Normalized Membership Functions Research Articles

АНАЛИЗ МОДЕЛЕЙ АДАПТИВНЫХ НЕЙРО-НЕЧЕТКИХ СИСТЕМ

The article deals with the study of basic methods for models of adaptive neuro-fuzzy systems. Based on the analysis, the strengths of neural networks and fuzzy logic were found, that became powerful tools for solving complex modeling and forecasting issues. There is studying and analyzing the adaptive neural network, which is a class of neural networks that have the ability to change their structure and parameters in the process of learning and adaptation to new data and conditions and besides the article studies the Gaussian membership function, also known as the normal membership function or the Gauss-type membership function, which is a valuable tool in the f ield of fuzzy logic and fuzzy systems. The paper provides as well an analysis of the generalized Bell membership function, also known as the Bell type membership function or Bell function, which plays an important role in the field of fuzzy logic and fuzzy systems. Furthermore it analyzes the Tsukamoto model, which is one of the main models of fuzzy logic. The author opted to choose the Co-Active Neuro-Fuzzy Inference System model, which is an adaptive neuro-fuzzy system that combines neural networks and fuzzy logic for processing data with uncertainty and fuzziness. With the further implementation of the combined model based on the abovelisted models based on the STL of the C++ language, thus the neural network model is obtained, the model with versatility, that is achieved by using a combination of those models. That will facilitate its easy modification and adaptation to various tasks.

Fuzzy evaluation of substation switching misoperation risk based on CRITIC-G1 method

In order to prevent the risk of misoperation in the process of substation switching operation more effectively, a fuzzy evaluation method based on the combination weighting model of CRITIC-G1 method is proposed. Firstly, on the basis of taking into account the effectiveness and feasibility, the key factors affecting the safety and correctness of the substation switching operation are comprehensively considered, and the risk evaluation index system of the substation switching misoperation is established; Then, the index volatility and conflict obtained by the CRITIC method is used to modified the G1 subjective weighting method, and the weight value is calculated; Furthermore, the normal membership function is introduced to realize the fuzzy evaluation. Finally, 5 substations belonging to different operation teams are taken as examples to verify the rationality and effectiveness of the proposed method.

New results on Hessian matrices and stabilization for stochastic T–S models via line integral

This paper is concerned with the stability analysis and stabilization for Itô stochastic T–S models via the line integral approach. Unlike the deterministic case, stochastic stability analysis of this model needs to handle the Hessian matrix of the line integral function. Since the Hessian matrix is dependent on the partial derivatives of membership-based functions, representing and estimating it are complicated and challenging and no relevant result has been reported so far. The novel representation of the Hessian matrix is skillfully provided which includes a part of the partial derivatives of membership-based functions. By giving an assumption on the partial derivatives of normalized membership functions, together with the property of matrix with rank one, an upper bound of the symmetric sum of partial derivative dependent part of the Hessian matrix is achieved. Thus a more general result of stochastic stability for the underlying model can be established by the line integral function. In terms of the obtained condition and a decoupling technique, a convergent cone complementarity linearization-like algorithm is provided for the controller design. Numerical examples are given to illustrate the effectiveness of the proposed scheme.

Skewed normal cloud modified whale optimization algorithm for degree reduction of S-λ curves

Whale optimization algorithm (WOA) is a new meta-heuristic algorithm for mathematical description of the foraging behavior of whales. Cloud model (CM) is an effective model to describe cognitive uncertainty, at the same time, it can comprehensively describe the randomness and fuzziness in uncertain phenomena. Skew normal distribution (SND) can better describe the biological behavior situation when the environment changes. In this paper, a new skew normal cloud model (SNCM) is established in the first place by combining the CM with the SND and the skew normal membership function to describe the fuzziness and randomness of environmental change. Secondly,because of the randomness and fuzziness of the whales’ foraging behavior, in order to improve the exploration and exploitation ability of WOA,an emendatory WOA based on the skew normal cloud (SNC) is proposed through the use of the SNCM to modify the shrinking encircling and spiral update strategies of WOA, and the adaptive position and skewness parameters in the SNC are designed to increase the exploration capability in prophase and the exploitation capability in anaphase. Lastly,the experimental results of the complex CEC2017 test set verify the effectiveness of the emendatory WOA under different strategies and different dimensions and different improved WOA algorithm and other representative heuristic algorithms. Three degree reduction of S − λ curves verify the practicability of the the modified WOA in the field of curve design.

A Fuzzy Lyapunov Function Approach to Positive Ll Observer Design for Positive Fuzzy Semi-Markovian Switching Systems With Its Application

This paper concerns a positive G1 observer for positive nonlinear semi-Markovian switching systems (MSSs) via the expansion of Taylor formula and the fuzzy Lyapunov function approach, in which semi-Markovian switching parameters, positivity, Takagi-Sugeno (T-S) fuzzy, and external disturbance are all considered in a unified framework. A fuzzy Lyapunov function approach with less conservativeness is introduced into the research of positive systems. In the system under consideration, positive S-MSSs with the semi-Markovian process can describe more complex systems in a practical control process. The main motivation of this paper is that the practical system subject to positivity and abrupt changes can be described by positive nonlinear S-MSSs, which always needs to consider the external disturbance. First, by using the normalized membership function approach, positive nonlinear S-MSSs can be represented by local positive T-S fuzzy S-MSSs. Second, by constructing the fuzzy Lyapunov function, some sufficient conditions are proposed for stochastic stability and G1-gain performance analysis, respectively. Then, a positive G1 observer in a novel standard linearprogramming condition is designed to guarantee the resulting closed-loop augmented system is positive and stochastically stable with a required G1-gain performance. Finally, a practical example about the epidemiological model is introduced to show the effectiveness of the main theory.

Filter for Positive Stochastic Nonlinear Switching Systems With Phase-Type Semi-Markov Parameters and Application

In this article, the issue of positive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {L}_{1}$ </tex-math></inline-formula> filter design is investigated for positive nonlinear stochastic switching systems subject to the phase-type semi-Markov jump process. Many complicated factors, such as semi-Markov jump parameters, positivity, T–S fuzzy strategy, and external disturbance, are taken into consideration. Practical systems under positivity constraint conditions and unpredictable structural changes are characterized by positive semi-Markov jump systems (S-MJSs). First, by the key properties of the supplementary variable and the plant transformation technique, phase-type S-MJSs are transformed into Markov jump systems (MJSs), which means that, to an extent, these two kinds of stochastic switching systems are mutually represented. Second, with the help of the normalized membership function, the associated nonlinear MJSs are transformed into the local linear MJSs with specific T–S fuzzy rules. Third, by choosing the linear copositive Lyapunov function (LCLF), stochastic stability (SSY) criteria are given for the corresponding system with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {L}_{1}$ </tex-math></inline-formula> performance. Some solvability conditions for positive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {L}_{1}$ </tex-math></inline-formula> filter are constructed under a linear programming framework. Finally, an epidemiological model illustrates the effectiveness of the theoretical findings.

Fuzzy Functional Observer-Based Finite-Time Adaptive Sliding-Mode Control for Nonlinear Systems With Matched Uncertainties

This article is concerned with the fuzzy functional observer-based finite-time adaptive sliding-mode control for nonlinear systems with the partly unmeasurable states and some matched uncertainties. First, considering that the upper bound of the uncertain function exists but unknown, a fuzzy functional observer (FFO) with an adaptive compensator is constructed. Second, an FFO-based fuzzy integral sliding mode controller (ISMCr) is designed such that the closed-loop fuzzy systems are finite-time bounded with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> performance over the reaching phase, the sliding phase, and the whole finite-time interval, respectively. To reduce the conservatism and increase the solution space of linear matrix inequality conditions, the fuzzy Lyapunov functional approach and equivalent fuzzy relaxed matrices technique are developed by introducing some relaxed matrices in the derivative of the fuzzy normalized membership function. Compared with the common Luenberger-type observer-based approach, the gain matrices of ISMCr depend on the FFO designed, which also enhances the flexibility of controller design. Finally, a simulation example with some comparison is given to show the effectiveness of the proposed method.

Research on Active Intelligent Perception Technology of Vessel Situation Based on Multisensor Fusion

In this paper, we focus on the safety supervision of inland vessels. This paper especially aims at studying the vessel target detection and dynamic tracking algorithm based on computer vision and the target fusion algorithm based on multisensor. For the vessel video target detection and tracking, this paper analyzes the current widely used methods and theories. Additionally, facing the application scenarios and characteristics of inland vessels, a comprehensive vessel video target detection algorithm is proposed in this paper. It is combined with a three-frame difference method based on Canny edge detection and a background subtraction method based on mixed Gaussian background modeling. Besides, for the multisensor target fusion, the processing method of laser point cloud data and automatic identification system (AIS) data is analyzed in this paper. Based on the idea of fuzzy mathematics, this paper proposes a method for calculating the fuzzy correlation matrix with normal membership function, which realizes the fusion of vessel track features of laser point cloud data and AIS data under dynamic video correction. Finally, through this method, a set of vessel situation active intelligent perception systems based on multisensor fusion was developed. Experiments show that this method has better environmental applicability and detection accuracy than traditional manual detection and any single monitoring method.

A Novel Method to Evaluate the Effect of Slope Blasting under Impact Loading

In this paper, the quantitative evaluation of the explosion effect based on the fuzzy comprehensive evaluation method is proposed to describe the qualitative evaluation results. The selected state characteristic parameters are expressed by two kinds of membership functions, fuzzy normal and triangular distribution membership functions, and preliminary evaluation results are obtained. The validity index of the maximum membership principle is used to assess the accuracy of the evaluation results of two algorithms, and a relevant approaching degree is chosen to optimize the results. The entire evaluation process selects eleven indicators to form an evaluation set, including the boulder yield, root rate, flying distance of flyrock, explosive consumption, postcracking distance, detonator unit consumption, vibration velocity, loose coefficient, cast distance, throw rate, and blasted volume per meter of hole. Part of the indicator parameters are derived from field test monitoring, and another part of the indicator parameters are derived from numerical simulation. The simulation process uses the user-defined material interface function provided by LS-DYNA. And the numerical model of slope blasting is established by embedding the evolution relationship of tensile and compressive damage into the elastoplastic constitutive material. The evaluation method proposed in this paper is used to evaluate the postexplosion effect of Zijin Mountain gold-copper mine slope cast blasting. The results demonstrate that the fuzzy normal distribution membership function can correlate the state characteristic information and evaluation index effectively, and the working condition after explosion can be reflected accurately. Additionally, the influencing factors can be ranked by the importance degrees according to the calculated value of the evaluation index.

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

The mathematical structure of some natural phenomena of nonlinear physical and engineering systems can be described by a combination of fuzzy differential equations that often behave in a way that cannot be fully understood. In this work, an accurate numeric-analytic algorithm is proposed, based upon the use of the residual power series, to investigate the fuzzy approximate solution for a nonlinear fuzzy Duffing oscillator, along with suitable uncertain guesses under strongly generalized differentiability. The proposed approach optimizes the approximate solution by minimizing a residual function to generate r-level representation with a rapidly convergent series solution. The influence, capacity, and feasibility of the method are verified by testing some applications. Level effects of the parameter r are given graphically and quantitatively, showing good agreement between the fuzzy approximate solutions of upper and lower bounds, that together form an almost symmetric triangular structure, that can be determined by central symmetry at r = 1 in a convex region. At this point, the fuzzy number is a convex fuzzy subset of the real line, with a normalized membership function. If this membership function is symmetric, the triangular fuzzy number is called the symmetric triangular fuzzy number. Symmetrical fuzzy estimates of solutions curves indicate a sense of harmony and compatibility around the parameter r = 1. The results are compared numerically with the crisp solutions and those obtained by other existing methods, which illustrate that the suggested method is a convenient and remarkably powerful tool in solving numerous issues arising in physics and engineering.

A peak-to-peak filtering for continuous Takagi-Sugeno fuzzy systems by a local method

We investigate the peak-to-peak filtering problem for continuous Takagi-Sugeno fuzzy systems. First, a novel filter, where the premise variables depend partly on the output of the filter, is constructed and the corresponding filtering error system is obtained. Then, a new approach, which is called the local approach, is developed for the filtering error system. Here, the local approach means that the state trajectories of the filtering error system are restricted to a compact set such that the deviations between the normalized membership functions of the system and those of the filter have prescribed upper bounds. Then, the information regarding the aforementioned upper bounds is used for the filtering design. Furthermore, a peak-to-peak performance analysis method is given to achieve the objective of the disturbance attenuation. Unlike in some existing filtering schemes, the information regarding the aforementioned upper bounds is fully utilized in the design for a more relaxed result. Furthermore, on the basis of the proposed peak-to-peak filtering scheme, the influence of the persistent disturbance on the filtering error can be reduced. Finally, two examples are given to demonstrate the validity of the filter scheme presented.

Fuzzy Observer-Based Repetitive Tracking Control for Nonlinear Systems

This article is concerned with the periodic tracking control problem for nonlinear systems. First, the Takagi-Sugeno (T-S) fuzzy model is employed to describe the nonlinear control systems. Second, considering the partly unmeasurable states of the system, a novel fuzzy observer-based repetitive controller, which is the mixed controller of the fuzzy observer-based controller and the fuzzy repetitive controller, is designed to deal with the periodic tracking control problem. To reduce the conservatism and increase the feasible solution space of the stabilization conditions, a new fuzzy relaxed matrix technique is developed by introducing some relaxed matrices in the derivative of the fuzzy normalized membership function. Then, the fuzzy Lyapunov functional with an additional separation parameter and the augmented fuzzy matrix technique (the interactions of fuzzy observer subsystems) are proposed such that the delay-dependent stability condition of the closed-loop system in the form of linear matrix inequality is obtained with less conservatism. It is worth noting that due to introducing an additional parameter in the fuzzy Lyapunov functional, the fuzzy controller and fuzzy observer can be separately designed, which largely enhances the flexibility of design with low computational complexity. Finally, three examples are provided to illustrate the effectiveness and less conservatism of the proposed method.

Stability Analysis and Dynamic Output Feedback Control for Nonlinear T-S Fuzzy System with Multiple Subsystems and Normalized Membership Functions

This paper addresses the stability analysis and dynamic output-feedback control problems for a class of nonlinear Takagi-Sugeno (T-S) fuzzy systems with multiple subsystems and normalized membership functions. First, the switching control law of the membership function is proposed based on the membership function for the nonlinear T-S fuzzy subsystems. Secondly, the relaxation parameter is introduced into this switching control law to guarantee a minimal dwell time between two consecutive switching. Then, based on the proposed switching control law of the membership function and relaxation parameter, the dynamic output feedback controller with the estimate algorithm is designed to estimate the attraction domain. By introducing the new switched Lyapunov-Krasovskii functional, it can be seen that the solutions of the resultant closed-loop system converge to an adjustable bounded region. Compared with the previous works, the developed controller in this paper is flexible and smooth, which only uses the system output. And the results are further extended to the mobile robot case and the chemical process case. Finally, two simulation examples are performed to show the effectiveness of the theoretical results.

Network-Based T-S Fuzzy Dynamic Positioning Controller Design for Unmanned Marine Vehicles.

This paper is concerned with a Takagi-Sugeno (T-S) fuzzy dynamic positioning controller design for an unmanned marine vehicle (UMV) in network environments. Network-based T-S fuzzy dynamic positioning system (DPS) models for the UMV are first established. Then, stability and stabilization criteria are derived by taking into consideration an asynchronous difference between the normalized membership function of the T-S fuzzy DPS and that of the controller. The proposed stabilization criteria can stabilize states of the UMV. The dynamic positioning performance analysis verifies the effectiveness of the networked modeling and the controller design.

Networked non-fragile H∞ static output feedback control design for vehicle dynamics stability: A descriptor approach

This paper deals with the problem of networked non-fragile H∞ static output feedback controller design for continuous Takagi–Sugeno T–S fuzzy systems with uncertainties and measurement noise under unreliable communication links. The packet dropouts and network-induced delays, which are two typical network constraints of data transmission through network, are considered in control design. Due to the introduction of the noise measurement and in order to simplify the control design, the descriptor model is adopted. For the multiplicative case of controller gain variations, we suggest a new solution of the fragility problem by developing the non-fragile design schemes ensuring the robustly admissibility for the resulting network closed-loop systems. By considering an appropriate Lyapunov-Krasovskii functional (LKF), and using the deviation bounds of asynchronous normalized membership functions, we propose a new delay-dependent sufficient conditions formulated in LMI constraints which can be easily solved using convex optimization tools. To demonstrate the validity and the effectiveness of the proposed approach, the vehicle dynamics stability and the comparison with existing results are considered.

H∞stabilization for sampling fuzzy systems with asynchronous constraints on membership functions

This paper investigates the problem of $H_\infty $ stabilization for nonuniform sampling fuzzy systems. To achieve better performance, a parallel distributed compensation scheme is proposed to design a nonlinear controller. By taking the deviation bounds of asynchronous normalized membership functions into account, the $H_\infty $ stability criterion is first obtained, which is dependent on the deviation bounds of asynchronous normalized membership functions. To reduce the computational complexity, a method for eliminating slack variables and a structured vertex separator for reducing the number of linear matrix inequalities are provided, and then an $H_\infty $ stabilization criterion with less complexity and less conservatism is obtained. Finally, an illustrative example is given to show the effectiveness of the proposed method and the significant improvement on the existing results.

An Improved Model of Variable Fuzzy Sets with Normal Membership Function for Crane Safety Evaluation

Crane safety evaluation is significant for reducing the casualties and economic losses. Various evaluation methods have been applied to evaluate crane safety. However, when index standard with respect to every level is described in terms of intervals, existing crane safety evaluation models are not ideal. Although variable fuzzy sets (VFS) method has successfully solved similar problems, its evaluation processes are rather complex complicated and tedious. In this paper, we provided an improved VFS evaluation method with normal membership function for solving crane safety problem, in which the evaluation criteria are described in terms of intervals. To demonstrate our presented method, the improved variable fuzzy sets (IVFS) method was applied to two numerical cases including an application to crane safety assessment. It is shown that our proposed method possesses the merit to simplify assessment processes of traditional VFS and can be applied to assess crane safety when criteria values are defined as interval numbers.

Event-triggered [formula omitted]∞ control for T–S fuzzy nonlinear systems and its application to truck–trailer system

This paper addresses the problem of event-triggered H∞ control for a class of T–S fuzzy nonlinear systems. An improved event-triggered scheme (ETS) characterized by discrete sampling is proposed, where the time-derivative of the membership function is not required. To get conservative conditions, the deviation bound of asynchronous normalized membership functions is considered. By utilizing the non-quadratic fuzzy line-integral Lyapunov functions and a free-matrix-based integral inequality, novel criteria for stabilization analysis of T–S fuzzy nonlinear systems are established. Finally, a truck–trailer system is provided to show the effectiveness of the proposed theories.

Multi-instant switching control of nonlinear networked systems under unreliable wireless digital channels

Integrating wireless networks in control systems allow users to take benefit from modular and flexible system design, but the analysis and design of such systems pose new challenges such as more serious packet loss. This study develops a new multi-instant switching controller of nonlinear networked systems under unreliable wireless digital channels. The proposed controller can be viewed as an improvement of previous results concerned with the homogenous polynomials approach, since it presents a denser subdivision of the joint space of multi-instant normalized membership functions and therefore essentially yields an efficient multi-instant switching mechanism for the first time. As a result, more efficient results can be obtained in the study. Finally, an illustrative example is provided to demonstrate the effectiveness of the multi-instant switching controller.

L 2-Gain Control for T-S Fuzzy Systems Over an Event-Triggered Communication Network Using Delay Decomposition and Deviation Bounds of Membership Functions

This paper is concerned with networked \(L_2\)-gain control for a T-S fuzzy system over an event-triggered communication network. Taking an event-triggered communication scheme and network-induced delays into consideration, the networked control system is described by an asynchronous T-S fuzzy system with an interval time-varying delay. Due to variation characteristic of network-induced delays, the interval is decomposed into N subintervals and the jumping among these subintervals is governed by a Markov chain. A new relaxation method, which fully utilizes the convexity of normalized membership functions and the deviation bounds of asynchronous normalized membership functions, is proposed and a stochastic Lyapunov–Krasovskii functional is constructed to derive some delay-dependent criteria on \(L_2\)-gain performance analysis and controller design of the asynchronous T-S fuzzy system. An illustrative example is provided to show that the proposed criteria are of less conservatism and less computational complexity than some existing results, and are effective in achieving a prescribed \(L_2\)-gain performance.