Some convergence theorems of fuzzy concave integral on fuzzyσ-algebra

In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some important properties of such integral are discussed.Finally,various kinds of convergence theorems of a sequence of fuzzy concave integrals are proved.

Fuzzy Henstock-Kurzweil Triple Integral on a Type 1 Quasi-Fuzzy Parallelepipedal Domain

In this article, we propose by using the Hausdorff distance Simpson’s rule for the triple integral of a fuzzy-valued function and the error bound of this method, one of the variables of which is fuzzy. In addition, thin δ-fine partitions are introduced. The integration domain is a quasi-fuzzy parallelipiped. A numerical example is presented in order to show the application and the significance of the method.

Robust Fuzzy Tracking Control for Nonlinear Networked Control Systems with Integral Quadratic Constraints

This paper investigates the robust tracking control problcm for a class of nonlinear networked control systems （NCSs） using the Takagi-Sugeno （T-S） fuzzy model approach. Based on a time-varying delay system transformed from the NCSs, an augmented Lyapunov function containing more useful information is constructed. A less conservative sufficient condition is established such that the closed-loop systems stability and time-domain integral quadratic constraints （IQCs） are satisfied while both time-varying network- induced delays and packet losses are taken into account. The fuzzy tracking controllers design scheme is derived in terms of linear matrix inequalities （LMIs） and parallel distributed compensation （PDC）. Furthermore, robust stabilization criterion for nonlinear NCSs is given as an extension of the tracking control result. Finally, numerical simulations are provided to illustrate the effectiveness and merits of the proposed method.

Emotional inference by means of Choquet integral and λ-fuzzy measurement in consideration of ambiguity of human mentality

Research on human emotions has started to address psychological aspects of human nature and has advanced to the point of designing various models that represent them quantitatively and systematically. Based on the findings, a method is suggested for emotional space formation and emotional inference that enhance the quality and maximize the reality of emotion-based personalized services. In consideration of the subjective tendencies of individuals, AHP was adopted for the quantitative evaluation of human emotions, based on which an emotional space remodeling method is suggested in reference to the emotional model of Thayer and Plutchik, which takes into account personal emotions. In addition, Sugeno fuzzy inference, fuzzy measures, and Choquet integral were adopted for emotional inference in the remodeled personalized emotional space model. Its performance was evaluated through an experiment. Fourteen cases were analyzed with 4.0 and higher evaluation value of emotions inferred, for the evaluation of emotional similarity, through the case studies of 17 kinds of emotional inference methods. Matching results per inference method in ten cases accounting for 71% are confirmed. It is also found that the remaining two cases are inferred as adjoining emotion in the same section. In this manner, the similarity of inference results is verified.

Fuzzy Integral Based Information Fusion for Water Quality Monitoring Using Remote Sensing Data

To improve the monitoring precision of lake chlorophyll a (Chl-a), this paper presents a fusion method based on Choquet Fuzzy Integral (CFI) to estimate the Chl-a concentration. A group of BPNN models are designed. The output of multiple BPNN model is fused by the CFI. Meanwhile, to resolve the over-fitting problem caused by a small number of training sets, we design an algorithm that fully considers neighbor sampling information. A classification experiment of the Chl-a concentration of the Taihu Lake is conducted. The result shows that, the proposed approach is superior to the classification using a single neural network classifier, and the CFI fusion method has higher identification accuracy.

Fuzzy integral based measurement and assessment of medium and small enterprises'product innovation

A fuzzy integral based way of measurement assessment has been established by using linguistic varia-bles and combining fuzzy integral with hierarchy analysis for measurement of medium and small enterprises'product innovation. The conclusions drawn from analyses made with 20 medium and small enterprises providebases for governments to formulate applicable policies and for medium and small enterprises to enhance theirproduct innovation.

DESIGNING OF TWO NEW FUZZY CONTROLLERS WITH ERROR INTEGRAL AND THEIR APPLICATIONS IN FOLLOWING CONTROL OF WARSHIP-GUNS

Two new fuzzy controllers are designed, one's control rules are adjusted byerror integral, another are with feedforward of error integral. If only the control rules andparameters are selected felicitously, the static errors of the system will be eliminated. The testresults show that the controllers can assuredly improve static performances of the controlledsystem.

Supplier Evaluation Model Based on Fuzzy Measures and Choquet Integral

To identify interactions among evaluation criteria and describe their importance,a new identification method making use of a fuzzy measure is presented.The relative weight and interaction degree of every evaluation criteria pair are obtained by using the diamond pairwise comparison method.Based on comparison results,the maximum eigenvector method of analytic hierarchy process （AHP）,the hierarchical clustering method,and the phi（s） transformation are utilized to generate values of the fuzzy measure for each subset of the evaluation criterion set.Overall evaluation on each supplier is aggregated by Choquet integral with respect to the fuzzy measure.Finally,an illustrative example demonstrates the practical feasibility and validity of the proposed method.

Lebesgues-Stieltjes Integrals of Fuzzy Stochastic Processes with Respect to Finite Variation Processes

Let be a fuzzy stochastic process and be a real valued finite variation process. We define the Lebesgue-Stieltjes integral denoted by for each by using the selection method, which is direct, nature and different from the indirect definition appearing in some references. We shall show that this kind of integral is also measurable, continuous in time t and bounded a.s. under the Hausdorff metric.

Unique Solution of Integral Equations via Intuitionistic Extended Fuzzy b-Metric-Like Spaces

In this manuscript,our goal is to introduce the notion of intuitionistic extended fuzzy b-metric-like spaces.We establish some fixed point theorems in this setting.Also,we plot some graphs of an example of obtained result for better understanding.We use the concepts of continuous triangular norms and continuous triangular conorms in an intuitionistic fuzzy metric-like space.Triangular norms are used to generalize with the probability distribution of triangle inequality in metric space conditions.Triangular conorms are known as dual operations of triangular norms.The obtained results boost the approaches of existing ones in the literature and are supported by some examples and applications.

Adaptive fuzzy integral sliding mode pressure control for cutter feeding system of trench cutter

A nonlinear pressure controller was presented to track desired feeding pressure for the cutter feeding system(CFS) of trench cutter(TC) in the presence of unknown external disturbances.The feeding pressure control of CFS is subjected to unknown load characteristics of rock or soil; in addition,the geological condition is time-varying.Due to the complex load characteristics of rock or soil,the feeding velocity of TC is related to geological conditions.What is worse,its dynamic model is subjected to uncertainties and its function is unknown.To deal with the particular characteristics of CFS,a novel adaptive fuzzy integral sliding mode control(AFISMC) was designed for feeding pressure control of CFS,which combines the robust characteristics of an integral sliding mode controller and the adaptive adjusting characteristics of an adaptive fuzzy controller.The AFISMC feeding pressure controller is synthesized using the backstepping technique.The stability of the overall closed-loop system consisting of the adaptive fuzzy inference system,integral sliding mode controller and the cutter feeding system is proved using Lyapunov theory.Experiments are conducted on a TC test bench with the AFISMC under different operating conditions.The experimental results demonstrate that the proposed AFISMC feeding pressure controller for CFS gives a superior and robust pressure tracking performance with maximum pressure tracking error within ?0.3 MPa.

Solution and Analysis of the Fuzzy Volterra Integral Equations viaHomotopy Analysis Method

Homotopy Analysis Method(HAM)is semi-analytic method to solve the linear and nonlinear mathematical models which can be used to obtain the approximate solution.The HAM includes an auxiliary parameter,which is an efficient way to examine and analyze the accuracy of linear and nonlinear problems.The main aim of this work is to explore the approximate solutions of fuzzy Volterra integral equations(both linear and nonlinear)with a separable kernel via HAM.This method provides a reliable way to ensure the convergence of the approximation series.A new general form of HAM is presented and analyzed in the fuzzy domain.A qualitative convergence analysis based on the graphical method of a fuzzy HAM is discussed.The solutions sought by the proposed method show that the HAM is easy to implement and computationally quite attractive.Some solutions of fuzzy second kind Volterra integral equations are solved as numerical examples to show the potential of the method.The results also show that HAM provides an easy way to control and modify the convergence area in order to obtain accurate solutions.

On AP-Henstock Integrals of Interval-Valued Functions and Fuzzy-Number-Valued Functions

In 2000, Wu and Gong [1] introduced the thought of the Henstock integrals of inter-valvalued functions and fuzzy-number-valued functions and obtained a number of their properties. The aim of this paper is to introduce the thought of the AP- Henstock integrals of interval-valued functions and fuzzy-number-valued functions which are extensions of [1] and investigate a number of their properties.

Fuzzification of Feynman Path Integral and Its Effect on Field Theory and Quantum Gravity—Reformation and Redevelopment of Quantum Theory

The quantum probability theory of fuzzy event is suggested by using the idea and method of fuzzy mathematics, giving the form of fuzzy event path integral, membership degree amplitude, fuzzy field function, Green function, physical quantity and fuzzy diagram. This theory reforms quantum mechanics, making the later become its special case. This theory breaks unitarity, gauge invariance, probability conservation and information conservation, making these principles become approximate ones under certain conditions. This new theory, which needs no renormalization and can naturally give meaningful results which are in accordance with the experiments, is the proper theory to describe microscopic high-speed phenomenon, whereas quantum mechanics is only a proper theory to describe microscopic low-speed phenomenon. This theory is not divergent under the condition of there being no renormalization and infinitely many offsetting terms, thereby it can become the theoretical framework required for the quantization of gravity.

Fuzzy Henstock-Kurzweil Triple Integral

In this article, we use the Hausdorf distance to treat triple Simpson’s rule of the Henstock triple integral of a fuzzy valued function as well as the error bound of the method. We also introduce δ-fine subdivisions for a Henstock triple integral and numerical example is presented in order to show the application and the consequence of the method.

On Henstock-Stieltjes Integrals of Interval-Valued Functions and Fuzzy-Number-Valued Functions

In this paper we introduce the notion of the Henstock-Stieltjes (HS) integrals of interval-valued functions and fuzzy-number-valued functions and discuss some of their properties.

Fuzzy Integral Theory and Its Application in Fault Intelligent Integrated Diagnosis

In this paper, by employing the idea of 'dispersion first, then concentration'' adhering to the biological perceptual system, an idea of the multi-symptom-domain based fault consensus diagnosis is developed. From the point of group decision-making, the method based on neural networks to realize this diagnosis idea is studied, and a particular multi-symptom-domain based diagnosis strategy is proposed, which is based on fuzzy integral theory. Finally, a case study is given. The research results show that the proposed diagnosis strategy is available and more efficient than conventional methods.

Selection of the best initial orbital elements of satellite based on fuzzy integration evaluation method

The fuzzy integration evaluation method （FIEM） is studied in order to select the best orbital elements from the multi-group initial orbits determined by a satellite TT＆C （Tracking, Telemetry and Control） center with all kinds of data sources. By employing FIEM together with the experience of TT＆C experts, the index system to evaluate the selection of the best initial orbits is established after the data sources and orbit determination theories are studied. Besides, the concrete steps in employing the method are presented. Moreover, by taking the objects to be evaluated as evaluation experts, the problem of how to generate evaluation matrices is solved. Through practical application, the method to select the best initial orbital elements has been proved to be flexible and effective The originality of the method is to find a new evaluation criterion （comparing the actually tracked orbits） replacing the traditional one （comparing the nominal orbits） for selecting the best orbital elements.

Integrated guidance and control of guided projectile with multiple constraints based on fuzzy adaptive and dynamic surface

Based on fuzzy adaptive and dynamic surface(FADS),an integrated guidance and control(IGC)approach was proposed for large caliber naval gun guided projectile,which was robust to target maneuver,canard dynamic characteristics,and multiple constraints,such as impact angle,limited measurement of line of sight(LOS)angle rate and nonlinear saturation of canard deflection.Initially,a strict feedback cascade model of IGC in longitudinal plane was established,and extended state observer(ESO)was designed to estimate LOS angle rate and uncertain disturbances with unknown boundary inside and outside of system,including aerodynamic parameters perturbation,target maneuver and model errors.Secondly,aiming at zeroing LOS angle tracking error and LOS angle rate in finite time,a nonsingular terminal sliding mode(NTSM)was designed with adaptive exponential reaching law.Furthermore,combining with dynamic surface,which prevented the complex differential of virtual control laws,the fuzzy adaptive systems were designed to approximate observation errors of uncertain disturbances and to reduce chatter of control law.Finally,the adaptive Nussbaum gain function was introduced to compensate nonlinear saturation of canard deflection.The LOS angle tracking error and LOS angle rate were convergent in finite time and whole system states were uniform ultimately bounded,rigorously proven by Lyapunov stability theory.Hardware-in-the-loop simulation(HILS)and digital simulation experiments both showed FADS provided guided projectile with good guidance performance while striking targets with different maneuvering forms.

Existence of supremum and infimum in E^-and relations of (M) integral and (G) integral

Proves the existence of supremum and infimum in the noncompact fuzzy number space E ～, redefines E ～ valued (M) integral, and then discusses the relations between (M) integral and (G) integral of E ～ valued functions.