Continuous-time Takagi-Sugeno Research Articles

Robust $$H_{\infty }$$ Control of Takagi–Sugeno Systems with Actuator Saturation

The robust static output feedback control for continuous-time Takagi–Sugeno systems subject to actuator saturation is solved here, including $$H_{\infty }$$ performance guarantees. Based on a polytopic model of the saturation, sufficient conditions are proposed for designing these controllers in terms of Linear Matrix Inequalities. With the aid of some special derivations, bilinear matrix inequalities are converted into a set of linear matrix inequalities which can be solved easily without requiring iterative algorithms or equality constraints, moreover, the output matrix of the considered system does not require to be full row rank. Finally, some examples are presented to show the validity of the proposed methodology.

Further Study on Stabilization for Continuous-Time Takagi-Sugeno Fuzzy Systems With Time Delay.

In the recently published paper, a switching method has been proposed to deal with the time derivative of the membership functions and less conservative results can be obtained due to this method; however, this method is based on the assumption that the switching times are finite. In this article, this method is further studied and the average dwell-time (ADT) switching technique is applied to ensure the stability if there is no such assumption. In addition, an algorithm is proposed to find the switching controller gains. The final simulation demonstrates the effectiveness of the developed new results.

Stability and stabilization of T–S fuzzy systems with variable delays via new Bessel–Legendre polynomial based relaxed integral inequality

This paper investigates the delay-range-dependent (DRD) stability problem in regards to continuous time Takagi–Sugeno (T–S) fuzzy system with variable delays. First, a new integral inequality Lemma referred herein as higher order polynomial based relaxed integral inequality (HOPBRII) is proposed to reduce the estimation gap of the current work with that of Bessel–Legendre polynomial based inequality (BLPBI). Next, a suitable Lyapunov–Krasovskii functional (LKF) is constructed with delay product terms and then using the proposed inequality, suitable condition for DRD stability and stabilization condition of the considered T–S fuzzy system are derived in a linear matrix inequality (LMI) framework. To analyse the advantage theoretically and less conservatism of the proposed method, a new DRD stability condition using similar LKFs and the BLPBI approach is further derived herein. Finally, the efficacy of the proposed stability and stabilization conditions are validated through the solutions of four numerical examples.

Improved Delay-Range-Dependent Stability Condition for T–S Fuzzy Systems with Variable Delays Using New Extended Affine Wirtinger Inequality

In this paper, a new integral inequality lemma along with an appropriate Lyapunov–Krasovskii functional (LKF) is proposed for fuzzy time-delay system to enhance the delay upper bound estimate. The proposed lemma is referred to hereafter as Extended Affine Wirtinger inequality. The novelty of the lemma is twofold, it has the ability to integrate uncertain delay information utilizing the convex combination of certain and uncertain delay intervals involved in the proposed LKF, and it is compatible to derive delay-range-dependent (DRD) stability conditions for continuous time Takagi–Sugeno (T–S) fuzzy time-delay system. One noteworthy advantage of the proposed stability condition in a linear matrix inequality (LMI) framework is that it requires less number of matrix variables compared to the existing integral inequalities of the similar type, thus reducing the computational burden too. The efficacy of the proposed DRD stability condition over existing conditions is validated numerically by solving three examples related to fuzzy time-delay system.

Finsler-based Sampled-data Controller Design for Takagi-Sugeno Systems

Abstract This paper investigates the sampled-data control of continuous-time Takagi-Sugeno (T-S) fuzzy systems. The closed-loop dynamics is rewritten as a T-S system with input time-varying delays. In this context, asynchronous membership functions appears in the closed-loop dynamics. Thus, to reduce the conservatism of design conditions involving mismatch membership functions, a dedicated relaxation scheme is proposed. Then, from a convenient Lyapunov-Krasovskii function and the application of the Finsler’s Lemma, new LMI-based conditions are proposed for the design of sampled-data Parallel-Distributed-Compensation (PDC) controllers. An example is provided to illustrate the effectiveness of the proposed design methodology in simulation, as well as to highlight their conservatism improvement regarding to previous related results from the literature.

Stability and Stabilizability of TS-based Interconnected Nonlinear Systems

This paper aims to establish stability conditions for large scale nonlinear systems with unknown bounded interconnection terms. These conditions are then used to design of stabilizing controllers for such systems. Takagi-Sugeno (TS) fuzzy modeling is used to describe the nonlinear interconnected systems. Further, the drawbacks of using slack variables by adding null term and the Lyapunov fuzzy function in terms of increasing conservatism are highlighted. Unlike the continuous time TS fuzzy systems, the null product technique, when applied to continuous-time TS fuzzy large-scale systems (CFLSS) leads to more conservative results. Numerical examples are presented to illustrate our theoretical findings.

Robust $H_{\infty }$ Control for T–S Fuzzy Systems With State and Input Time-Varying Delays via Delay-Product-Type Functional Method

This paper investigates the problem of robust $H_{\infty }$ control for a class of nonlinear systems with state and input time-varying delays. The nonlinearity is presented by a continuous-time Takagi-Sugeno (T-S) fuzzy model with parameter uncertainties. A sufficient asymptotic stability condition is first proposed by constructing a delay-product-type augmented Lyapunov–Krasovskii functional and utilizing an extended reciprocally convex matrix inequality together with a Wirtinger-based integral inequality. Then, a state feedback controller is derived that guarantees the closed-loop fuzzy system being asymptotically stable with an $H_{\infty }$ performance index. Finally, four numerical examples are given to reveal the effectiveness and merits of the developed new design techniques.

Further Study on Observer Design for Continuous-Time Takagi-Sugeno Fuzzy Model With Unknown Premise Variables via Average Dwell Time.

This article further studies the problem of observer design for the continuous-time Takagi-Sugeno (T-S) fuzzy system with unmeasurable premise variables. A membership function-dependent Lyapunov function is designed to obtain the observer-based controller. Different from the existing results, a switching method is proposed to deal with the time derivative of membership functions. Several problems such as, too many parameters, and the small local stabilization region in the existing papers are solved by applying the switching method. In addition, two algorithms are designed to obtain the controller gains and observer gains. In the end, two examples are provided to demonstrate the effectiveness of the proposed approach.

Exponential Stabilization of Takagi–Sugeno Fuzzy Systems With Aperiodic Sampling: An Aperiodic Adaptive Event-Triggered Method

In this paper, we study the exponential stabilization problem for continuous-time Takagi–Sugeno fuzzy systems subject to aperiodic sampling. By aiming to transmission reduction, an appropriate aperiodic event-triggered communication scheme with adaptive mechanism is put forward, which covers the existing periodic mechanisms as special cases. For the sake of reduction in design conservativeness, both the available information of sampling behavior and threshold error are fully acquired by constructing a novel time-dependent Lyapunov functional. Then, a new exponential stability criterion is presented to establish the quantitative relationship among the adaptive adjusted event threshold, the decay rate, the upper bound, and the lower bound of variable sampling period, simultaneously. By resorting to a matrix transformation, the corresponding stabilization criterion is further derived by which the sampled-data controller can be obtained. Finally, two illustrative examples are provided to demonstrate the virtue and applicability of proposed design method.

Further study on local analysis of continuous‐time T‐S fuzzy models with bounded disturbances

This study further addresses the problem of computing the bounds of disturbances for continuous-time Takagi-Sugeno (T-S) fuzzy models. First, in order to bound the time derivatives of the membership functions, less conservative results, which overcome some drawbacks of the previous results such as special structure of free variable, redundant restrictive conditions and parameters, are obtained by applying the generalised Lyapunov function and non-quadratic control law. Second, if the disturbances are bounded by energy, some conditions are derived to get two local regions (a smaller and a bigger) that are contained in a specific compact set C, and guarantee that all the trajectories starting from the smaller region will remain in the bigger one. Finally, two examples are given to illustrate the effectiveness of the results.

Asynchronous Switched Observers Design for Switched Takagi-Sugeno Systems Subject to Output Disturbances

Abstract This paper deals with the design of asynchronous switched observers for a class of switched continuous-time Takagi-Sugeno (TS) systems subject to output disturbances. The considered switched TS observer is proposed for the continuous state estimation when the active switched modes of the system are unknown or not precisely known. The design conditions, obtained via switched Lyapunov-like function candidates, are expressed as Linear Matrix Inequality (LMI) and satisfy H∞ disturbance attenuation performances. The effectiveness of the proposed approach is illustrated through the simulation of an academic example.

선적분 퍼지 리아푸노프 함수를 이용한 연속시간 T-S 퍼지 시스템에 대한 정적 출력궤환 제어

The problem of static output feedback control design for continuous-time Takagi-Sugeno (T-S) fuzzy systems is addressed. The proposed stabilizing condition is based on a line-integral fuzzy Lyapunov function and a non-parallel distributed compensation (non-PDC). The existence condition of the stabilizing static output feedback control law is represented in terms of linear matrix inequalities (LMIs) without any transformation matrices and equality conditions. The proposed sufficient stabilizing condition is less conservative than previous results in the literature. An example is illustrated to verify this.

LMI and SDP technique for stability analysis of nonlinear delay systems subject to constraints

The aim of the study is to present an estimation of the domain of attraction (DA) for nonlinear delay systems. The proposed approach is based on the assumption that on a subset of instant states the system is represented by a continuous-time Takagi–Sugeno (TS) system with delay. Because of the constraints defining the subset under consideration, an important issue is to estimate the set of initial points such that the trajectories starting at those points remain to satisfy the constraints. Using Lyapunov functions with the Razumikhin technique, we describe the problem of constructing an inner estimation of the DA in terms of invariant ellipsoids. As the result, the problem reduces to linear matrix inequalities and semidefinite programs which can be solved numerically. The method is also applied to constructing some outer estimate for the attractor of a nonlinear delay system subject to asymptotic stability for an approximating TS system. The resulting ellipsoidal estimate for the attractor depends on the estimate of the approximation error.

Finite-Frequency Static Output Feedback H∞ Control of Continuous-Time T–S Fuzzy Systems

This paper is concerned with finite-frequency static output feedback (SOF) [Formula: see text] control for a class of continuous-time Takagi–Sugeno (T–S) fuzzy systems. With the aid of the generalized Kalman–Yakubovich–Popov (GKYP) lemma, sufficient conditions for the existence of the finite-frequency SOF [Formula: see text] control are presented. The bilinear matrix inequalities are converted to a set of linear matrix inequalities, with the aid of some special derivations. Two practical examples are given to demonstrate the effectiveness of the proposed method.

Bibo Stabilisation of Continuous–Time Takagi–Sugeno Systems under Persistent Perturbations and Input Saturation

Abstract This paper presents a novel approach to the design of fuzzy state feedback controllers for continuous-time non-linear systems with input saturation under persistent perturbations. It is assumed that all the states of the Takagi-Sugeno (TS) fuzzy model representing a non-linear system are measurable. Such controllers achieve bounded input bounded output (BIBO) stabilisation in closed loop based on the computation of inescapable ellipsoids. These ellipsoids are computed with linear matrix inequalities (LMIs) that guarantee stabilisation with input saturation and persistent perturbations. In particular, two kinds of inescapable ellipsoids are computed when solving a multiobjective optimization problem: the maximum volume inescapable ellipsoids contained inside the validity domain of the TS fuzzy model and the smallest inescapable ellipsoids which guarantee a minimum *-norm (upper bound of the 1-norm) of the perturbed system. For every initial point contained in the maximum volume ellipsoid, the closed loop will enter the minimum *-norm ellipsoid after a finite time, and it will remain inside afterwards. Consequently, the designed controllers have a large domain of validity and ensure a small value for the 1-norm of closed loop.

A New Approach to Stability and Stabilization Analysis for Continuous-Time Takagi–Sugeno Fuzzy Systems With Time Delay

Till now, there are lots of stability and stabilization results about Takagi–Sugeno (T–S) fuzzy systems with time delay, but most of them are independent of the analysis of membership functions. Since the membership functions are an essential component to make a fuzzy system different from others, the conditions without its information are conservative. In this brief paper, a new Lyapunov–Krasovskii functional is designed to investigate the stability and stabilization of continuous-time T–S fuzzy systems with time delay. Different from the existing results in the literature, the integrand of the Lyapunov–Krasovskii functional in this paper depends not only on the integral variable, but also on the membership functions, and thus, the information of the time derivative of membership functions can also be used to reduce the conservativeness of finding the maximum delay bounds. Utilizing the information of the time derivative of membership, a number of controllers are designed according to their sign, and then, a switching idea is applied to stabilize the fuzzy system. In the end, two examples are given to illustrate the feasibility and validity of the design and analysis.

A novel Lyapunov–Krasovskii functional approach to stability and stabilization for T–S fuzzy systems with time delay

Abstract This paper is concerned with the problem of the stability and stabilization for continuous-time Takagi–Sugeno(T–S) fuzzy systems with time delay. A novel Lyapunov–Krasovskii functional which includes fuzzy line-integral Lyapunov functional and membership-function-dependent Lyapunov functional is proposed to investigate stability and stabilization of T–S fuzzy systems with time delay. In addition, switching idea which can avoid time derivative of membership functions is introduced to deal with derivative term. Relaxed Wirtinger inequality is employed to estimate integral cross term. Sufficient stability and stabilization criteria are derived in the form of matrix inequalities which can be solved using the switching idea and LMI method. Several numerical examples are given to demonstrate the advantage and effectiveness of the proposed method by comparing with some recent works.

Notice of Violation of IEEE Publication Principles: Dissipativity-Based Fuzzy Integral Sliding Mode Control of Continuous-Time T-S Fuzzy Systems

This paper is concerned with dissipativity-based fuzzy integral sliding mode control (FISMC) of continuous-time Takagi-Sugeno (T-S) fuzzy systems with matched/unmatched uncertainties and external disturbance. To better accommodate the characteristics of T-S fuzzy models, an appropriate integral-type fuzzy switching surface is put forward by taking the state-dependent input matrix into account, which is the key contribution of the paper. Based on the utilization of Lyapunov function and property of the transition matrix for unmatched uncertainties, sufficient conditions are presented to guarantee the asymptotic stability of corresponding sliding mode dynamics with a strictly dissipative performance. A FISMC law is synthesized to drive system trajectories onto the fuzzy switching surface despite matched/unmatched uncertainties and external disturbance. A modified adaptive FISMC law is further designed for adapting the unknown upper bound of matched uncertainty. Two practical examples are provided to illustrate the effectiveness and advantages of developed FISMC scheme.

Design of distributed sampled‐data fuzzy controller for a class of non‐linear hyperbolic PDE systems: input delay approach

This study addresses the problem of distributed sampled-data fuzzy controller design for a class of non-linear distributed parameter systems which are described by first-order hyperbolic partial differential equations (PDEs). To achieve this goal, first, the non-linear system is modelled by a continuous-time Takagi–Sugeno first-order hyperbolic PDE fuzzy model. Subsequently, the authors design a new distributed sampled-data fuzzy controller that generates a zero-order hold sampled-data control signal appropriate for the PDE systems. Then, a new Lyapunov–Krasovskii functional is suggested to provide the stability analysis conditions of the closed-loop control system. Moreover, the stabilisation conditions are obtained and converted to linear matrix inequalities using some new null terms. The proposed technique has removed the structural constraints on the convection and Lyapunov matrices. Finally, the proposed approach is applied on a biological system and a non-isothermal plug flow reactor.

Notice of Violation of IEEE Publication Principles: Fuzzy-Affine-Model-Based Memory Filter Design of Nonlinear Systems With Time-Varying Delay

This paper studies the piecewise-affine memory $\mathscr {H}_{\infty }$ filtering problem for nonlinear systems with time-varying delay in a delay-dependent framework. The nonlinear plant is characterized by a continuous-time Takagi–Sugeno fuzzy-affine model with parametric uncertainties. The purpose is to develop a new approach for filter synthesis procedure with less conservatism. Specifically, by constructing a novel Lyapunov–Krasovskii functional, together with a Wirtinger-based integral inequality, reciprocally convex inequality and S-procedure, an improved criterion is first attained for analyzing the $\mathscr {H}_{\infty }$ performance of the filtering error system, and then via some linearization techniques, the piecewise-affine memory filter synthesis is carried out. It is shown that the existence of desired filter gains can be explicitly determined by the solution of a convex optimization problem. Finally, simulation studies are presented to reveal the effectiveness and less conservatism of the developed approaches. It is anticipated that the proposed scheme can be further extended to the analysis and synthesis of continuous-time fuzzy-affine dynamic systems with integrated communication delays in the networked circumstance.