D-FNN Based Modeling and BP Neural Network Decoupling Control of PVC Stripping Process

PVC stripping process is a kind of complicated industrial process with characteristics of highly nonlinear and time varying. Aiming at the problem of establishing the accurate mathematics model due to the multivariable coupling and big time delay, the dynamic fuzzy neural network (D-FNN) is adopted to establish the PVC stripping process model based on the actual process operation datum. Then, the PVC stripping process is decoupled by the distributed neural network decoupling module to obtain two singleinput-single-output (SISO) subsystems (slurry flow to top tower temperature and steam flow to bottom tower temperature). Finally, the PID controller based on BP neural networks is used to control the decoupled PVC stripper system. Simulation results show the effectiveness of the proposed integrated intelligent control method.


Introduction
With VCM as a raw material, the suspension method to produce polyvinyl chloride (PVC) resin is a kind of typical intermittent chemical production process [1].The traditional polymerization monomer conversion ratio is controlled to the scope 85%-90%.Unreacted vinyl chloride monomer is recovered by self-pressure, and 1%-2% of the monomers are still remained in PVC slurry.In the production, vinyl chloride monomer must be getting a further removal and recovery due to its some toxicity.The removal of residual vinyl chloride in polyvinyl chloride can improve the quality of polyvinyl chloride, reduce the cost of production, and solve the problem of environmental pollution caused in plastic processing and usage.At present, the stripping technique is very effective among all the removal methods of redundant VCM monomers in PVC slurry [2].Now the research on the control methods in PVC industry mostly is in the stage of polymerization and distillation, where many advanced control techniques, such as neural network, fuzzy control, expert system, and predictive control methods, have been widely used [3][4][5].Li [3] proposed a BP neural network decoupling controller applied to the temperature control of rectifying column.The simulation results show that the algorithm has strong robustness.Xiao et al. [4] established steady-state optimization model of expanded bed energy consumption and designed feedback control system of multivariable state based on artificial neural network for the drying process of fluidized bed, effectively removing the decoupling association between the variables and achieving the dynamic optimization control of bed temperature.The actual operation results show that this system has strong robustness and achieve the minimum energy consumption control indicators of expanded bed.Khairiyah et al. [5] proposed the predictive control system based on neural network applied to the control of the rectifying tower, which makes the system have a smaller overshoot and shorter response time.
However, the research on the advanced control methods for the PVC stripping process is lack.The stripping process has characteristics of highly nonlinear, strong coupling, and time varying.It is difficult to establish a precise mathematical model and achieve good control effect by using traditional control methods.

Mathematical Problems in Engineering
Dynamic fuzzy neural network (DFNN) is a hybrid model of fuzzy theory and neural network method, whose function is equivalent to the TSK fuzzy system [6][7][8][9][10].Based on the problem complexity and precision demand, D-FNN model can be constructed combining the system prior knowledge.An improved fuzzy neural network modeling method [11] was proposed to realize the soft sensor of molten index in the polymerization process, in which a hybrid learning method based on LM algorithm and a steepest descend method to train the network parameters exist.Simulation results show that the model has the merits of being not sensitive to initial values, fast convergence velocity, and high forecasting precision.The particle swarm algorithm (PSO) is adopted to optimize the parameters of the fuzzy neural network to set up a soft-sensor model of acrylonitrile yield [12]. Li and Lee [13] proposed an interval type-2 fuzzy system integrated with an observer-based hierarchical fuzzy neural controller, which can greatly reduce the number of adjusted parameters.Wu et al. proposed the maximum likelihood estimators used in nonparametric maximum likelihood fuzzy neural networks (MFNNs) for nonlinear regression problems and simple weight updating rules based on gradient descent and iteratively reweighted least squares (IRLS) will be derived [14].Nasr and Chtourou [15] proposed a self-organizing map-based initialization for hybrid training based on a twostage learning approach for feedforward neural networks.The weights between input and hidden layers are firstly adjusted by Kohonen algorithm with fuzzy neighborhood, whereas the weights connecting hidden and output layers are adjusted using gradient descent method.Hung-Ching proposed an adaptive self-constructing fuzzy neural network (ASCFNN) controller for a real inverted pendulum system, where the Mahalanobis distance (M-distance) method in the structure learning is also employed to determine if the fuzzy rules are generated/eliminated or not [16].D-FNN is also successfully applied in many fields, such as function approximation, permanent-magnet synchronous motor drive control [17,18].
In this paper, the D-FNN model of the PVC stripping process has been established firstly.Then the neural network decoupling modules and BP neural network PID controller are applied in the stripping temperature control.The effectiveness of the proposed intelligent control strategy is verified by the simulation experiments.The paper is organized as follows.In Section 2, the technique flowchart of PVC stripping process is introduced.The stripping process model based on dynamic fuzzy neural networks is presented in Section 3. In Section 4, the neural network decoupling controller of stripping process is summarized.In Section 5, simulation results are introduced in detail.Finally, the conclusion illustrates the last part.

Technique Flowchart of PVC Stripping Process
In the polymerization reaction process of vinyl chloride, according to the different techniques and resin grades, the polymerization conversion rate of vinyl chloride is generally controlled within the scope of (80%, 90%).Even the unreacted monomer is disposed by the autoclave recovery and precipitated groove blowing, and there still remains small amount of vinyl chloride monomer in resin products, which will influence the products quality and increase the hazards when using the produced plastic.In addition, vinyl chloride emission into atmosphere per year not only causes a huge waste of raw materials, but also causes serious environment pollution.Currently, among all the approaches to the removal of excess VCM monomer in PVC slurry, the stripping technique is very effective.Firstly, the stripping mechanism of PVC slurry is introduced in detail.In the PVC suspension to be stripping treated in the off chute of PVC polymerization, the mass fraction of PVC is about 25%-30%, the VCM is 2%-4%, and the rest is water.The mass approximation allocation ratio of VCM in gassed is about 1 : 1000 : 100.Because of the different amount distribution of VCM in the three phases, the slurry itself has the characteristic of the diffused concentration gradient.In PVC slurry, the VCM in the solid PVC either by parsing or by spreading must pass through the film layer to a low concentration of VCM in water for spreading through the pores in PVC particles.Moreover, the solubility of VCM in water changes within the variation of temperature.In different temperature, when the VCM in water reaches to solubility, the part which diffused from PVC into water will break the static pressure of the liquid water layer and diffuse into the gas phase.With the above mechanism, the heat exchange makes the slurry's temperature increase so that the VCM will be parsed.
There are two main tower stripping devices.What we research in this paper is the sieve wear weir plate stripper.The stripping technique of the PVC slurry is shown in Figure 1.
After the polymerization reaction, the PVC slurry is threw into the top of stripper from the polymerization vessel through the inlet slurry pump.But before entering the stripper, they need to carry through the heat exchange with the slurry stripped from the bottom of stripper and then go into the top of stripper.PVC slurry sprinkles evenly on the tray by sprinklers and flow from the top to bottom along the strays from the first layer stray.The low pressure steam enters from the stripper bottom, rises through the orifice of the tray from the bottom to top, and countercurrent contacts with the declining PVC slurry at each plate for heat and mass transfer.Finally, it will resolve internal vinyl chloride and the residue in the surface of the resin particles.Steam and parsed vinyl chloride gas are discharged from the top of the stripper.After the condensation by overhead condenser, it enters into the condensate sink, then separates condensate, and controls the level of condensate sink.The excess condensate will be discharged to the centrifuge mother liquor water recovery unit, and the separate vinyl chloride gas will be recovered to vinyl chloride gas cabinets.The key parameters influencing the PVC stripping process can mainly include temperature, steam, slurry flow rate, and pressure.Table 1 gives the parameters values in the PVC stripper controller of a certain type resin.
Through the analysis of PVC stripping process, it has the characteristics of strong nonlinear and coupling.Because of lack of effective mathematical model and strong coupling between the variables, it is difficult to achieve satisfactory The data used for modeling all derived from the history data recorded in the operating process of PVC stripping tower.Each data sample includes the following elements: top tower temperature, bottom tower temperature, top tower pressure, tower differential pressure, slurry flow rate, and steam flow rate.According to the analysis of stripping technique, the key factors which influence vinyl chloride removed from PVC are the top tower temperature and the bottom tower temperature.So there are two mainly important control factors, which are the steam flow rate and the slurry flow rate during the process regulation.Therefore, the D-FNN model is built in which the steam flow rate and slurry flow rate are the input variables, and the top tower temperature and the bottom tower temperature are the output variables.

Stripping Process Modelling Based on Dynamic Fuzzy Neural Networks
According to the analysis of the stripping process, the changeable slurry flow or the steam flow rate not only influences the top tower temperature but also the bottom tower temperature.Consequently, the coupling exists in the PVC stripping process.The coupling object model needs to be built and the stripping process needs to be decoupled.Suppose the dynamic system model is described as the following nonlinear equation: where  is the top tower temperature or bottom tower temperature,  1 is the steam flow rate,  2 is the slurry flow rate, and  is the sample time (20 s).
The D-FNN control strategy of the PVC stripping process is put forward shown in Figure 2, whose inputs variables are the steam flow rate and the slurry flow rate and output variables are the top tower temperature and the bottom tower temperature.

Structure of D-FNN.
The structure diagram of D-FNN [19][20][21][22][23] is shown in Figure 3.In Figure 3,  1 ,  2 , . . .,   is the input linguistic variables, y is the system output, MF  represents the jth membership function of the ith input variable,   is the jth fuzzy rule,   is the jth normalized node,   is the connection weight of the jth rule, and u refers to the numbers of the system rules.Then each layer of the D-FNN is described in detail.
Layer 1 is the input layer, whose each node represents an input linguistic variable.
Layer 2 is the membership function layer, whose each node represents a membership function (MF) defined in the followed Gaussian function: where  = 1, 2, . . ., ,  = 1, 2, . . .,   is the jth membership function,   is the center of the jth Gaussian membership function,   is the width of the jth Gaussian membership function of   , and  is the number of input variables.Layer 3 is the T-norm layer, whose each node represents a possible IF-part of fuzzy rules.For the jth rule   , its output is defined as follows: where  = 1, . . ., ,  = ( 1 , . . .,   ) and   =  1 , . . .,   is the center of the jth RBF unit.Layer 4 is the normalization layer, where the output of the   node is defined as  Layer 5 is the output layer, whose each node is an output variable, a summation of input signals described as follows: where y is the output variable and   is the weight of the kth rule.
For the TSK model, it can be represented as follows: Then ( 2)-( 5) are fed into (5) to obtain

Learning Algorithm of D-FNN. Each node in layer 3 of the D-FNN represents an RBF unit or the IF-part for fuzzy rules.
If the number of fuzzy rules is required to be identified, the D-FNN structure is not predefined.So a new learning algorithm for the D-FNN is adopted to automatically determine the fuzzy rules satisfying the required system performances.

Rule Generation Criteria.
If there are only few rules, the system cannot fully represent the input-output state space.But if there are too many rules, the system can be unnecessarily complicated, and the generic ability of D-FNN can become worse because of the greatly increased calculation burden.So the system errors are the principal elements to determine whether the new rules can be added.
The error criterion can be described as follows.For the ith observation point (  ,   ), where   is the input vector and   is the desired output, compute the overall D-FNN output   according to (7) Find the minimum distance by (11).If  min >   , a new fuzzy rule should be added.Otherwise, the observation   can be represented by the nearest existed RBF unit.Here,   is the effective radius of the accommodation boundary  min = arg min (  ()) . (11)

Hierarchical Learning Strategy.
The concept of hierarchical learning is that the accommodation boundary of each RBF unit is not fixed but adjusted dynamically in the following style.Firstly, the accommodation boundaries are set larger for achieving rough but global learning.Then, they are gradually reduced for fine adjustment.Inspired by this idea, a simple method based on monotonically decreasing function is proposed to reduce both the effective radius of each RBF unit and error index gradually [22].To be more specific, the discussed above   and   are not constant.They are changeable according to the following equations: where  max is the predefined maximum error,  min is the desired D-FNN accuracy,  (0 <  < 1) is the convergence constant,  (0 <  < 1) is the decay constant, and  max and  min are the largest and the smallest length of the input space, respectively.
The key idea of the hierarchical learning is to firstly find the most troublesome positions, which have large errors between the desired and the actual outputs but are not properly covered by existing fuzzy rules.This stage is called the coarse learning.When   and   reach  min and  min , respectively, this stage is named the fine learning.

Allocation of Precedent Parameters.
After a fuzzy rule is obtained, the problem is how to determine its parameters.The width of the RBF unit has great significant on its generalization.If it is less than the distance between adjacent inputs, the RBF unit does not generalize well and D-FNN will not give meaningful outputs in response to inputs.However, if the width is too large, the output of the RBF nodes may always be large irrespective of inputs.Hence, the initial parameters of the new produced rule are allocated based on the following equations: where  ( > 1) is an overlap factor.What we needed to pay attention to is that when the first pattern ( 1 ,  1 ) enters the D-FNN, it is selected as the first fuzzy rule:  1 =  1 ,  1 =  0 , where  0 is a preset constant.It is worth highlighted that only when ‖  ‖ >   ,  min >   , one fuzzy rule needs to be added.

Determination of Decision Parameters.
According to the criteria of rule generation, suppose  fuzzy rules are generated based on the  observation datum.So the output of  nodes can be obtained according to (4), which can be represented as follows: Then, for any input   ( 1 ,  2 , . . .,   ), the system output   can be calculated by (5), which can be described in more compact form: Suppose the ideal output is  = ( 1 ,  2 , . . .,   ) ∈ R  .The relation between Ψ ∈ R (+1)× and  ∈ R  is represented as follows: Find an optimal coefficient vector based on The optimal  * is in the following form: where Ψ  is the transposition of Ψ and

Simulation Results.
The samples are derived from the actual operation datum of the certain chemical corporation, in which 250 groups of input-output data have been selected to identify the system randomly.The other 80 groups of inputoutput data are used to test the identification effect.Then the D-FNN models are obtained, in which the steam flow rate and slurry flow rate are inputs variables, and the top tower temperature and the bottom tower temperature are output variables separately.The identification effect and test results are shown in Figures 4 and 5.It can be seen form Figures 4  and 5 that the output curves of the D-FNN model and the actual curves of the tower temperature match very well, and the errors are within ±0.1, which can meet the requirement of the process optimization control.In Figure 6,   is the system input,   is the system output,   is the controller output vector,   is the output vector of the neural network decoupling module, and    is the control input vector ( = 1, 2).The intelligent decoupling control system of the stripping process can be divided into two stages.

Training Stage.
The neural network PID controller is disconnected and the system is in open-loop training state.Firstly, the neural decoupling module ND 1 is trained and ND 2 is maintained as no input.If decoupling is successful, the channel between the slurry flow rate and the tower top temperature will not have output ( 1 is 0).Then  1 is set as the training index function of ND 1 .The connection weights of ND 1 's each layer are adjusted.Similarly, ND 2 is trained in the same way.Finally, the whole stripper controlled object is decoupled into two SISO systems (slurry flow rate, the tower top temperature, and the steam flow rate, the tower bottom temperature).

Control Stage
. NN 1 and NN 2 are used to online optimize the PID controller parameters for realizing the closedloop control for the two SISO systems.After training of ND 1 and ND 2 , the weights are maintained unchanged.At this time, the entire stripper controlled object is decoupled into two SISO systems.Then two BPNN PID controllers (NN 1 and NN 2 ) are, respectively, adopted to realize the closed-loop control of two SISO systems.NN 1 and NN 2 will be used to online adjust the PID controller's parameters   ,   , and   .

Structure of Neural Network Decoupling Module. The neural network decoupling module consists of 𝑛 × (𝑛 − 1)
SISO neural network ( is the number of the object inputs), whose structure is shown in Figure 7.Each neural network decoupling module is single-input-single-output (SISO) subsystem, which is responsible for only one channel decoupling.So its structure and indicator function are relatively simple because the total objective function has been decomposed into single objective functions and the decoupling modules' interconnection problem needs not be considered.At the same time, the dispersion of the objective function is also easy to achieve parallel distributed processing in real time.In addition, the decoupling structure does not change the dynamic characteristics of the controlled process main channel.It will make the design problem easy in engineering.As long as the reasonable structure and parameters of controller   are chosen, the required dynamic characteristics can be obtained.
The PVC stripping process is double input and double output coupling system.The structure of the neural decoupling module is shown in Figure 7.
The decoupling module is composed of two different neural networks ND 1 and ND 2 , which are dynamic time delay TDNN networks.Each network has an input terminal and an output terminal.The decoupling module outputs can be expressed as where u is the output of neural network decoupling module,  is sampling frequency, and  1 ,  2 ,  1 , and  2 are the positive integers, which are determined according to the controlled process' order.The decoupling module in this paper adopts a three-layer neural network.The neural network's excitation function is the hyperbolic tangent function () = (1 −  − )/(1 +  − ).Its derivative is () = 1/(2 × (1 −  2 ())). 1 () and  2 () are the outputs of the decoupling module.After superimposing the controller outputs  1 and  2 , the system control output is obtained by  1 =   1 +  1 and  2 =   2 +  2 .

Training Algorithm of Neural Network Decoupling Module.
For the neural decoupling module shown in Figure 8, theoretically speaking, the expected output of the neural decoupling module should be viewed as the target function.
But the NN expected output is unknown.In order to avoid this contradiction, the neural decoupling module and the controlled object are viewed as a generalized object.Thus the neural network training can adopt the system output as the performance function.For neural decoupling modules ND 1 and ND 2 , the performance index functions are defined as follows: Mathematical Problems in Engineering According to the BP learning algorithm [24,25], the network weights are updated based on the following equation: where  is the learning rate,  is the momentum coefficient,  is the weights of the corresponding neural network, and  is the number for training samples.
For ND 1 and ND 2 , differentiate the performance index function to obtain For the convenience of discussion, the subscripts are omitted to get the universal formula: where Δ is the change of weight.
In order to adopt BP learning algorithm, the partial derivative / needs to be calculated.However, the learning rate is an adjustable parameter.The partial derivatives multiply the learning rate's product deciding the change of weights.Therefore, the absolute value of the partial derivatives is not very important.If the partial derivative's symbols are known, it is recorded as sign(/).Adjust the learning rate to obtain Therefore, (23) can be written as In the actual industrial production, there are many controlled processes which have monotone increasing or decreasing characteristics.That is to say, y is the monotonically increasing or decreasing function of   .When  is the monotone increasing of   , sign(/) = 1, or it is −1.
For the output layer, For the middle layer, Put ( 28) and ( 29) into (23); the weight of the output layer and middle layer is updated by the following equations.
For the output layer, +  * Δ  ( − 1) . (30) For the middle layer, The algorithm procedure is described as follows.
(1) Connect the neural network decoupling module ND 1 and ND 2 with the controlled process.
(2) Initialize the neural decoupling module weights with the smaller random values.
(3) Give the input signals and make the controller output signal correspond to the input signal of the neural decoupling module.(4) According to the selected samples and time interval, adopt (26) to calculate the output of the neural network   ,  (1) ,  (0) , and  values of the process variables changed with time (set   = 0 at the begin of calculation).
(5) According to (30) and (31), calculate the weight variances of the output layer and middle layer and update weights  (1) and  (2) .
The above process does not involve each variable's subscript.In practice, the training of network weights can be completed individually according to the above procedure.When ND 1 , ND 2 , . . ., ND  are determined, the whole system can be viewed as  single variable nonlinear systems.Then each loop can be designed the corresponded controller.

PID Controller Based on BP Neural
Network.The structure of PID controller based on BP neural network is shown in Figure 9.According to the system's operation states, the PID controller parameters are adjusted to achieve the optimization performance.The neural units in the output layer are in correspondence with three PID controller adjustable parameters   ,   , and   .Through self-learning of neural network, the weighted coefficients are adjusted to make the neural network output corresponding to the PID controller parameters with a certain optimal control law.
Three layers BP neural network [26] is used for the PVC stripping process.The input layer have 3 neurons: input , output , and error error: where the subscripts , , and , respectively, represent the input layer, the hidden layer, and the output layer.
The hidden layer has 5 neurons.The input of the hidden layer is calculated by The output of the hidden layer is calculated by The activation function of the hidden layer neuron adopts the symmetrical sigmoid function () = tanh() = (  −  − )/(  +  − ).
The output layer has 3 neurons.The input of the output layer is calculated by The output of the output layer is calculated by The output layer nodes are in accordance with three adjustable parameters   ,   , and   .Since   ,   , and   cannot be negative, the output layer neuron's activation function is not negative sigmoid function: The performance index function is selected as () = 1/(2 × (() − ()) 2 ).The network weights   and   are revised according to the gradient descent method.That is to say, according to (), the weighted coefficients are searched and adjusted in the negative gradient direction with an inertia which make search rapidly converge to global minima.
From the output layer to the hidden layer, where  is the learning rate and  is the inertial coefficient: Because ()/() is unknown, it is approximated by the symbol function sign(()/()).The problem of calculation accuracy can be solved by adjusting the learning rate  to compensate.
For the output layer, Thus the learning algorithm of the output layer weights is described as Similarly, the learning algorithm of the hidden layer weights is described as 2 . (41) The algorithm procedure is described as follows.
(1) Determine the structure of BP neural network.That is to say, determine the number of the input layer nodes and the hidden layer nodes.Given all initial weighting coefficients, select the learning rate  and the inertial coefficient .
(3) Calculate each layer's input and output of the neural network.The output in BPNN is the three adjustable parameters of the PID controller.
(4) Calculate the PID output according to the incremental algorithm.
(5) Train BP neural network to online adjust the weights for realizing the self-tuning of the PID controller parameters.

Simulation Experiments
In According to the technique of the PVC stripping process, the optimum of the stripper tower top temperature is 85 ∘ C. Thus in our simulation, the tower top temperature is 85 ∘ C. The temperature should not exceed 110 ∘ C, otherwise it will make the PVC resin decomposed and metamorphic.Therefore the upper limit is 110 ∘ C. The optimum of the stripper tower bottom temperature is 107 ∘ C and its upper limit is 113 ∘ C. According to the actual possible working conditions of the stripping process, this paper considers the following situations in the simulation.
(1) Without any disturbance, the tracking performance of the tower top temperature and tower bottom temperature is verified, which is shown in Figure 10.
It can be seen form Figure 10 that the control system can track the input changes and has a high response performance.
(2) The random disturbance Δ 1 = 0.01 is added on the tower top temperature and the simulation results are shown in Figure 11.The random disturbance Δ 2 = 0.01 is added on the tower bottom temperature and the simulation results are shown in Figure 12.By adding the disturbance signals on the tower top temperature and tower bottom temperature, the system decoupling effects are verified.It can be seen form Figures 10 and 11 that the neural network decoupling modules improve the performance of the control system and the transaction between two loops can be controlled effectively.Based on the above simulation results, the following conclusions can be made.(1) The tracking performance of the system is better and is able to adapt to the change of set points.The response time and other parameters are satisfactory.(2) The coupling degree between the slurry flow rate, the tower top temperature, and steam flow rate, the tower bottom temperature, has been significantly reduced and the decoupling is achieved.(3) The system has certain robustness for the external disturbances.It can overcome the temperature disturbance of the tower top temperature and the tower bottom temperature.

Conclusions
In the paper, the D-FNN was used to set up the PVC stripping process model.Then the strategy of combination of the neural network decoupling module and the BP neural network PID controller is put forward for controlling the stripping process temperature.The simulation results verify the effectiveness of the proposed PVC stripping intelligent control strategy.

Figure 2 :
Figure 2: Modeling of stripping process based on D-FNN.

Figure 4 :
Figure 4: Simulation results of D-FNN modeling of tower top temperature.

Figure 5 :
Figure 5: Simulation results of D-FNN modeling of tower bottom temperature.

Figure 9 :
Figure 9: PID controller based on BP neural network.
order to verify the effectiveness of the D-FNN model and the proposed intelligent decoupling control strategy based on neural network, the D-FNN model is used in neural network decoupling control strategy.Based on the tower top temperature D-FNN model and tower bottom temperature D-FNN model, the neural network decoupling modules ND 1 and ND 2 and neural network controller NN 1 and NN 2 are designed.The structure and the parameters of all neural networks are set as follows.The structure of ND 1 and ND 2 is 5-4-1, the momentum factor  ND = 0.005, and the learning rate  ND = 0.9.The structure of NN 1 and NN 2 is 3-5-3, the inertia  NN = 0.25, and learning rate  NN = 0.05.

Figure 10 :
Figure 10: Simulation results of step input without disturbance.

Figure 11 :
Figure 11: Simulation results with tower top temperature disturbance.

Figure 12 :
Figure 12: Simulation results with tower bottom temperature disturbance.

Table 1 :
Parameters of PVC string tower.
. Define the error as follows:           =       −       .Gaussian function has good locality characteristic because its output is monotonic decreasing along with the increasing distance between the output and the centers.When the membership function of input variables is described by Gaussian function, the whole input space is divided by a series of Gaussian function.If a new sample is located in some Gaussian function, that is to say within accommodation boundary, the new sample can be represented by the existed Gaussian function, without new Gaussian unit produced by D-FNN.The accommodation boundary criterion is described as follows.For the ith observation point (  ,   ), calculate the distance   () between the observation   and the center   of the existed RBF units by () =        −        ,  = 1, 2, . . ., .