Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 8594738, 10 pages

https://doi.org/10.1155/2017/8594738

## Fuzzy Modeling for Uncertainty Nonlinear Systems with Fuzzy Equations

Departamento de Control Automatico, CINVESTAV-IPN (National Polytechnic Institute), Mexico City, Mexico

Correspondence should be addressed to Wen Yu; xm.vatsevnic.lrtc@wuy

Received 19 July 2016; Revised 26 October 2016; Accepted 21 December 2016; Published 22 January 2017

Academic Editor: Rosana Rodriguez-Lopez

Copyright © 2017 Raheleh Jafari and Wen Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The uncertain nonlinear systems can be modeled with fuzzy equations by incorporating the fuzzy set theory. In this paper, the fuzzy equations are applied as the models for the uncertain nonlinear systems. The nonlinear modeling process is to find the coefficients of the fuzzy equations. We use the neural networks to approximate the coefficients of the fuzzy equations. The approximation theory for crisp models is extended into the fuzzy equation model. The upper bounds of the modeling errors are estimated. Numerical experiments along with comparisons demonstrate the excellent behavior of the proposed method.

#### 1. Introduction

Since the uncertainties in the nonlinear systems can be transformed into fuzzy set theory [1], the fuzzy systems are good models for uncertainty systems. The fuzzy models are based on fuzzy rules. These fuzzy rules give information of the uncertain nonlinear systems. Any nonlinear system can be approximated by several piecewise linear systems (Takagi-Sugeno fuzzy model [2]) or known nonlinear systems (Mamdani fuzzy model [3]). The uncertain nonlinear systems can be modeled by the fuzzy models with simple linear or nonlinear models.

The nonlinear systems can be also modeled with difference equations and algebraic systems. Interpolation methodology has been broadly utilized for function approximation as well as system identification [4, 5]. In [6], the fuzzy polynomial interpolation is applied for system modeling. The coefficients of the polynomials are the fuzzy numbers [7], such that the uncertainties are interpolated with the fuzzy set theory. The theory problem associated with polynomial interpolation is discussed in [8]. It concludes that the interpolation of the function includes time complexity at data points.

In [9], two-dimensional polynomial interpolation is demonstrated. The constraint associated with multivariable interpolation has been investigated in [6], where the Newton-form interpolation is employed. In [10], the multivariate Vandermonde matrix is utilized. Smooth function approximation has been broadly implemented currently [11, 12]. It yields a model by utilizing Lagrange interpolating polynomials at the points of product grids [7, 13]. However if it involves uncertainties in the interpolation points, the above suggested techniques will not work appropriately.

The fuzzy equation can be regarded as a generalized form of the fuzzy polynomial. Compared with the normal fuzzy systems, the fuzzy equations are more easy to be applied, because the uncertainties are direct fuzzy parameters of the fuzzy equations. However, these parameters are not easy to be obtained. There are several approaches to construct the fuzzy equations. Reference [14] utilized the parametric form of fuzzy numbers and restored the original fuzzy equations using crisp linear systems. In [15], the extension principle is implemented and it suggests that the coefficients can be either real or complex fuzzy numbers. Whatsoever, the validation of the solution is not assured. Reference [16] prescribed the homeotypic analysis methodology. Reference [17] inducted Newton’s technique. In [18], the solution of fuzzy equations is extracted using the fixed point methodology. One of the well-known methods is termed as -level [19]. By using the method of superimposition of sets, fuzzy numbers can be solved. Recently, fuzzy fractional differential and integral equations have been extensively studied in [20–22]. However, the above methods are very complex.

The numerical solution associated with fuzzy equation can be fetched using the iterative technique [23], interpolation technique [24], and Runge-Kutta technique [25]. It can also be implemented to fuzzy differential equations [26]. These methods are also difficult to be applied. Both neural networks as well as fuzzy logic are considered to be the universal estimators which can estimate any nonlinear function to any notified precision [27]. Current outcomes demonstrate that the fusion methodology of these two different techniques appears to be highly efficient for nonlinear systems identification [28]. Neural networks can also be implemented for resolving the fuzzy equation. In [29], the simple fuzzy quadratic equation is resolved by the neural network method. References [30, 31] elaborated the outcomes of [29] into fuzzy polynomial equation. In [32, 33], the solution of dual fuzzy equations is obtained by neural networks. A matrix pattern associated with the neural learning has been quoted in [34]. However, these techniques are not general; they cannot give the fuzzy coefficients directly with neural networks [35, 36].

We use the neural network method to approximate the coefficients of the fuzzy equations as in our previous paper [37]. In this paper, the standard backpropagation method is modified, such that the fuzzy numbers in the fuzzy ideations can be trained. The approximation theory of the crisp models is extended into the fuzzy equations. The upper bounds of the modeling errors with fuzzy equations are estimated. Finally, we use some real examples to show the effectiveness of our approximation method.

#### 2. Nonlinear System Modeling with Fuzzy Equations

A general discrete-time nonlinear system can be described asHere we consider as the input vector, is regarded as an internal state vector, and is the output vector. and are noted as generalized nonlinear smooth functions . Define and Supposing is nonsingular at the instance , , this will create a path towards the following model:where is a nonlinear difference equation exhibiting the plant dynamics and and are computable scalar input and output, respectively. The nonlinear system which is represented by (2) is implied as a NARMA model. The input of the system with incorporated nonlinearity is considered to be

Taking into consideration the nonlinear systems as mentioned in (2), it can be simplified as the following linear-in-parameter model:where is considered to be the linear parameter and is nonlinear function. The variables related to this function are quantifying input and output.

Many nonlinear systems can be expressed by linear-in-parameter models such as Lagrangian mechanical systems. The parameters of these models are uncertain and the uncertainties satisfy the fuzzy set theory [1]. In this way, the inconvenience problems in nonlinear modeling such as complexity and uncertainty are solved by the fuzzy logic theory and linear-in-parameter structure. The modeling process with the fuzzy equation is to find the fuzzy coefficients of the linear-in-parameter model such that the fuzzy equation can represent the uncertain nonlinear system.

We assume that the model of the nonlinear system (4) has uncertainties in the parameter . The following definitions will be used in this paper.

*Definition 1 (fuzzy number). *A fuzzy number is a function ; in such a way, is normal, (there prevail in such a way that ); is convex, , , ; is upper semicontinuous on ; that is, , , , , is a neighborhood; the set is compact.

In order to demonstrate the fuzzy numbers, the membership functions are utilized. The most widely discussed membership functions are noted to be the triangular functionand trapezoidal function

On a par with crisp variable, the fuzzy variable possesses three essential operations: , , and . They signify these operations: sum, subtract, and multiply.

The fuzzy variable which contains the dimension of is dependent on the membership function, (5) has three variables, and (6) has four variables. In order to define consistency operations, we first apply -level operation to the fuzzy number.

*Definition 2 (-level). *The -level associated with a fuzzy number is stated aswhere , .

Therefore Since , is bounded mentioned as . The -level of between and is explained as

*Definition 3 (fuzzy operations). *If , , the fuzzy operations are as follows:

Sum: Subtract:Multiply:Scalar multiplication: .

*Definition 4 (absolute value). *Absolute value of a triangular fuzzy number is

Now we utilize the following fuzzy equation to model the uncertain nonlinear system (1):Because is fuzzy number, we apply the fuzzy operation .

Taking into consideration a particular case, has polynomial form,Equation (15) is termed as fuzzy polynomial.

Modeling with fuzzy equation (or fuzzy polynomial) can be regarded as fuzzy interpolation. In this paper, we utilize the fuzzy equation (14) to model the uncertain nonlinear system (1), in such a manner that the output related to the plant can approach the desired output ,

This modeling object can be regarded as a way to detect for the following fuzzy equation:where

#### 3. Fuzzy Parameter Estimation with Neural Networks

We design a neural network to represent the fuzzy equation (14); see Figure 1. The input to the neural network is and the output is the fuzzy number The weights are The main idea is to detect appropriate weight of neural network in such a manner that the output of the neural network converges to the desired output .