Fuzzy Lyapunov-Krasovskii Function Research Articles

Further Stability Criteria for Sampled-Data-Based Interval Type-2 Fuzzy Systems via a Refined Two-Side Looped-Functional Method

This article investigates the stability and stabilization problem of aperiodic sampled-data nonlinear systems in the framework of interval type-2 (IT-2) fuzzy models. First, by introducing two adjustable parameters and splitting the sampling intervals into four nonuniform intervals, a refined two-side looped-functional method is constructed to fully utilize inner state information during the whole aperiodic sampling interval. Simultaneously, the positive definiteness constraint for the individual matrix in Lyapunov–Krasovskii functional can be further relaxed. Then, via constructing a novel fuzzy Lyapunov–Krasovskii functional (FLKF) together with a fuzzy-dependent-switching scheme, the available features of fuzzy membership functions (FMFs) can be further taken into consideration to increase the design flexibility. Consequently, the stability condition and corresponding controller design approach for aperiodic sampled-data IT-2 fuzzy systems can be obtained with less design conservatism and larger sampling intervals. Finally, two simulation examples are employed to demonstrate the validity and superiority of the proposed method.

Fuzzy Sampled-Data Control for DFIG-Based Wind Turbine With Stochastic Actuator Failures

The stabilization of fault-tolerant control problem for the doubly fed induction generator (DFIG)-based wind turbine (WT) model has been investigated with stochastic actuator faults by incorporating a Takagi-Sugeno (T-S) fuzzy technique. To depict such stochastic actuator faults, the proposed model is incorporated with Markovian switching parameters for denoting various faulty modes. Besides, a novel fault-tolerant fuzzy switching control with sampled-data (SD) inputs has also been employed for eliminating the stochastic faults so that the stabilization of the system and an optimal H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> level could be ensured simultaneously. Furthermore, the stability conditions are derived by employing fuzzy Lyapunov-Krasovskii functionals (LKFs), whose matrices are membership dependent functions. Finally, the proposed control method for the stabilization problem is validated through a numerical simulation performed on the DFIG-based WT model.

Robust Stability and Stabilization Conditions for Nonlinear Networked Control Systems With Network-Induced Delay via T–S Fuzzy Model

Robust stability and stabilization problems for a class of discrete-time nonlinear systems are studied in this article. The control signal is transmitted to the nonlinear systems through a lossy communication channel in which time-varying network-induced delay occurs. A Takagi–Sugeno (T–S) fuzzy model is used to represent the nonlinear systems. Using membership function boundary information, a novel membership-function-dependent (MFD) analysis approach is presented. Unlike most previous results, the proposed MFD analysis approach does not need to increase the number of slack variables to reduce conservatism. Combining auxiliary matrices and this approach, we propose some new fuzzy Lyapunov–Krasovskii functionals (FLKFs) that do not require the restrictions of positive definiteness imposed on Lyapunov matrices. Using the FLKF and MFD analysis approach, a sufficient condition in the form of linear matrix inequalities (LMIs) is developed to guarantee the stability of T–S fuzzy time-varying delay systems. Based on the results for stability, less conservative stabilization conditions are also derived. Finally, numerical examples show that less conservative results can be obtained by the proposed approach. The inverted pendulum system with state delay or network-induced delay is also studied to verify the effectiveness of the suggested approach.

T–S Fuzzy Model-Based Single-Master Multislave Teleoperation Systems With Decentralized Communication Structure and Varying Time Delays

For the teleoperation system consisting of single-master and multislave (SMMS) manipulators, the Takagi–Sugeno (T–S) fuzzy technique has been adopted in this article to represent the nonlinear system and to approximate its nonlinearities effectively. With respect to the master–slave interactions, a decentralized leaderless communication topology has been considered for the control design with time delays in signal transmission. The corresponding forward and backward delays among the master and slaves were assumed to be asymmetric and varying with time. Moreover, to improve the tracking performance of the master and slaves, the decentralized controller is implemented with force feedback strategy to form the closed-loop teleoperation system. For the stability analysis, the fuzzy Lyapunov–Krasovskii functionals with some membership function dependent matrices were employed to guarantee the asymptotical stability of the proposed T–S fuzzy SMMS teleoperation system. By employing free-matrix-based inequality, some sufficient conditions were derived and expressed in terms of linear matrix inequalities. Finally, numerical simulations are performed on flexible joint manipulators to illustrate the validity of the proposed strategy.

Fuzzy dissipative and observer control for wind generator systems: a fuzzy time-dependent LKF approach

This paper examines the network-based dissipative and fuzzy observer control for wind generator systems (WGSs) by using time-dependent fuzzy Lyapunov–Krasovskii functional (LKF) approach. The main aim of this work is to design the desired dissipative and observer control gains. An effective time-dependent fuzzy LKF is constructed, and using a new mathematical analysis method dissipative and fuzzy observer controller performance are designed for wind generator systems. The results are represented in the form of linear matrix inequalities. Takagi–Sugeno fuzzy model-based algorithm is considered to increase the efficiency of the system and guarantee almost convergence. To show the effectiveness of the proposed novel method, simulations for WGS are provided.

Robust $$ H_{\infty } $$ Fault-Tolerant Control for Discrete-Time Nonlinear System with Actuator Faults and Time-Varying Delays Using Nonlinear T–S Fuzzy Models

In this paper, the problem of fault estimation and fault-tolerant control for a class of nonlinear discrete-time system state time-varying delay and actuator fault is investigated. This class of systems is represented through the Takagi–Sugeno (T–S) fuzzy model with nonlinear functions satisfying some sector-bounded conditions. By adding these nonlinear functions in the local sub-models, the observer and controller can be designed with fewer rules and less computation burden. The method proceeds in two steps: first, a full-order fuzzy fault estimation observer (FFEO) design is proposed to estimate the actuator faults and the nonlinear functions in the T–S models. Second, based on the online fault estimation, a dynamic output feedback fault-tolerant controller (DOFTC) is then designed to compensate the effect of faults by stabilizing the closed-loop system. Furthermore, sufficient less conservative delay-dependent conditions for the existence of the desired FFEO and DOFTC are given in terms of linear matrix inequalities by employing the fuzzy Lyapunov–Krasovskii function and free-weighting approach. Finally, a practical example is given to show the effectiveness and advantages of the proposed approach.

Event-triggered reliable dissipative filtering for the delay nonlinear system under networked systems with the sensor fault

ABSTRACTThis paper investigates the problem of fuzzy filter design for a class of delayed nonlinear system under random sensor faults with an event-triggered (ET) mechanism. (1) To estimate the dynamics of nonlinear plant, a T–S fuzzy model is manipulated. Random variables are disclosed to express the sensor fault. (2) To take some advantages over existing one, a variable ET mechanism is offered in networked systems (NSs). Under the ET mechanism, sensor data are released only when the plant's measurement (sampled) violate specific threshold of the event condition. (3) Another purpose of this article is to design filters involving system state delays. Then, by using a novel fuzzy Lyapunov–Krasovskii function approach with free weighting matrix technique, dissipative filter design of ET delay networked control systems is proposed. We consider both the sensor fault and ET scheme simultaneously. The simulation example is given to demonstrate the effectiveness of design method.

Delay Dependent Local Stabilization Conditions for Time-delay Nonlinear Discrete-time Systems Using Takagi-Sugeno Models

We propose convex conditions for stabilization of nonlinear discrete-time systems with time-varying delay in states through a fuzzy Takagi-Sugeno (T-S) modeling. These conditions are developed from a fuzzy Lyapunov-Krasovskii function and they are formulated in terms of linear matrix inequalities (LMIs). The results can be applied to a class of nonlinear systems that can be exactly represented by T-S fuzzy models inside a specific region called the region of validity. As a consequence, we need to provide an estimate of the set of safe initial conditions called the region of attraction such that the closed-loop trajectories starting in this set are assured to remain in the region of validity and to converge asymptotically to the origin. The estimate of the region of attraction is done with the aid of two sets: one dealing with the current state, and the other concerning the delayed states. Then, we can obtain the feedback fuzzy control law depending on the current state, x k , and the maximum delayed state vector, xk−d. It is shown that such a control law can locally stabilize the nonlinear discrete-time system at the origin. We also develop convex optimization procedures for the computation of the fuzzy control gains that maximize the estimates of the region of attraction. We present two examples to demonstrate the efficiency of the developed approach and to compare it with other approaches in the literature.

Event-triggered fuzzy filtering in networked control system for a class of non-linear system with time delays

ABSTRACTThis paper focuses on the design of H∞ filter and event-triggered scheme for a class of continuous non-linear networked control systems (NCSs) based on fuzzy system appeared with time delays. First, we consider the discrete event-triggered (ET) scheme to make efficient utilisation of bandwidth. Under this ET-scheme, sensor releases the data only when our sampled-data plant violates the specific event-triggered condition. Second, the T-S fuzzy system is used to model the non-linear NCSs. Another purpose of this paper is to design filters involving delays. Such filters have a more general form than the delay-free filters that have been mostly considered in the traditional studies. By using the time-delay system, co-design of event-triggered scheme and H∞ filter for the delayed NCSs is presented in a unified frame work. To choose the latest data packet and discard the dis-ordered packet logic, zero-order hold is inserted between the fuzzy filter and event generator. Then, by using the novel fuzzy Lyapunov–Krasovskii function approach with free-weighting matrix technique, H∞ filter design of event-triggered delay NCSs is proposed. Finally, to show the effective result of our co-design method, a tunnel-diode example is given.

Delay-dependent stabilization of discrete-time interval type-2 T–S fuzzy systems with time-varying delay

This study focuses on the stabilization problem of discrete-time interval type-2 (IT2) Takagi–Sugeno (T–S) fuzzy systems with time-varying delay. A new IT2 fuzzy state feedback controller that does not share the same membership functions and number of rules with the IT2 T–S fuzzy model is designed to close the feedback loop. A novel fuzzy Lyapunov–Krasovskii function (FLKF) is introduced based on the delay partitioning technique to derive the existence conditions of the IT2 fuzzy state feedback controller. The proposed FLKF does not require all the involved symmetric matrices to be positive definite. Several advanced inequalities and the boundary information on membership functions are used to achieve delay-dependent stabilization conditions in terms of linear matrix inequalities. Two numerical examples are provided to show the effectiveness of the proposed approach.

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.

Local stability and local stabilization of discrete-time T–S fuzzy systems with time-delay

The goal of this paper is to propose linear matrix inequality (LMI) formulations to analyze the local stability and local stabilization of discrete-time nonlinear systems represented by Takagi–Sugeno (T–S) fuzzy systems with time-delay. In order to estimate the domain of attraction (DA) of the time-delay systems, the so-called fuzzy Lyapunov-Krasovskii functionals are used to characterize the subsets of the DA as sublevel sets of the Lyapunov- Krasovskii functionals. We use the information of the variation rates of the membership functions to obtain less conservative results. Finally, examples are given to illustrate the proposed methods.

H ∞ Fuzzy filtering for nonlinear singular systems with time-varying delay

This paper considers the $H_{\infty}$ fuzzy filtering problem for continuous-time nonlinear singular systems with time-varying delay through the T-S fuzzy model approach. Firstly, by combining a reciprocally convex combination lemma and the fuzzy Lyapunov-Krasovskii function method, a new bounded real lemma (BRL) is proposed such that the resultant closed-loop systems are admissible and satisfy the prescribed $H_{\infty}$ disturbance attenuation. Then, by using the matrix decoupling technique, we translate the BRL into another form, which separates the coefficient matrices of systems and Lyapunov matrices. On the basis of such a new form of BRL, the fuzzy filter design problem is solved by checking the feasibility of a series of linear matrix inequalities (LMIs), and the filter gains can also be provided explicitly. Numerical examples are presented to show the reduction of the conservativeness compared to some published results in the literature.

Finite-time H∞ control for a class of nonlinear system with time-varying delay

This paper is concerned with the finite-time H ∞ control for a class of nonlinear system with time-varying delay, which can be represented by Takagi–Sugeno (T–S) fuzzy system. First, a new integral inequality, which is less conservative than other existing ones, is established by introducing some free fuzzy weighting matrices. Based on a new fuzzy Lyapunov–Krasovskii function, delay-dependent finite-time boundedness (FTB) criterion and H ∞ finite-time boundedness ( H ∞ FTB) criterion are derived for the open-loop fuzzy time-delay system. Then, some delay-dependent finite-time stabilization conditions are given by designing a memory state feedback controller, which is based on the so-called parallel distributed compensation (PDC) scheme. Finally, an example is provided to illustrate the validity of the proposed design approach.

Stability and Stabilization of Discrete-Time T–S Fuzzy Systems With Stochastic Perturbation and Time-Varying Delay

This paper is concerned with the problems of stability analysis and stabilization for a class of discrete-time Takagi-Sugeno (T-S) fuzzy systems with stochastic perturbation and time-varying state delay. By means of the delay-partitioning method and slack variables, a novel fuzzy Lyapunov-Krasovskii function is constructed to reduce the conservatism of stability conditions. Those conditions are converted to finite linear matrix inequalities, which can be readily solved by standard numerical software. Then, the delay-dependent stabilization approach, which is based on a nonparallel distributed compensation scheme, is introduced for the closed-loop fuzzy systems. Finally, illustrative examples are provided to illustrate the feasibility and effectiveness of the proposed methods.

Stability analysis and decentralized H∞ control for time-delay fuzzy interconnected systems via fuzzy Lyapunov-Krasovskii functional

This paper is concerned with the stability analysis and H∞ fuzzy controller design of continuous-time fuzzy interconnected systems with time-varying delays. The fuzzy interconnected system consists of N interconnected time-delay Takagi-SugenoT-S fuzzy subsystems and the delay-dependent stability analysis is based on the fuzzy weighting-dependent Lyapunov-Krasovskii functionals. It is shown that the delay-dependent stability of the time-delay fuzzy interconnected systems can be established if a fuzzy weighting-dependent Lyapunov-Krasovskii functional can be constructed, and moreover, the functional can be obtained by solving a set of linear matrix inequalities LMIs that are numerically feasible. The decentralized H∞ controllers are also designed by solving a set of LMIs based on these fuzzy weighting-dependent Lyapunov-Krasovskii functionals. It is demonstrated via numerical examples that the stability and controller synthesis results based on fuzzy weighting-dependent Lyapunov-Krasovskii functionals are less conservative than those based on the common quadratic Lyapunov-Krasovskii functionals.

Delay-dependent [formula omitted] fuzzy observer-based control for discrete-time nonlinear systems with state delay

This paper studies the H ∞ fuzzy observer-based control problem for discrete-time nonlinear systems with state delay via a fuzzy Lyapunov–Krasovskii functional (LKF) approach. The Takagi and Sugeno (T–S) fuzzy model is employed to represent a discrete-time nonlinear system with state delay. Based on a non-parallel distributed compensation (PDC) scheme, a new fuzzy LKF is first constructed to derive a delay-dependent condition such that the closed-loop fuzzy system is asymptotically stable and satisfies a prescribed level of H ∞ disturbance attenuation. In the derivation process, two free fuzzy weighting matrices are introduced to express the relationship between the present state and delay state. Then, a novel linear matrix inequality (LMI) approach is proposed for the design of a delay-dependent H ∞ fuzzy observer-based controller. The control and observer matrices can be solved directly by using the existing LMI optimization techniques. Finally, a numerical example is given to illustrate the effectiveness of the proposed design method.