Continuous-time Fuzzy Systems Research Articles

H∞ fault detection and control of Takagi–Sugeno continuous-time conic-type nonlinear systems

In this study, control and fault detection are examined for Takagi–Sugeno (T–S) fuzzy continuous-time systems with conic-type nonlinearity. In a conic-type dynamic system, Takagi–Sugeno fuzzy models are applied. A conic-type non-linearity must lie in an n-dimensional hyper-sphere with an uncertain centre to satisfy the restrictive condition. A combined error system of estimation and control error is obtained by combining the observations and the controller. Our main goal is to design an observer and controller for fault detection that minimizes the impact of unknown inputs and faults on the system in terms of the H ∞ performance. The Lyapunov function is used to construct the residual signal using an observer and a controller. Using Linear Matrix Inequalities (LMIs), we can express the performance conditions for H ∞ . The scheme's effectiveness is demonstrated with numerical examples.

On the state estimation for nonlinear continuous-time fuzzy systems

On the state estimation for nonlinear continuous-time fuzzy systems

New Results of Observer Design for Continuous-Time Fuzzy Systems: A Switching Technique

It is observed that if the premise variables are differentiable, the time derivatives of membership functions are positive or nonpositive. Based on the above fact, the switching observer gains and controller gains are designed to guarantee that the fuzzy system is globally exponentially asymptotically stable by applying the average dwell-time (ADT) technique. In addition, two methods are proposed to find the controller gains and observer gains by dividing the matrices into <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3\times 1$ </tex-math></inline-formula> blocks or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2\times 2$ </tex-math></inline-formula> blocks. Finally, two examples are given to verify the conclusion.

Filter Design for Positive T–S Fuzzy Continuous-Time Systems With Time Delay Using Piecewise-Linear Membership Functions

This article focuses on the filtering problem and stability analysis for positive Takagi-Sugeno (T-S) fuzzy systems with time delay under L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -induced performance. Due to the importance of estimation of system states but the few filter design results on positive nonlinear systems, it is an attractive and meaningful topic well worth studying. In order to fully exploit and take advantage of the positivity of positive T-S fuzzy systems, many commonly used methods, for instance free-weighting matrix approach and similarity transformation are probably not suitable for positive systems. To address the hard-nut-to-crack problem, an auxiliary variable is introduced so that the augmentation approach can be employed to carry out the positivity and stability analysis of filtering error systems. In addition, another obstacle that cannot be ignored is the existence of nonconvex terms in the stability and positivity conditions. For getting around this barrier, some iterative linear matrix inequality algorithms have been proposed in the literature. However, considering the weakness that these methods cannot guarantee the convergence to a numerical solution and the iterative process is exhaustive, we present an effective matrix decoupling method to convert the nonconvex conditions into convex ones in this article. Furthermore, a linear copositive Lyapunov function, which incorporates the positivity of system states and time delay at the same time is chosen so that the positivity characteristic of filtering error systems can be captured further. However, because of plenty of valuable information of membership functions being ignored, hence, the obtained results are conservative. For the sake of relaxing the conservativeness, the advanced piecewise-linear membership functions approximate method is utilized to facilitate the stability and positivity analysis. Therefore, the relaxed stability and positivity conditions, which are cast as sum of squares (SOS) are obtained and can be solved numerically. Finally, the effectiveness of the designed fuzzy filtering strategy with satisfying L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -induced performance are demonstrated by a simulation example.

Analysis and synthesis conditions for T-S fuzzy continuous-time systems with partially matched premises

Abstract A recognized challenge in the Takagi-Sugeno (T-S) fuzzy controller design concerns the use of different membership functions (MF) for controller and system. Most of the conditions available in the literature require that the controller’s MF matche the system’s one. Therefore, the implementation of such controllers may lead to unsafe operational conditions whenever such a match is lost. The main contribution of this paper is to provide new convex formulations for both stability analysis and controller design for T-S fuzzy systems under unmatched MF. We assume the same number of premises, yielding conditions called partially matched premises. To reduce conservatism, we use the Lyapunov approach, and we write the MF of the controller from the MF of the system. Two examples serve to compare our approach with others found in the literature. The achieved results suggest that our method outperforms the others.

A novel event-based fuzzy control approach for continuous-time fuzzy systems

This paper investigates the event-based fuzzy control problem for continuous-time interval type-2 (IT2) fuzzy networked control systems (NCSs). A new Lyapunov–Krasovskii functional is introduced in the stability analysis, where the integrands of the Lyapunov–Krasovskii functional are dependent of membership functions. Applying the information of time-derivatives of membership functions, a novel switching event-based fuzzy controller is designed to ensure that the closed-loop IT2 fuzzy system with mismatched membership functions is asymptotically stable. Different from the existing event-based fuzzy control results, both of the information of membership functions and the time-derivatives of membership functions can be utilized in the stability analysis to reduce the conservativeness. Finally, a simulation example is given to show the effectiveness and advantages of the design and analysis.

A Unitary Construction of Takagi-Sugeno Fuzzy Fault Detection Filters

The unitary approach to fault detection filters design, intended to Takagi-Sugeno fuzzy continuous-time systems, is presented in the paper. Based on fuzzy observers guaranteing asymptotic convergence of the state estimation error, the matrix gains of all linear components of the fuzzy filter are designed to obtain the same singular values of the filter transfer function matrix. Related to the fuzzy state control, and for systems under influence of actuator faults, the problems are revealed for a three-tank system plant.

Extended dissipative filtering of fuzzy systems with time-varying delay under imperfect premise matching

ABSTRACTThis paper investigates the extended dissipative filter design problems for continuous-time fuzzy systems with time-varying delays under imperfect premise variables based on a unified performance index−extended dissipative. Attention is focused on solving the H∞, L2 − L∞, passive and dissipative filtering problems for fuzzy systems with time-varying delays under this unified framework. Based on the unified performance index, a new delay-dependent filter design approach in terms of linear matrix inequalities is obtained by employing Lyapunov–Krasovskii functional method together with a novel efficient integral inequality. The designed filter can guarantee the filtering error system satisfy the prescribed H∞, L2 − L∞, passive and dissipative performance by tuning the weighting matrices in the conditions. Moreover, in this paper, the fuzzy filter does not need to share the same membership function with fuzzy model, which can enhance design flexibility and robust property of the fuzzy filter system. Finally, two examples are provided to illustrate the effectiveness and significant improvement of the method proposed in this paper.

Relaxed formulation of the design conditions for Takagi-Sugeno fuzzy virtual actuators

Abstract The H∞ norm approach to virtual actuators design, intended to Takagi-Sugeno fuzzy continuous-time systems, is presented in the paper. Using the second Ljapunov method, the design conditions are formulated in terms of linear matrix inequalities in adapted bounded real lemma structures. Related to the static output controller, and for systems under influence of single actuator faults, the design steps are revealed for a three-tank system plant.

Non-fragile H∞ output feedback control design for continuous-time fuzzy systems

In this paper, we investigate the problem of non-fragile H∞ fuzzy control design for continuous Takagi Sugeno (T–S) fuzzy systems with uncertainties, external disturbance and unmeasurable state variables. For the case of controller and observer gain additive variations, we propose a new solution of the fragility problem by developing the non-fragile design schemes ensuring the asymptotic stability and H∞ performance for the resulting closed loop systems. By considering a fuzzy Lyapunov function and by introducing slack variables, we propose the new sufficient stabilization conditions formulated in LMI constraints which can be easily solved using the convex optimization tools. The effectiveness the proposed results are illustrated through three numerical examples.

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.

New delay-dependent stability analysis for fuzzy time-delay interconnected systems

This paper investigates the delay-dependent stability of continuous-time fuzzy interconnected systems with time-varying delays. The fuzzy interconnected system consists of interconnected continuous-time Takagi–Sugeno fuzzy subsystems. A stability criterion based on fuzzy Lyapunov–Krasovskii functional is proposed to guarantee the asymptotic stability of fuzzy time-delay interconnected systems. In the derivation process, some free fuzzy weighting matrices are introduced among the terms of the system equation to yield less conservative results. The proposed stability criterion can be obtained by solving a set of linear matrix inequalities that are numerically feasible. Finally, two numerical examples are provided to demonstrate the effectiveness of the present method.

Robust sampled-data stabilization of uncertain fuzzy systems via input delay approach

This paper is concerned with robust sampled-data stabilization for uncertain fuzzy systems. The system is modeled as a continuous-time fuzzy system, while the control input is a zero-order sampled-data signal. When a zero-order control input is considered, the closed-loop system can be seen as an input delay system. Hence, an input delay system approach is introduced for sampled-data control design. Sufficient stability conditions for the closed-loop system are given in terms of linear matrix inequalities (LMIs). Under the assumption that sampling interval is not greater than some prescribed number, stability conditions for fuzzy time-delay systems are given by using Jensen’s Inequality, which simplifies conditions with less matrix variables. A design method of robust sampled-data state feedback controller for uncertain fuzzy systems is then proposed. Numerical examples are given to illustrate our robust sampled-data state feedback control design and to show the effectiveness over other existing results.

Nonfragile $H_{\infty}$ Filtering of Continuous-Time Fuzzy Systems

This paper is concerned with the problem of nonfragile <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> filtering for a class of continuous-time fuzzy systems. Attention is focused on the design of a filter such that the filtering error system preserves a prescribed <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance, where the filter to be designed is assumed to have gain variations. By using the quadratic Lyapunov function approach and adding slack matrix variables, sufficient conditions for designing the nonfragile <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> filter are proposed in terms of solutions to a set of linear matrix inequalities (LMIs). Finally, a simulation example will be given to show the efficiency of the proposed design methods.

Finite-Time Control of Fuzzy Continuous-Time Systems with Exogenous Disturbance

In the paper, new results on finite-time control of fuzzy continuous nonlinear systems subject to exogenous disturbance are presented. Applying the Lyapunov function theory, some sufficient conditions including relaxed ones are established for finite-time stability by fuzzy controller laws.

Relaxações convexas de convergência garantida para o projeto de controladores para sistemas nebulosos de Takagi-Sugeno

Este artigo fornece condições convexas com convergência garantida para o projeto de controladores por realimentação de estados que estabilizam quadraticamente e também asseguram desempenhos ótimos H2 e H∞ sob estabilidade quadrática para sistemas nebulosos de Takagi-Sugeno contínuos no tempo. As condições propostas são formuladas como desigualdades matriciais lineares (LMIs) dependentes de parâmetros com variáveis de folga provenientes do Lema de Finsler e com parâmetros pertencentes ao simplex unitário. A lei de controle nebulosa é dada por um ganho de realimentação de estados que é um polinômio homogêneo de grau g, que tem como caso particular o compensador paralelo distribuído quando g = 1. Propriedades algébricas dos parâmetros do sistema e resultados recentes sobre positividade de polinômios são usados para construir relaxações LMIs que, diferentemente da maioria das relaxações da literatura, convergem assintoticamente para a solução sempre que esta existir. Graças ao grau de liberdade obtido pelo uso de variáveis de folga, as condições apresentadas no artigo representam avanços em relação a condições recentemente publicadas baseadas no Teorema de Pólya, e podem ser vistas como uma alternativa a técnicas baseadas na relaxação de formas quadráticas. A eficiência numérica em termos de precisão e esforço computacional é demonstrada por meio de comparações com outros métodos da literatura.

Sample-data Decentralized Reliable H∞ Hyperbolic Control for Uncertain Fuzzy Large-scale Systems with Time-varying Delay

This paper studies the problem of sampled-data reliable H ∞ hyperbolic control for uncertain continuous-time fuzzy large-scale systems with time-varying delay. Firstly, the fuzzy hyperbolic model (FHM) is used to establish the model for certain complex large-scale systems, then according to the Lyapunov direct method and the decentralized control theory of large-scale systems, linear matrixine qualities (LMIs)-based conditions arederived toguarantee the H ∞ performance not only when all control components are operating well, but also in the presence of some possible actuator failures. Moreover, the precise failure parameters of the actuators are not required, and the requirements are only the lower and upper bounds of failure parameters. The conditions are dependent on the upper bound of time-delay, and not dependent on the derivative of time-varying delay. Therefore, the obtained results are less conservative. Finally, two examples are provided to illustrate the design procedure and its effectiveness.

Robust guaranteed cost control of uncertain fuzzy systems under time-varying sampling

In practice, the system is often modeled as a continuous-time fuzzy system, while the control input is applied only at discrete instants. This system is called a sampled-data control system. In this paper, robust guaranteed cost control for uncertain sampled-data fuzzy systems is discussed. A guaranteed cost control where a quadratic cost function is bounded by a certain scalar, not only stabilizes a system but also considers a control performance. A typical sampled-data control is the zero-order input, which can be represented as a piecewise-continuous delay. Here we take a delay system approach to the sampled-data guaranteed cost control problem. The closed-loop system with a sampled-data state feedback controller becomes a system with time-varying delay. First, guaranteed cost control performance conditions for the closed-loop system are given in terms of linear matrix inequalities (LMIs). Such conditions are derived by using Leibniz–Newton formula and free weighting matrix method for fuzzy systems under the assumption that sampling time is not greater than some prescribed scalar. Then, a design method of robust guaranteed cost state feedback controller for uncertain sampled-data fuzzy systems is proposed. Examples are given to illustrate our robust sampled-data guaranteed cost control design.

Robust [formula omitted] control of uncertain fuzzy systems under time-varying sampling

Robust H ∞ sampled-data control for uncertain fuzzy systems is discussed. In many practical situations, a system is modeled as a continuous-time fuzzy system, while the control input is the zero-order hold, which can be represented as a piecewise-continuous delay. Here we take a delay system approach to the H ∞ sampled-data control problem. The closed-loop system with a sampled-data state feedback controller becomes a system with time-varying delay. First, H ∞ performance conditions for the closed-loop system are given in terms of linear matrix inequalities (LMIs). Such conditions are derived by using the Leibniz–Newton formula and free-weighting matrix method for fuzzy time-delay systems under the assumption that sampling time is not greater than some prescribed number. Then, a design method for a robust H ∞ sampled-data state feedback controller for uncertain fuzzy systems is proposed. Numerical examples are given to illustrate our robust H ∞ sampled-data state feedback control design.

New Results on $H_{\infty}$ Filtering for Fuzzy Time-Delay Systems

This paper is concerned with the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> fuzzy filtering problem for a class of continuous-time fuzzy systems with time-varying delays. The objective is to design a stable filter guaranteeing the asymptotic stability and a prescribed <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> performance of the filtering error system. Motivated by the parallel distributed compensation technique, we design a new filter model in this paper. Filter parameter matrices can be obtained from the solution of a convex optimization problem in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> fuzzy filter is given. Two numerical examples are provided to demonstrate the effectiveness and less conservativeness of the proposed design approach.