comprehensive analysis of a new fuzzy rule interpolation method the first published result in fuzzy rule interpolation was the spl alpha cut based fuzzy rule interpolation termed as kh fuzzy rule interpolation originally devoted for complexity reduction a modified version of the kh approach has been presented by yam et al 1999 which eliminates the subnormality problem while at the same time intending to maintain the advantageous computational properties of the original method this paper presents a comprehensive analysis of the new method which includes detailed comparison with the original kh fuzzy rule interpolation method concerning the explicit functions of the methods preservation of piecewise linearity and stability the fuzziness of the conclusion with respect to the fuzziness of the observation is also investigated in comparison with several interpolation techniques all these comparisons shows that the new method preserves the advantageous properties of the kh method and alleviates its most significant disadvantage the problem of subnormality

weak efficiency for multiobjective variational problems abstract sometimes to locate efficient solutions for multiobjective variational problems mvps is quite costly so in this paper we tackle the study of weakly efficient solutions for mvps a new concept of weak vector critical point which generalizes other ones already existent and a new class of pseudoinvex functions are introduced we will apply a new approach to prove that the new class of pseudoinvex functions is equivalent to the class of functions whose weak vector critical points are weakly efficient solution for mvps

an automated multilevel substructuring method for eigenspace computation in linear elastodynamics we present an automated multilevel substructuring amls method for eigenvalue computations in linear elastodynamics in a variational and algebraic setting amls first recursively partitions the domain of the pde into a hierarchy of subdomains then amls recursively generates a subspace for approximating the eigenvectors associated with the smallest eigenvalues by computing partial eigensolutions associated with the subdomains and the interfaces between them we remark that although we present amls for linear elastodynamics our formulation is abstract and applies to generic h1 elliptic bilinear forms in the variational formulation we define an interface mass operator that is consistent with the treatment of elastic properties by the familiar steklov poincare operator with this interface mass operator all of the subdomain and interface eigenvalue problems in amls become orthogonal projections of the global eigenvalue problem onto a hierarchy of subspaces convergence of amls is determined in the continuous setting by the truncation of these eigenspaces independent of other discretization schemes the goal of amls in the algebraic setting is to achieve a high level of dimensional reduction locally and inexpensively while balancing the errors associated with truncation and the finite element discretization this is accomplished by matching the mesh independent amls truncation error with the finite element discretization error our report ends with numerical experiments that demonstrate the effectiveness of amls on a model problem and an industrial problem

on efficiency and duality for multiobjective variational problems abstract the concept of efficiency pareto optimum is used to formulate duality for multiobjective variational problems wolfe and mond weir type duals are formulated under generalized f convexity assumptions on the functions involved weak and strong duality theorems are proved

generating an interpretable family of fuzzy partitions from data in this paper we propose a new method to construct fuzzy partitions from data the procedure generates a hierarchy including best partitions of all sizes from n to two fuzzy sets the maximum size n is determined according to the data distribution and corresponds to the finest resolution level we use an ascending method for which a merging criterion is needed this criterion is based on the definition of a special metric distance suitable for fuzzy partitioning and the merging is done under semantic constraints the distance we define does not handle the point coordinates but directly their membership degrees to the fuzzy sets of the partition this leads to the introduction of the notions of internal and external distances the hierarchical fuzzy partitioning is carried independently over each dimension and to demonstrate the partition potential they are used to build fuzzy inference system using a simple selection mechanism due to the merging technique all the fuzzy sets in the various partitions are interpretable as linguistic labels the tradeoff between accuracy and interpretability constitutes the most promising aspect in our approach well known data sets are investigated and the results are compared with those obtained by other authors using different techniques the method is also applied to real world agricultural data the results are analyzed and weighed against those achieved by other methods such as fuzzy clustering or discriminant analysis

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

about the use of fuzzy clustering techniques for fuzzy model identification abstract in this work we present an alternative approach to generate fuzzy rules with a functional consequent associated to the tsk fuzzy model in our case using fuzzy clustering algorithms that look for linear behaviours in the product space of the input output data we analyse different methods to generate the associated fuzzy rules using in some cases multidimensional reference fuzzy sets in the product space of the input variables and in other cases fuzzy sets in each of the different dimensions in any case the rules being generated correspond to a tsk fuzzy model

multi level substructuring combined with model order reduction methods abstract in many fields of engineering problems linear time invariant dynamical systems lti systems play an outstanding role they result for instance from discretizations of the unsteady heat equation and they are also used in optimal control problems often the order of lti systems is a limiting factor since it becomes easily very large as a consequence these systems cannot be treated efficiently without model reduction algorithms in this paper a new approach for the combination of model order reduction methods and recent multi level substructuring mls techniques is presented similar multi level substructuring methods have already been applied successfully to huge eigenvalue problems up to several millions of degrees of freedom however the presented approach does not make use of a modal analysis like former algorithms instead the original system is decomposed in smaller lti systems which are treated with recent model reduction methods furthermore the error which is induced by this substructuring approach is analysed and numerical examples based on the oberwolfach benchmark collection are given in this paper

duality for multiobjective variational problems abstract wolfe and mond weir type duals for multiobjective variational problems are formulated under convexity assumptions on the functions involved weak and converse duality theorems are proved to relate properly efficient solutions of the primal and dual problems a close relationship between these variational problems and nonlinear multiobjective programming problems is also indicated

a continuous time generalization of gordan s transposition theorem abstract a certain infinite dimensional generalization of gordan s transposition theorem is obtained as a consequence of a strong duality principle in continuous time programming some implications and corollaries of the main theorem are also discussed

fuzzy control of model car this paper discusses fuzzy control of a model car fuzzy control rules are derived by modelling an expert s driving actions experiments are performed using a model car with a sensing unit and a micro computer

fuzzy partition and fuzzy rule base abstract in this paper we show that a rule base can be represented by mathematical formulae to this end we introduce a mathematical definition of fuzzy rule base by means of fuzzy number partitions based on this definition both linear and nonlinear fuzzy rule base interpolations are investigated

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

brief paper positivity preserving h model reduction for positive systems this paper is concerned with the model reduction of positive systems for a given stable positive system our attention is focused on the construction of a reduced order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed h performance based upon a system augmentation approach a novel characterization on the stability with h performance of the error system is first obtained in terms of linear matrix inequality lmi then a necessary and sufficient condition for the existence of a desired reduced order model is derived accordingly furthermore iterative lmi approaches with primal and dual forms are developed to solve the positivity preserving h model reduction problem finally a compartmental network is provided to show the effectiveness of the proposed techniques

a class of nondifferentiable continuous programming problems abstract optimality conditions duality and converse duality results are obtained for a class of continuous programming problems with a nondifferentiable term in the integrand of the objective function the proofs are based on a fritz john theorem for constrained optimization in abstract spaces the results generalize various well known results in variational problems with differentiable functions and also give a dynamic analogue of certain nondifferentiable programming problems

multiobjective h sub 2 h sub spl infin control for a linear time invariant system with several disturbance inputs and controlled outputs we show how to minimize the nominal h sub 2 norm performance in one channel while keeping bounds on the h sub 2 norm or h sub spl infin norm performance implying robust stability in the other channels this multiobjective h sub 2 h sub spl infin problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems we show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller if it exists we reveal how the novel trick of optimizing the trace norm of the youla parameter over certain convex constraints allows one to design a nearly optimal controller whose youla parameter is of the same order as the optimal one

a generalized concept for fuzzy rule interpolation the concept of fuzzy rule interpolation in sparse rule bases was introduced in 1993 it has become a widely researched topic in recent years because of its unique merits in the topic of fuzzy rule base complexity reduction the first implemented technique of fuzzy rule interpolation was termed as spl alpha cut distance based fuzzy rule base interpolation despite its advantageous properties in various approximation aspects and in complexity reduction it was shown that it has some essential deficiencies for instance it does not always result in immediately interpretable fuzzy membership functions this fact inspired researchers to develop various kinds of fuzzy rule interpolation techniques in order to alleviate these deficiencies this paper is an attempt into this direction it proposes an interpolation methodology whose key idea is based on the interpolation of relations instead of interpolating spl alpha cut distances and which offers a way to derive a family of interpolation methods capable of eliminating some typical deficiencies of fuzzy rule interpolation techniques the proposed concept of interpolating relations is elaborated here using fuzzy and semantic relations this paper presents numerical examples in comparison with former approaches to show the effectiveness of the proposed interpolation methodology

notes on the approximation rate of fuzzy kh interpolators this paper investigates the approximation behaviour of the koczy hirota kh interpolative fuzzy controllers first in accordance with the remarks in fuzzy sets and systems 125 1 2002 105 it is pointed out that it is a fuzzy generalization of the shepard operator shepard operator has thoroughly studied by approximation theorist since the mid 1970s exploiting the aforementioned relationship we establish analog results on the approximation rate of kh controllers the optimal order and class of approximation saturation problem are determined for certain values of the exponent corresponding results on the modified alpha cut based interpolation method being an improvement of the kh interpolator are also provided the results offer trade off facilities between approximation accuracy and the number of rules as a consequence the necessary and sufficient number of rules can be determined for a prescribed accuracy

construction of a neurofuzzy network capable of extrapolating and interpolating with respect to the convex hull of a set of input samples in bb r n the problem of regression estimation is considered with a specific regard for the distinction between interpolation and extrapolation a neurofuzzy network named nfech is proposed that is capable of extrapolating and interpolating with respect to the convex hull of a finite set of input samples x sub ropfn the geometrical construction of the proposed network is explained both mathematically and graphically the illustrations explain how the particular parts of the construction work and also show the final surfaces of the obtained models the method is tested on artificial datasets generated from mathematical functions according to various statistical distributions also comparisons to the commonly used radial basis function rbf multilayered perceptron mlp neural networks and to fuzzy rule interpretation fri fuzzy rule extrapolation fre approach are presented

parallel fuzzy inference based on a level sets and generalized means abstract in this paper a parallel fuzzy inference method is proposed in which inference consequences are unified on the basis of level sets and generalized means it has the following advantages over conventional methods 1 it can control the degree to which the fuzziness and specificity of given facts are reflected to those of unified inference consequences 2 it can deduce unified inference consequences in the form of normal and convex fuzzy sets which can thus be treated as fuzzy numbers and 3 it effectively matches systems that include fuzzy set operations based on the extension principle this paper first reviews the generalized mean and describes the computational steps of the proposed inference method then the properties of this method are investigated and the control mechanism of the fuzziness and specificity in unified inference consequences reflecting those in given facts are presented the efficient inference computations are also provided taking advantage of the level set based scheme of the proposed inference method next a learning algorithm is derived for the proposed inference method based on the error back propagation by feeding fuzzy exemplar patterns it can automatically adjust the above mentioned degree of fuzziness and specificity as well as the fuzzy sets in conditional propositions the simulation studies show the feasibility of the proposed inference method

optimality conditions and duality for multiobjective variational problems with generalized b invexity we extend the concepts of b type i and generalized b type i functions to the continuous case and we use these concepts to establish sufficient optimality conditions and duality results for multiobjective variational programming problems

interpolative reasoning with insufficient evidence in sparse fuzzy rule bases abstract rule based fuzzy approximate reasoning uses various techniques of modified modus ponens the observation is in most cases not identical with any of the antecedents in the rules however a conclusion still can be computed by using some combination of all consequents where an overlapping of observation and antecedent is present if the rule base is sparse i e it contains insufficient information on the total state space it might occur that an observation has absolutely no overlapping with any of the antecedents and so not even a single rule is fired i e no conclusion can be computed on the basis of modus ponens in such a case interpolative reasoning in the strict sense can be applied some kind of weighted average of the flanking rules is calculated this technique can be extended to a form of extrapolation when the observation is not flanked from both sides linear interpolation and extrapolation is presented and then the idea is extended to arbitrary approximation

a similarity measure for fuzzy rulebases based on linguistic gradients fuzzy rulebases are an approximate representation of some interesting system as such they could potentially be used for indexing and searching candidate solutions in case based reasoning cbr systems in a variety of application areas however there is currently no method for directly and efficiently determining the similarity between two rulebases which is a necessary element of their use in any cbr system we propose a method for measuring similarity between two linguistic fuzzy rulebases based on the granular computing technique of linguistic gradients the proposed similarity measure is based on comparing the linguistic structure of two rulebases using the linguistic gradient operator to reveal that structure our algorithm operates at the level of linguistic rulebases rather than a defuzzified reasoning surface and thus belongs to the computing with words paradigm in a validation experiment we compare our new method with the root mean square difference between reasoning surfaces for 603 pairs of fuzzy rulebases drawn from the literature a spearman correlation analysis shows that our new linguistic method is consistent with the numerical rms results while being theoretically and empirically faster than computing an accurate rms difference

difference distance and similarity as a basis for fuzzy decision support based on prototypical decision classes the use of prototypical decision classes as a basis for fuzzy decision support is outlined it is shown that given certain assumptions about the structure of both numeric and non numeric linguistic variables trapezoidal fuzzy sets can be used to model linguistic terms in particular the notion of a fuzzy index is introduced to model sets of linguistic terms for which there is no formal measurement scale the similarity between prototypes and input cases is measured as a function of the difference or distance between fuzzy sets the difference measure is itself a fuzzy set to reflect the underlying uncertainties in the original linguistic model similarity profiles for different decision classes are output as trapezoidal fuzzy sets the approach can be applied to the rapid development of decision aids using standard business software

a methodology to model fuzzy systems using fuzzy clustering in a rapid prototyping approach in this paper we present different approaches to the problem of fuzzy rules extraction by using fuzzy clustering they try to investigate the use of fuzzy clustering as a technique that let us obtain a first approximation of the fuzzy model in a rapid prototyping approach the key ideas here are to generate in a first step a collection of fuzzy rules from numerical data pairs subsequently these rules can be converted either in a collection of linguistic rules in order to obtain a descriptive approach or in an inference machine so that we obtain an approximative approach this can be considered as a quasi descriptive approach because it assigns a weight to the fuzzy rules like in the descriptive approaches but also it generates the fuzzy sets directly in the product space of the input variables like in approximative approaches both by using the information disclosed by means of a fuzzy clustering process

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

a fuzzy clustering based rapid prototyping for fuzzy rule based modeling this paper presents different approaches to the problem of fuzzy rules extraction by using fuzzy clustering as the main tool within these approaches we describe six methods that represent different alternatives in the fuzzy modeling process and how they can be integrated with a genetic algorithms these approaches attempt to obtain a first approximation to the fuzzy rules without any assumption about the structure of the data because the main objective is to obtain an approximation the methods we propose must be as simple as possible but also they must have a great approximative capacity and in that way we work directly with fuzzy sets induced in the variables input space the methods are applied to four examples and the errors obtained are specified in the different cases

stability of interpolative fuzzy kh controllers the classical approaches in fuzzy control zadeh and mamdani deal with dense rule bases when this is not the case i e in sparse rule bases one has to choose another method fuzzy rule interpolation proposed first by koczy and hirota internat j approx reason 9 1993 197 225 offers a possibility to construct fuzzy controllers kh controllers under such conditions the main result of this paper shows that the kh interpolation method is stable it also contributes to the application oriented use of balazs shepard interpolation operators investigated extensively by researchers in approximation theory the numerical analysis aspect of the result contributes to the well known problem of finding a stable interpolation method in the following sense we would like to approximate a function that is only known at distinct points e g where it can be measured since measurement errors cannot be eliminated in practice it is a natural requirement that the interpolation method should be stable independently from the selection of measurement points the classical interpolation methods generally do not fulfil this condition only with certain strong restrictions concerning the measurement points if we neglect the classical form of the approximating function polynomial and trigonometrical we can produce better behaving approximation the kh controller approximating function being a simple fractional function fulfils the stability condition this can also be interpreted as kh controllers are universal approximators in the space of continuous functions with respect to e g the supremum norm

a schur method for balanced truncation model reduction it is shown that a not necessarily balanced state space realization of the moore reduced model can be computed directly without balancing via projections defined in terms of arbitrary bases for the left and right eigenspaces associated with the large eigenvalues of the product pq of the reachability and controllability grammians two specific methods for computing these bases are proposed one based on the ordered schur decomposition of pq and the other based on the cholesky factors of p and q the algorithms perform reliably even for nonminimal models

assessing fuzzy sets and the potential of possibility theory the alpha cut is generalized to depend on relations other than greater than or equal to various measures of fuzziness and specificity are also generalized as assessment functions a consistent notation and terminology are developed possibility theory is introduced and the assessment functions are recast as potential functions the nonspecificity function is seen to have special relevance

similarity rules and gradual rules for analogical and interpolative reasoning with imprecise data similarity rules and gradual rules in the form of functional mappings of imprecise data are proposed to implement analogical and interpolative reasoning the rules are deduced from the property of the horizontal graduality of fuzzy sets and the horizontal graduality is represented by introducing a horizontal difference function as an alternative to the proposed extensions of distance between fuzzy concepts a thorough discussion of the main characteristics of fuzzy sets and horizontal difference function is given an analogical reasoning algorithm based on these characteristics and a set of intuitively sound axioms are proposed the algorithm is consistent with the intuitive concepts of analogical reasoning expressed by the axioms it guarantees a convex and normal output regardless of the shape of the precondition and action patterns and input data when some axiomatic conditions are not satisfied the algorithm relaxes the less important ones and produces an approximate solution the fuzziness cardinality and shape of an output fuzzy set are easily controlled some steps of the algorithm can be carried out in parallel numerical examples confirm the theoretical analysis of the properties of the algorithm some extensions for interpolative reasoning are discussed the proposed rules are particularly well suited for systems which support qualitative reasoning and qualitative modelling

comparison of fuzzy reasoning methods l a zadeh e h mamdani and m mizumoto et al have proposed methods for fuzzy reasoning in which the antecedent involves a fuzzy conditional proposition if x is a then y is b with a and b being fuzzy concepts mizumoto et al have investigated the properties of their methods in the case of generalized modus ponens this paper deals with the properties of their methods in the case of generalized modus tollens and investigates the other new fuzzy reasoning methods obtained by introducing the implication rules of many valued logic systems finally the properties of syllogism and contrapositive are investigated under each fuzzy reasoning method

fuzzy systems and approximation abstract the basic motivation of using fuzzy rule based systems especially for control purposes is to deduce simple and fast approximations of the unknown or too complicated models fuzzy rule based systems have become very popuar because of their transparency and easiness of tuning and modification recently some results concerning the explicit functions implemented by realistic fuzzy controllers presented the class of functions that could be implemented in this way some parallel results on the other hand attempted to prove that the main advantage of using fuzzy systems was the suitability for approximation with arbitrary accuracy in their universality the explicit formulas and some very recent theoretical results made it clear however that fuzzy systems were not really good approximators as realistic fuzzy controllers could generate only very rough approximations of given transference functions in connection with approximation the question can be asked whether there is an optimal fineness roughness of a fuzzy rule base that controls a certain action with roughness gives minimal time complexity as an example a target tracking problem was chosen cat and mouse or hawk and sparrow problem where the antagonistic criteria of minimizing inference time by the given rule base and minimizing action time search for the target with given uncertainty provided by the rule model were examined under certain assumptions the solution of this optimization problem leads to nontrivial rule base sizes these results have also practical applicability since if a fine enough model of the system is known it is always possible to generate a rougher version of the same by applying the model transformation technique offered by rule interpolation with levels

directional interpolation approach to h optimization and robust stabilization it is shown that under a mild condition h infty optimization and robust stabilization are equivalent to a directional interpolation problem which is a matrix extension of the classical pick nevanlinna interpolation problem a classical iterative method which is an extension of the schur nevanlinna algorithm is given for solving the problem this method does not require the inner outer factorization nor the balanced realization of the original plant a circuit theoretical parameterization of all solutions is derived that is expected to enhance the physical insight to the h infty optimal control and robust stabilization this parameterization has the degree much less than the one obtained previously

fuzzy rule interpolation for multidimensional input spaces with applications a case study fuzzy rule based systems have been very popular in many engineering applications however when generating fuzzy rules from the available information this may result in a sparse fuzzy rule base fuzzy rule interpolation techniques have been established to solve the problems encountered in processing sparse fuzzy rule bases in most engineering applications the use of more than one input variable is common however the majority of the fuzzy rule interpolation techniques only present detailed analysis to one input variable case this paper investigates characteristics of two selected fuzzy rule interpolation techniques for multidimensional input spaces and proposes an improved fuzzy rule interpolation technique to handle multidimensional input spaces the three methods are compared by means of application examples in the field of petroleum engineering and mineral processing the results show that the proposed fuzzy rule interpolation technique for multidimensional input spaces can be used in engineering applications

mixed type duality for multiobjective variational problems the concept of mixed type duality has been extended to the class of multiobjective variational problems a number of duality relations are proved to relate the efficient solutions of the primal and its mixed type dual problems the results are obtained for convex generalized convex functions these studies have been generalized to the case of invex generalized invex functions our results apparently generalize a fairly large number of duality results previously obtained for finite dimensional nonlinear programming problems under various convexity assumptions

ordering distance and closeness of fuzzy sets abstract in dense rule bases where the observation usually overlaps with several antecedents in the rule base various algorithms are used for approximate reasoning and control if the antecedents are located sparsely and the observation does not overlap as a rule with any of the antecedents function approximation techniques combined with the resolution principle lead to applicable conclusions this kind of approximation is possible only if a new concept of ordering and distance i e a metric in the state space and a partial ordering among convex and normal fuzzy sets cnf sets is introduced so the fuzzy distance of two cnf sets can be defined and by this distance closeness and similarity of cnf sets as well