Premise Membership Functions Research Articles

Fault estimation and fault tolerant control for interval type-2 Takagi–Sugeno fuzzy systems via membership-function-dependent approach

This paper investigates the fault estimation (FE)-based fault tolerant control (FTC) technique to achieve the desired control performance for the nonlinear systems suffering from uncertainties, external disturbance and actuator faults using the interval type-2 (IT2) Takagi–Sugeno (T–S) fuzzy model. In this work, an IT2 fuzzy observer is built to simultaneously estimate both the system states and actuator faults, upon which a fault tolerant controller is proposed to guarantee the asymptotical stability of the closed-loop system with a prescribed H_{infty } performance level. Considering the bidirectional robustness interactions between the observer and FTC system, an integrated design technique is developed to address observer and FTC units together in one step to realize the required robustness within the whole closed-loop FTC system. By utilizing Lyapunov stability theory combined with the matrix inequality convexification techniques, a membership-function-dependent (MFD) FTC strategy is proposed where the information of membership functions is taken into account in the analysis for relaxation of the stability conditions. Additionally, to offer greater design flexibility and lower implementation cost to the fault tolerant controller, the imperfect premise matching (IPM) scheme is adopted, such that the premise membership functions of the fault tolerant controller can be chosen differently from those of IT2 fuzzy model. Finally, simulation results are provided to validate the effectiveness of the proposed FTC strategy.

Stabilization Analysis and Impulsive Controller Design for Positive Interval Type-2 Polynomial Fuzzy Systems

This article investigates the polynomial fuzzy impulsive control problem for positive nonlinear systems subject to parameter uncertainties. The positive nonlinear systems are represented as positive interval type-2 (IT2) polynomial fuzzy-model-based systems, while the parameter uncertainties are captured by the IT2 membership functions (MFs). Considering that the controller being designed needs to be implementable and practical, in addition to ensuring the positivity and stability of the system, there are still some potential requirements for the controller, such as lower implementation cost and control cost. To reduce the implementation cost, the IT2 polynomial fuzzy impulsive controller is designed under the imperfect premise matching concept that the premise MFs and the number of rules of fuzzy controller are different from those of fuzzy model. To reduce the control cost, the impulsive controller that can tolerate larger impulse control interval (ICI) is designed by the following two novel methods. The first one is the proposed impulse-time-dependent discretized polynomial copositive Lyapunov function whose Lyapunov variable is designed as the interpolation polynomial function of time. The second one is the advanced IT2 membership-function-dependent (IT2MFD) analysis method which is improved by a novel footprint of uncertainty partitioning method proposed by this paper, so that this advanced IT2MFD method can alleviate the negative impact of parameter uncertainties on the ICI. Finally, a simulation example using lipoprotein metabolism and potassium ion transfer model as the nonlinear system to be controlled is used to verify the effectiveness of the proposed impulsive control methods.

Membership-Function-Dependent Control Design of Interval Type-2 Sampled-Data Fuzzy-Model-Based Output-Feedback Tracking Control System

This article focuses on the stability analysis and control design of an interval type-2 sampled-data fuzzy-model-based output-feedback (IT2SDFMBOF) tracking control system, which is formed by the interval type-2 (IT2) Takagi–Sugeno (T–S) fuzzy model, the stable reference model, and the interval type-2 sampled-data output-feedback (IT2SDOF) fuzzy controller. The design goal is to establish a proper IT2SDOF fuzzy controller that is able to drive the states of the nonlinear plant subject to uncertainties to track those of the stable reference model, and the <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 index is adopted to optimize the tracking control performance. To improve the robustness of the IT2SDOF fuzzy controller against uncertainties of the nonlinear plant, IT2 fuzzy sets are utilized. Considering that the tracking control system is sampled-data (SD) type, a two-sided looped-functional-based approach exploiting the information on the interval from time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t_k$</tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> and the interval from time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t_{k+1}$</tex-math></inline-formula> is applied to enhance the stability analysis. The imperfect premise matching (IPM) concept permits the number of rules and the premise membership functions of the IT2 T-S fuzzy model and the IT2SDOF fuzzy controller to be different, which can be employed to promote the design flexibility. The tracking control system with the SD type and IPM design will cause the membership grades between the IT2 T-S fuzzy model and the IT2SDOF fuzzy controller to be mismatched. In addition, the membership-function-dependent stability analysis approach that can introduce the boundary information of membership functions into the stability analysis is used to relax the stability conditions. The relaxed stability conditions subject to the <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 building on the Lyapunov stability theory are developed in terms of linear matrix inequalities. Simulation results demonstrate the effectiveness of the proposed IT2SDFMBOF tracking control system.

Memory‐based dynamic event‐triggered control for networked interval type‐2 fuzzy systems subject to DoS attacks

SummaryIn this article, the memory‐based dynamic event‐triggered controller design issue is investigated for networked interval type‐2 (IT2) fuzzy systems under non‐periodic denial‐of‐service (DoS) attacks. For saving limited network bandwidth, a novel memory‐based dynamic event‐triggered mechanism (DETM) is proposed to schedule data communication. Unlike existing event‐triggered generators, the developed memory‐based DETM can utilize a series of newly released signals and further save network resources by introducing interval dynamic variables. Moreover, to improve design flexibility, an IT2 fuzzy controller with freely selectable fuzzy rule number and premise membership functions (MFs) is synthesized. Then, a new switched time‐delay system with imperfectly matched MFs is established under the consideration of memory‐based DETM and DoS attacks simultaneously. Besides, based on the property of MFs, the boundary information of membership grades and slack matrices are introduced in the stability analysis. Furthermore, by using a piecewise Lyapunov–Krasovskii method, membership‐functions‐dependent criteria are deduced to ensure the asymptotic stability of built fuzzy switched systems. Finally, the effectiveness of proposed control strategies is demonstrated by simulation examples.

Membership-Function-Dependent Control Design and Stability Analysis of Interval Type-2 Sampled-Data Fuzzy-Model-Based Control System

This article investigates the design and stability analysis of the interval type-2 (IT2) sampled-data (SD) fuzzy-model-based (FMB) control system with the optimal guaranteed cost performance. An IT2 Takagi–Sugeno (T-S) fuzzy model is applied to describe the dynamics of the nonlinear systems where the parameter uncertainties are captured by the lower and upper membership functions. To conduct the stability analysis for the SD FMB control system, a looped-functional approach taking the advantage of the information about the sampling periods is employed. Because of the SD control strategy, the state will be sampled at each sampling instant and the control signal generated by the IT2SD fuzzy controller will be kept by the zero-order holder during the sampling period, which will result in mismatched membership grades between IT2 T-S fuzzy model and IT2SD fuzzy controller that leads to the complexity in carrying out stability analysis. Thanks to the imperfect premise matching concept, which allows the difference on the number of rules and the premise membership functions between model and controller, the design of the IT2SD fuzzy controller can be more flexible. To further relax the stability conditions and minimize the upper bound of the guaranteed cost index, the membership-function-dependent stability analysis approach which can make use of the features of the IT2 membership functions is adopted. The performance of the control system can also be adjusted through the choice of the weighting matrices in the cost function. The stability conditions building on the Lyapunov stability theory and the performance conditions building on the concept of the guaranteed cost control in the shape of linear matrix inequalities are established to assure the system stability and acquire the optimal guaranteed cost performance. The proposed IT2SDFMB control design is tested on the inverted pendulum system and the simulation results verify the effectiveness of the proposed approach.

Stability Analysis and Estimation of Domain of Attraction for Positive Polynomial Fuzzy Systems With Input Saturation

In this paper, the stability and positivity of positive polynomial fuzzy model based (PPFMB) control system are investigated, in which the positive polynomial fuzzy model and positive polynomial fuzzy controller are allowed to have different premise membership functions from each other. These mismatched premise membership functions can increase the flexibility of controller design; however, it will lead to the conservative results when the stability is analyzed based on the Lyapunov stability theory. To relax the positivity/stability conditions, the improved Taylor-series-membership-functions-dependent (ITSMFD) method is introduced by introducing the sample points information of Taylor-series approximate membership functions, local error information and boundary information of substate space of premise variables into the stability/positivity conditions. Meanwhile, the ITSMFD method is extended to the PPFMB control system with input saturation to relax the estimation of domain of attraction. Finally, simulation examples are presented to verify the feasibility of this method.

Membership-Function-Dependent Stabilization of Event-Triggered Interval Type-2 Polynomial Fuzzy-Model-Based Networked Control Systems

In this article, the stability analysis and control synthesis of interval type-2 (IT2) polynomial-fuzzy-model-based networked control systems are investigated under the event-triggered control framework. The nonlinear dynamics in the plant is efficiently represented by an IT2 polynomial fuzzy model that the IT2 membership functions are utilized to capture the uncertainties in the plant. An event-triggered IT2 polynomial fuzzy controller is then designed to stabilize the nonlinear model subject to uncertainties. The stability conditions of the closed-loop control system are summarized in the form of sum-of-squares. Under the imperfectly premise matching (IPM) concept, the membership-function-dependent (MFD) approach is applied to endow the polynomial fuzzy controllers with more flexibility in terms of number of rules and premise membership functions. In the MFD approach under the IPM concept, both the number of rules and the shape of membership functions in the fuzzy models and controllers can be different. Also, the information of IT2 membership functions of the polynomial fuzzy model and controller is considered and adopted to further relax the stability conditions. Furthermore, the intrinsic mismatched issue of the premise variables of the fuzzy model and controllers due to the event-triggering mechanism is handled by the MFD approach. A detailed simulation example is provided to verify the effectiveness of the proposed event-based control strategy.

A Novel FEM Based T-S Fuzzy Particle Filtering for Bearings-Only Maneuvering Target Tracking.

In this paper, we propose a novel fuzzy expectation maximization (FEM) based Takagi-Sugeno (T-S) fuzzy particle filtering (FEMTS-PF) algorithm for a passive sensor system. In order to incorporate target spatial-temporal information into particle filtering, we introduce a T-S fuzzy modeling algorithm, in which an improved FEM approach is proposed to adaptively identify the premise parameters, and the model probability is adjusted by the premise membership functions. In the proposed FEM, the fuzzy parameter is derived by the fuzzy C-regressive model clustering method based on entropy and spatial-temporal characteristics, which can avoid the subjective influence caused by the artificial setting of the initial value when compared to the traditional FEM. Furthermore, using the proposed T-S fuzzy model, the algorithm samples particles, which can effectively reduce the particle degradation phenomenon and the parallel filtering, can realize the real-time performance of the algorithm. Finally, the results of the proposed algorithm are evaluated and compared to several existing filtering algorithms through a series of Monte Carlo simulations. The simulation results demonstrate that the proposed algorithm is more precise, robust and that it even has a faster convergence rate than the interacting multiple model unscented Kalman filter (IMMUKF), interacting multiple model extended Kalman filter (IMMEKF) and interacting multiple model Rao-Blackwellized particle filter (IMMRBPF).

Observer-based dissipative output feedback control for network T–S fuzzy systems under time delays with mismatch premise

This paper deals with the problem of non-parallel distribution compensation networked control systems (NCSs) under event-triggered strategy based on the fuzzy system for a class of nonlinear systems with time-varying delays and mismatching premise membership functions. The T–S fuzzy model under event-triggered strategy is provided in unified framework, in which (1) a delay fuzzy observer-based with the mismatch premise is employed to approximate the immeasurable states of the existing system; (2) a delay fuzzy observer-based controller is constructed, by considering the same premises as in observer; and (3) event-triggered strategy is considered to mitigate the effect of communication burden. Another derivation of this paper is to introduce the notation of extended dissipativity which provides the $$H_{\infty }$$ , $$L_{2}-L_{\infty }$$ and dissipative performances. Furthermore, in this regard, by taking into account a novel fuzzy Lyapunov–Krasovskii functional in conjunction with free weighting matrices, containing mode-dependent integral terms such that the resulting system is stable with the desired performance. Then, desired observer-based controller is presented in the account of LMI. Finally, we validate our results with a truck–trailer example.

T–S Fuzzy-Model-Based Output Feedback Tracking Control With Control Input Saturation

This paper investigates the output feedback tracking control for a fuzzy-model-based (FMB) control system when the control input is saturated, where the FMB is developed based on a T–S fuzzy model and a fuzzy controller. The controller is employed to close the feedback loop and generate the system to trace the trajectory of the states of a stable reference model subject to $H_\infty$ performance. To enhance the fuzzy controller design flexibility, the number of rules and premise membership functions can be adjusted. Stability analysis for the FMB control system is performed based on Lyapunov stability theory. To address the control input saturation problem, linear sectors are created by local linear upper and lower bounds to include the possible control area. Hence, the nonlinear saturation problem can be tackled by the stability analysis of linear sectors. The membership-functions-dependent technique is used to bring the information and address the nonlinearity of embedded membership functions into the stability analysis. The numerical simulation example demonstrates the effectiveness of the proposed approach and discusses the effect of $H_\infty$ performance and control input saturation rate according to the tracking result.

Stability Analysis of Positive Polynomial Fuzzy-Model-Based Control Systems With Time Delay Under Imperfect Premise Matching

This paper deals with the stability and positivity analysis of polynomial-fuzzy-model-based (PFMB) control systems with time delay, which is formed by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop, under imperfect premise matching. To improve the design and realization flexibility, the polynomial fuzzy model and the polynomial fuzzy controller are allowed to have their own set of premise membership functions. A sum-of-squares-based stability analysis approach using the Lyapunov stability theory is employed to investigate the positivity and stability of the PFMB control systems and synthesize the polynomial fuzzy controller. In order to relax the stability results, we propose two methods: first, membership functions are considered as symbolic variables in the stability analysis; and second, the property of the membership functions and the boundary information of the membership functions are considered in the stability analysis. A simulation example is given to illustrate the effectiveness of the proposed approach.

State Feedback Control for Interval Type-2 Fuzzy Systems With Time-Varying Delay and Unreliable Communication Links

This study explores the stabilization problem of nonlinear systems subject to parameter uncertainties, time-varying delay, and data-packet dropouts. An interval type-2 Takagi-Sugenofuzzy model is employed to represent nonlinear systems subject to parameter uncertainties. The lower and upper membership functions of the system enable parameter uncertainties to be captured effectively. To enhance the flexibility of design, the proposed state feedback controller does not share the premise membership functions of the model. Unlike the majority of existing results, the present study simultaneously considers parameter uncertainties, time-varying delay, and data-packet dropouts in nonlinear control systems. A novel finite-sum inequality that includes well-known inequalities as special cases, is developed to derive new stability conditions in the form of linear matrix inequalities. Several examples of simulation are used to demonstrate the effectiveness of the proposed approach.

Event-Triggered Dynamic Output Feedback Control for Networked T-S Fuzzy Systems With Asynchronous Premise Variables

This paper investigates the problem of event-triggered dynamic output feedback control for networked Takagi-Sugeno (T-S) fuzzy systems with asynchronous premise variables. Our attention is focused on the design of an event-triggered fuzzy dynamic output feedback controller that guarantees that the closed-loop system is asymptotically stable. First, with consideration of the limited bandwidth and restricted network resource, an event-triggered communication scheme is proposed to save the limited network resource. Second, different from the existing results in the literature, a fuzzy dynamic output feedback controller with asynchronous premise variables is constructed such that the premise membership functions are not necessarily the same as the networked T-S fuzzy systems. Then, by using the Wirtinger inequality and some slack matrices, a less conservative stability and stabilization conditions for the existence of fuzzy dynamic output feedback controller are derived in the form of linear matrix inequalities, which can be solved by the standard MATLAB toolbox. Finally, an example is given to demonstrate the effectiveness of the designed event-triggered fuzzy dynamic output feedback controller.

A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis

This paper reviews the stability analysis of continuous-time fuzzy-model-based (FMB) control systems, with emphasis on state-feedback control techniques, which is an essential issue received a great deal of attention in the fuzzy control community. It gives an overview of the milestones and the trend of developments and achievements for the past decades. Focusing on the stability analysis of FMB control systems, it summarizes the issues in the four fundamental and essential aspects, namely, the types of membership-function matching, types of Lyapunov functions, types of stability analysis and the techniques of stability analysis are discussed. To start with systematic discussion, the FMB control systems are categorized into three types of membership-function matching, namely, perfectly, partially and imperfectly matched premises, regarding the premise membership functions and the number of rules used in the fuzzy model and fuzzy controller which forms the FMB control system. The features of each category are thoroughly discussed from theoretical to practical point of view. Various types of Lyapunov functions available in the literature for conducting stability analysis and their characteristics are then reviewed. Especially, the focus of this paper is to promote the concept of membership-function-dependent (MFD) stability analysis, which makes use of the information of membership functions aiming to relax the stability conditions compared with the dominant membership-function-independent (MFI) stability analysis in the literature. The techniques for MFI and MFD stability analysis are then discussed in details, which provide some solid ideas to analyze the stability of FMB control systems. More importantly, it sheds light on the fact that MFD stability analysis demonstrates a greater potential than the MFI one for relaxing the conservativeness of stability analysis results, which points a promising research direction for this topic. The purposes of this paper are to provide a comprehensive update for the stability analysis of FMB control systems to the researchers in the field and serve as a quick guide for the potential researchers who want to enter the field.

Control design for interval type‐2 polynomial fuzzy‐model‐based systems with time‐varying delay

In this study, the problems of stabilisation for interval type-2 polynomial fuzzy systems with time-varying delay and parameter uncertainties are investigated. The objective is to design a state-feedback interval type-2 polynomial fuzzy controller such that the closed-loop control system is asymptotically stable. The conditions for the existence of such a controller are delay dependent and membership function dependent in terms of sum-of-squares (SOS). Based on a basic lemma to deal with the delay terms, the authors formulate and solve the problem with more flexibility due to imperfect premise matching that the number of rules and premise membership functions are not necessary the same between the interval type-2 polynomial fuzzy model and interval type-2 polynomial fuzzy controller. Piecewise linear membership functions approximations enclosing the original lower and upper membership functions are employed to facilitate the stability analysis. A numerical example indicates the effectiveness of the derived results.

Stability and Stabilization Analysis of Positive Polynomial Fuzzy Systems With Time Delay Considering Piecewise Membership Functions

This paper is concerned with the stability analysis of positive polynomial-fuzzy-model-based control systems with time delay. Each of the polynomial fuzzy model and the polynomial fuzzy controller is allowed to have its own set of premise membership functions, which can improve the design and realization flexibility. Conditions showing the positivity of the system are first obtained. Then, to deal with the difficulty resulting from the mismatched premise membership functions to the stability/stabilization analysis, approximated membership functions representing the original ones are constructed, and the information of membership functions is brought into the stability conditions. This way, the stability conditions can be relaxed, since the extra knowledge on the system is considered. Stability conditions are obtained by using the sum-of squares approach based on the Lyapunov stability theory. A simulation example is provided to verify the effectiveness of this method.

A New Fuzzy H∞ Filter Design for Nonlinear Time-Delay Systems with Mismatched Premise Membership Functions

This paper is concerned with the fuzzy H∞ filter design issue for nonlinear systems with time-varying delay. To overcome the limitations of conventional methods with matched preconditions, the filter to be designed and the T-S fuzzy model are assumed to have different premise membership functions and different number of rules. Hence, greater design flexibility can be obtained. Moreover, in order to reduce conservatism, a novel integral inequality which is tighter than the traditional inequalities derived from the Leibniz-Newton formula is applied. Furthermore, the proposed filter design approach also considered the information about the membership functions, which can help further relax the derived results. Finally, two numerical examples are provided to demonstrate the effectiveness and superiority of the proposed method.

Polynomial Fuzzy-Model-Based Control Systems: Stability Analysis via Approximated Membership Functions Considering Sector Nonlinearity of Control Input

This paper presents the stability analysis of polynomial fuzzy-model-based (PFMB) control systems, in which both the polynomial fuzzy model and the polynomial fuzzy controller are allowed to have their own set of premise membership functions. In order to address the input nonlinearity, the control signal is considered to be bounded by a sector with nonlinear bounds. These nonlinear lower and upper bounds of the sector are constructed by combining local bounds using fuzzy blending such that local information of input nonlinearity can be taken into account. With the consideration of imperfectly matched membership functions and input nonlinearity, the applicability of the PFMB control scheme can be further enhanced. To facilitate the stability analysis, a general form of approximated membership functions representing the original ones is introduced. As a result, approximated membership functions can be brought into the stability analysis leading to relaxed stability conditions. The sum-of-squares approach is employed to obtain the stability conditions based on Lyapunov stability theory. Simulation examples are presented to demonstrate the feasibility of the proposed method.

Two-Step Stability Analysis for General Polynomial-Fuzzy-Model-Based Control Systems

This paper investigates the stability of a polynomial-fuzzy-model-based (PFMB) control system formed by a nonlinear plant represented by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop. Three cases of polynomial fuzzy controllers are proposed for the control process with the consideration of a matched/mismatched number of rules and/or premise membership functions, which demonstrate different levels of controller complexity, design flexibility, and stability analysis results. A general polynomial Lyapunov function candidate is proposed to investigate the system stability. Unlike the published work, there is no constraint on the polynomial Lyapunov function candidate, which is independent of the form of the polynomial fuzzy model. Thus, it can be applied to a wider class of PFMB control systems and potentially produces more relaxed stability analysis results. Two-step stability conditions in terms of sum-of-squares (SOS) are obtained to numerically find a feasible solution. To facilitate the stability analysis and relax the stability analysis result, the boundary information of membership functions is taken into account in the stability analysis and incorporated into the SOS-based stability conditions. Simulation examples are given to illustrate the effectiveness of the proposed approach.

Linear matrix inequalities‐based membership‐function‐dependent stability analysis for non‐parallel distributed compensation fuzzy‐model‐based control systems

This study investigates the stability of fuzzy-model-based (FMB) control system, which is formed by a Takagi–Sugeno (T–S) fuzzy model representing a non-linear plant and a fuzzy controller connected in a closed loop. The proposed fuzzy controller considers a higher design flexibility that the premise membership functions and the number of fuzzy rules are allowed to be not the same as those of the T–S fuzzy model. By employing simple membership functions and a smaller number of fuzzy rules, the complexity of the fuzzy controller can be reduced to facilitate its implementation. Based on the Lyapunov stability theory, a membership-function-dependent (MFD) stability analysis is proposed to develop stability conditions in the form of linear matrix inequalities (LMIs) to determine the system stability and facilitate the control synthesis. The information of the membership functions is brought to the MFD LMI-based stability conditions, which are dedicated to the FMB control system under consideration. It is thus the proposed MFD stability analysis result is more relaxed compared with the existing membership-function-independent stability analysis results. A simulation example is given to illustrate the effectiveness of the proposed approach.