Asymptotically Necessary and Sufficient Quadratic Stability Conditions of T-S Fuzzy Systems Using Staircase Membership Function and Basic Inequality

Asymptotically necessary and sufficient quadratic stability conditions of Takagi-Sugeno (T-S) fuzzy systems are obtained by utilizing staircase membership functions and a basic inequality. The information of the membership functions is incorporated in the stability analysis by approximating the original continuous membership functions with staircase membership functions. The stability of the T-S fuzzy systems was investigated based on a quadratic Lyapunov function. The asymptotically necessary and sufficient stability conditions in terms of linear matrix inequalities were derived using a basic inequality. A fuzzy controller was also designed based on the stability results. The derivation process of the stability results is straightforward and easy to understand. Case studies confirmed the validity of the obtained stability results.

Ingredients-based Methodology and Fuzzy Logic Combined Short-Duration Heavy Rainfall Short-Range Forecasting:An Improved Scheme

Short-duration heavy rainfall(SHR),as delineated by the National Meteorological Center of the China Me-teorological Administration,is characterized by hourly rainfall amounts no less than 20.0 mm.SHR is one of the most common convective weather phenomena that can cause severe damage.Short-range forecasting of SHR is an important part of operational severe weather prediction.In the present study,an improved objective SHR forecasting scheme was developed by adopting the ingredients-based methodology and using the fuzzy logic approach.The 1.0°×1.0°National Centers for Environmental Prediction(NCEP)final analysis data and the ordinary rainfall(0.1-19.9 mm h-1)and SHR observational data from 411 stations were used in the improved scheme.The best lifted index,the total precipitable water,the 925 hPa specific humidity(Q 925),and the 925 hPa divergence(DIV 925)were selected as predictors based on objective analysis.Continuously distributed membership functions of predictors were obtained based on relative frequency analysis.The weights of predictors were also objectively determined.Experiments with a typhoon SHR case and a spring SHR case show that the main possible areas could be captured by the improved scheme.Verification of SHR forecasts within 96 hours with NCEP global forecasts 1.0°×1.0°data initiated at 08:00 Beijing Time during the warm seasons in 2015 show the results were improved from both deterministic and probabilistic perspectives.This study provides an objectively feasible choice for short-range guidance forecasts of SHR.The scheme can be applied to other convective phenomena.

Predictive functional control based on fuzzy T-S model for HVAC systems temperature control

In heating, ventilating and air-conditioning （HVAC） systems, there exist severe nonlinearity, time-varying nature, disturbances and uncertainties. A new predictive functional control based on Takagi-Sugeno （T-S） fuzzy model was proposed to control HVAC systems. The T-S fuzzy model of stabilized controlled process was obtained using the least squares method, then on the basis of global linear predictive model from T-S fuzzy model, the process was controlled by the predictive functional controller. Especially the feedback regulation part was developed to compensate uncertainties of fuzzy predictive model. Finally simulation test results in HVAC systems control applications showed that the proposed fuzzy model predictive functional control improves tracking effect and robustness. Compared with the conventional PID controller, this control strategy has the advantages of less overshoot and shorter setting time, etc.

T-S Fuzzy Stochastic Bilinear Model and Fuzzy Controller Design Based on Switching Piecewise Lyapunov Functions

Based on a piecewise quadratic lyapunov function （PQLF）, this paper presents stochastic stability analysis and synthesis methods for ItO and discrete T-S fuzzy bilinear stochastic systems. Two improved stochastic stability conditions have been established in terms of linear matrix inequalities （LMIs）. It is shown that the stability in the mean square for T-S fuzzy bilinear stochastic systems can be established if a PQLF can be constructed. Considering the established stability criterion, the controller can be designed by solving a set of （LMIs）, and the closed loop system is asymptotically stable in the mean square. Two illustrative examples are provided to demonstrate the effectiveness of the results proposed in this paper.

Fuzzy Set-Membership Filtering for Discrete-Time Nonlinear Systems

In this article,the problem of state estimation is addressed for discrete-time nonlinear systems subject to additive unknown-but-bounded noises by using fuzzy set-membership filtering.First,an improved T-S fuzzy model is introduced to achieve highly accurate approximation via an affine model under each fuzzy rule.Then,compared to traditional prediction-based ones,two types of fuzzy set-membership filters are proposed to effectively improve filtering performance,where the structure of both filters consists of two parts:prediction and filtering.Under the locally Lipschitz continuous condition of membership functions,unknown membership values in the estimation error system can be treated as multiplicative noises with respect to the estimation error.Real-time recursive algorithms are given to find the minimal ellipsoid containing the true state.Finally,the proposed optimization approaches are validated via numerical simulations of a one-dimensional and a three-dimensional discrete-time nonlinear systems.

Statistic PID Tracking Control for Non-Gaussian Stochastic Systems Based on T-S Fuzzy Model

A new robust proportional-integral-derivative （PID） tracking control framework is considered for stochastic systems with non-Gaussian variable based on B-spline neural network approximation and T-S fuzzy model identification. The tracked object is the statistical information of a given target probability density function （PDF）, rather than a deterministic signal. Following B-spline approximation to the integrated performance function, the concerned problem is transferred into the tracking of given weights. Different from the previous related works, the time delay T-S fuzzy models with the exogenous disturbances are applied to identify the nonlinear weighting dynamics. Meanwhile, the generalized PID controller structure and the improved convex linear matrix inequalities （LMI） algorithms are proposed to fulfil the tracking problem. Furthermore, in order to enhance the robust performance, the peak-to-peak measure index is applied to optimize the tracking performance. Simulations are given to demonstrate the efficiency of the proposed approach.

Fuzzy logic controller design with unevenly-distributed membership function for high performance chamber cooling system

Fuzzy logic controller adopting unevenly-distributed membership function was presented with the purpose of enhancing performance of the temperature control precision and robustness for the chamber cooling system.Histogram equalization and noise detection were performed to modify the evenly-distributed membership functions of error and error change rate into unevenly-distributed membership functions.Then,the experimental results with evenly and unevenly distributed membership functions were compared under the same outside environment conditions.The experimental results show that the steady-state error is reduced around 40% and the noise disturbance is rejected successfully even though noise range is 60% of the control precision range.The control precision is improved by reducing the steady-state error and the robustness is enhanced by rejecting noise disturbance through the fuzzy logic controller with unevenly-distributed membership function.Moreover,the system energy efficiency and lifetime of electronic expansion valve(EEV) installed in chamber cooling system are improved by adopting the unevenly-distributed membership function.

On Development of Fuzzy Controller: The Case of Gaussian and Triangular Membership Functions

In recent years, the use of Fuzzy set theory has been popularised for handling overlap domains in control engineering but this has mostly been within the context of triangular membership functions. In actual practice however, such domains are hardly triangular and in fact for most engineering applications the membership functions are usually Gaussian and sometimes cosine. In an earlier paper, we derived explicit Fourier series expressions for systematic and dynamic computation of grade of membership in the overlap and non-overlap regions of triangular Fuzzy sets. In another paper, we extended the methodology to cover cases of cosine, exponential and Gaussian Fuzzy sets by presenting explicit Fourier series representation for encoding fuzziness in the overlap and non-overlap domains of Fuzzy sets. This current paper presents the development of a “Fuzzy Controller” device, which incorporates the formal mathematical representation for computing grade of membership of Gaussian and triangular Fuzzy sets. It is shown that triangular approximation of Gaussian membership function in Fuzzy control can lead to wrong linguistic classification which may have adverse effects on operational and control decisions. The development of the Fuzzy controller demonstrates that the proposed technique can indeed be incorporated in engineering systems for dynamic and systematic computation of grade of membership in the overlap and non-overlap regions of Fuzzy sets;and thus provides a basis for the design of embedded Fuzzy controller for mission critical applications.

Stability and stabilization of discrete T-S fuzzy time-delay system based on maximal overlapped-rules group

The problems of stability and stabilization for the discrete Takagi-Sugeno（T-S） fuzzy time-delay system are investigated.By constructing a discrete piecewise Lyapunov-Krasovskii function（PLKF） in each maximal overlapped-rules group（MORG）,a new sufficient stability condition for the open-loop discrete T-S fuzzy time-delay system is proposed and proved.Then the systematic design of the fuzzy controller is investigated via the parallel distributed compensation control scheme,and a new stabilization condition for the closed-loop discrete T-S fuzzy time-delay system is proposed.The above two sufficient conditions only require finding common matrices in each MORG.Compared with the common Lyapunov-Krasovskii function（CLKF） approach and the fuzzy Lyapunov-Krasovskii function（FLKF） approach,these proposed sufficient conditions can not only overcome the defect of finding common matrices in the whole feasible region but also largely reduce the number of linear matrix inequalities to be solved.Finally,simulation examples show that the proposed PLKF approach is effective.

Optimization of Membership Function for Fuzzy Control Based on Genetic Algorithm and Its Applications

In this paper, a simple and practicable algorithm for optimization of membership function (MF) is proposed. As it is known that MF is very important in the fuzzy control. Unfortunately, to find, especially to optimize MF is always rather complex even difficult. So, to study and develop an effectual aglorithm for MF optimization is a good topic. Allow for the inner advantages of genetic algorithm (GA), it is adopted in the algorithm .The principle and executive procdeure are first presented. Then it is applied in the fuzzy control system of a typical plant. Results of real time run show that the control strategy is encouraging, and the developed algorithm is practicable.

A modeling approach for fuzzy programming with echelon form membership functions

In order to solve fuzzy mathematical programming with soft constraints,the initial models must first be converted into crisp models.Membership functions are employed to describe the fuzzy right-hand side parameters needed to achieve this conversion.In some cases,echelon form membership functions(EFMFs)are required to depict the actual fuzzy situation.However,due to their discrete properties,fuzzy programming problems with such membership functions cannot be modeled by traditional methods.Motivated by these challenges,this paper introduces a novel absolute value representation modeling approach to formulate fuzzy programming using EFMFs.This approach can translate a discrete model to a continuous one which can then be easily solved.Finally,by means of a numerical example,the effectiveness of our new approach is demonstrated.

Generating Type 2 Trapezoidal Fuzzy Membership Function Using Genetic Tuning

Fuzzy inference system(FIS)is a process of fuzzy logic reasoning to produce the output based on fuzzified inputs.The system starts with identifying input from data,applying the fuzziness to input using membership functions(MF),generating fuzzy rules for the fuzzy sets and obtaining the output.There are several types of input MFs which can be introduced in FIS,commonly chosen based on the type of real data,sensitivity of certain rule implied and computational limits.This paper focuses on the construction of interval type 2(IT2)trapezoidal shape MF from fuzzy C Means(FCM)that is used for fuzzification process of mamdani FIS.In the process,upper MF(UMF)and lower MF(LMF)of the MF need to be identified to get the range of the footprint of uncertainty(FOU).This paper proposes Genetic tuning process,which is a part of genetic algorithm(GA),to adjust parameters in order to improve the behavior of existing system,especially to enhance the accuracy of the system model.This novel process is a hybrid approach which produces Genetic Fuzzy System(GFS)that helps to enhance fuzzy classification problems and performance.The approach provides a new method for the construction and tuning process of the IT2 MF,based on the FCM outcomes.The result is compared to Gaussian shape IT2 MF and trapezoid IT2 MF generated by the classic GA method.It is shown that the proposed approach is able to outperform the mentioned benchmarked approaches.The work implies a wider range of IT2 MF types,constructed based on FCM outcomes,and an optimum generation of the FOU so that it can be implemented in practical applications such as prediction,analytics and rule-based solutions.

Energy Price Forecasting Through Novel Fuzzy Type-1 Membership Functions

Electricity price forecasting is a subset of energy and power forecasting that focuses on projecting commercial electricity market present and future prices.Electricity price forecasting have been a critical input to energy corporations’strategic decision-making systems over the last 15 years.Many strategies have been utilized for price forecasting in the past,however Artificial Intelligence Techniques(Fuzzy Logic and ANN)have proven to be more efficient than traditional techniques(Regression and Time Series).Fuzzy logic is an approach that uses membership functions(MF)and fuzzy inference model to forecast future electricity prices.Fuzzy c-means(FCM)is one of the popular clustering approach for generating fuzzy membership functions.However,the fuzzy c-means algorithm is limited to producing only one type of MFs,Gaussian MF.The generation of various fuzzy membership functions is critical since it allows for more efficient and optimal problem solutions.As a result,for the best and most improved results for electricity price forecasting,an approach to generate multiple type-1 fuzzy MFs using FCM algorithm is required.Therefore,the objective of this paper is to propose an approach for generating type-1 fuzzy triangular and trapezoidal MFs using FCM algorithm to overcome the limitations of the FCM algorithm.The approach is used to compute and improve forecasting accuracy for electricity prices,where Australian Energy Market Operator(AEMO)data is used.The results show that the proposed approach of using FCM to generate type-1 fuzzy MFs is effective and can be adopted.

Properties of Fuzzy Entropy Based on the Shape Change of Membership Function

Modification of a fuzzy partition often leads to the change of fuzziness of a fuzzy system. Researches on the change of fuzzy entropy of a fuzzy set, responding to shape alteration of membership function, therefore, play a significant role in analysis of the change of fuzziness of a fuzzy system because a fuzzy partition consists of a set of fuzzy sets which satisfy some special constraints. This paper has shown several results about entropy changes of a fuzzy set. First, the entropies of two same type of fuzzy sets have a constant proportional relationship which depends on the ratio of the sizes of their support intervals. Second, as for Triangular Fuzzy Numbers （TFNs）, the entropies of any two TFNs which can not be always the same type, also, have a constant proportional relationship which depends on the ratio of the sizes of their support intervals. Hence, any two TFNs with the same sizes of support intervals have the same entropies. Third, concerning two Triangular Fuzzy Sets （TFSs） with same sizes of support intervals and different heights, the relationship of their entropies lies on their height. Finally, we point it out a mistake that Chen＇s assertion that the entropy of resultant fuzzy set of elevation operation is directly to that of the original one while elevation factor just acts as a propartional factor. These results should contribute to the analysis and design of a fuzzy system.

New Approaches to the Prognosis and Diagnosis of Breast Cancer Using Fuzzy Expert Systems

Breast cancer remains a significant global health challenge, necessitating effective early detection and prognosis to enhance patient outcomes. Current diagnostic methods, including mammography and MRI, suffer from limitations such as uncertainty and imprecise data, leading to late-stage diagnoses. To address this, various expert systems have been developed, but many rely on type-1 fuzzy logic and lack mobile-based applications for data collection and feedback to healthcare practitioners. This research investigates the development of an Enhanced Mobile-based Fuzzy Expert system (EMFES) for breast cancer pre-growth prognosis. The study explores the use of type-2 fuzzy logic to enhance accuracy and model uncertainty effectively. Additionally, it evaluates the advantages of employing the python programming language over java for implementation and considers specific risk factors for data collection. The research aims to dynamically generate fuzzy rules, adapting to evolving breast cancer research and patient data. Key research questions focus on the comparative effectiveness of type-2 fuzzy logic, the handling of uncertainty and imprecise data, the integration of mobile-based features, the choice of programming language, and the creation of dynamic fuzzy rules. Furthermore, the study examines the differences between the Mamdani Inference System and the Sugeno Fuzzy Inference method and explores challenges and opportunities in deploying the EMFES on mobile devices. The research identifies a critical gap in existing breast cancer diagnostic systems, emphasizing the need for a comprehensive, mobile-enabled, and adaptable solution by developing an EMFES that leverages Type-2 fuzzy logic, the Sugeno Inference Algorithm, Python Programming, and dynamic fuzzy rule generation. This study seeks to enhance early breast cancer detection and ultimately reduce breast cancer-related mortality.

Describing Fuzzy Membership Function and Detecting the Outlier by Using Five Number Summary of Data

One of the most important activities in data science is defining a membership function in fuzzy system. Although there are few ways to describe membership function like artificial neural networks, genetic algorithms etc.;they are very complex and time consuming. On the other hand, the presence of outlier in a data set produces deceptive results in the modeling. So it is important to detect and eliminate them to prevent their negative effect on the modeling. This paper describes a new and simple way of constructing fuzzy membership function by using five-number summary of a data set. Five states membership function can be created in this new method. At the same time, if there is any outlier in the data set, it can be detected with the help of this method. Usually box plot is used to identify the outliers of a data set. So along with the new approach, the box plot has also been drawn so that it is understood that the results obtained in the new method are accurate. Several real life examples and their analysis have been discussed with graph to demonstrate the potential of the proposed method. The results obtained show that the proposed method has given good results. In the case of outlier, the proposed method and the box plot method have shown similar results. Primary advantage of this new procedure is that it is not as expensive as neural networks, and genetic algorithms.

Observer-Based Fuzzy Control Design for Discrete-Time T-S Fuzzy Bilinear Stochastic,Systems with Infinite-Distributed Delays

This paper is concerned with the problem of observer-based fuzzy control design for discrete-time T-S fuzzy bilinear stochastic systems with infinite-distributed delays. Based on the piecewise quadratic Lyapunov functional （PQLF）, the fuzzy observer-basedcontrollers are designed for T-S fuzzy bilinear stochastic systems. It is shown that the stability in the mean square for discrete T-S fuzzy bilinear stochastic systems can be established if there exists a set of PQLF can be constructed and the fuzzy observer-based controller can be obtained by solving a set of nonlinear minimization problem involving linear matrix inequalities （LMIs） constraints. An iterative algorithm making use of sequential linear programming matrix method （SLPMM） to derive a single-step LMI condition for fuzzy observer-based control design. Finally, an illustrative example is provided to demonstrate the effectiveness of the results proposed in this paper.

基于线性T-S模型的模糊系统放宽的稳定性条件

研究了基于 T-S 线性模型的模糊控制系统稳定性,对放宽的稳定性条件进行了分析,给出并证明了一个不同的放宽稳定性的充分条件。利用现代控制理论中的状态反馈方法,结合并行分布补偿的思想,可以借助求解一组线性矩阵不等式(LMI)等来分析闭环模糊控制系统的稳定性。

基于T-S模型的稳定自适应FNN控制器的设计

对一类不确定非线性系统 ,提出一种基于 T- S模型的自适应 FNN控制器 .首先用权值固定的 FNN作为非线性系统的近似模型 ,然后再应用自适应 FNN逼近建模误差 ,并引入滑模项增加控制器的鲁棒性 .通过稳定性理论设计自适应律 ,保证了系统的全局稳定 。