Xiao-Jun Zeng

Approximation theory of fuzzy systems/spl minus/SISO case

In this paper, the approximation properties of MIMO fuzzy systems generated by the product inference are discussed. We first give an analysis of fuzzy basic functions (FBFs) and present several properties of FBFs. Based on these properties of FBFs, we obtain several basic approximation properties of fuzzy systems: 1) basic approximation property which reveals the basic approximation mechanism of fuzzy systems; 2) uniform approximation bounds which give the uniform approximation bounds between the desired (control or decision) functions and fuzzy systems; 3) uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions; and 4) universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect. The similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds. >

Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making

The hesitant fuzzy linguistic term sets (HFLTSs), which can be used to represent an expert’s hesitant preferences when assessing a linguistic variable, increase the flexibility of eliciting and representing linguistic information. The HFLTSs have attracted a lot of attention recently due to their distinguished power and efficiency in representing uncertainty and vagueness within the process of decision making. To enhance and extend the applicability of HFLTSs, this paper investigates and develops different types of distance and similarity measures for HFLTSs. The paper first proposes a family of distance and similarity measures between two HFLTSs. Then a variety of weighted or ordered weighted distance and similarity measures between two collections of HFLTSs are proposed and analyzed for discrete and continuous cases respectively. After that, the application of these measures to multi-criteria decision making problems is given. Based on the proposed distance and similarity measures, the satisfaction degrees for different alternatives are established and are then used to rank alternatives in multi-criteria decision making. Finally a practical example concerning the evaluation of the quality of movies is given to illustrate the applicability and advantage of the proposed approach and the differences between the proposed distance and similarity measures.

Approximation accuracy analysis of fuzzy systems as function approximators

This paper establishes the approximation error bounds for various classes of fuzzy systems (i.e., fuzzy systems generated by different inferential and defuzzification methods). Based on these bounds, the approximation accuracy of various classes of fuzzy systems is analyzed and compared. It is seen that the class of fuzzy systems generated by the product inference and the center-average defuzzifier has better approximation accuracy and properties than the class of fuzzy systems generated by the min inference and the center-average defuzzifier, and the class of fuzzy systems defuzzified by the MoM defuzzifier. In addition, it is proved that fuzzy systems can represent any linear and multilinear function and explicit expressions of fuzzy systems generated by the MoM defuzzified method are given.

Qualitative decision making with correlation coefficients of hesitant fuzzy linguistic term sets

The hesitant fuzzy linguistic term set (HFLTS) is a new and flexible tool in representing hesitant qualitative information in decision making. Correlation measures and correlation coefficients have been applied widely in many research domains and practical fields. This paper focuses on the correlation measures and correlation coefficients of HFLTSs. To start the investigation, the definition of HFLTS is improved and the concept of hesitant fuzzy linguistic element (HFLE) is introduced. Motivated by the idea of traditional correlation coefficients of fuzzy sets, intuitionistic fuzzy sets and hesitant fuzzy sets, several different types of correlation coefficients for HFLTSs are proposed. The prominent properties of these correlation coefficients are then investigated. In addition, considering that different HFLEs may have different weights, the weighted correlation coefficients and ordered weighted correlation coefficients are further investigated. Finally, an application example concerning the traditional Chinese medical diagnosis is given to illustrate the applicability and validation of the proposed correlation coefficients of HFLTSs in the process of qualitative decision making.

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 Kolmogorovs 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

T–S-Fuzzy-Model-Based Approximation and Controller Design for General Nonlinear Systems

This paper presents a novel approach to control general nonlinear systems based on Takagi-Sugeno (T-S) fuzzy dynamic models. It is first shown that a general nonlinear system can be approximated by a generalized T-S fuzzy model to any degree of accuracy on any compact set. It is then shown that the stabilization problem of the general nonlinear system can be solved as a robust stabilization problem of the developed T-S fuzzy system with the approximation errors as the uncertainty term. Based on a piecewise quadratic Lyapunov function, the robust semiglobal stabilization and H∞ control of the general nonlinear system are formulated in the form of linear matrix inequalities. Simulation results are provided to illustrate the effectiveness of the proposed approaches.This paper presents a novel approach to control general nonlinear systems based on Takagi-Sugeno (T-S) fuzzy dynamic models. It is first shown that a general nonlinear system can be approximated by a generalized T-S fuzzy model to any degree of accuracy on any compact set. It is then shown that the stabilization problem of the general nonlinear system can be solved as a robust stabilization problem of the developed T-S fuzzy system with the approximation errors as the uncertainty term. Based on a piecewise quadratic Lyapunov function, the robust semiglobal stabilization and H∞ control of the general nonlinear system are formulated in the form of linear matrix inequalities. Simulation results are provided to illustrate the effectiveness of the proposed approaches.

Short-Term and Midterm Load Forecasting Using a Bilevel Optimization Model

During the last decade, neural networks have emerged as one of the most powerful and accurate nonlinear models for load forecasting. However, using neural networks requires users to have in-depth knowledge to determine the model structure and parameters, which limits their wide application. To overcome this weakness, this paper proposes an integrated approach which combines a self-organizing fuzzy neural network (SOFNN) learning method with a bilevel optimization method. SOFNNs can automatically determine both the model structure and parameters, while the bilevel optimization method automatically selects the best pre-training parameters to ensure that the best fuzzy neural networks be identified. Therefore, the proposed approach is able to automatically identify the best fuzzy neural network for a given forecasting task and is much easier to use in practice. The proposed approach is tested on real-load data from the Southern Power Network of Hebei Province, China, and on the EUNITE competition data. Results show the proposed approach improves existing load forecasting models.

An enhanced consensus reaching process in group decision making with intuitionistic fuzzy preference relations

Group decision making (GDM) with intuitionistic fuzzy preference relations (IFPRs) has been an important and active research topic recently, in which one of the most challenging issues is how to reach the group consensus so as to get the best decision. As the uniform consensus is often unachievable in practice, in order to achieve the consensus, the existing method needs to remove the experts with the most different opinions from the decision group. It has two drawbacks: the first is the loss of the valuable judgments and opinions of the removed experts. This is especially harmful in practice where most experts or decision makers often have the biased knowledge in the sense of in-depth expertise in some aspects and naive views in other aspects. The second is demotivating the experts in GDM. To overcome these weaknesses in the existing method, this paper presents an enhanced consensus reaching process for GDM with IFPRs, which only removes some opinions of an expert for alternative(s) instead of removing the expert from the decision group. A numerical example concerning the selection of outstanding PhD students for China Scholarship Council is given to show the feasibility and effectiveness of the enhanced consensus reaching process.

Decomposition property of fuzzy systems and its applications

This paper presents the decomposition property of fuzzy systems using a simple, constructive, decomposition procedure. That is, by properly dividing the input space into sub-input spaces, a general fuzzy system is decomposed into several sub-fuzzy systems which are the simplest fuzzy systems in the sub-input spaces. Based on the decomposition property of fuzzy systems, the analysis of fuzzy systems can be divided into two steps: first, analyze the properties of the simplest fuzzy systems, and then, use the decomposition property to extend the results to general fuzzy systems. Using this idea, two applications of the decomposition property are given. The first is the application to the representation capability analysis of fuzzy systems. The second is the application to the analysis of a class of nonlinear control systems. Then, based on the piecewise affine fuzzy-system model, the existence condition and the design of a stable control for a class of single-input single-output (SISO) nonlinear systems are presented.

Output tracking of constrained nonlinear processes with offset-free input-to-state stable fuzzy predictive control

This paper develops an efficient offset-free output feedback predictive control approach to nonlinear processes based on their approximate fuzzy models as well as an integrating disturbance model. The estimated disturbance signals account for all the plant-model mismatch and unmodeled plant disturbances. An augmented piecewise observer, constructed by solving some linear matrix inequalities, is used to estimate the system states and the lumped disturbances. Based on the reference from an online constrained target generator, the fuzzy model predictive control law can be easily obtained by solving a convex semi-definite programming optimization problem subject to several linear matrix inequalities. The resulting closed-loop system is guaranteed to be input-to-state stable even in the presence of observer estimation error. The zero offset output tracking property of the proposed control approach is proved, and subsequently demonstrated by the simulation results on a strongly nonlinear benchmark plant.