introduction to fuzzy partial differential equations this paper considers solutions to elementary fuzzy partial differential equations we follow the same strategy as in buckley and feuring fuzzy sets and systems to appear which is 1 first check to see if the buckley feuring method produces a solution and 2 if the buckley feuring method does not give a solution then see if the seikkala procedure generates a solution examples are presented showing the buckley feuring solution and the seikkala solution

fuzzy convex set based pattern classification for analysis of mammographic microcalcifications abstract there are many different criteria for the comparative analysis of pattern classifiers they include generalization ability computational complexity and understanding of the feature space in some applications such as the medical diagnostic systems it is crucial to use reliable tools whose behavior is always predictable so that the risk of misdiagnosis is minimized in such applications the use of the popular feedforward backpropagation bp neural network algorithm can be seen as questionable this is because it is not inherent for the backpropagation method to analyze the problem s feature space during training which can sometimes result in inadequate decision surfaces a novel convex set based neuro fuzzy algorithm for classification of difficult to diagnose instances of breast cancer is described in this paper with its structural approach to feature space the new method offers rational advantages over the backpropagation algorithm the classification performance computational and structural efficiencies are analyzed and compared with that of the bp network a 20 dimensional set of difficult to diagnose mammographic microcalcifications was used to evaluate the neuro fuzzy pattern classifier nfpc and the bp methods in order to evaluate the learning ability of both methods the relative size of training sets was varied from 40 to 90 the comparative results obtained using receiver operating characteristic roc analysis show that the ability of the convex set based method to infer knowledge was better than that of backpropagation in all of the tests performed making it more suitable for use in real diagnostic systems

training anfis as an identifier with intelligent hybrid stable learning algorithm based on particle swarm optimization and extended kalman filter this paper proposes a novel hybrid learning algorithm with stable learning laws for adaptive network based fuzzy inference system anfis as a system identifier the proposed hybrid learning algorithm is based on the particle swarm optimization pso for training the antecedent part and the extended kalman filter ekf for training the conclusion part lyapunov stability theory is used to study the stability of the proposed algorithm comparison results of the proposed approach pso algorithm for training the antecedent part and recursive least squares rlss or ekf algorithm for training the conclusion part with the other classical approaches such as gradient descent resilient propagation quick propagation levenberg marquardt for training the antecedent part and rlss algorithm for training the conclusion part are provided moreover it is shown that applying pso a powerful optimizer to optimally train the parameters of the membership function on the antecedent part of the fuzzy rules in anfis system is a stable approach which results in an identifier with the best trained model stability constraints are obtained and different simulation results are given to validate the results also the stability of levenberg marquardt algorithms for anfis training is analyzed

a differential quadrature based approach a la picard for systems of partial differential equations associated with fuzzy differential equations departing from a numerical method designed to solve ordinary differential equations in this manuscript we extend such approach to solve problems involving fuzzy partial differential equations the method proposed in this work is a non recursive technique that combines differential quadrature rules and a picard like scheme in order to obtain general solutions of systems of partial differential equations derived from a general fuzzy partial differential model the property of stability and a bound for the hausdorff distance are established under suitable conditions several numerical examples are provided in order to show the effectiveness of the proposed technique

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

constructing narmax models using armax models this paper outlines how it is possible to decompose a complex non linear modelling problem into a set of simpler linear modelling problems local armax models valid within certain operating regimes are interpolated to construct a global narmax non linear narmax model knowledge of the system behaviour in terms of operating regimes is the primary basis for building such models hence it should not be considered as a pure black box approach but as an approach that utilizes a limited amount of a priori system knowledge it is shown that a large class of non linear systems can be modelled in this way and indicated how to decompose the systems range of operation into operating regimes standard system identification algorithms can be used to identify the narmax model and several aspects of the system identification problem are discussed and illustrated by a simulation example

on fuzzy solutions for partial differential equations in this study we investigate heat wave and poisson equations as classical models of partial differential equations pdes with uncertain parameters considering the parameters as fuzzy numbers the fuzzy solution is built from fuzzification of the deterministic solution the continuity of the zadeh extension is used to obtain qualitative properties on regular a cuts of the fuzzy solution we prove the stability with respect to the initial boundary data and show that as time goes to zero the diameter of the fuzzy solution converges to zero and as a consequence to the cylindrical surface determined by the curve of the degree of membership numerical simulations are used to obtain a graphical representation of the fuzzy solution and a defuzzification of this solution is obtained using the center of gravity method we theoretically show that the surface obtained by defuzzification with the plane determined by fixing time is indeed the solution of the same initial boundary problem for this time point for the heat and poisson equations and in a particular case for the wave equation the deterministic solution and the defuzzified surface intercept are numerically compared using the euclidean distance

numerically solution of fuzzy differential equations by adomian method in this paper a method for numerical approximation of fuzzy first order initial value problem is presented buckly and feruing fuzzy sets syst 110 2000 43 have solved fuzzy first order initial value problem they solved problem in crisp case and then extend the solution to fuzzy case in this paper we use adomian method to find the approximate solution in crisp case and extend it to fuzzy case using this method we solve problems that the classical methods could not be applied for them in crisp case

universal approximation using radial basis function networks there have been several recent studies concerning feedforward networks and the problem of approximating arbitrary functionals of a finite number of real variables some of these studies deal with cases in which the hidden layer nonlinearity is not a sigmoid this was motivated by successful applications of feedforward networks with nonsigmoidal hidden layer units this paper reports on a related study of radial basis function rbf networks and it is proved that rbf networks having one hidden layer are capable of universal approximation here the emphasis is on the case of typical rbf networks and the results show that a certain class of rbf networks with the same smoothing factor in each kernel node is broad enough for universal approximation

a high order recurrent neuro fuzzy system with internal dynamics application to the adaptive noise cancellation a high order recurrent neuro fuzzy system ho rnfs is suggested in this paper suitable for modeling highly complex nonlinear temporal processes feedback connections are introduced in the network including the context and the feedback nodes that serve as a means to memorize the firing history the feedback paths in the firing loop are implemented through finite impulse response fir synaptic filters leading to a higher order network with enhanced temporal capabilities the inference mechanism of the ho rnfs is implemented by means of dynamic fuzzy rules where multiple steps ahead predictions are provided for the internal variables at the consequent part its structure is organized in an on line fashion using a concurrent structure and parameter algorithm structure learning generates dynamically the input and output clusters of the rules while parameter learning adjusts the network weights the ho rnfs is compared to the recurrent self organizing neural fuzzy inference network rsonfin being a special case of the suggested network the experimental setup includes a benchmark temporal system and the adaptive noise cancellation problem extensive experimentation reveals that ho rnfs exhibits superior speech enhancement performance as contrasted to rsonfin when complex noise passages are considered

probability and fuzziness we discuss some relationships between probability theory and statistics on one hand and the theory of fuzzy sets on the other hand we develop various statistical techniques for the analysis of imprecise data and for inference based on imprecise data we study the estimation of fuzzy parameters testing problems with fuzzy probabilities and confidence estimation under uncertainty finally we indicate some possible applications to the assessment of credibility in expert systems

the composition and validation of heterogeneous control laws we present a method for creating and validating a nonlinear controller by the composition of heterogeneous local control laws appropriate to different operating regions like fuzzy logic control these methods apply even in the presence of incomplete knowledge of the structure of the system the boundaries of the operating regions or even the control action to take unlike fuzzy logic control these methods can be analyzed by a combination of classical and qualitative methods each operating region of the system has a classical control law which provides high resolution control and can be analyzed by classical methods operating regions are defined by fuzzy set membership functions the global control law is the weighted average of the local control laws where the weights are provided by the operating region membership functions a heterogeneous control law can be analyzed even in the presence of incomplete knowledge by representing it as a qualitative differential equation and using qualitative simulation to predict the set of possible behaviors of the system by expressing the desired guarantee as a statement in a modal temporal logic the validity of the guarantee can be automatically checked against the set of possible behaviors we demonstrate heterogeneous controllers and our qualitative methods for proving their properties first for a simple level controller for a water tank and second for a highly nonlinear chemical reactor

recurrent neuro fuzzy networks for nonlinear process modeling a type of recurrent neuro fuzzy network is proposed in this paper to build long term prediction models for nonlinear processes the process operation is partitioned into several fuzzy operating regions within each region a local linear model is used to model the process the global model output is obtained through the centre of gravity defuzzification which is essentially the interpolation of local model outputs this modeling strategy utilizes both process knowledge and process input output data process knowledge is used to initially divide the process operation into several fuzzy operating regions and to set up the initial fuzzification layer weights process i o data are used to train the network network weights are such trained so that the long term prediction errors are minimized through training membership functions of fuzzy operating regions are refined and local models are learnt based on the recurrent neuro fuzzy network model a novel type of nonlinear model based long range predictive controller can be developed and it consists of several local linear model based predictive controllers local controllers are constructed based on the corresponding local linear models and their outputs are combined to form a global control action by using their membership functions this control strategy has the advantage that control actions can be calculated analytically avoiding the time consuming nonlinear programming procedures required in conventional nonlinear model based predictive control the techniques have been successfully applied to the modeling and control of a neutralization process

multiobjective design of water distribution systems under uncertainty 1 the water distribution system wds design problem is defined here as a multiobjective optimization problem under uncertainty the two objectives are 1 minimize the total wds design cost and 2 maximize wds robustness the wds robustness is defined as the probability of simultaneously satisfying minimum pressure head constraints at all nodes in the network decision variables are the alternative design options for each pipe in the network the sources of uncertainty are future water consumption and pipe roughness coefficients uncertain variables are modeled using probability density functions pdfs assigned in the problem formulation phase the corresponding pdfs of the analyzed nodal heads are calculated using the latin hypercube sampling technique the optimal design problem is solved using the newly developed rnsgaii method based on the nondominated sorting genetic algorithm ii nsgaii in rnsgaii a small number of samples are used for each fitness evaluation leading to significant computational savings when compared to the full sampling approach chromosome fitness is defined here in the same way as in the nsgaii optimization methodology the new methodology is tested on several cases all based on the new york tunnels reinforcement problem the results obtained demonstrate that the new methodology is capable of identifying robust pareto optimal solutions despite significantly reduced computational effort

hierarchical fuzzy cmac for nonlinear systems modeling since the fuzzy cerebellar model articulation controller fcmac uses linguistic variables it is highly intuitive and easily comprehended despite the fcmac s good local generalization capability for approximating nonlinear functions and fast learning a normal fcmac requires huge memory and its dimension increases exponentially with the number of inputs in order to overcome the memory explosion problem this paper proposes two types of hierarchical fcmac hfcmac another contribution of the paper is that we give stable learning algorithms for these two hfcmacs backpropagation like approach is applied to train each block with a time varying learning rate which is obtained by the input to state stability technique

fuzzy identification using fuzzy neural networks with stable learning algorithms in general fuzzy neural networks cannot match nonlinear systems exactly unmodeled dynamic leads parameters drift and even instability problem according to system identification theory robust modification terms must be included in order to guarantee lyapunov stability this paper suggests new learning laws for mamdani and takagi sugeno kang type fuzzy neural networks based on input to state stability approach the new learning schemes employ a time varying learning rate that is determined from input output data and model structure stable learning algorithms for the premise and the consequence parts of fuzzy rules are proposed the calculation of the learning rate does not need any prior information such as estimation of the modeling error bounds this offer an advantage compared to other techniques using robust modification

on using a cuts to evaluate fuzzy equations abstract consider an equation y f x1 xn where the extension principle is used to find y f x 1 x n when we subtitute fuzzy numbers x i for the xi 1 i n if x i denotes an cut of x i then we argue that for only certain simple equations will one obtain y equal to f x 1 x n where f is evaluated using interval arithmetic

robust optimization for water distribution systems least cost design 1 the objective of the least cost design problem of a water distribution system is to find its minimum cost with discrete diameters as decision variables and hydraulic controls as constraints the goal of a robust least cost design is to find solutions which guarantee its feasibility independent of the data i e under model uncertainty a robust counterpart approach for linear uncertain problems is adopted in this study which represents the uncertain stochastic problem as its deterministic equivalent robustness is controlled by a single parameter providing a trade off between the probability of constraint violation and the objective cost two principal models are developed uncorrelated uncertainty model with implicit design reliability and correlated uncertainty model with explicit design reliability the models are tested on three example applications and compared for uncertainty in consumers demands the main contribution of this study is the inclusion of the ability to explicitly account for different correlations between water distribution system demand nodes in particular it is shown that including correlation information in the design phase has a substantial advantage in seeking more efficient robust solutions

analysis and design of hierarchical fuzzy systems in this letter the hierarchical fuzzy systems are analyzed and designed in the analysis part we prove that the hierarchical fuzzy systems are universal approximators and analyze the sensitivity of the fuzzy system output with respect to small perturbations in its inputs in the design part we derive a gradient descent algorithm for tuning the parameters of the hierarchical fuzzy system to match the input output pairs the algorithm is simulated for two examples and the results show that the algorithm is effective and the hierarchical structure gives good approximation accuracy

learning and convergence analysis of neural type structured networks a class of feedforward neural networks structured networks has recently been introduced as a method for solving matrix algebra problems in an inherently parallel formulation a convergence analysis for the training of structured networks is presented since the learning techniques used in structured networks are also employed in the training of neural networks the issue of convergence is discussed not only from a numerical algebra perspective but also as a means of deriving insight into connectionist learning bounds on the learning rate are developed under which exponential convergence of the weights to their correct values is proved for a class of matrix algebra problems that includes linear equation solving matrix inversion and lyapunov equation solving for a special class of problems the orthogonalized back propagation algorithm an optimal recursive update law for minimizing a least squares cost functional is introduced it guarantees exact convergence in one epoch several learning issues are investigated

identification and control of dynamic systems using recurrent fuzzy neural networks proposes a recurrent fuzzy neural network rfnn structure for identifying and controlling nonlinear dynamic systems the rfnn is inherently a recurrent multilayered connectionist network for realizing fuzzy inference using dynamic fuzzy rules temporal relations are embedded in the network by adding feedback connections in the second layer of the fuzzy neural network fnn the rfnn expands the basic ability of the fnn to cope with temporal problems in addition results for the fnn fuzzy inference engine universal approximation and convergence analysis are extended to the rfnn for the control problem we present the direct and indirect adaptive control approaches using the rfnn based on the lyapunov stability approach rigorous proofs are presented to guarantee the convergence of the rfnn by choosing appropriate learning rates finally the rfnn is applied in several simulations time series prediction identification and control of nonlinear systems the results confirm the effectiveness of the rfnn

approximation capabilities of hierarchical fuzzy systems derived from practical application in location analysis and pricing and based on the approach of hierarchical structure analysis of continuous functions this paper investigates the approximation capabilities of hierarchical fuzzy systems by first introducing the concept of the natural hierarchical structure it is proved that continuous functions with natural hierarchical structure can be naturally and effectively approximated by hierarchical fuzzy systems to overcome the curse of dimensionality in both the number of rules and parameters then based on kolmogorov s theorem it is shown that any continuous function can be represented as a superposition of functions with the natural hierarchical structure and can then be approximated by hierarchical fuzzy systems to achieve the universal approximation property further the conditions under which the hierarchical fuzzy approximation is superior to the standard fuzzy approximation in overcoming the curse of dimensionality are analyzed

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

learning control using fuzzified self organizing radial basis function network this note describes an approach to integrating fuzzy reasoning systems with radial basis function rbf networks and shows how the integrated network can be employed as a multivariable self organizing and self learning fuzzy controller in particular by drawing some equivalence between a simplified fuzzy control algorithm sfca and a rbf network we conclude that the rbf network can be interpreted in the context of fuzzy systems and can be naturally fuzzified into a class of more general networks referred to as fbfn with a variety of basis functions not necessarily globally radial synthesized from each dimension by fuzzy logical operators on the other hand as a result of natural generalization from rbf to sfca we claim that the fuzzy system like rbf is capable of universal approximation next the fbfn is used as a multivariable rule based controller but with an assumption that no rule base exists leading to a challenging problem of how to construct such a rule base directly from the control environment we propose a simple and systematic approach to performing this task by using a fuzzified competitive self organizing scheme and incorporating an iterative learning control algorithm into the system we have applied the approach to a problem of multivariable blood pressure control with a fbfn based controller having six inputs and two outputs representing a complicated control structure

model selection and validation methods for non linear systems the theory of hypothesis testing is used to select a model with the correct structure and the relation of such a method to the aic and fpe criteria is investigated parametric validation and correlation validation methods are developed for non non linear difference equation models several shortcomings of traditional methods especially when applied to non linear systems are described

iterative solvers for a spectral galerkin approach to elliptic partial differential equations with fuzzy coefficients mathematical models of physical systems often contain parameters with an imprecisely known and uncertain character it is quite common to represent these parameters by means of random variables numerous methods have been developed to compute accurate approximations to solutions of equations with such parameters this approach however may not be entirely justified when the uncertainty is due to vagueness or incomplete knowledge for such cases alternative uncertainty representations using tools from imprecise probability theory have been suggested among those the fuzzy representation is probably most popular in this paper we consider numerical methods for solving partial differential equations with fuzzy coefficients we demonstrate that spectral expansion methods quite common in the random variable approach can also be used effectively for solving fuzzy equations we motivate the use of chebyshev polynomials in the spectral representation and apply a galerkin projection to convert the fuzzy prob

a two phase decomposition method for optimal design of looped water distribution networks a two phase decomposition method is proposed for the optimal design of new looped water distribution networks as well as for the parallel expansion of existing ones the main feature of the method is that it generates a sequence of improving local optimal solutions the first phase of the method takes a gradient approach with the flow distribution and pumping heads as decision variables and is an extension of the linear programming gradient method proposed by alperovits and shamir 1977 for nonlinear modeling the technique is iterative and produces a local optimal solution in the second phase the link head losses of this local optimal solution are fixed and the resulting concave program is solved for the link flows and pumping heads these then serve to restart the first phase to obtain an improved local optimal solution the whole procedure continues until no further improvement can be achieved some applications and extensions of the method are also discussed

new solutions for first order linear fuzzy difference equations abstract in this paper we study the different formulations of fuzzy difference equation x n w x n 1 q where w q are positive fuzzy numbers using a generalization of division for fuzzy numbers we investigate the existence uniqueness and global behavior of the solution

a neuro fuzzy system modeling with self constructing rule generationand hybrid svd based learning we propose an approach for neuro fuzzy system modeling a neuro fuzzy system for a given set of input output data is obtained in two steps first the data set is partitioned automatically into a set of clusters based on input similarity and output similarity tests membership functions associated with each cluster are defined according to statistical means and variances of the data points included in the cluster then a fuzzy if then rule is extracted from each cluster to form a fuzzy rule base second a fuzzy neural network is constructed accordingly and parameters are refined to increase the precision of the fuzzy rule base to decrease the size of the search space and to speed up the convergence we develop a hybrid learning algorithm which combines a recursive singular value decomposition based least squares estimator and the gradient descent method the proposed approach has advantages of determining the number of rules automatically and matching membership functions closely with the real distribution of the training data points besides it learns faster consumes less memory and produces lower approximation errors than other methods

a locally recurrent fuzzy neural network with support vector regression for dynamic system modeling this paper proposes a new recurrent model known as the locally recurrent fuzzy neural network with support vector regression lrfnn svr that handles problems with temporal properties structurally an lrfnn svr is a five layered recurrent network the recurrent structure in an lrfnn svr comes from locally feeding the firing strength of each fuzzy rule back to itself the consequent layer in an lrfnn svr is a takagi sugeno kang t s k type consequent which is a linear function of current states regardless of system input and output delays for the structure learning a one pass clustering algorithm clusters the input training data and determines the number of network nodes in hidden layers for the parameter learning an iterative linear svr algorithm is proposed to tune free parameters in the rule consequent part and feedback loops the motivation for using svr for parameter learning is to improve the lrfnn svr generalization ability this paper demonstrates lrfnn svr capabilities by conducting simulations in dynamic system prediction and identification problems with noiseless and noisy data in addition this paper compares simulation results from the lrfnn svr with other recurrent fuzzy models

adaptive filter design using type 2 fuzzy cerebellar model articulation controller this paper aims to propose an efficient network and applies it as an adaptive filter for the signal processing problems an adaptive filter is proposed using a novel interval type 2 fuzzy cerebellar model articulation controller t2fcmac the t2fcmac realizes an interval type 2 fuzzy logic system based on the structure of the cmac due to the better ability of handling uncertainties type 2 fuzzy sets can solve some complicated problems with outstanding effectiveness than type 1 fuzzy sets in addition the lyapunov function is utilized to derive the conditions of the adaptive learning rates so that the convergence of the filtering error can be guaranteed in order to demonstrate the performance of the proposed adaptive t2fcmac filter it is tested in signal processing applications including a nonlinear channel equalization system a time varying channel equalization system and an adaptive noise cancellation system the advantages of the proposed filter over the other adaptive filters are verified through simulations

evolutionary algorithm based learning of fuzzy neural networks part 2 recurrent fuzzy neural networks fuzzy neural networks fnn as opposed to neuro fuzzy systems whose main task is to process numerical relationships can process both numerical measurement based information and perception based information in spite of great importance of fuzzy feed forward and recurrent neural networks for solving wide range of real world problems today there are no effective training algorithm for them currently there are two approaches for training of fnn first approach is based on application of the level sets of fuzzy numbers and the back propagation bp algorithm the second approach involves using evolutionary algorithms to minimize error function and determine the fuzzy connection weights and biases the method based on the second approach was proposed by the authors and published in part 1 of this paper r a aliev b fazlollahi r vahidov genetic algorithm based learning of fuzzy neural networks part 1 feed forward fuzzy neural networks fuzzy sets and systems 118 2001 351 358 in contrast to the bp and other supervised learning algorithms evolutionary algorithms do not require nor use information about differentials and hence they are most effective in case where the derivative is very difficult to obtain or even unavailable however the main deficiency of the existing fnn based on the feed forward architecture is its adherence to static problems in case of dynamic or temporal problems there is a need for recurrent fuzzy neural networks rfnn designing efficient training algorithms for rfnn has recently become an active research direction in this paper we propose an effective differential evolution optimization deo based learning algorithm for rfnn with fuzzy inputs fuzzy weights and biases and fuzzy outputs the effectiveness of the proposed method is illustrated through simulation of benchmark forecasting and identification problems and comparisons with the existing methods the suggested approach has also been used for real applications in an oil refinery plant for petrol production forecasting

some remarks on a new dq based method for solving a class of volterra integro differential equations in this paper a numerical picard like method which combines successive approximations with integral and differential quadrature is considered to solve some volterra integro differential equations where rational functions are involved under certain conditions the method provides solutions in explicit form without any recursive computation and for some cases it is possible to give a formula to compute the related error several numerical examples are proposed to illustrate the behaviour of the method by comparing it where possible with some existing methods

a ts type maximizing discriminability based recurrent fuzzy network for classification problems this work proposes a takagi sugeno ts type maximizing discriminability based recurrent fuzzy network mdrfn that can classify highly confusable patterns the discriminative capability plays a significant role in determining classification performance to increase the discriminative capability the proposed mdrfn considers minimum classification error mce and minimum training error mte in mce the weights are updated by maximizing the discrimination among different classes in mte the parameter learning adopts the gradient descent method to reduce the cost function therefore the novelty of mdrfn is that it not only minimizes the cost function but maximizes the discriminative capability as well the effectiveness of the proposed mdrfn is demonstrated by three temporal classification problems in the experiments other rfns including the singleton type recurrent neural fuzzy network srnfn ts type rfn trfn and simple rfn srfn are compared analysis results indicate that the proposed mdrfn exhibits excellent classification performance

back propagation through adjoints for the identification of nonlinear dynamic systems using recurrent neural models in this paper back propagation is reinvestigated for an efficient evaluation of the gradient in arbitrary interconnections of recurrent subsystems it is shown that the error has to be back propagated through the adjoint model of the system and that the gradient can only be obtained after a delay a faster version accelerated back propagation that eliminates this delay is also developed various schemes including the sensitivity method are studied to update the weights of the network using these gradients motivated by the lyapunov approach and the adjoint model the predictive back propagation and its variant targeted back propagation are proposed a further refinement predictive back propagation with filtering is then developed where the states of the model are also updated the convergence of this scheme is assured it is shown that it is sufficient to back propagate as many time steps as the order of the system for convergence as a preamble convergence of online batch and sample wise updates in feedforward models is analyzed using the lyapunov approach

fuzzy probabilistic design of water distribution networks 1 the primary aim of this paper is to present a fuzzy probabilistic approach for optimal design and rehabilitation of water distribution systems combining aleatoric and epistemic uncertainties in a unified framework the randomness and imprecision in future water consumption are characterized using fuzzy random variables whose realizations are not real but fuzzy numbers and the nodal head requirements are represented by fuzzy sets reflecting the imprecision in customers requirements the optimal design problem is formulated as a two objective optimization problem with minimization of total design cost and maximization of system performance as objectives the system performance is measured by the fuzzy random reliability defined as the probability that the fuzzy head requirements are satisfied across all network nodes the satisfactory degree is represented by necessity measure or belief measure in the sense of the dempster shafer theory of evidence an efficient algorithm is proposed within a monte carlo procedure to calculate the fuzzy random system reliability and is effectively combined with the nondominated sorting genetic algorithm ii nsgaii to derive the pareto optimal design solutions the newly proposed methodology is demonstrated with two case studies the new york tunnels network and hanoi network the results from both cases indicate that the new methodology can effectively accommodate and handle various aleatoric and epistemic uncertainty sources arising from the design process and can provide optimal design solutions that are not only cost effective but also have higher reliability to cope with severe future uncertainties

fuzzy differential equations we will concentrate on the second order linear constant coefficient ordinary differential equation for x in interval i i can be 0 t for t 0 or i can be 0 the initial conditions are y 0 0 y 0 1 we assume g is continuous on i

some new results on system identification with dynamic neural networks nonlinear system online identification via dynamic neural networks is studied in this paper the main contribution of the paper is that the passivity approach is applied to access several new stable properties of neuro identification the conditions for passivity stability asymptotic stability and input to state stability are established in certain senses we conclude that the gradient descent algorithm for weight adjustment is stable in an l sub spl infin sense and robust to any bounded uncertainties

calculating insurance claim reserves with fuzzy regression claim provisions are crucial for the financial stability of insurance companies this is why actuarial literature has proposed numerous claim reserving methods which are usually based on statistical concepts however the mutant and uncertain behaviour of insurance environments does not make advisable to use a wide data base when calculating claim reserves on the one hand this involves a considerable loss in reliability of statistical methods but on the other it makes the use of fuzzy set theory very attractive in this paper then we propose a claim reserving method that combines the extension by ishibuchi and nii of tanaka s classical fuzzy regression method with sherman s claim reserving scheme

self learning fuzzy controllers based on temporal backpropagation a generalized control strategy that enhances fuzzy controllers with self learning capability for achieving prescribed control objectives in a near optimal manner is presented this methodology termed temporal backpropagation is model sensitive in the sense that it can deal with plants that can be represented in a piecewise differentiable format such as difference equations neural networks gmdh structures and fuzzy models regardless of the numbers of inputs and outputs of the plants under consideration the proposed approach can either refine the fuzzy if then rules of human experts or automatically derive the fuzzy if then rules if human experts are not available the inverted pendulum system is employed as a testbed to demonstrate the effectiveness of the proposed control scheme and the robustness of the acquired fuzzy controller

an orthogonal least squares method for recurrent fuzzy neural modeling abstract this paper presents an orthogonal least squares ols based modeling method named dynamic ols d ols for generating recurrent fuzzy models a dynamic neuron based fuzzy neural network is proposed comprising generalized takagi sugeno kang tsk fuzzy rules whose consequent parts consist of dynamic neurons with local output feedback from an arbitrarily large set of candidate dynamic neurons the d ols method selects automatically the most important ones thus in the resulting model the consequent part of each fuzzy rule contains dynamic neurons with different time delays the proposed dynamic model equipped with the learning algorithm is applied to two temporal problems where the effectiveness of the suggested method as well as the advantages of the resulting dynamic model are demonstrated

nl sub q theory checking and imposing stability of recurrent neural networks for nonlinear modeling it is known that many discrete time recurrent neural networks such as e g neural state space models multilayer hopfield networks and locally recurrent globally feedforward neural networks can be represented as nl sub q systems sufficient conditions for global asymptotic stability and input output stability of nl sub q systems are available including three types of criteria 1 diagonal scaling 2 criteria depending on diagonal dominance 3 condition number factors of certain matrices the paper discusses how narendra s 1990 1991 dynamic backpropagation procedure which is used for identifying recurrent neural networks from i o measurements can be modified with an nl sub q stability constraint in order to ensure globally asymptotically stable identified models an example illustrates how system identification of an internally stable model corrupted by process noise may lead to unwanted limit cycle behavior and how this problem can be avoided by adding the stability constraint

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection