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FUZZY PNN ALGORITHM AND ITSAPPLICATION TONONLINEARPROCESSESTAECHON AHN a & SANGMOON RYU aa School of Electrical & Computer Engineering,Wonkwang University , 344-2, Shinyong-dong,Iksan-city, Chollabuk-do, 570-749, KoreaPublished online: 30 May 2007.

To cite this article: TAECHON AHN & SANGMOON RYU (2001) FUZZY PNN ALGORITHM AND ITSAPPLICATION TONONLINEARPROCESSES, International Journal of General Systems, 30:4, 463-478, DOI: 10.1080/03081070108960725

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FUZZY PNN ALGORITHM AND ITS. APPLICATION TO NONLINEAR

PROCESSES

TAECHON AHN* and SANGMOON RYU

School of Electrical & Computer Engineering. Wonkwang University, 344-2, Shinyong-dong,

Iksan-city, Chollabuk-do. 570-749, Korea

(Received 25 March 1997; Revised I5 March 2000; In jinaljorm 15 March 2000)

In this paper, a fuzzy Polynomial Neural Network (PNN) algorithm is proposed to estimate the structure and parameters of fuzzy model, using the PNN based on Group Method of Data Handling (GMDH) algorithm. The new algorithm uses PNN algorithm and fuzzy reasoning in order to identify the premise structure and parameter of fuvy implications rules, and the least square method in order to identify the optimal con- sequence parameters. Both time series data for the gas furnace and data for the NO, emission process of gas turbine power plants are used for the purpose of evaluating the performance of the fuzzy PNN. The simulation results show that the proposed technique can produce the fuzzy model with higher accuracy and feasibility than other works achieved previously. This algorithm will be applied to limited data processes with several inputs.

Keywork PNN; fuzzy PNN; GMDH; NOx emission; fuzzy model; gas turbine plant; gas furnace

1 INTRODUCTION

Recently, many researchers have had much interest in various methods for system modeling. Among them, mathematical modeling methods

Corresponding author. Tel: +82-653-850-6344. Fax: +82-653-853-2196. E-mail: tcahn@wonnms.wonkwang.ac.kr.

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such as regression techniques were widely used to identify and to predict the linear systems based on input-output data. However, the mathematical models to express dynamic analysis of nonlinear real systems have had lots of problems in the selection of variables con- structing the model among many input-output variables and of model structure. In general, higher-order equations require too many data against estimating all system parameters in mathematical models. To solve the problem, the PNN based on GMDH was first introduced by lvankhnenko (1968).

The GMDH has been used to synthesize the PNN, namely, the building blocks of modeling methodology. This approximation technique based on the perceptron principle with neural network- type architecture has been applied to modeling, identification and prediction of the input-output relationship of a nonlinear process system with limited data sets. Fuzzy modeling is another highly skilled technique, using trial and error, to properly describe the static and dynamics of nonlinear process system. As known, fuzzy mod- eling has been widely investigated and successively used for indus- trial applications. Therefore, two methods have had the advantages in the performance of systems with intense non-linearity. However, they have a little problem against accuracy and feasibility, to some extent.

In this paper, a fuzzy PNN algorithm is proposed to estimate the structure and parameters of fuzzy model, using the PNN based on the GMDH algorithm and the fuzzy modeling method. The new algo- rithm fuses the PNN algorithm and fuzzy inference by replacement of each neuron of the PNN with fuzzy implications rules, in order to model the nonlinear process system with limited data sets, namely, to identify the premise structure and parameters of fuzzy implications rules. The premise fuzzy membership of each input variable uses Gaussian functions obtained by heuristic. The consequence utilizes the simplified inference consisting of constants and the linear inference consisting of regression polynomials. The optimal consequence para- meters are obtained by the least square method. The fuzzy PNN is applied to both time series data for gas furnace and data for the NO, emission process of gas turbine power plants, for the purpose of evaluating its performance. The results are compared-to conventional methods, in details.

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FUZZY PNN ALGORITHM

2 THE PNN MODELING

The highly complex nonlinear systems have limited the success of pro- cess modeling from measured data. Recently, PNNs (Ivankhnenko, 1968) as well as fuzzy systems (Madala et al., 1994; Tong, 1980; Pedrycz, 1984; Hayashi et al., 1990; Tanaka, 1993) have emerged as a more attractive alternative to physical models and empirical statistical methods. The PNNs, however, are still in the first stage of their research on the complex process modeling. The inherent property of the PNN is to model complex systems using simple building blocks.

The design of a variable control strategy requires the availability of a reasonable accurate model of the process system. Such models had not been available for the complex process model due to the process dimensionality and the complexity of the interacting physical phe- nomena. Therefore the PNN strategy is employed to address the complex process modeling via GMDH. The PNN methodology is implemented on the random input-output sets of measured training and testing data obtained from the system.

In this paper, the inputkoutput data for the gas furnace and the NO, emission process of gas turbine are becoming available and open- up the possibility for robust modeling tools that adopt a modeling paradigm that is based on a PNN.

2.1 The Structure of PMV

The PNN based on the perceptron principle with neural network-type architecture (Ivankhnenko, 1968; Sang, 1996) is used to model the input-output relationship of a complex process system. At each layer, new generations of complex equations are constructed from simple forms. Survival of the fittest principle (appropriate thresholds) deter- mines the equations that are passed on to the next layer and those that are discarded, that is, only the best combination of input prop- erties (new variables) are allowed to pass through to the next layer. The model obtained after each layer is progressively more complex than the model at the preceding layers. To avoid an overfit, the data sample is divided into (a) the training set, which is used for the gen- eration of several computing alternative models and (b) the testing set,

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466 TAECHON AHN AND SANGMOON RYU

which is used to test the accuracy of model generated and for the selection of the best models at each layer.

The P N N uses the mean squared difference between the measured output data and model output data as predictability. Then, an optimal model of the system is related in the viewpoint of this error.

The number of layers is increased until the newer models begin to have poorer powers of predictability than their predecessors. This indicates overspecialization of the system. The final model is defined as a function of two or three variables. The network result is a very sophisticated model obtained from a very limited data set. In a PNN technique, a simple form function is usually combined at each node of a polynomial neural network to obtain a more complex form.

This function as an approximation represents the current model for the given training and testing sets of input-output data. This approx- imation is written as a regression polynomial of second degree like equation (2.1) in a case of combining two inputs at each node;

where y is the output and Xi and 4 are the two inputs. A , B, C and D are coefficients of filer, respectively.

The outputs obtained from each of these nodes are then combined to obtain a higher-degree polynomial so that the best model may be achieved which represents the input-output data. The degree of the polynomial increases by the number of selected inputs a t each layer. Here, a complex polynomial is obtained from the above process, called an Ivakhnenko polynomial. The process is stopped when the perform- ance of testing set satisfies a given criterion. This function usually has a formula, such as;

where Xi, Xj and Xk are nodal input variables, and y is the output of an individual neuron (node). A , B,, CU, and DVk are the coefficients of the lvankhnenko polynomial (1968).

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FUZZY PNN ALGORITHM 467

TABLE I The node equations considered for polynomial neural network synthesis

Degree Input

1 2 3

1 Linear Bi-linear Tri-linear 2 Quadratic Bi-quadratic Tri-quadratic 3 Cubic Bi-cubic Tricubic

In this paper, we considered nine linear types of model equations as shown in Table I, in order to obtain an optimal model.

trilinear = xo + wlxl + ~ 2 x 2 + ~ 3 x 3

triquadratic = trilinear + ~ 4 x 1 ~ 2 + ~ 5 x 1 ~ 3 + ~ 6 ~ 2 x 3 + w, 4 + wsx: + w9x:

tricubic = triquadratic + wloxlx2x3 + W I IX: + wlzxi + ~ 1 3 ~ :

2.2 The Algorithm of PNN

The PNN systhesis activities have focused over the past years on the development of self-organizing, minimal polynomial networks with good generation capabilities. Searching for the optimal configuration in the space of all possible polynomial neural networks is untract- able and requires a set of heuristic. A series of heuristic is, therefore, defined to prune the search as follows;

(a) Construct a observation matrix of each layer, by taking all of previous observation variables, and solving the. regression poly- nomial like equation (2.2) in the least square sense. If previous observation variables are X1, XZ, . . . , Xn, n (n - 1)/2! observation variables are at least generated to predict the output y.

(b) Define a modeling criterion to select the new observation variables that best predict and estimate the dependent variable y . The cri- terion called the Performance Index (PI) is applied to the testing set of the system.

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(c) Pick all columns of the observation matrix such that PI < R, where R is some threshold value chosen a priori. A new ob- servation matrix is built by taking the satisfied coIumns in the training set, and is replaced at the previous matrix, and then new observation variables are formed. One should note that the number of variables saved may be less than or greater than the original number. Also note that the test of goodness of fit PI is summed over the observations in the testing set.

(d) Find the smallest of PIS' (PI,;,). If the value of PIi, is less than the value of PI^, from the previous generation, the above pro- cedure is repeated. If the value of PI^, is greater than the pre- ceding value of PI,,,, it is assumed that the smallest value has reached its minimum, and the PNN process of building block is stopped. The polynomial obtained from the preceding generation is used as the optimal approximation (minimum Ivakhnenko polynomial). The topology of the PNN can evolve into the mini- mization of the PIS', according to the heuristic rules.

The PNN leads to self-organizing heuristic hierarchical models of high degree with automatic elimination of undesirable variable interactions. In contrast with the conventional regression technique, this scheme has several distinct advantages. A smaller data set is required, the computational time and resources are reduced and the final structure of the PNN does not need to be specified. In addition, high-order regression often leads to a severely ill conditioned system of equations. However, the PNN avoids this by constantly eliminating variables and variable interactions at each layer, and heIps to reduce linear dependencies. Therefore, complex systems can be modeled with- out specific knowledge of the system or massive amounts of data.

3 THE STRUCTURE AND ALGORITHM OF FUZZY PNN

In this section, a fuzzy PNN algorithm is proposed to estimate the structure and parameters of fuzzy model, using the PNN based on the GMDH algorithm and the fuzzy modeling method. The new algo- rithm fuses the PNN algorithm and fuzzy inference by replacement of each neuron of the PNN with fuzzy implications rules, in order to

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FUZZY PNN ALGORITHM 469

model the nonlinear process system with limited data sets, namely, to identify the premise structure and parameters of fuzzy implications rules. The differences between proposed fuzzy PNN and conventional PNN are also considered. While conventional PNN obtains the output using the second order equation of two variables, fuzzy P N N extract the output from each node of conventional PM\J, using fuzzy models with fuzzy implications rules. Each node is operated as a small fuzzy system. Overally, the structure of fuzzy PNN is like Fig. 1.

The premise of fuzzy membership function of each node is expressed by Gaussian functions obtained from heuristics. This function is selected by the c-means clustering and simplex methods. The con- sequence of each node is expressed by constants in' fuzzy PNN using the simplified inference and by regression polynomials in fuzzy PNN using linear inference. If premise input variables and parameters are given, the optimal consequence parameters which minimizes per- formance index can be determined by least-square method in a similar way of fuzzy systems (Oh, 1994). As known in Fig. I , the algorithm and structure of fuzzy PNN are almost like those of the PNN except using the fuzzy system in each node.

f : fuzzy rule ,,Zk FIGURE 1 Basic fuzzy PNN by simplified inference.

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3.1 The Fuzzy PNN by Simplified Inference

The consequence part of the simplified inferences expressed by con- stants is given as equation (3.1).

R' : If x; is Ail , . . . ,xk is Aik, Then y = a;

where R' is the i-th fuzzy rule, xi is input variable, Aik is a membership function of fuzzy sets, a; is a constant, n is the number of the fuzzy rules, y* is the inferred value, pi is the premise fitness of R' and f i i is the normalized premise fitness of pi.

If input variables and parameters of the premise are given, the optimal consequence parameters that minimize PI, can be determined in consequence parameter identification, using the least-square method. PI is the criterion that uses the mean squared differences between the output data of original system and the output data model. It can be defined by equation (3.2), in general.

where is output of fuzzy model, k is number of input variables, and m is total number of data.

The consequence parameters can be estimated by least-square method, when the input-output data set is given as XI;, xzi,. . . , xki - yi (i = 1,2 , . . . , m). Equation (3.3) provides the minimal estimated values of a by least-square method.

where

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FUZZY PNN ALGORITHM 47 1

3.2 Fuzzy PNN by Regression Polynomial Inference

The consequence part of the inferences expressed by regression poly- nomials in the Table I is given as equation (3.4) (linear inference).

R' : I f xi is A i l , ..., xk is Aik, Then y = f ; . ( x l , . . . , xk) (3 .4)

n n n

where R' is the i-th fuzzy rule, xk is input variable, Aik is a membership function of fuzzy sets, A , Bi, Cv and Diik are coefficients of regression polynomial, n is the number of the fuzzy rules, y* is the inferred value, pi is the premise fitness of R' and ,& is the normalized premise fitness of pi.

The consequence parameters can be estimated by least-square method in a similar way of equations (3.2) and (3.3).

4 SIMULATIONS AND RESULTS

In this section, both time series data for gas furnace and data for the NO, emission process are considered for the purpose of evaluating the performance of the fuzzy PNN. These systems use PI as criterion. Varying the nodal polynomial, the results of fuzzy PNN are analyzed. Tri-quadratic type shows the best performance among the nine nodal polynomials in Table I. As shown in two systems, fuzzy PNN gives

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a very sophisticated model from a very limited data set and/or a sys- tem with intense non-linearity. Fuzzy PNN model also has more performance in regression polynomial reasoning method than sim- plified reasoning method. Therefore, this model provides very good results in performance, when the number of membership function is properly chosen in modeling system.

4.1 Gas Furnace

The fuzzy PNN modeling is applied to the time series data of gas furnace utilized by Box and Jenkins (Oh, 1995), in order to test the feasibility and effectiveness. Input-output data of 296 pairs is used to model the gas furnace. While the delayed terms of gas flow rate, u ( t )

and burned carbon dioxide density, y ( r ) are set to input variables such as u( t - l), u(t - 2), u ( t ) , y ( t - 3), y ( t - 2) and y ( t - l), y ( t ) is set to the output variable. Gaussian is used as the premise membership

FIGURE 2 FPNN model structures for gas furnace in the cases of (a) linear (b) bi- quadratic and (c) tri-linear polynomial.

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FUZZY PNN ALGORITHM

- o : Real Data - * : Model Output

44 1 1 0 50 10 0 15 0 200 25 0 300

Data Nanber

FIGURE 3 Comparison of the output of fuzzy PNN model with real data in gas furnace.

function of the FPNN. In this process, though we use all the inputs possible, the optimal model is chosen in case of only 4 inputs such as

. u ( t ) , y ( t - 3), y( t - 2) and y ( t - I). The flow rate of methane gas, u,(t) used in laboratory process,

changes from -2.5 to +2.5, but u ( t ) used in real process, from 0.5 to 0.7 by relation like equation (4.1).

The FPNN's structures of modeling for gas furnace are shown in the cases of linear, bi-quadratic and tri-linear polynomials, respectively, in Fig. 2. For each FPNN structure, the PIS are calculated, changing the number of membership functions.

The optimal output of identified model using the fuzzy PNN is compared with real measured data, in Fig. 3. The identification error of the fuzzy PNN is also compared with other fuzzy modeling methods in view of PI, in Table 11.

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TABLE I1 Comparison of identification error with conventional f ~ p y modeling methods in gas furnace

Model PI

Tong's model (1980) Pedrycz's model (I 984) Xu's model (1987) Sugeno's model (199 1) Oh's model (1994) FNN (Ahn, 1997) PNN (our model) FPNN (our model)

4.2 NO, Emission Process of Gas Turbine Power Plaot

The density of NO, emissions is also modeled using the data of gas turbine power plants (Vachtsevanos et al., 1995). Till now, almost NO, emission process use mathematical model in order to obtain regulation data from control process. However, a mathematical model

( c )

FIGURE 4 FPNN model structures for turbine power plants in the (b) bi-quadratic and (c) tri-linear polynomials.

-

cases of linear

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FUZZY PNN ALGORITHM 475

does not design the relationships between variables of the NO, emis- sion process and parameters of its model, accurately and effectively. The accurate modeling of the NO, emission process should be re- quired vigorously. The accurate model can provide control informa- tion for the control operators so that they should treat and predict NO, emission efficiently.

A NO, emission process of GE gas turbine power plant in Virginia, U.S.A., is chosen as a model. The modeling by fuzzy PNN is done in the NO, emission process, using the measured real data of gas turbine power plant.

We really measured input variables such as AT (Ambient Temper- ature at site), CS (Compressor Speed), LPTS (Low Pressure Turbine Speed) and CDP (Compressor Discharge Pressure) and TET (Turbine Exhaust Temperature), and output variable such as NO,. Gaussian is used as the premise membership function of the FPNN.

In this process, we use all the inputs. The FPNN's structures of modeling for NO, emission process are shown in a similar way of Fig. 4, in the cases of linear, biquadratic and trilinear polynomials, respectively. For each FPNN structure, the PIS are calculated under the change of the number of membership functions, in a similar way of Table 11.

- : Real Data : Mod$ a t p u t

FIGURE 5 Comparison of identification error with conventional modeling methods in turbine power plants.

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TAECHON AHN AND SANGMOON RYU

TABLE 111 Comparison of identification error with conventional modeling methods in turbine power plants

Model PI

Fuzzy model (Sugeno et al . , 1991) 2.2172 GE model (Vachtsevanos er a / . . 1995) 0.0439 GT model (Vachtsevanos er a / . , 1995) 0.0402 FNN (Ahn, 1997) 0.0520 PNN (our model) 0.2009 FPNN (our model) 0.0002

The optimal output of identified model using the fuzzy PNN is compared with real measured data in Fig. 5. The identification error of the fuzzy PNN is also compared with other fuzzy modeling methods in view of PI, in Table 111.

5 CONCLUSIONS

In this paper, the fuzzy PNN was proposed to combine conventional PNN to fuzzy inference, in order to model the nonlinear system with small data sets, in 3-layer structure. Both data for gas furnace of time series and data for gas turbine power plant of NO, emission process are used for the purpose of evaluating the' performance of the pro- posed modeling method.

Some results are drawn from computer simulation as follows:

(a) The fuzzy PNN gives a more sophisticated model and a more accurate prediction than the PNN or fuzzy modeling methods, from a very limited data set and/or a system with intense non- linearity;

(b) When the number of membership function is properly chosen, fuzzy PNN has more excellent results than the PNN;

(c) The fuzzy PNN with regression polynomial inference can obtain more accurate results in performance than fuzzy PNN with sim- plified inference;

(d) The fuzzy PNN can easily give author information that determines the number of layers on the basis of some prescribed small quantity; and

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FUZZY P N N ALGORITHM 477

(e) The fuuzzy P W can contribute to develop the modeling of emis- sion pattern for air pollutant in power plants and the high value- added products like the simulator and modal-based controller.

In future, fuzzy P N N with fuzzy c-means clustering will be studied, in case that the number of input variables goes up drastically.

Acknowledgment

This paper was supported by Fund of Korea Electric Company (19961, under the Supervision of the Electrical Engineering & Science Research Instituze.

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