An IV-QR Algorithm for Neuro-Fuzzy Multivariable ... - IEEE Xplore

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An IV-QR Algorithm for Neuro-Fuzzy Multivariable Online Identification Ginalber Serra and Celso Bottura, Member, IEEE

Abstract—In this paper, a new algorithm for neuro-fuzzy identification of multivariable discrete-time nonlinear dynamic systems, more specifically applied to consequent parameters estimation of the neuro-fuzzy inference system, is proposed based on a decomposed form as a set of coupled multiple input and single output (MISO) Takagi-Sugeno (TS) neuro-fuzzy networks. An on-line scheme is formulated for modeling a nonlinear autoregressive with exogenous input (NARX) recurrent neuro-fuzzy structure from input-output samples of a multivariable nonlinear dynamic system in a noisy environment. The adaptive weighted instrumental variable (WIV) algorithm by QR factorization based on the numerically robust orthogonal Householder transformation is developed to modify the consequent parameters of the TS multivariable neuro-fuzzy network. Index Terms—Fuzzy systems, instrumental variable, multivariable identification, neural networks.

I. INTRODUCTION

I

NTELLIGENT identification is a significant approach for modeling complex, uncertain and highly nonlinear dynamic systems. The models to be identified are defined on multidimensional spaces and represent input-output mappings which may be finite, infinite, discrete, or continuous. These models are also used for simulations, predictions, analysis of the system’s behavior, design of controllers (model based control), and so forth. The research on fuzzy systems had been developed in two main directions. The first direction is the linguistic or qualitative approach, in which the fuzzy system is constructed from a collection of rules (proposals). The second direction is a quantitative approach and is related to classic and modern systems theory. The combination of qualitative and quantitative informations, which constitutes the main motivation for the use of intelligent systems, gave origin to interesting contributions on identification of systems. In [1], a new method to generate a multivariable fuzzy inference system is presented. In that approach, a given sample set is decomposed into a sample sets cluster, from which results fuzzy rules as well as membership functions for each input variable, independently of the other variables, by solving a single input multiple output fuzzy system extracted from the set cluster. The computational complexity of the multivariable system, as shown, can be reduced to a single variable by using a parallel processing multi-CPU system. In Manuscript received December 19, 2004; revised September 1, 2005 and December 25, 2005. This work was supported by CAPES and CNPq. The authors are with the Department of Machines, Components and Intelligent Systems, State University of Campinas, Campinas 13083-852, Brazil (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TFUZZ.2006.879997

[2], that approach is extended to generate a more stable fuzzy system by using a Fourier method, covering a wider range of more robust applications. For nonlinear dynamic systems identification purposes, Takagi-Sugeno (TS) neuro-fuzzy networks are also widely investigated, since they provide good interpolation and generalization characteristics [3]–[5]. Identifying TS neuro-fuzzy networks consists in two parts: structure modeling (determining the number of rules and the input variables involved), and parameters optimization (optimizing the consequent parameters). To be applicable to real world problems, the noise treatment in the consequent parameter optimization must be highly efficient. In [6], an adaptive noise cancellation algorithm using a dynamic neuro-fuzzy network structure is proposed, where the number of fuzzy rules from the radial basis function (RBF) neurons and the input-output space clustering are adaptively determined so that rapid adaptation is attained. In [7], the fuzzy model identification from noisy data set problem is addressed. The classical results within the Frisch scheme [8] are extended to piecewise linear fuzzy modeling and used in a series of guidelines to achieve noise rejection when identifying fuzzy models from data. In [9], a new and interesting approach to building Sugeno models is proposed, where the premise of the model is determined by a fuzzy discretization technique and an orthogonal estimator is provided to identify the consequence of the model. However, that approach is not used in a noisy environment, which for such it would have the necessity of the noise model. In [10], the effect of the white noise in systems identification using the Mamdani fuzzy and neuro-fuzzy system are investigated. Conclusions about the choice of the antecedent and consequent membership functions used in the attenuation of the noise on the experimental data for better representation of the input-output nominal relation are obtained. However, the effect of noise on the experimental data for nonlinear dynamic discrete time systems identification and the implementation of enough robust algorithms to estimate the consequent parameters of the T-S neuro-fuzzy inference system are still open problems. Our paper is inserted in this context. For a linear model identification view point, for low levels of noise the least squares (LS) method may produce excellent estimates of the model parameters. For larger levels of noise, methods such as generalized least squares (GLS) [11], extended least squares (ELS) [12] and prediction error (PE) [13], [14] can be used, although they are inevitably dependent upon the accuracy of the noise model. To obtain consistent parameter estimates without modeling the noise, the instrumental variable (IV) method can be used. By choosing proper instrument variables it provides a way to obtain consistent estimates with certain optimal properties [13], [15], [16]. In a nonlinear model identification view point, using TS neuro-fuzzy inference system, all these methods

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could be applied to estimate the consequent parameters of the inference system in a noisy environment. In particular, as a simple example, when applying GLS, ELS or PE, the th rule of the neuro-fuzzy network, which includes the past values of the prediction error in its formulation, would be

(1) where , are parameters of the deterministic model and are parameters of the noise model in the consequent inference system, to be determined. In this context, methods such as GLS, ELS or PE would increase the complexity of the TS neuro-fuzzy network in two ways: 1) computational complexity, due to the parameters of the noise model that would be estimated and 2) structural complexity, because the noise would be a new linguistic input variable. For the structural complexity the number of rules increases exponentially, contributing to the well known problem of curse of dimensionality (rules explosion). Furthermore, this structure should not be used for clustering based identification, as the regression vector, including the prediction error past values, cannot be directly constructed from the data [4]. To avoid this complexity problem and to obtain good estimates of the consequent parameters of the TS neuro-fuzzy network in a noisy environment, the instrumental variable method can be used. In this paper a new algorithm for TS neuro-fuzzy network identification of multivariable discrete-time nonlinear dynamic systems in a noisy environment, more specifically applied to consequent parameters estimation of the TS neuro-fuzzy inference system, is proposed. From the input and output data, fuzzy regions are adequately defined by a clustering method, where each region can be regarded as a local linear aproximation of the nonlinear multivariable discrete-time nonlinear dynamic system, and a submodel is obtained for each fuzzy region by the proposed algorithm. The paper is organized as follows. In Section II, the T-S neuro-fuzzy network is used to represent the multivariable nonlinear discrete-time plant. The TS neuro-fuzzy structure as well as the consequent parameters estimate problem in a noisy environment are presented. Section III addresses the derivation procedures of the proposed instrumental variable identification algorithm. Some convergence conditions for identification in a noisy environment are introduced and the adopted on-line scheme for consequent parameters optmization by QR factorization based on the numerically robust orthogonal Householder transformation are analysed. A comparative analysis is established in order to substantiate as well as highlight the comments and advantages of the proposed algorithm. Section IV presents simulation results for the neuro-fuzzy on-line identification of a multivariable nonlinear discrete-time plant. Finally, brief concluding remarks and future works suggestions are presented in Section V.

Fig. 1. General configuration of the T-S neuro-fuzzy network.

II. T–S NEURO-FUZZY NETWORK The neuro-fuzzy network is based on T-S fuzzy model and membership functions of gaussian type, so that the algorithm can utilize all information contained in the training data set to calculate the consequent parameters of the network. The TS neuro-fuzzy network is composed of a fuzzy THEN rule base that IF partitions the input space, so-called universe of discourse, into fuzzy regions described by the rule antecedents in which consequent functions are valid [3]. The consequent of each rule is usually a simple functional expression of the antecedent th TS rule, where is the linguistic variables. The number of rules, has the following form:

(2) The vector contains the antecedent linguistic variables, which has its own universe of discourse partitioned by fuzzy sets representing the corresponding belinguistic terms. The linguistic variable with a truth value delongs to the fuzzy set fined by a membership function , with , where is the number of partitions of the universe of discourse for the linguistic . The truth value for the complete rule , variable using an aggregation operator, or -norm, AND, denoted by , is given by (3) where is some point in , and is the truth value of associated to a fuzzy set partitioning . Among the different -norms available, in this paper the algebraic product will be used, and (4)

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The normalized truth value for the rule is defined by

is a straightforward extension of (6) and is given by

(5)

The response of the TS neuro-fuzzy network consists in a weighted sum of the consequent functions, i.e., a convex combination of the local functions (models)

(9)

or (10) with (11)

(6) The general configuration of the TS neuro-fuzzy network is shown in Fig. 1, where the layers I and II are the antecedent in (2) and the layers III and IV are (3), (5), and (6), respectively. Such model can be seen as a linear parameter varying (LPV) system [15]. In this sense, a TS neuro-fuzzy network can be considered as a mapping from the antecedent (input) space to a convex region (politope) in the space of the local submodels defined by the consequent parameters. This property simplifies the analysis of TS neuro-fuzzy networks in a robust linear system framework for identification, controllers design with desired closed loop characteristics and stability analysis [17], [18].

and is given as (3). This NARX network represents multiple input and single output (MISO) systems directly and multiple input and multiple output (MIMO) systems in a decomposed form as a set of coupled MISO neuro-fuzzy networks. B. Consequent Parameters Estimation Problem The inference formula of the TS neuro-fuzzy network in (10) can be expressed as

A. Neuro-Fuzzy Structure Model The nonlinear autoregressive with exogenous input (NARX) structure model, which can represent the observable and controllable modes of a large class of discrete-time nonlinear systems, will be used in this paper. This structure is used in most nonlinear identification methods such as neural networks, radial basis functions, cerebellar model articulation controller (CMAC), and also fuzzy models [4], [19]–[21]. The NARX structure establishes a relation between the collection of past input-output data and the predicted output

(7) where denotes discrete time samples, is the time delay of the and are integers related to the nonlinear dynamic system, dynamic systems’order. In terms of rules, the structure is given by

(12) which is linear in the consequent parameters: , input-output data pairs a set of available, the following vetorial form is obtained

(13) where

,

, , and are the normalized membership degree matrix of (5), the data matrix, the output vector, the approximation error vector and the parameters vector, respectively. As initial , and conditions for this structure, we have: . A number of different techniques can be used when the variables associated with the unknown parameters are exactly known quantities. In practice, and in the present context, the basic relationship between the parameters is still in the form are no longer exactly known in (13), but the elements of quantities and can only be observed with error so that the observed value can be expressed as (14)

(8) where, at the where , and are the consequent parameters to be determined. The inference formula of the TS neuro-fuzzy network

and . For

th sampling, is the data vector with noise, is the

SERRA AND BOTTURA: AN IV-QR ALGORITHM FOR NEURO-FUZZY MULTIVARIABLE ONLINE IDENTIFICATION

data vector with exactly known quantities, e.g., free noise is a noise vector associated with the input-output data, is a disturbance noise. observation of , and Assume and statistically independent. The normal equations are formulated as

of

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Applying Slutsky’s theorem and assuming that the elements and converge in probability, we have (20)

Thus (15) and multiplying by

gives

(16) Noting that

(21) with . Hence, the asymptotic analysis of the consequent parameters estimation of TS neuro-fuzzy network is based in a weighted sum of the covariance matrices of and . Similarly

(22) Combining (22) and (21) with (20), results (17) and

(23)

(18) where is the parameters error, is the parameters vector estimation. Taking the probability in the limit as

(19) with

with . Provided that the input continues to excite the process and, at the same time, the coefficients in the submodels from the consequent are not all zero, then the output will exist for all observation intervals. As a result, the cowill also be nonsingular and its inverse variance matrix will exist. Thus, the only way in which the asymptotic error can identically zero at the limit. But, in general, be zero is for and are correlated and the asymptotic error will not be zero and the least squares estimates will be asymptotically biased to an extent determined by the relative ratio of noise to signal variances [15]. III. INSTRUMENTAL VARIABLE QR IDENTIFICATION ALGORITHM To overcome this bias error and inconsistence problem, generating a vector of variables, the instrumental variables vector, which is independent of the noise inputs and correlated with from the system is required. If this is possible, data vetor

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then it will be easy to see that the choice of this vector becomes effective to remove the asymptotic bias from the consequent parameters estimates. The least squares estimation results according to

Proof: Developing the left side of (26), results

Because the chosen variables are independent of the disturbance and will be zero in the limit. noise, the product between Hence

Using the new variables vector of the form the last equation can be placed as

, 3) Condition 3:

(27) Proof:

(24) where is a vector with the order of , associated to the dyis the normalized degree namic system behavior, and of activation, as in (5), associated to . As a consequence, three conditions must be satisfyied. 1) Condition 1:

(25) Proof: Developing the vector product at the left side of (25), results

As is a scalar, and, by definition, the chosen variables are independent of the noise inputs, the inner product between and will be zero. Thus, taking the limit, results

According to (25), this expression can be simplifyied to the form

Due to correlation between has the following property:

2) Condition 2:

(26)

and

, this covariance matrix

(28)

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and

Now, (24) can be expressed in the form

Fig. 2. Performance analysis block diagram.

Therefore, provided is non-singular, the limit value of the parameter error, in probability, is

is the extended instrumental variwhere able matrix with rows given by , is the and is extended data matrix with rows given by the output vector. The model can be obtained by the following two approaches: • Global approach: In this approach all linear consequent parameters are estimated simultaneously, minimizing the criterion:

(29) and the estimates are asymptotically unbiased, as required. An effective choice of the vector would be the one based on the delayed input-output sequence

where is the applied delay. Other methods of fuzzy instrumental variables can be seen in [22]. In a structural model situation, the IV normal equations must be formulated as

(30) or, with

,

(34) • Local approach: In this approach the consequent parameters are estimated for each rule , and, hence, independently of each other, minimizing a weighted local criteria : set

(35) has rows given by and is the normalized where membership degree diagonal matrix according to . The motivation for this work is the development of an algorithm that provides frequent estimates of the parameters by properly processing the I/O data on-line and adapts itself to variations on the parameters with time. The interest problem may be placed as

(36) (31)

Equation (36) can be written as

so that the IV estimate is obtained as

(37) where

, with . The scalar , called forgetting factor, is used to place less weight on past data. Developing both sides in (37), results

(38) (32) for

and, in vectorial form, the interest problem may be placed as

(33)

and . It is worth emphasizing that the resulting order of the matrix and of the vector are lower than the matrix and vector orders, respectively, is equal to the number of parameters because

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to be estimated, implying less computational effort and, in consequence, greater speed to solve for . Generically, for and results: .. .

.. .

.. .

(39)

(40)

.. . Letting

and .. .

, results .. .

and (48) where is an orthogonal matrix, the upper triangular matrix is given by and is a resulting vector. Hence, the minimizer of (46) may be found by back substitution. For automatic initial by solving estimation, the algorithm receives an initial batch data and updating is obtained by simple acquisition of input and output data and insertion of it into summations of the matrix and of the vector . Thus, for the th sample .. .

(41)

.. .

(49)

and .. .

(42)

where . From (41) and (42), it can be seen that the matrix and the vector are summations that depend of the actual and immediately former values, based on the dimension of the problem, the input and output measurements. These structures imply in generating, directly, i.e., in each sample, and , without need of a priori batch of matricial operations, as in (37), with the advantage that the order is lower to application of the QR factorization. An orthogonal Householder matrix has the following form:

(43)

.. .

(50)

The proposed algorithm may be summarized as follows: general methodology of algorithm 1) Define a number of input/output measures to automatically do the initial estimation; 2) Generate the

matrix and the

vector from (39)–(40);

3) Apply QR factorization by Householder orthogonal transformation to generate (48); 4) Solve (48) by back substitution ;

where and [23], [24]. Householder transformations are often used to annul a block of elements in matrices or vectors by appropriately selecting the Householder vector in (43). If is a nonzero vector and is an unit vector with 1 in the th position, when (44) then (45) The vectors and are identifiable except for the th element. Thus, the problem may be placed as that of finding the solution (46) Applying the QR factorization with the Householder orthogonal transformation results (47)

5) Obtain the new input/output measures ; 6) Generate the new (50);

matrix and the

vector from (49) and

7) Go to 3. Once the proposed algorithm is already formulated, some simulations will be presented, without less of generality, in order to illustrate the previous comments and its advantages. The plant to be identified consists on a second order highly nonlinear discrete-time system

(51) which is a benchmark problem in neural and fuzzy modeling is the output and is the [5], [21], where is a white noise with zero mean applied input. In this case and and variance . The TS model has two inputs and , and the antecedent part of the fuzzy model a single output (the fuzzy sets) is designed based on the evolving clustering

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method (ECM) proposed in [25]. The model is composed of rules of the form

(52) are gaussian fuzzy sets. The linguistic input variable where is omitted in the antecedent, because, in this case, it is used all the universe of discourse with truth value equal to 1. The use of this approach is efficient in cases where the dinamic system is linear in the input . Experimental data sets of points each are created from . This means that the noise applied (51), with of the output nominal value, take values between 0 and which is an acceptable practical noise percentage. These data sets are presented to the proposed algorithm, for obtaining a IV neuro-fuzzy model (model IV), and to the LS based algorithm, for obtaining a LS neuro-fuzzy model (model LS). The models are obtained by the global and local approaches. The noise influence is analized according to the difference between the outputs of the neuro-fuzzy models, obtained from the noisy experimental data, and the plant output without noise. The antecedent parameters and the neuro-fuzzy models structure are the same in the experiments, while the consequent parameters are obtained by the proposed method and by the LS method. Thus, due to the obtained results, these algorithms and accuracy conclusions will be derived about the proposed algorithm performance in the presence of noise. Two criteria, widely used in experimental data analysis, are applied to avaliate the obtained neuro-fuzzy models fit: variance accounted for (VAF)

(53) where is the nominal plant output, is the neuro-fuzzy model output and means signal variance, and mean square error (mse) (54) is the nominal plant output, is the neuro-fuzzy where is the number of points. Once obtained model output and these values, a comparative analysis will be established between the proposed algorithm, based on IV, and the algorithm based on LS according to the approaches presented above. Fig. 2 shows a representative block diagram for this performance analysis. Fig. 3 shows the performance of the TS models obtained off-line according to (34) and (35). For these results, the number of points is 500, the proposed algorithm used equal to 0.99; the number of rules is 4, the structure is the presented in (52) and the antecedent parameters are obtained by the ECM method for both algorithms. From Fig. 3 it can be clearly seen that the proposed algorithm performs better than the LS algorithm for the two approaches as it is more robust to noise. This is due , to to the chosen instrumental variable matrix, with satisfy the convergence conditions as well as possible. In the global approach, for low noise variance, both algorithms presented similar performance with VAF and mse of 99.50% and

Fig. 3. Comparative analysis.

0.0071 for the proposed algorithm and of 99.56% and 0.0027 for the LS based algorithm, respectively. However, when the noise variance increases, the chosen instrumental variable matrix satisfy the convergence conditions, and as a consequence the proposed algorithm becomes more robust to the noise with VAF and mse of 98.81% and 0.0375. On the other hand, the LS-based algorithm presented VAF and mse of 82.61% and 0.4847, respectively, that represents a poor performance. Similar analysis can be applied to the local approach: increasing the noise variance, both algorithms present good performances where the VAF and mse values increase too. This is due to the polytope property (see Section II), where the obtained models can represent local approximations giving more flexibility curves fitting. The proposed algorithm presented VAF and mse values of 93.70% and 0.1701 for the worst case and of 96.3% and 0.0962 for the better case. The LS based algorithm presented VAF and mse values of 92.4% and 0.2042 for the worst case and of 95.5% and 0.1157 for the better case. The worst case of noisy data set was still used by the algorithm proposed in [9] based on the structure presented in (52), where the VAF and MSE values were 92.6452% and 0.1913, respectively. These results, considering the local approach, show that it has an intermediate performance between the proposed method in this paper and the LS-based algorithm. In general, we conclude that the proposed method has favorable results compared with existing techniques. To ilustrate the obtained neuro-fuzzy models , quality, Fig. 4 shows part of the noisy output data for as well as the neuro-fuzzy model outputs for both algorithms and the noise free plant output for both approaches objectiving model validation. Clearly, it can be observed the output estimates accuracy for the proposed method for both approaches. For the global approach, the VAF and MSE values are 96.5% and 0.09 for the proposed method and of 81.4% and 0.52 for the LS-based algorithm, respectively. For the local approach, the VAF and mse values are 96.0% and 0.109 for the proposed method and of 95.5% and 0.1187 for the LS based algorithm, respectively.

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Fig. 4. Models validation. Fig. 5. Multivariable recurrent neuro-fuzzy network.

IV. NEURO-FUZZY MIMO IDENTIFICATION The proposed algorithm is applied to a MIMO nonlinear plant, with two inputs and two outputs, widely used as benchmark example in system identification using neural networks and fuzzy systems [21]

(55) In this case, the TS multivariable neuro-fuzzy network is composed by two coupled TS MISO recurrent neuro-fuzzy networks , and and a single output of three inputs , as shown in Fig. 5, where and are the input and output of each TS MISO recurrent neuro-fuzzy network, respectively. This neuro-fuzzy network structure presents conections or feedback of information and it is able to represent complex and nonlinear dynamic behavior with a few number of parameters. The rule base, for each TS MISO recurrent neurofuzzy network, is of the form

(56) are Gaussian fuzzy sets. As in the previous where are omitted in case, the linguistic input variables the antecedent, considering all the universe of discourse equal to 1. Thus, the multivariable recurrent neuro-fuzzy mapping is of the form

(57) represents the th MISO recurrent neuro-fuzzy netwhere work. An experimental data set of 300 points is created from and uniformly distributed in (55), with random inputs , with meaning that the noise level the interval

Fig. 6. Membership function in the layer II of the multivariable recurrent neuro-fuzzy network.

applied to outputs takes values between 0 and 15% from its nominal values, which is an acceptable practical percentage of noise. This data set is presented to the proposed algorithm based on the local approach: The consequent parameters are estimated for each rule , independently of each other, minimizing a set of weighted local criteria

(58) where and . In this application, the number of rules is 4 for each MISO recurrent neuro-fuzzy network, the antecedent parameters are obtained by the ECM method, 30 points are used for the initial and . estimate, Fig. 6 shows the linguistic variables partitions obtained by the ECM method. These partitions correspond to the membership functions of the layer II in the neuro-fuzzy network. Fig. 7

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Fig. 7. Outputs tracking by the multivariable recurrent neuro-fuzzy network. Fig. 8. Validation of the obtained multivariable recurrent neuro-fuzzy network.

show the tracking of the nonlinear plant outputs in a noisy environment obtained on-line by the local approach. Fig. 8 shows the performance of the TS recurrent neuro-fuzzy network during the validation procedure, which is based on the assumption that the TS model is a lattice recurrent one, as shown in Fig. 5, using a new experimental data set, so that the one step ahead predictions are made based on the past values of the predicted outputs. For (55), the local approach obtained TS multivariable recurrent neuro-fuzzy network rules base is

Fig. 9. Parameters estimates of the multivariable recurrent neuro-fuzzy network.

The recursive parameters estimate results are shown in Fig. 9. It can be seen that the algorithm is sensitive to the multivariable nonlinear plant behavior, the parameters estimates are consistent and converge rapidly. This is particularly so for the parameters which show that the plant is linear for the inputs. V. CONCLUSION

As expected, the proposed method provides unbiased and suficiently accurate estimates of the consequent parameters and, as a consequence, high speed of convergence of the multivariable recurrent neuro-fuzzy network to the nonlinear plant behavior in a noisy environment. These characteristics are very important in multivariable adaptive control design applications.

An algorithm for neuro-fuzzy identification of multivariable nonlinear discrete-time dynamic systems in a noisy environment was proposed. The neuro-fuzzy network structure as well as a LPV systems view point interpretation were presented. Convergence conditions for identification in a noisy environment were introduced. Based on the TS neuro-fuzzy network, an on-line scheme was developed to form a NARX recurrent neuro-fuzzy structure from input-output samples of a multivariable nonlinear

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dynamic system where the consequent parameters were modified by an adaptive WIV algorithm based on the numerically robust orthogonal Householder transformation. Simulation results have shown that with the proposed methodology a multivariable TS recurrent neuro-fuzzy network rules base for a multivariable nonlinear plant was efficiently obtained in a noisy environment. For future work, the application of the proposed algorithm in a multivariable indirect adaptive control design should be considered. REFERENCES [1] L. Chen, N. Tukuda, X. Zhang, and Y. He, “A new scheme for an automatic generation of multi-variable fuzzy systems,” Fuzzy Sets Syst., vol. 120, pp. 323–329, 2001. [2] L. Chen and N. Tukuda, “An efficient algorithm for automatically generating multivariable fuzzy systems by Fourier series method,” IEEE Trans. Syst., Man, Cybern., vol. 32, no. 5, pp. 622–629, Oct. 2002. [3] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its aplication to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. 15, pp. 116–132, Jan. 1985. [4] H. Hellendoorn and D. Driankov, Fuzzy Model Identification: Selected Approaches. New York: Springer-Verlag, 1997. [5] S. E. Papadakis and J. B. Theocaris, “A GA-based fuzzy modeling approach for generating TSK models,” Fuzzy Sets Syst., vol. 131, pp. 121–152, 2002. [6] M. J. Er, Z. Li, and Q. Chen, “Adaptive noise cancellation using enhanced dynamic fuzzy neural networks,” IEEE Trans. Fuzzy Syst., vol. 13, pp. 331–342, Jun. 2005. [7] S. Simani, C. Fantuzzi, R. Rovatti, and S. Beghelli, “Noise rejection in parameters identification for piecewise linear fuzzy models,” in Proc. IEEE Conf. Fuzzy Syst., 1998, vol. 1, pp. 378–383. [8] R. Frich, Statistical Confluence Analysis by Means of Complete Regression Systems. Oslo, Sweden: Univ. Oslo, Economic Inst., 1934, vol. 5. [9] L. Wang and R. Langari, “Building Sugeno type models using fuzzy discretization and orthogonal parameter estimation techniques,” IEEE Trans. Fuzzy Syst., vol. 3, pp. 454–458, Nov. 1995. [10] P. J. C. Branco and J. A. Dente, “Noise effects in fuzzy modeling systems: Three case studies,” in Computational Intelligence and Applications. Danvers, MA: World Scientific and Engineering Society Press, 1999, pp. 103–108. [11] T. Söderström and P. G. Stoica, System Identification. Englewood Cliffs, NJ: Prentice-Hall, 1989. [12] L. Ljung and T. Söderström, Theory and Practice of Recursive Identification. Cambridge: MIT Press, 1987. [13] L. Ljung, System Identification: Theory for the User, 2 ed. Englewood Cliffs, NJ: Prentice-Hall, 1999. [14] P. C. Young, Recursive Estimation and Time-Series Analysis: An Introduction. New York: Springer-Verlag, 1984. [15] C. P. Bottura and G. L. O. Serra, “An algorithm for fuzzy identification of nonlinear discrete-time systems,” in Proc. 43rd IEEE Conf. Decision Contr., Dec. 2004, pp. 5421–5426. [16] T. Söderström and P. G. Stoica, “Instrumental variable methods for system identification,” in Lecture Notes in Control and Information Sciences. New York: Springer-Verlag, 1983. [17] R. E. King, Computational Intelligence in Control Engineering. New York: Marcel Dekker, 1999. [18] C. P. Bottura and G. L. O. Serra, “Neural gain scheduling multiobjective genetic fuzzy PI control,” in Proc. IEEE Int. Symp. Intell. Contr., Taipei, Taiwan, Sep. 2004, pp. 483–488.

[19] J. Sjöberg, Q. Zhang, L. Ljung, A. Benveniste, B. Delyon, P. Glorennec, H. Hjalmarsson, and A. Juditsky, “Nonlinear black-box modeling in system identification: A unified overview,” Automatica, vol. 31, pp. 1691–1724, 1995. [20] M. Brown and C. Harris, Neurofuzzy Adaptive Modelling and Control. Englewood Cliffs, NJ: Prentice-Hall, 1994. [21] K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Trans. Neural Netw., vol. 1, pp. 4–27, Mar. 1990. [22] C. P. Bottura and G. L. O. Serra, “Fuzzy instrumental variable concept and identification algorithm,” in Proc. IEEE Conf. Fuzzy Syst., 2005, pp. 1062–1067. [23] R. V. Patel, A. J. Laub, and P. M. V. Dooren, Numerical Linear Algebra Techniques for Systems and Control. New York: IEEE, 1994. [24] G. H. Golub and C. R. Van Loan, Matrix Computations, 3 ed. London, U.K.: The John Hopkins Univ. Press, 1996. [25] N. K. Kasabov and Q. Song, “DENFIS: Dynamic evolving neuralfuzzy inference system and its application for time-series prediction,” IEEE Trans. Fuzzy Syst., vol. 10, pp. 144–154, Apr. 2002. Ginalber Serra was born in São Luis, Brazil, in 1976. He received the B.Sc. and M. Sc. degrees in electrical engineering from Federal University of Maranhão, Maranhão, Brazil, in 1999 and 2001, respectively. He received the Ph.D. degree in electrical engineering from State University of Campinas (UNICAMP), Campinas, Brazil, in September 2005. He is currently working as a Postdoctoral Researcher working on multivariable neuro-fuzzy adaptive control of nonlinear systems with the Department of Machines, Components and Intelligent Systems, UNICAMP. His research interests include fuzzy systems, neural networks and genetic algorithms applied to identification and control of nonlinear systems. Dr. Serra has served as Reviewer for many prestigious international journals, including the IEEE TRANSACTIONS ON FUZZY SYSTEMS, IEEE TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS—PART B: CYBERNETICS, Institute of Electronics Engineers CONTROL THEORY AND APPLICATIONS, and for papers from IEEE Conferences.

Celso Bottura (M’74) was born in Catanduva, Brazil, in 1938. He received the B.Sc. degree in aeronautical engineering from ITA, Brazil, in 1962, the M.Sc. degree in mechanical engineering from Purdue University, West Lafayette, IN, in 1964, and the Ph.D. degree in electrical engineering from the State University of Campinas (UNICAMP), Campinas, Brazil, in 1973. From 1984 to 1985, he was a Fullbright Visiting Scholar with the Department of Economics, University of California, Los Angeles. He is currently a Full Professor and a Member of the Laboratory for Intelligent Systems and Control, School of Electrical and Computer Engineering, UNICAMP. He has already published three books in his area of interest, all of them in Portuguese. His research interests are in the areas of multivariable control, parallel and distributed computation, control of electrical machines and drives, control of systems with rational expectations, robust control, intelligent systems and control, modeling, estimation, and control. Dr. Bottura is a former President of Sociedade Brasileira de Automática (SBA)—the Brazilian Society of Automatic Control.