On the Stability of Takagi–Sugeno Fuzzy Systems With Time-Varying Uncertainties

This paper studies the problems of stability analysis of Takagi-Sugeno free fuzzy systems with time-varying uncertainties. In our prior study, we represented the time-varying uncertainty incurred in characteristic interval matrices in terms of the stability of Takagi-Sugeno free fuzzy systems with consequent parameter uncertainties. Based on Mayer's convergent theorem for powers of single interval matrix and its generalization, we further proposed some sufficient conditions for the Takagi-Sugeno free fuzzy system with time-varying uncertainties to be globally asymptotically stable. In this paper, we propose the notion of simultaneously nilpotent interval matrices to investigate the Takagi-Sugeno free fuzzy system with time-varying uncertainties to be strongly stable within steps, where relates to the dimension of interval matrices. Moreover, a unique situation for the deterministic Takagi-Sugeno free fuzzy system to be strongly stable within steps is derived as well, where relates to the dimension of characteristic matrices for the deterministic Takagi-Sugeno free fuzzy system.     

Dynamic Output Feedback-Fault Tolerant Controller Design for Takagi–Sugeno Fuzzy Systems With Actuator Faults

This paper addresses the problem of robust fault estimation and fault tolerant control (FTC) for Takagi–Sugeno (T–S) fuzzy systems. A fuzzy-augmented fault estimation observer (AFEO) design is proposed to achieve fault estimation of T–S models with actuator faults. Furthermore, based on the information of online fault estimation, an observer-based dynamic output feedback-fault tolerant controller (DOFFTC) is designed to compensate for the effect of faults by stabilizing the closed-loop system. Sufficient conditions for the existence of both AFEO and DOFFTC are given in terms of linear matrix inequalities. Simulation results of an inverted pendulum system are presented to illustrate the effectiveness of the proposed method.      

Relaxed LMI-Based Stability Conditions for Takagi–Sugeno Fuzzy Control Systems Using Regional-Membership-Function-Shape-Dependent Analysis Approach

This paper presents relaxed stability conditions for Takagi-Sugeno (T-S) fuzzy-model-based control systems. It is assumed that the stability conditions are represented by some inequalities in the form of a p-dimensional fuzzy summation. To investigate the system stability, the inequalities are expanded to n-dimensional fuzzy summation (n ges p). The boundary and regional informations of membership functions are then utilized for relaxation of stability analysis results. Two analysis approaches are proposed in this paper. The first approach is called the global-membership-function-shape-dependent approach, in which the lower and upper bounds of the membership functions, and its products from 2 to n in the full operating domain, are considered in the stability analysis. The second approach is named as the regional-membership-function-shape-dependent approach in which the operating region is partitioned to subregions, and the boundary information of membership functions on each operating subregion is employed to facilitate the stability analysis. In both approaches, by the help of the boundary and/or regional information of the membership functions, some inequality constraints in the form of multidimensional fuzzy summation containing some slack matrices are constructed. Stability conditions in the form of linear matrix inequalities (LMIs) are derived. Numerical examples are given to demonstrate the effectiveness of the proposed stability conditions.     

New Robust Stability Conditions and Design of Robust Stabilizing Controllers for Takagi–Sugeno Fuzzy Time-Delay Systems

In this paper, we consider robust stability and stabilization of uncertain Takagi-Sugeno fuzzy time-delay systems where uncertainties come into the state and input matrices. Some stability conditions and robust stability conditions for fuzzy time-delay systems have already been obtained in the literature. However, those conditions are rather conservative and do not guarantee the stability of a wide class of fuzzy systems. This is true in case of designing stabilizing controllers for fuzzy time-delay systems and it thus leads to a conservative fuzzy controller design as well. We first consider rather relaxed robust stability conditions of uncertain fuzzy systems. To this end, we introduce an auxiliary system to the original fuzzy time-delay system to obtain generalized delay-dependent stability conditions. Such an auxiliary system has some arbitrary matrices that generalize not only the system representation but also delay-dependent stability conditions. Conditions we obtain here are delay-dependent conditions that depend on the upper bound of time-delay, and are given in terms of linear matrix inequalities (LMIs). Then, we compare our delay-dependent stability conditions with other conditions in the literature, and show that our conditions guarantee the stability of a wider class of systems than others. Next, we consider the robust stabilization problem with memoryless and delayed state feedback controllers. Based on our generalized robust stability conditions, we obtain delay-dependent sufficient conditions for the closed-loop system to be robustly stable, and give a design method of robustly stabilizing controllers. Finally, we give three examples that illustrate our results.     

Simultaneous Structure Identification and Fuzzy Rule Generation for Takagi–Sugeno Models

One of the main attractions of a fuzzy rule-based system is its interpretability which is hindered severely with an increase in the dimensionality of the data. For high-dimensional data, the identification of fuzzy rules also possesses a big challenge. Feature selection methods often ignore the subtle nonlinear interaction that the features and the learning system can have. To address this problem of structure identification, we propose an integrated method that can find the bad features simultaneously when finding the rules from data for Takagi–Sugeno-type fuzzy systems. It is an integrated learning mechanism that can take into account the nonlinear interactions that may be present between features and between features and fuzzy rule-based systems. Hence, it can pick up a small set of useful features and generate useful rules for the problem at hand. Such an approach is computationally very attractive because it is not iterative in nature like the forward or backward selection approaches. The effectiveness of the proposed approach is demonstrated on four function-approximation-type well-studied problems.      

Improved L2 Gain Performance Controller Synthesis for Takagi–Sugeno Fuzzy System

In this paper, an improved L2 gain performance controller synthesis is proposed for Takagi-Sugeno (T-S) fuzzy system. The T-S fuzzy controller can be easily derived by a three-step procedure with the linear matrix inequalities (LMIs) technique. First, a new T-S fuzzy model structure is presented, which includes the original T-S fuzzy plant with stable pre- and post filters on the input and output of the original plant. Second, by using this structure the L2 gain performance controller design problem can be easily transformed into standard LMIs formulation. Compared with the previous results, it not only gives us a simple structure of the T-S fuzzy controller, but also provides us effective LMIs-based conditions, which include a small number of unknown matrix variables; consequently, less value of L2 gain performance of the closed-loop system can be obtained. Third, an augmented T-S fuzzy controller which guarantees L2 gain performance is obtained for the original T-S fuzzy plant, which is composed of the T-S controller derived from step two and two stable pre- and post filters. Finally, some numerical examples are demonstrated to show the effectiveness of the proposals.     

Delay-Distribution-Dependent Stability and Stabilization of T–S Fuzzy Systems With Probabilistic Interval Delay

In this paper, we are concerned with the problem of stability analysis and stabilization control design for Takagi–Sugeno (T–S) fuzzy systems with probabilistic interval delay. By employing the information of probability distribution of the time delay, the original system is transformed into a T–S fuzzy model with stochastic parameter matrices. Based on the new type of T–S fuzzy model, the delay-distribution-dependent criteria for the mean-square exponential stability of the considered systems are derived by using the Lyapunov–Krasovskii functional method, parallel distributed compensation approach, and the convexity of some matrix equations. The solvability of the derived criteria depends not only on the size of the delay but also on the probability distribution of the delay taking values in some intervals. The revisions of the main criteria in this paper can also be used to deal with the case when only the information of variation range of the delay is considered. It is shown by practical examples that our method can lead to very less conservative results than those by other existing methods.      

FLEXFIS: A Robust Incremental Learning Approach for Evolving Takagi–Sugeno Fuzzy Models

In this paper, we introduce a new algorithm for incremental learning of a specific form of Takagi–Sugeno fuzzy systems proposed by Wang and Mendel in 1992. The new data-driven online learning approach includes not only the adaptation of linear parameters appearing in the rule consequents, but also the incremental learning of premise parameters appearing in the membership functions (fuzzy sets), together with a rule learning strategy in sample mode. A modified version of vector quantization is exploited for rule evolution and an incremental learning of the rules' premise parts. The modifications include an automatic generation of new clusters based on the nature, distribution, and quality of new data and an alternative strategy for selecting the winning cluster (rule) in each incremental learning step. Antecedent and consequent learning are connected in a stable manner, meaning that a convergence toward the optimal parameter set in the least-squares sense can be achieved. An evaluation and a comparison to conventional batch methods based on static and dynamic process models are presented for high-dimensional data recorded at engine test benches and at rolling mills. For the latter, the obtained data-driven fuzzy models are even compared with an analytical physical model. Furthermore, a comparison with other evolving fuzzy systems approaches is carried out based on nonlinear dynamic system identification tasks and a three-input nonlinear function approximation example.      

Observer-Based Stabilization of T–S Fuzzy Systems With Input Delay

This paper discusses the stabilization of Takagi-Sugeno (T-S) fuzzy systems with bounded and time-varying input delay. The robust stabilization via state feedback is first addressed, and delay-dependent stabilization conditions are proposed in terms of LMIs. Observer-based feedback stabilization is also discussed for T-S fuzzy input delay systems without uncertainties. A separate design principle is developed. Some illustrative examples are given to show the effectiveness and the feasibility of the proposed methods.     

Stabilization of Takagi–Sugeno Model via Nonparallel Distributed Compensation Law

This paper addresses the stabilization of nonlinear systems, which is represented by a Takagi–Sugeno (T–S) model. Based on the extended nonquadratic Lyapunov function and the nonparallel distributed compensation law, three new results are obtained by using appropriate slack matrices, collection matrices, and the higher dimensional collection matrix. The first two results are less conservative, and computationally less expensive, than some of the existing results. The third result combines the procedures of the first two results, and is less conservative, but is computationally more expensive than the first two results. The effectiveness of the new results is validated by two numerical examples.      

Relaxed Stability and Performance Conditions for Takagi–Sugeno Fuzzy Systems With Knowledge on Membership Function Overlap

This correspondence presents a relaxation of some earlier linear matrix inequality (LMI) conditions, which allow setting up less conservative stability or performance conditions for Takagi-Sugeno fuzzy models. Unlike the previous literature, this correspondence takes into account the knowledge of the membership functions' shape by considering bounds on them and their cross products (interpreted as an overlap measure), introducing auxiliary LMI variables. Numerical examples illustrate the achieved improvements     

Simultaneous Design of Parallel Distributed Output Feedback and Anti‐windup Compensators for Constrained Takagi‐Sugeno Fuzzy Systems

This paper is devoted to developing a novel approach to deal with constrained continuous‐time nonlinear systems in the form of Takagi‐Sugeno fuzzy models. Here, the disturbed systems are subject to both input and state constraints. The one‐step design method is used to simultaneously synthesize the dynamic output feedback controller and its anti‐windup strategy. A parameter‐dependent version of the generalized sector condition is used together with Lyapunov stability theory to derive linear matrix inequality design conditions. Based on this result and for different design specifications, the synthesis of an anti‐windup based dynamic output feedback controller is expressed on the form of convex optimization problems. A physically motivated example is given to illustrate the effectiveness of the proposed method.     

Stability Analysis and Stabilization for Discrete-Time Fuzzy Systems With Time-Varying Delay

This paper is concerned with the problems of stability analysis and stabilization for discrete-time Takagi–Sugeno fuzzy systems with time-varying state delay. By constructing a new fuzzy Lyapunov function and by making use of novel techniques, an improved delay-dependent stability condition is obtained, which is dependent on the lower and upper delay bounds. The merit of the proposed stability condition lies in its reduced conservatism, which is achieved by avoiding the utilization of some bounding inequalities for the cross products between two vectors. Then, a delay-dependent stabilization approach based on a parallel distributed compensation scheme is developed for both state feedback and observer-based output feedback cases. The proposed stability and stabilization conditions are formulated in terms of linear matrix inequalities, which can be solved efficiently by using existing optimization techniques. Two illustrative examples are provided to demonstrate the effectiveness of the results proposed in this paper.      

Stability Analysis of Takagi–Sugeno Fuzzy Cellular Neural Networks With Time-Varying Delays

This correspondence investigates the global exponential stability problem of Takagi-Sugeno fuzzy cellular neural networks with time-varying delays (TSFDCNNs). Based on the Lyapunov-Krasovskii functional theory and linear matrix inequality technique, a less conservative delay-dependent stability criterion is derived to guarantee the exponential stability of TSFDCNNs. By constructing a Lyapunov-Krasovskii functional, the supplementary requirement that the time derivative of time-varying delays must be smaller than one is released in the proposed delay-dependent stability criterion. Two illustrative examples are provided to verify the effectiveness of the proposed results     

On Delay-Dependent Approach for Robust Stability and Stabilization of T–S Fuzzy Systems With Constant Delay and Uncertainties

This paper investigates robust stability analysis and stabilization of delay and uncertain systems approximated by a Takagi-Sugeno (T-S) fuzzy model. An innovative approach is proposed to develop delay-dependent stability criteria of the systems, which makes use of less-redundant information to construct Lyapunov function, employs an integral equation method to handle the cross-product terms, and alleviates the requirements of the bounding technique and model transformations that have been popularly adopted in many existing references. This leads to significant improvement in the stability performance with far fewer unknown variables in the stability computation. From the derived stability criteria, a new memoryless state-feedback control is further developed. The controller gain and the maximum allowable delay bound of the closed-loop control system can be obtained simultaneously by solving an optimization problem. Numerical examples are also given to demonstrate the theoretical results.     

New relaxed stabilization conditions for discrete‐time Takagi–Sugeno fuzzy control systems

This paper develops relaxed stabilization conditions for discrete‐time Takagi–Sugeno fuzzy control systems based on the extended nonquadratic Lyapunov function, the nonparallel distributed compensation law, and the convexity of the fuzzy blending coefficients. Three new main results are proposed to further reduce the conservatism by fully exploring the slack matrix technique and introducing new slack matrices and extra collection matrices. The new stabilization conditions are gradually less and less conservative, and more advantageous than the existing results by overall consideration of the conservatism and the computational efforts. A well‐known numerical case and a practical case are carried out to demonstrate the effectiveness of the proposed stabilization conditions.     

时延网络控制系统的稳定性分析

为保证闭环系统的全局渐进稳定性,系统的时延必须是有界的。本文中讨论了具有网络诱导时延的网络控制系统的模型,同时给出了最大允许传输间隔以及一个实际例子。实践证明本文中给出的方法在简单的系统中是十分有效的。接着提出了一个减小网络诱导时延的算法,仿真结果表明了该方法的有效性。     

Identification of nonlinear systems by Takagi–Sugeno fuzzy logic grey box modeling for real-time control

Identification of nonlinear systems by fuzzy models has been successfully applied in many applications. Fuzzy models are capable of approximating any real continuous function to a chosen accuracy. An algorithm for real-time identification of nonlinear systems using Takagi–Sugeno's fuzzy models is presented in this paper. A Takagi–Sugeno fuzzy system is trained incrementally each time step and is used to predict one-step ahead system output. Ability of the proposed identifier to capture the nonlinear behavior of a synchronous machine is illustrated. Effectiveness of the proposed identification technique is demonstrated by simulation and experimental studies on a power system.     

T-S模糊时变时滞系统的稳定性分析

考虑具有时变时滞的Takagi-Sugeno模糊系统稳定性分析的问题.基于Lyapunov 稳定性理论,定义一个新的Lyapunov-Krasovskii泛函,使用恰当的界定技术处理Lyapunov-Krasovskii泛函求导过程中产生的积分项,以线性矩阵不等式的形式给出一个决策变量少、运算效率高、保守性小的时滞相关稳定性准则;此外,在理论推导过程中避免了一些现有文献理论推导过程中出现的问题.最后通过给出的数例验证了该稳定性准则的有效性.     

Sensor fault tolerant control of nonlinear Takagi–Sugeno systems. Application to vehicle lateral dynamics

This paper presents a new scheme for sensor fault tolerant control for nonlinear systems based on the Takagi–Sugeno modeling. First, a structured residual generator aimed at detecting and isolating sensor faults is designed. A bank of observers controlled either by only one system output or a set of outputs is then implemented, leading to a set of state estimates. The parallel distributed compensation structure is adopted to design the fault tolerant controller. The novelty in this paper is that the estimated state used in the controller is a weighted state vector obtained from all the estimated states provided by the different observers. The weighting functions depend on the residual vector signals delivered by the residual generator. They are designed to avoid crisp switches in the control law. Indeed, the interesting feature of the proposed approach is to avoid the commonly used switching strategy. For each residual component, the greater its magnitude is, the less the weight affected to the corresponding state estimate is. Consequently, the controller only uses estimations computed on the basis of healthy measurements. The closed‐loop stability is studied with the Lyapunov theory, and the obtained conditions are expressed as a set of linear matrix inequalities. The proposed residual generation and fault tolerant controller are applied to a vehicle lateral dynamics affected by sensor faults. Copyright © 2015 John Wiley & Sons, Ltd.