Hongxing Li

Beijing Normal University | bnu · Department of Mathematics

In this paper, we study the interval type-2 fuzzy system based on NT method. We establish an interval type-2 fuzzy system based on a set of input-output data and reveal the probability significance of the interval type-2 fuzzy system. We prove that the crisp output of the interval type-2 fuzzy system based on NT method can be regarded as the condit...

In this paper, a new multi-class classification technology combining fuzzy neural network and ensemble learning technology is proposed. Several fuzzy neural networks, including fuzzy TS neural network and fuzzy wavelet neural network, are considered as sub-classifiers. Restricted Boltzmann machines and gradient descent algorithm are used to train t...

The problem is discussed about approximating a sequence of intuitionistic fuzzy numbers or aggregating the sequence of intuitionistic fuzzy numbers and then approximating the output of aggregation with condition of preserving the width. An interesting conclusion is obtained.

In this paper, we utilize generalized Bernstein polynomials to construct fuzzy system. Different from traditional Bernstein polynomials, partition of interval on input variable can be chosen as non-equidistant division. We prove that generalized Bernstein fuzzy systems are universal approximators to a given continuous function and its high-order de...

In this paper, we build up a connection between intuitionistic fuzzy special set (IFSS) and intuitionistic fuzzy set (IFS). Firstly, by using the concept of IFSS, we present the concept of intuitionistic nested set (INS) and show that an IFS can be determined by an INS. Secondly, we introduce the concept of lambda-cut sets of IFS and show that lamb...

In this paper, a novel Error-compensated MArginal LINEarization (EMALINE) fuzzy modeling method is proposed. This method models a group of data information to a piecewise linear fuzzy system with high accuracy within a given error bound. It is proved that the fuzzy system generalized by the EMALINE method possesses universal approximation capabilit...

In this paper, a dynamic fuzzy inference marginal linearization (DFIML) method is proposed for modeling nonlinear dynamic systems. This method can transfer a group of input-output data into a time-variant fuzzy system with variable coefficients. It is shown that solutions of time-variant fuzzy systems generalized by DFIML method are universal appro...

Wang et al [1] have put forward the problem of upgrade of topology and obtained a series of systematic theories and applicable results. Li et al [2] have put forwarded the problem of upgrade of group and also obtained a series of results. Consequently, the problem of upgrade of mathematic structure has drawn more attention of researchers. However,...

The connections between Zadeh fuzzy set and three-valued fuzzy set are established in this paper. The concepts of interval-valued level cut sets on Zadeh fuzzy set are presented and new decomposition theorems and representation theorems of Zadeh fuzzy set are established based on new cut sets. Firstly, four interval-valued level cut sets on Zadeh f...

In this paper, a method of constructing joint probability density function and fuzzy system using non-singleton fuzzifier is proposed based on a set of input-output data. Firstly, we create fuzzy relations using bounded product implication operator. With the fuzzy relations, we get the joint probability density function of binary random variable. S...

The design and response abilities of time-varying fuzzy systems are discussed. Firstly, based on non-singleton fuzzifier and dynamic fuzzy inference method, the response function of time-varying fuzzy system is obtained. Then, a sufficient and necessary condition for time-varying fuzzy system possessing interpolation property is given. It is proved...

In this paper, a class of time-variant fuzzy systems is designed and analyzed. Firstly, a novel model of dynamic fuzzy inference (DFI) is proposed, and the corresponding DFI method is obtained. Then, the response abilities of time-variant fuzzy systems constructed by the DFI method are analyzed. Some sufficient conditions for time-variant fuzzy sys...

Based on a set of input - output data, we first build a joint probability density function of certain two-dimensional random variables utilizing fuzzy inference rules, then propose its marginal probability density function and numerical characteristics. Finally, it is pointed out that the fuzzy system constructed with center of gravity defuzzifier...

In this paper, we provide a method of constructing joint probability density function and fuzzy system by using fuzzy inference based on a set data of input-output. Firstly, we build fuzzy relation and derive the joint probability density function. Secondly, we discuss the marginal density function and numerical characteristic of the joint probabi...

In this paper, we established decomposition theorems and representation theorems of interval-valued intuitionistic fuzzy sets(IVIFS) by use of cut sets of interval-valued intuitionistic fuzzy sets. We have shown that each kind of cut sets corresponds to two kinds of decomposition theorems and representation theorems, thus eight kinds of decompositi...

The response properties of fuzzy systems generalized by non-singleton fuzzifier method are discussed. Firstly, the fuzzy system with non-singleton fuzzifier, CRI method and centroid defuzzifier is discussed. A sufficient and necessary condition of the corresponding fuzzy system possessing interpolation properties is deduced. Furthermore, some suffi...

With the applications of topological upgrading [1], some problems of upgrading mathematical structure to its power set have caused wide concerns. This paper studies the upgrading from group structure to its power set and gives the definition, examples and ways of construction of power groups, then shows the relationship between power groups and com...

This paper shows by an example that Lemma 2 in Wang and Elhag [On the normalization of interval and fuzzy weights, Fuzzy Sets and Systems 157 (2006) 2456–2471] is false and presents the modified lemma and the correct proof of Theorem 5.

In this paper we study some single-machine scheduling problems with learning effects where the actual processing time of a job serves as a function of the total actual processing times of the jobs already processed and of its scheduled position. We show by examples that the optimal schedules for the classical version of problems are not optimal und...

A new training algorithm for hierarchical hybrid fuzzy–neural networks (HHFNN) based on Gaussian membership function is proposed in this paper. This new algorithm adjusts the widths of Gaussian membership functions of the IF parts of fuzzy rules in the lower fuzzy sub-systems, and updates the weights and bias terms of the upper neural network by gr...

Based on different emphases, some general information content measures of fuzzy relations are given. Distinguished from the measures related to fuzzy relations proposed before, these measures try to estimate the information conveyed by multiple-domain fuzzy relations without preassigned probability distribution. Since fuzzy rules can be fully captu...

In this paper, the concepts of interval-valued level cut sets on Zadeh fuzzy sets are presented and new decomposition theorems of Zadeh fuzzy sets based on new cut sets are established. Firstly, four interval-valued level cut sets on Zadeh fuzzy sets are introduced, which are generalizations of the normal cut sets on Zadeh fuzzy sets and have the s...

In paper [1], the definition of fuzzy set similarity measure and it's application to fuzzy inference are given. So we introduce similarity measure of fuzzy set-valued, which is called fuzzy similarity measure. Fuzzy similarity measure is a generalized similarity measure, the axiomatic description is given, and some examples of fuzzy similarity meas...

The information content of rules is categorized into inner mutual information content and outer impartation information content. Actually, the conventional objective interestingness measures based on information theory are all inner mutual information, which represent the confidence of rules and the mutual information between the antecedent and con...

In this paper, Fuzzy Similarity Inference (FSI) is investigated. First, the axiomatic definition of the fuzzy similarity measure is introduced. It is a generalization of the similarity measure. Some formulas are given to compute the fuzzy similarity of two fuzzy sets. Second, a novel fuzzy reasoning, called fuzzy similarity inference, is proposed....

In this paper, an optimal adaptive H-infinity tracking control design method via wavelet network for a class of uncertain nonlinear systems with external disturbances is proposed to achieve H-infinity tracking performance. First, an alternate tracking error and a performance index with respect to the tracking error and the control effort are introd...

This paper studies the problem of solving a matrix equation over an arbitrary (not necessarily complete) Brouwerian lattice. A criterion for the solvability and a method for finding all solutions of the equation are obtained. Over an arbitrary self-dual Brouwerian lattice, an equivalent condition for the unique solution, a necessary and sufficient...

In this paper, we introduce an actual time-dependent and position-dependent learning effect into single-machine scheduling problems. The learning effect of a job is assumed to be a function of the sum of actual processing times of the jobs already processed and the job’s scheduled position. We show that under certain conditions, the problems with c...

This paper studies an application of the incline matrix theory in the dynamic analysis of incline-valued fuzzy bidirectional associative memories (L-FBAMs, for short). It is proved that the strong convergence and the strong stability of an L-FBAM are equivalent to the existence of indices and the convergence in finite steps of the product matrices...

For a class of uncertain nonlinear systems, a design method of robust fuzzy control systems with the state estimators is developed in this paper. This design method guarantees each local closed-loop systems poles within a specified disc and steady-state variances to be less than a set of given upper bounds. T-S fuzzy models with uncertainties are u...

This paper proves that the maximum order-index of n × n matrices over an arbitrary commutative incline equals (n − 1)2 + 1. This is an answer to an open problem “Compute the maximum order-index of a member of Mn(L)”, proposed by Cao, Kim and Roush in a monograph Incline Algebra and Applications, 1984, where Mn(L) is the set of all n × n matrices ov...

Purpose – The purpose of this paper is to study the relationship between the property of distributivity and Zadeh's operators. Design/methodology/approach – The results are established by studying the relationship between the condition of distributivity and Zadeh's operations. Findings – When the t‐conorm T* and t‐norm T are Zadeh's “intersection...

This work studies the problem of solving a sup-T composite finite fuzzy relation equation, where T is an infinitely distributive strong pseudo-t-norm. A criterion for the equation to have a solution is given. It is proved that if the equation is solvable then its solution set is determined by the greatest solution and a finite number of minimal sol...

A new fuzzy inference method, called a VWSI (variable weighted synthesis inference) method, is presented by applying the principle of variable weighted synthesis in factor spaces theory to fuzzy inference. The analysis for response abilities of fuzzy systems constructed by VWSI algorithms indicates that such fuzzy systems have a characteristic of i...

In this paper, the probability significance of fuzzy systems is revealed. It is pointed out that COG method, a defuzzification technique used commonly in fuzzy systems, is reasonable and is the optimal method in the sense of mean square. Based on different fuzzy implication operators, several typical probability distributions such as Zadeh distribu...

Inclines are additively idempotent semirings in which products are less than or equal to either factor. Thus they generalize Boolean algebra, fuzzy algebra and distributive lattice. This paper studies the nilpotent incline matrices in detail. It is proved that an incline matrix is nilpotent if and only if it has index and the zero vector is its uni...

This paper mainly deals with a new kind of fuzzy relation equations based on inner transformation. First, the inner projection of a fuzzy relation and the inner transformation by a fuzzy relation are introduced, and also what is a fuzzy relation equation based on inner transformation is described. Second, the discriminative matrix and the discrimin...

This paper shows by examples that Theorems 4.1–4.3, 4.5 in [Fuzzy Sets and Systems 132 (2002) 113] and Theorems 2.1, 2.2, 3.1 in [Fuzzy Sets and Systems 139 (2003) 673] about pseudo-t-norms and implication operators are false, and gives a necessary condition for them being true.

This paper investigates the semigroup of incline Hall matrices. An incline Hall matrix means an incline matrix which is greater than or equal to an incline invertible matrix. It is proved that any incline Hall matrix has index. Some necessary and sufficient conditions for an incline Hall matrix to be regular in the semigroup of incline Hall matrice...

Inclines are the additively idempotent semirings in which the products are less than or equal to factors. Thus inclines generalize Boolean algebra, fuzzy algebra and distributive lattice. And the Boolean matrices, the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices (i.e., the matrices over inclines). In...

Inclines are the additively idempotent semirings in which products are less than or equal to either factor. Thus inclines generalize Boolean algebra, fuzzy algebra and distributive lattice. This paper studies the standard eigenvectors of incline matrices (i.e., matrices over inclines) in detail. As the main results, for an incline matrix with index...

The structure of pan-operations, involving a pan-addition and a pan-multiplication which form a commutative isotonic semiring, as defined by Wang and Klir, are discussed. Based on semi-group theory, special properties of identity element and the idempotent of pan-addition and pan-multiplication, the case of a non-idempotent pan-addition are studied...

Inclines are the additively idempotent semirings in which products are less than or equal to factors. Thus inclines generalize Boolean algebra, fuzzy algebra and distributive lattice. And the Boolean matrices, the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices (i.e., the matrices over inclines).This pa...

Variable weights analysis and its applications in fuzzy systems and especially in fuzzy decision making are researched in detail in this paper. After the background of variable weights is summarized, an empirical variable weights formula is discussed. Based on the empirical formula, the principles of variable weights are explored, and an axiomatic...

Mathematical essence and structures of the feedforward neural networks are investigated in this paper. The interpolation mechanisms of the feedforward neural networks are explored. For example, the well-known result, namely, that a neural network is an universal approximator, can be concluded naturally from the interpolative representations. Finall...

Several interesting analytic and geometric properties of the fuzzy controllers are discussed. From these discussions, some kind of measure of nonlinearity of the fuzzy controllers is proposed. A detailed study of the geometric structures of the fuzzy control functions is also carried out and the result is compared with that of linear control.

Based on the results obtained in the earlier paper [1], the fuzzy clustering method based on perturbation (FCMBP) is further studied and several important results are obtained. First, since the fuzzy similarity matrix equation X2 = X is one of the most important basic components of the FCMBP approach, the structure of the solutions of this matrix e...

A kind of linearization method in fuzzy control system modeling is proposed, in order to deal with the nonlinear model with variable coefficients. The method can turn a nonlinear model with variable coefficients into a linear model with variable coefficients in the way that the membership functions of the fuzzy sets in fuzzy partitions of the unive...

Interpolation functions, which can represent fuzzy logic system (FLS) models, are investigated. Since interpolation is a basic tool for function approximation and since FLS has been shown to be a function approximator, interpolation functions can be established for these models. To establish and to explore these interpolations, some features, which...

Mathematical essence and structures of feedforward neural networks are researched in detail in this paper. First of all, interpolation mechanism of Feedforward neural networks is exposed, so we can more clearly understand why a feedforward network is of approximation. For example, a well-known conclusion for arbitrarily a continuous function, there...

A kind of stable adaptive fuzzy control of a nonlinear system is implemented based on the variable universe method proposed first by H.-X. Li [“The mathematical essence of fuzzy controls and fine fuzzy controllers”, in: P. P. Wang (ed.), Advances in machine intelligence and soft computing. Vol. IV. Durham: Bookwrights Press, 55–74 (1997)]. First of...

Fuzzy logic systems and feedforward neural networks are equivalent in essence. First, interpolation representations of fuzzy logic systems are introduced and several important conclusions are given. Then three important kinds of neural networks are defined, i. e. linear neural networks, rectangle wave neural networks and nonlinear neural networks....

A new idea, output-back fuzzy logic systems, is proposed. It is proved that outputback fuzzy logic systems must be equivalent to feedback neural networks. After the notion of generalized fuzzy logic systems is defined, which contains at least a typical fuzzy logic system and an output-back fuzzy logic system, one important conclusion is drawn that...

Demonstrates that fuzzy logic systems and feedforward neural networks are equivalent in essence. First, we introduce the concept of interpolation representations of fuzzy logic systems and several important conclusions. We then define mathematical models for rectangular wave neural networks and nonlinear neural networks. With this definition, we pr...

New extension principles in fuzzy set theory are proposed. Fuzzy set category by the use of these extension principles is defined. Finally, the rationality of the extension principles is proved.

The internal relations between fuzzy controllers and PID controllers are revealed. First, it is pointed out that a fuzzy controller with one input and one output is just a piecewise P controller. Then it is proved that a fuzzy controller with two inputs and one output is just a piecewise PD (or I) controller, with interaction between P and D (or PI...

Adaptive fuzzy controllers by means of variable universe are proposed based on interpolation forms of fuzzy control. First, monotonicity of control rules is defined, and it is proved that the monotonicity of interpolation functions of fuzzy control is equivalent to the monotonicity of control rules. This means that there is not any contradiction am...

The internal relationship between fuzzy controllers and PID controllers is discussed. First, we point out that a fuzzy controller with one input and one output is a piecewise P controller. Then it is proved that a fuzzy controller with two inputs and one output is a piecewise PD (or I) controller with mutual affection between P and D (or PI). final...

The fuzzy control algorithms used commonly at present are all regarded as some interpolation functions, which is in essence equivalent to discrete response functions to be fitted. This means that fuzzy control method is similar to finite element method in mathematical physics, which is a kind of direct manner or numerical method in control systems.