles.

Keywords

Industrial textiles data fitting multi-agent textile industry

Introduction

To improve the quality of industrial textile products and ensure the accuracy of the real-time collected textile production data, it becomes the core issue problem for textile industry to be resolved [1]. For these, the countries with advanced textile production technology, such as Cuba, India, Germany, and Japan [2], have successfully adopted some advanced techniques [3] and systems [4], i.e. production data-processing, a data base for determining the evaporative resistance of clothing, and have proposed some advanced data fitting methods for industrial textile data, for instance, numerical simulation of textile flexibility testing [5], and fitting approach of the fatigue tensile response of textile composite materials. Meanwhile, they have also constructed some database management systems to facilitate data processing, for example, electronic textile systems [6], and simulation environment for electronic textiles. In general, the methods and systems above have ensured the correctness of industrial textile data. However, we know from the literature, data fitting and processing methods above are mainly concentrated on the textile property test, clothing permeability test, and moisture treatment of wool fabric, etc., reversely, data fitting approach oriented to the real-time collected industrial textile data is relatively small [7].

In China, to study data fitting and data processing method for industrial textile data, it is relatively late [8]. In 1999, an improved mathematical simulation of the coupled diffusion of moisture and heat in wool was proposed. Then, in 2008, a mathematical modeling and numerical simulation of moisture transfer in textile assemblies was put forward [9]. However, the research that improves equipment utilization and ensures the accuracy of industrial textile data is generated in 2008, for example, simulating the drawing of spunbonding nonwoven process using an artificial neural network technique [10], and numerical study of heat and moisture transfer in textile materials by a finite volume method was proposed, respectively. Although domestic textile scholars have made some significant contributions in data fitting and processing methods, they all go in for the study on textile materials testing, moisture treatment of wool fabric, etc., and cannot carry out the research on data fitting approach of industrial textiles. In 2010, data fitting method for industrial textile data was proposed.

For the textile industry, it is necessary to study data fitting method of industrial textiles, because the accuracy of production data is the basis of improving productivity and getting best profit, that is, only the correct processed production data can meet the requirements of quality control of industrial textiles. Thus, the specific treatment object is real-time collected production data, which is from all of the plant-level manufacturing workshops. The processing method is to filter, diagnose, verify, and compute production data, so as to store the processed data in the plant-level data center [11]. The aim is to make the processed data accurately reflect operating capacity of the equipment and realize data integration [12]. However, in the actual manufacturing process, there is a lot of the error factors, such as human, machine, and environment, etc., all of them have negative effect on the accuracy of production data [13]. Therefore, to ensure the accuracy and correctness of data processing result, we ought to design and develop a common software system, make the system determine whether data fitting result is ideal.

Definition

Def. 1 Object. Agent _i = {A₁, A₂, … , A_n}, where A_i is an object of the ith agent, of which, A_i (i = 1, 2, … , n) denotes a member of the agent’s object, and represents a specific agent object.

Def. 2 Task. The task is the problem which needs to be solved by agent _i . According to the nature properties of the problem domain, we define the task as a set T, and T = {T₁, T₂,… , T_m}, such that T_i (i = 1, 2, … , m) represents a specific task [14].

Def. 3 Ability. According to the performance level of the task, for each agent, the ability is defined as X = {X₁, X₂, … , X_n}, where X is a n-dimensional ability vector space, and X_i (i = 1,2, … , m) is the ith-dimensional ability vector.

Def. 4 Resources. Resources is some external affairs of the ith agent to complete the task T_i, and its goal is to meet the operating requirements of agent _i . Thus, we define the resources to be a set S, and S = {S₁, S₂, … , S_n}. Through the resources S_i, agent A_i (i = 1, 2, … , n) can be invoked and called by other agent _i or coordination task T_i (i = 1, 2, … , m).

Based on the above, an agent is composed of multiple agent’s objects, while all objects can be formed a set A, and defined as A = {A₁, A₂, … , A_n}, where A_i (i = 1,2, … , n) denotes the ith object. In the collaborative process among multiple objects, we define each object complete the task T_i = {T₁, T₂, … , T_n} [15]. Thus, one agent can be partitioned into several sub-agents, and one sub-agent need to complete the task T_i. According to the ability X_i and resources R_i, when single sub-agent cannot achieve the task T_i, the task T_i is commonly accomplished by multiple sub-agents.

Collaborative process of multi-agent

The collaborative process of multi-agent is a complex process, which requires a number of participants, sub-services, and constraint conditions, the goal is to make the system resources be optimum. Furthermore, the entire collaborative process must make multiple agents share data resources, and accomplish the assigned sub-task together. The collaborative flowchart is shown in Figure 1.

Figure 1.

The collaborative process among multiple agents.

Figure 1, each agent has own knowledge base, mechanism library, property, and communication module to complete various tasks, the main roles of each agent are marked in Figure 1, respectively. Meanwhile, the oval denotes different function modules and dashed line indicates logical relationships between function modules. The work principle among agents is the following.

In Figure 1, we divide an agent into multiple sub-agents, that is, resources agent, task agent, monitor agent, equipment agent, scheduling agent, and system management agent, etc. To complete various sub-tasks, each agent has special knowledge base, mechanism library, properties, and communication module. The main functions of each agent are marked. Of which, the oval denotes the functions of the modules and dashed line indicates the logical relationship among different function modules. Thus, work principle for multiple sub-agents is the following.

Resources agent. Resources agent is responsible for receiving external task and determining whether the task can be performed and the condition is satisfied. When the task is classified, resource agent issues the tender to all equipment agents. At this time, according to the priority rules, each equipment agent will select equipment involved in the tender and form alternative processing route, while resource agent will schedule required resources to all equipment agents and make them go into production workflow to complete special task. After the task ends, the result is returned to monitoring agent, when monitoring agent receives the message [16].

Task agent. Task agent is first used to evaluate and schedule the received outside task; its specific task is to gather the task information, such as emergency degree, processing procedure, and work time, etc. Then, task agent is in charge of the task information which is submitted to resource agent. After resource agent receives the task information, it needs to confirm the information. When the operation process above is successfully executed, resources agent begins to assess each task and calls colony algorithm to calculate the stimulating degree of each task. Then, resources agent returns stimulating degree information to task agent. Finally, the stimulating degree information is accepted by function modules of production planning.

Monitor agent. Monitor agent is mainly to simulate alternative production planning, which is reported by task agent. In the setting object function condition, we aim at the object, use genetic algorithm to solute the task, and make final selection processing route be returned to task agent. So, task agent begins to call monitor agent and make monitor agent be charged of dealing with equipment failure, adding new equipment, emergency task, and human control, etc. Then, task agent implements the special task.

Equipment agent. After the bid of resources agent was received by equipment agent, it begins to assess production capacity (i.e. processing expertise, free time, threshold size of the different manufacturing processes, etc.) of the corresponding equipment and then returns capacity information to resources agent. Resource agent determines whether it can complete the task. If the condition is success, all agents go into production processing, and the final certain production planning task is implemented. Furthermore, equipment failure information can be submitted to monitor agent at any time.

Scheduling process and algorithm

For textile industry, to meet the requirements of production management department or other workshop, data integration process is mainly composed of the following two steps. (1) Data interface is offered by the top database management system of the various workshops. Furthermore, according to the species consistency principle, production data can be linked by the common species. (2) When assortment data is linked successfully, sharing information of production data is extracted and stored in the upper-class integration management system of the textile industry.

Thus, the process of resources scheduling and algorithm implementation is the following.

(1) In general, task information includes the number, needed task processing, and stimulation degree of sub-tasks. Task agent receives the task from the task queue, and then submits the task information to monitor agent and resources agent, the goal is to get specific sub-task and needed resources. Monitor agent retrieves the task information from own knowledge base. In this case, if the task information cannot be found in knowledge base, then the workflow will turn to step (2). Else, the task information exists in the knowledge base and monitor agent will directly inform resource agent to schedule resources, call task agent to implement the task information, and start the equipment to produce as well as do without the associated operations to determine production planning. The benefit is to reduce system workload.

After resources agent receives the task information, it will comprehensively collect production capacity, equipment capability, and other information of the single device and determine whether the device can accept the task. If the task cannot be accepted, then the whole production process will exit.

Now, suppose that the current received task set is T = {T₁, T₂, … , T_i} [17], where i is the total number of the received task. For any one task T_m, the processing time set of each sub-task is {t_m₁, t_m₂, … , t_mr}. Meanwhile, suppose that the needed processing type set is L_m = {l_m₁, l_m₂, … , l_mr}, such that r is the sub-task number of the task T_m and the resources set is R = ${R_{1}, R_{2}, |, R_{j}}$ , where j is the number of the device. So, a sub-task corresponds to a processing operation and an operation possesses some resources. For any device R_p, the idle time set is s_p = {s_p₁, s_p₂, … , s_pt}, of which t is the number of the optional equipment R_h in the idle time section, and the processing procedure type set can be denoted as K_p = {k_p₁, k_p₂, … , k_pq}, here, q can be expressed as the device type number. Thus, the judgment for receiving the task can be described as follows.

The total capacity judgment is $\sum_{m = 1}^{i} \sum_{n = 1}^{r} t_{mn} \leq \sum_{m = 1}^{p} \sum_{n = 1}^{j} S_{mn}$ [18] and L₁ $\cup$ L₂ $\cup$ ··· $\cup$ L_i $\subseteq$ K₁ $\cup$ K₂ $\cup$ ··· $\cup$ K_j.

Based on the condition above, the ability judgment condition of the single device is t_mr ≤ maxs_pt, that is, the total time of all sub-tasks should be no more than the total sum of the idle time of all the alternative equipments. Meanwhile, the equipments (machines) within the various workshops should complete the required processing type of all parts, and the completion time for any sub-task should be no more than the longest idle time of the alternative equipment. If we decide to implement the task, then the workflow should return to step (3). Else, the workflow turns to step (2).

(2) By issuing the tender, resources agent will form alternative equipment list. Now, assumed that there have n alternative equipments, the objective function that is used to determine the winning equipment in the bidding mechanism can be denoted the following formula [19].

Min α_{1} (\sum_{i = 1}^{r} α_{2} M_{T_{mi}} + \sum_{i = 1}^{r - 1} N_{T_{mi + 1}}) + α_{3} (\sum_{i = 1}^{r} α_{4} O_{T_{mi}} + \sum_{i = 1}^{r - 1} P_{T_{mi + 1}}) .

where

α_{1}

α_{2}

α_{3}

and

α_{4}

is weighted coefficient, M_Tmi is processing cost, and denotes the tender equipment to complete sub-task T_mi, while N_Tmi_+ 1 is the cost that requires to pay the switching equipment cost from the task T_mi to T_mi_+ 1, such as equipment waste, transportation cost, etc. O_Tmi and P_Tmi_+ 1 are the completion time and switching time, respectively. The formula of the cost price is M_Tmi = δαt, where α is loading coefficient for the device. Generally, the greater the idle rate of the equipment is the smaller the load coefficient is, and the greater the successful likelihood of the tender is δ is processing cost per unit, and t is the expected time of the processed task. In short, through the contract net mechanism, the processing equipment can make the processing cost be reduced to minimization, and make the equipment loading be close to balance.

(3) After the task is decomposed or classified by resources agent, a sub-task begins to release the tender to the equipment. The principle is the following. First of all, we retrieve the latest ability information of all equipment agents from the knowledge base and estimate ability information to complete the sub-task. Then, we make task agent issue the tender to some special equipments, instead of all the equipments, the goal is to reduce network communication burden and computational burden of the subsequent processing tender. Here, the computing of the device is equivalent to get the threshold of a processing operation, while the setting of the threshold ought to be considered from the following four factors.

1. In the processing queue, make the waiting time of sub-task as short as possible.

2. Sub-task of the same parts should be continuously processed, the aim is to reduce time-consuming of the equipment switching task.

3. The priority to process the task should have high emergency degree and should be close to task completion time.

4. The priority should be given to the equipments with processing expertise and make them process good type sub-task. Here, for the same artifacts, good type indicates that processing time is relatively short.

The calculation formula of the threshold is θ_ij = θ′ _ij + ka, where a denotes the cutover or transmission time-consuming, that is, one device is switched among different production varieties, or a variety is produced in various devices. Thus, in the waiting queue, when the varieties need not be transferred to different devices, the value of k tends to zero [20].

Derivative of θ_ij is θ′ _ij = k₁ + k₂t_ij + k₃α_ij, for each produced sub-task, it has one value k₁, and the value k₁ of a priority produced sub-task is relatively small. t_ij is the time-consuming to complete the sub-task. α_ij is the strength coefficient to affect the task t_i, it comes from the device R_j. Meanwhile, the better R_j outputs a certain type of the process t_i the smaller α_ij is. The weighted coefficient k₂, k₃ depends on the time-consuming to complete sub-task, and the effect size of the specialty coefficient to act on the reaction threshold.

(4) Task agent is used to simultaneously calculate the stimulation degree S of each sub-task, and S = s₀ + k₄t. Of which, s₀ is the initial value, t is the task time in the waiting queue, k₄ is the weighted coefficient, and mainly used to determine the length of the waiting time, which affects the processed probability size of the task.

On this basis, we make all sub-tasks be packed into a queue to wait for calling. For example, in step (4), task agent calls resource agent, while resources agent judges the required resources of sub-task. In the actual production manufacturing process, the queue is called production queue. When combined with step (3), according to the formula P(θ_ij,S) = S²/(S²+ θ² _ij ) [21], resources agent can calculate the probability size. Furthermore, the greater the probability is, the higher the priority of the task in production queue is.

Now, suppose there are m queues and n equipments, we can see from the perspective of the permutations and combinations, they form $m * n$ alternative processing routes. In steps (4) and (5), resource agent will feedback alternative processing route to monitor agent, while monitor agent builds the objective function in terms of the optimization goal, which needs a dynamic scheduling system to achieve. For all production workshops in textile industry, the goal is to use the shortest time and minimum cost to complete the task.

(5) If monitor agent is called, it firstly begins to code for the received alternative processing route, and the processing line corresponds to one natural number-coded chromosomes. Then, monitor agent executes the iteration operation and makes the result generate initialization chromosomes. Thirdly, monitor agent does crossover operator to produce new chromosomes. Finally, the identical mutation operation is executed to create the new chromosomes, and the fitness calculation can do the formation of the population. Thus, fitness function called the objective function is shown as follows [22].

f (k) = ɛ_{1} U_{k} + ɛ_{2} V_{k} .

Of which

U_{k} = \sum_{i = 1}^{r} α_{2} M_{T_{mi}}^{(k)} + \sum_{i = 1}^{r - 1} N_{T_{mi + 1}}^{(k)}, V_{k} = \sum_{i = 1}^{r} α_{4} O_{T_{mi}}^{(k)} + \sum_{i = 1}^{r - 1} P_{T_{mi + 1}}^{(k)}, k = 1, |, m * n .

The resulting chromosome corresponding to the processing route is final selection route. According to the selection route, monitor agent informs resources agent to schedule the equipment to produce.

Based on the above, the algorithm we proposed has a lot of the weighted coefficients [23], while these coefficients need to be decided by the specific production requirements and actual situation of the textile industry. In addition, some causal factors and non-quantifiable factors need to be treated via production management experience of the operators. Therefore, the model we proposed has introduced man-machine interface, and the operator can modify the weighted coefficient by the experience and specific circumstances, the goal is to enhance the usefulness and applicability of the model.

Data fitting

With the restructuring and transformation of the textile industry of China, production manufacturing process among various workshops and divisions get complex. This change leads to the finished and semi-finished product processing being gradually altered, which is from simple process control of a single process to the complex process of multi-process and multi-index. Under this production process, we must ensure the correctness and accuracy of production data, because data is timely retrieved from the equipments (machines) of the various departments and workshops.

Prior to this, we have mainly used two kinds of mathematical algorithms (average filtering and weighted arithmetic average filtering) to match production data of industrial textiles. Meanwhile, production data includes: (1) dropper stopping time, weft stopping time, selvage stopping time, and other stopping time; (2) operating status, and standing status; and (3) roller pulse of the equipment.

As an example for the loom, the operating status and real-time production data is listed in Table 1.

Table 1.

Real-time production data of the loom.

Machine code	Speed	Warp stop time	Warp stop number	Weft stop time	Weft stop number	Selvage stop time	Selvage stop number	Other stop time	Other stop number	Sum stop time	Sum stop number	Real yield	Efficiency (%)	Assortment name
A025	534	51	18	31	41	0	0	5	3	87	62	66.24	101.61	104.5″C110 × 90 22
A026	439	68	29	90	132	0	0	3	4	161	165	56.95	84.09	C133 × 72 104.5″ 55
A027	445	51	19	126	106	2	1	2	2	181	128	54.69	79.65	C133 × 72 104.5″ 55
A028	438	26	8	91	42	0	0	6	2	123	52	62.56	78.02	C133 × 72 104.5″ 55
A029	443	43	24	24	23	0	0	2	1	69	48	71.33	92.53	C133 × 72 104.5″ 55
A030	434	40	17	104	119	0	1	6	3	150	140	57.94	72.97	C133 × 72 104.5″ 55
A031	443	64	31	57	41	6	2	5	3	132	77	61.85	86.52	C133 × 72 104.5″ 55
A032	434	86	23	45	32	5	2	0	1	136	58	60.04	89.63	C133 × 72 104.5″ 55
A033	437	57	13	0	0	0	0	41	30	98	43	52.88	88.25	104.5″C110 × 90 22
A034	437	104	36	51	33	6	1	5	5	166	75	44.71	83.88	104.5″C110 × 90 22
A036	463	136	61	32	28	2	2	4	3	174	94	58.00	68.48	C133 × 72 104.5″ 55
A037	442	115	20	80	88	5	1	6	6	206	115	50.55	62.49	C133 × 72 104.5″ 55
A038	438	37	14	39	43	0	0	3	3	79	60	69.08	86.25	C133 × 72 104.5″ 55
A039	440	46	20	73	77	0	0	6	3	125	100	62.54	77.65	C133 × 72 104.5″ 55
A040	440	24	14	21	23	0	0	5	4	50	41	73.84	105.71	C133 × 72 104.5″ 55
B450	399	48	20	41	30	0	0	0	1	89	51	63.37	102.35	105.5″130 × 70
B452	375	0	0	540	0	0	0	0	0	540	0	46.71	83.4	63″133 × 72(om)
B453	398	84	35	43	36	4	1	5	4	136	76	55.07	85.11	63″133 × 72(om)
B454	423	76	37	41	50	0	0	0	2	117	89	61.18	74.23	63″133 × 72T/R(om)
B455	424	19	10	30	47	1	1	2	6	52	64	70.75	91.31	63″133 × 73T/R(om)
B456	424	23	9	72	65	0	0	2	3	97	77	64.21	82.89	63″133 × 74T/R(om)
B457	424	39	19	29	29	0	0	4	6	72	54	69.85	87.56	105.5″130 × 70
B459	424	44	19	63	49	2	2	3	3	112	73	63.88	80.08	105.5″130 × 70
B460	400	0	0	540	0	0	0	0	0	540	0	48.96	83.65	105″142 × 90
B461	424	21	12	18	18	0	0	4	4	43	34	74.18	94.23	105.5″130 × 70
B463	424	72	29	28	31	0	0	2	3	102	63	65.32	88.77	105.5″130 × 70
B550	400	37	16	47	27	0	0	2	3	86	46	63.88	89.85	105.5″130 × 70
B551	397	92	36	84	88	0	0	0	0	176	124	50.89	64.23	105.5″130 × 70
B552	424	105	34	30	27	0	0	4	2	139	63	59.87	77.12	105.5″130 × 70

It can be seen from Table 1, the desired results cannot be obtained, and a large deviations between the collected production data in the monitoring system and actual production data of industrial textiles existed. Furthermore, real-time operating efficiency of the loom is slightly higher, and even some of them are beyond 100%. In the actual manufacturing process of the textile industry, this phenomenon is unreasonable. The root of the problem is the following.

First, data filtering algorithms is irrationally selected. Second, all factors completely failed to treat, because there probably exists some confounding factors to affect the correctness of production data. To solve this problem, the basic approach is to select appropriate data fitting method to process real-time acquisition production data of industrial textiles, the goal is to completely get rid of these confounding factors, which are also called white noise. Thus, in Table 1, aim at the fact that operation efficiency of the loom is beyond 100%, this is a phenomenon that data fitting algorithm is selected unreasonably, and production data is handled ineffectively. As a result, the collected production data cannot truly reflect actual operating efficiency of the loom. In this case, to ensure the correctness of production data, we proposed a new data fitting method, and its building process is the following.

Truncated monomial

In theory, the calculation formula of the valid work efficiency is shown as follows.

The valid work efficiency (VWE) = \frac{\sum t_{work} (m) - \sum t_{stop} (m)}{\sum t_{sum} (m)} .

The calculation formula that is the theory yield per hour (TYPH) of each eye (spindle) is the following.

TYPH (kg) = \frac{n \times D \times π \times E \times 60 \times tex}{1000 \times 1000 \times 1000} .

The calculation formula of real yield per hour (RYPH) of each eye (spindle) is shown as follows.

RYPH (kg) = TYPH * VWE .

In formula RYPH, n is front roller speed (r $*$ min⁻¹), D is roller diameter (mm), E is a draft ratio between the pressed axis and front roller, or an accident draft ratio between the pressed axis and spindle wing, and tex is the cotton or rove cotton yarn count.

We can see from the formula TYPH, D, π, tex are const, while roller speed n is variable, and E is from n. Thus, n not only is a key control index but also is an efficient evaluation index, which is to assess operating efficiency of the machine. This fully demonstrates that the entire operating process of the system is related with the roller speed n.

Now, suppose that roller speed pulse signal of the machine is in a custom-limited range [a, b], where a > 0, b > 0, and both hold b ≥ a, then we can divide the range [a, b] into n sub-intervals and make it become a = x₀ < x₁ < ··· < x_n = b. In the interval [x₀, x₁], k power polynomial can be written as follows.

S_{0} (x) = a_{0} + a_{1} x_{1} + \dots + a_{k} x_{k}, x \in [x_{0}, x_{1}] .

At this point, the coefficient in S₀(x) is arbitrarily selected, and the sum is k + 1. We can introduce the following truncated monomial notation [24].

(u)_{+}^{k} = u^{k}, u > 0 0, u \leq 0}

Using this notation, in the interval [x₁, x₂], the expression formula based on x₁ can be denoted a unified sub-node k-polynomial, which is S₁(x) = a₀ + a₁x + ··· + a_kx_k + b₁(x − x₁) ^k ₊, of which a₀, a₁, ··· , a_k and b₁ is the undetermined coefficients. They have k + 2, and are required k − 1 k-order continuous derivative at the point x₁, that is, there exists k constraint conditions, and there are 2(k + 1) − k = k + 2 freedom degrees. Obviously, in the internal [x₀, x₁], there exists the relationship S₁(x) = S₀(x).

In general,

S_{j} (x) = S_{_{-}_{1}} (x) + b_{j} (x - x_{j}) k + = S_{0} (x) + b_{1} (x - x_{1})_{+}^{k} + b_{2} (x - x_{2})_{+}^{k} + \dots + b_{j} (x - x_{j})_{+}^{k} .

From this, (1, x, … , x_k; (x − x_j) ^k ₊, j = 1, 2, … , n − 1) forms basic function group of k-spline function space, which is based on the node x₀, x₁, … , x_n. However, the basic functions have many defects, for example, they lead to non-banded matrix, poor condition number, etc. So, we need to search for another base.

Operator deduction

For numerical analysis, one of the important tasks is to seek the relationship and transforming between micro and plot. Based on the Section Truncated monomial, to derive and define the relationship among several basic operations, we introduce the following notation [25].

y_{r} = f (x_{r}), h = x_{_{+}_{1}} - x_{r}, {Iy}_{r} = y_{r} {Ey}_{r} = y_{_{+}_{1}}, y_{r} = y_{_{+}_{1}} - y_{r}, y_{r} = y_{r} - y_{_{-}_{1}} Δ y_{r} = y_{r + \frac{1}{2}} - y_{r - \frac{1}{2}}, μ y_{r} = \frac{1}{2} (y_{r + \frac{1}{2}} - y_{r - \frac{1}{2}}), D^{m} y = \frac{d^{m} y}{{dx}^{m}} .

Provided that $Δ^{0}$ , E⁰, $\nabla^{0}$ , and D⁰ is unchanged operator I, there exists Ey_r = y_r_+ 1 = y_r_+ 1 − y_r + y_r = (1 + $Δ$ )y_r_.

Therefore, E = 1 + $Δ$ [26], and

μ Δ y_{r} = Δ (μ y_{r}) = \frac{1}{2} (Δ y_{r + \frac{1}{2}} + Δ y_{r - \frac{1}{2}}) = \frac{1}{2} (Δ + \nabla) y_{r}

So, we can obtain $μ Δ = \frac{1}{2} (Δ + \nabla)$ .

Similarly, we can create the following equation of the operator.

Δ = E^{\frac{1}{2}} - E^{- \frac{1}{2}}, μ = \frac{1}{2} (E^{\frac{1}{2}} - E^{- \frac{1}{2}}), μ^{2} = 1 + \frac{1}{4} {\bar{Δ}}^{2} . E = 1 + \frac{1}{2} {\bar{Δ}}^{2} + \bar{Δ} \sqrt{1 + \frac{{\bar{Δ}}^{2}}{4}}, Δ \nabla = Δ - \nabla = {\bar{Δ}}^{2}

Note that Talor’s expansion expression is the following.

{Ey}_{r} = y_{_{+}_{1}} = y_{r} + hy'_{r} + \frac{h^{2}}{2!} y ″_{r} + \dots = (1 + hD + \frac{h^{2}}{2!} D^{2} + \dots) y_{r} = e^{hD} y_{r}

Hence, there is E = e^hD, and hD = lnE.

Basic spline function

In general, for a large number of functions f(x), we can obtain $\frac{d}{dx} \int f (x) dx = f (x)$ [27], and then $\frac{d}{dx} \approx \frac{\bar{Δ}}{h}$ .

Thus, f_{h} (x) = \frac{\bar{Δ}}{h} \int f (x) dx \approx \frac{d}{dx} \int f (x) dx = f (x) .

Here, the integral operation can turn unbounded function into bounded function, and alter rough function into smooth function. If dirac generalized function f(x) = $δ$ (x), then $δ$ _h (x) = $\frac{\bar{Δ}}{h} \int_{- \infty}^{x} δ (x) dx$ .

We can see from $δ$ _h (x), it looks like $δ$ (x), retains $\int δ_{h} (x) dx = \int δ (x) dx = 1$ , and exists nature property of $δ$ (x), that is $δ$ _h (x) = 0 when |x| $\frac{h}{2}$ [28]. Now, we re-implement integral and difference quotient operation for $δ$ _h (x), and successively do k + 1 times.

\underset{k + 1}{\underset{︸}{(\frac{\bar{Δ}}{h} \int) \dots (\frac{\bar{Δ}}{h} \int)}} δ (x) \underset{k + 1}{\underset{︸}{dx \dots dx}}

We name h = 1 and make difference quotient operator and integral operator can switch the order with each other. Then, we get ${\bar{Δ}}^{k + 1} \int \dots \int δ (x) dx \dots dx = {\bar{Δ}}^{k + 1} (\frac{x_{+}^{k}}{k!})$ and obtain the function $Ω_{k} (x) = {\bar{Δ}}^{k + 1} (\frac{x_{+}^{k}}{k!})$ , of which, k is non-negative integer.

By the operator, we first gain E $λ$ f(x) = f(x + $λ$ ) and $\bar{Δ}$ = (1−E⁻¹) $E^{, \frac{1}{2}}$ , and then get the following binomial expansion.

\bar{Δ} n = (1 - E - 1) n E \frac{n}{2} = \sum_{i = 0}^{n} (- 1) i (n i) E \frac{n}{2} - i .

So, Ω_{k} (x) = {\bar{Δ}}^{k + 1} (\frac{x_{+}^{k}}{k!}) = \frac{1}{k!} \sum_{j = 0}^{k + 1} (- 1)^{j} (k + 1 j) (x + \frac{k + 1}{2} - j)_{+}^{k}, k = 0, 1, 2 \dots .

We call $Ω_{k} (x)$ to be basic spline function, which has the following properties.

$Ω_{k} (x)$ is k-spline function, the node is ξ⁽ ^k ⁾ _j = $- \frac{k + 1}{2}$ + j, j = 0, 1, … , k + 1, and $Ω_{k}^{(i)} (x) |_{x = \pm \frac{k + 1}{2}}$ = 0, i = 0, 1, … , k−1. Furthermore, the amount of jump for function interruption is [ $Ω_{k}^{(k)} (ξ_{j} (k))$ ] = (−1) ^j $(k + 1 j)$ .

$Ω_{k} (x)$ = $Ω_{k} (- x)$ , the maximum value of $Ω_{k} (x)$ is obtained at the point x = 0, such that $Ω_{k} (x)$ ≥ 0, supp( $Ω_{k} (x)$ ) = ( $- \frac{k + 1}{2}$ , $\frac{k + 1}{2}$ ).

$\int_{- \infty}^{\infty} Ω_{k} (x) dx$ = 1. For any x, there exists $\sum_{n = - \infty}^{\infty} Ω_{k} (x + n)$ = 1.

$Ω_{k} (x) = \int_{- \infty}^{\infty} Ω_{k - 1} (x - t) Ω_{0} (t) dt = Ω_{k - 1} \times Ω_{0}$ .

Fourier transformation and inverse transformation of $Ω_{k} (x)$ is $\frac{1}{2 π} \int_{- \infty}^{\infty} {\tilde{Ω}}_{k} e^{i ξ x} d ξ$ , such that

{\tilde{Ω}}_{k} (ξ) = \int_{- \infty}^{\infty} e^{- i ξ x} Ω_{k} (x) dx = (\frac{\sin \frac{ξ}{2}}{\frac{ξ}{2}})^{k + 1} .

The function $Ω_{k} (x)$ is often referred to as B-spline function, by this function, we write B-spline curve function as s(t) = $\sum_{i} P_{i} Ω_{k} (t - i)$ .

High-precision basic spline function

We know from B-spline curve function, when we use differential operator D to approach the function, ${\bar{Δ}}_{h}$ (h = x_r_+ 1−x_r) can be taken, because it begins from δ(x), passes by ${\bar{Δ}}_{n}$ D⁻¹, and finally makes this polished operator execute k + 1 times, the basic spline function $Ω_{k} (x)$ is obtained [29].

In fact, we also use higher order approximation to fit differential operator D and obtain high-precision polished operator. To this end, the expansion formula of x is written as follows [30].

x = shx \sum_{m = 0}^{\infty} C_{m} (chx - 1)^{m}, of which C_{m} = - \frac{m}{2 m + 1} C_{m - 1} = (- 1)^{m} \frac{m!}{(2 m - 1)!!} .

To understand this expansion of x, we name $y = sh \frac{x}{2}$ , there exists x = 2log(y + $\sqrt{1 + y^{2}}$ ). Then, the expansion expression of x can be written in the following form.

2 log (y + \sqrt{1 + y^{2}}) = \sqrt{1 + y^{2}} \sum_{m = 0}^{\infty} (- 1)^{m} \frac{2^{2 m} (m!)^{2}}{(2 m + 1)!!} y^{2 m + 1} .

In the expansion expression of x, we use $\frac{hD}{2}$ instead of x, and use I to replace 1, then, we can get the following expression.

hD = 2 sh \frac{hD}{2} \sum_{m = 0}^{\infty} C_{m} (ch \frac{hD}{2} - I)^{m} 2 sh \frac{hD}{2} = e^{\frac{hD}{2}} - e^{\frac{- hD}{2}} = E^{\frac{h}{2}} - E^{- \frac{h}{2}} = {\bar{Δ}}_{h} ch \frac{hD}{2} = \frac{1}{2} (e^{\frac{hD}{2}} + e^{- \frac{hD}{2}}) = \frac{1}{2} (E^{\frac{h}{2}} + E^{- \frac{h}{2}}) = μ .

Hence, D = $\sum_{m = 0}^{\infty} C_{m} (μ - I)^{m} \frac{{\bar{Δ}}_{h}}{h}$ .

In this way, the first part of ${\bar{Δ}}_{h}^{(n)}$ is shown as follows.

{\bar{Δ}}_{h}^{(n)} = \sum_{m = 0}^{\infty} C_{m} (μ - I)^{m} \frac{{\bar{Δ}}_{h}}{h} P_{n} (μ) \frac{{\bar{Δ}}_{h}}{h} .

Thus, the formulas of ${\bar{Δ}}_{h}^{(n)}$ becomes n-level approximation of D, then we define ${\bar{Δ}}^{(n)}$ D⁻¹ as the n-polished operator (h = 1), and call ${\bar{Δ}}^{(n)}$ D⁻¹f(x) as a n—polished function of f(x). If this operator repeatedly polishes δ(x), then get we $Ω_{n, k} (x)$ = ( ${\bar{Δ}}^{(n)}$ D⁻¹) ^k $Ω_{n, 0} (x)$ = $P_{n}^{k + 1}$ ( $μ$ ) $Ω_{k} (x)$ .

Here, P₀( $μ$ ) = I, $Ω_{0, k} (x)$ = $Ω_{k} (x)$ [31]. In the end, we get much more accurate fitting function, which is shown as follows.

S (t) = \sum_{i} P_{i} Ω_{n, k} (t - i) .

In the actual manufacturing process, we apply the function S(t) to fit real-time collected production data of the loom again, the result is shown in Table 2.

Table 2.

Data fitting result with the function S(t).

Machine code	Speed	Warp stop time	Warp stop number	Weft stop time	Weft stop number	Selvage stop time	Selvage stop number	Other stop time	Other stop number	Sum stop time	Sum stop number	Real yield	Efficiency (%)	Assortment name
A025	530	51	18	31	41	0	0	5	3	87	62	66.24	84.76	104.5″C110 × 90 22
A026	440	68	29	90	132	0	0	3	4	161	165	56.95	70.91	C133 × 72 104.5″ 55
A027	440	51	19	126	106	2	1	2	2	181	128	54.69	67.17	C133 × 72 104.5″ 55
A028	440	26	8	91	42	0	0	6	2	123	52	62.56	78.02	C133 × 72 104.5″ 55
A029	440	43	24	24	23	0	0	2	1	69	48	71.33	88.13	C133 × 72 104.5″ 55
A030	440	40	17	104	119	0	1	6	3	150	140	57.94	72.97	C133 × 72 104.5″ 55
A031	440	64	31	57	41	6	2	5	3	132	77	61.85	76.34	C133 × 72 104.5″ 55
A032	440	86	23	45	32	5	2	0	1	136	58	60.04	75.59	C133 × 72 104.5″ 55
A033	430	57	13	0	0	0	0	41	30	98	43	52.88	82.70	104.5″C110 × 90 22
A034	430	104	36	51	33	6	1	5	5	166	75	44.71	69.98	104.5″C110 × 90 22
A036	440	136	61	32	28	2	2	4	3	174	94	58.00	68.48	C133 × 72 104.5″ 55
A037	440	115	20	80	88	5	1	6	6	206	115	50.55	62.49	C133 × 72 104.5″ 55
A038	440	37	14	39	43	0	0	3	3	79	60	69.08	86.25	C133 × 72 104.5″ 55
A039	440	46	20	73	77	0	0	6	3	125	100	62.54	77.65	C133 × 72 104.5″ 55
A040	440	24	14	21	23	0	0	5	4	50	41	73.84	91.68	C133 × 72 104.5″ 55
B450	400	48	20	41	30	0	0	0	1	89	51	63.37	84.38	105.5″130 × 70
B452	380	0	0	540	0	0	0	0	0	540	0	46.71	68.10	63″133 × 72(om)
B453	380	84	35	43	36	4	1	5	4	136	76	55.07	75.59	63″133 × 72(om)
B454	424	76	37	41	50	0	0	0	2	117	89	61.18	79.14	63″133 × 72T/R(om)
B455	424	19	10	30	47	1	1	2	6	52	64	70.75	91.31	63″133 × 73T/R(om)
B456	424	23	9	72	65	0	0	2	3	97	77	64.21	82.89	63″133 × 74T/R(om)
B457	424	39	19	29	29	0	0	4	6	72	54	69.85	87.56	105.5″130 × 70
B459	424	44	19	63	49	2	2	3	3	112	73	63.88	80.08	105.5″130 × 70
B460	400	0	0	540	0	0	0	0	0	540	0	48.96	83.65	105″142 × 90
B461	420	21	12	18	18	0	0	4	4	43	34	74.18	92.99	105.5″130 × 70
B463	420	72	29	28	31	0	0	2	3	102	63	65.32	81.95	105.5″130 × 70
B550	400	37	16	47	27	0	0	2	3	86	46	63.88	84.94	105.5″130 × 70
B551	400	92	36	84	88	0	0	0	0	176	124	50.89	68.10	105.5″130 × 70
B552	420	105	34	30	27	0	0	4	2	139	63	59.87	75.03	105.5″130 × 70

After being processed, we can see from Table 2, the roller speed pulse data makes the calculation results be close to the actual value of the devices, there exist inefficient machines, which are denoted with white, and the efficiency of the abnormal looms are lower than in Table 1. Being compared with the collected production data, we find that data fitting method is correct, the computation result is true, and the system model meets the actual requirement of production management. Meanwhile, the results in Table 2 shows that data fitting function can effectively suppress and remove the interference factors mixed in roller speed pulse and effectively ensure the useful part of the pulse.

Conclusions

According to the complexity of the business workflow of industrial textiles, we set up a networking monitoring system model based on multi-agent, the aim is to strengthen the interactivity role of both the user and the system, meet the requirements of production management, intelligent decision, and personalized service. In this system model, we proposed a new data fitting method for real-time collected industrial textiles data, which first uses the integral and difference quotient operators to obtain the basic spline function and B-spline curve by operator deduction and then makes use of high-order approximation method for differential operator D to get the precision spline fitting function. As verified by application, the accuracy of production data of industrial textiles is guaranteed effectively and the utilization rate of the equipment is increased substantially. Finally, on the basis of the local area network (LAN) of the textile industry, we developed a networking monitoring system. In this system, each agent is either independent or linked with each other; this characteristics makes the system’s maintenance and upgrading easy, overcomes the limitation of the existed monitoring system, effectively integrates system data resource, and solves the problems which are information isolation, complex production process and inefficient personalized service, and so on.

Footnotes

Funding

Project supported by Science Research Program for Education Department of Shaanxi Province of China (Grant No.11JK1055), and Guiding Program for Textile Industry Association of China (Grant No.2011081).

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