Relaxed Stability Conditions Research Articles

Novel stability conditions for discrete-time T–S fuzzy systems: A Kronecker-product approach

This paper is concerned with the issue of developing a novel strategy to reduce the conservatism of stability conditions for discrete-time Takagi–Sugeno (T–S) fuzzy systems. Unlike the previous ones which are almost quadratic with respect to the state vector, a new class of Lyapunov functions is proposed which is quadratic with respect to the Kronecker products of the state vector, thus including almost the existing ones found in the literature as special cases. By combining the characterizations of homogeneous matrix polynomials and the properties of membership functions, relaxed stability conditions are derived in the form of linear matrix inequalities which can be efficiently solved by the convex optimization techniques. Finally, a numerical example is provided to illustrate the effectiveness of the proposed approach.

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.

Observer-based [formula omitted] controller design for interval type-2 T–S fuzzy systems

This paper focuses on studying observer-based controller synthesis for nonlinear plant subject to parameter uncertainties, which is represented as interval type-2 T–S fuzzy model. An interval type-2 fuzzy augmented system is formed by an interval type-2 T–S fuzzy model, an interval type-2 fuzzy observer and an interval type-2 fuzzy controller connected in a closed loop. Based on the Lyapunov stability theory, the stability of the interval type-2 fuzzy augmented systems is investigated. Some matrix inequality techniques are applied to derive stability conditions in the form of linear matrix inequalities (LMIs). With the consideration of the information of lower and upper membership functions, more relaxed stability conditions are obtained to guarantee the system stability. Finally, two numerical example are provided to illustrate the effectiveness of the proposed conditions.

A novel stabilization condition for a class of T–S fuzzy time-delay systems

In this paper, a relaxed stabilization problem for a class of Takagi and Sugeno (T–S) fuzzy time-delay systems is explored. By utilizing homogeneous polynomials scheme and Pólya׳s theorem, a relaxed delay-dependent stabilization condition is proposed. In addition, a novel slack matrix scheme is presented for stabilization condition of a class of T–S fuzzy time-delay systems in terms of linear matrix inequalities (LMIs). Lastly, a well-know numerical example and a truck-trailer example are given to demonstrate that the proposed stabilization condition can provide a longer allowable delay time than some existing studies.

Further studies on stability and stabilization conditions for discrete-time T–S systems with the order relation information of membership functions

This paper is concerned with the stability and stabilization problem for discrete-time Takagi–Sugeno (T–S) fuzzy systems. By taking into account the order relation information of membership functions (MFs), a so-called pseudo-MF vector is obtained through the transformation matrix; an improved approach to constructing the transformation matrix is given in order to make the best of the information of the MFs. Based on the transformed T–S fuzzy systems, relaxed stability and stabilization conditions are derived via both quadratic and non-quadratic Lyapunov functions. Numerical examples are provided to illustrate the effectiveness of the proposed approach.

New relaxed stability and stabilization conditions for continuous-time T–S fuzzy models

This paper deals with the problems of stability and stabilization for continuous-time Takagi–Sugeno (T–S) models. By employing the order relation information of the membership functions (MFs), the transformation matrix and the pseudo-MF vector are obtained. The conventional continuous-time T–S fuzzy model is then transformed to a new compact T–S model, based on which relaxed stability and stabilization conditions are derived in the form of linear matrix inequalities. An improved approach to constructing the transformation matrix is presented, which could take full advantage of the order relation information of the MFs and further reduce the conservatism. Numerical examples are provided to illustrate the effectiveness of the proposed approach.

Reducing the conservatism of stability analysis for discrete-time T–S fuzzy systems based on a delayed Lyapunov function

In this study, we propose some relaxed stability conditions for discrete-time Takagi–Sugeno fuzzy systems. In contrast to existing methods in the literature which are only parameter-dependent on the current-instant normalized fuzzy weighting functions, a delayed Lyapunov function that is formulated in a higher order form of the multi-instant normalized fuzzy weighting functions than existing ones is proposed for achieving the mission of reducing conservatism. It is proved in theory that the method given in this paper includes a recent result in the literature as one special case. Finally, a simulation example is employed to show the effectiveness of the proposed method.

More relaxed non-quadratic stabilization conditions for TS fuzzy control systems using LMI and GEVP

In this paper, a new systematic approach is presented to further decrease the conservativeness in stability analysis condition and controller design. Non-quadratic Lyapunov function is utilized to derive stability conditions in terms of linear matrix inequalities. Also, the control problem is formulated in a generalized eigenvalue problem. Considering the concept of decay rate and control input constraint, a new systematic procedure is proposed to calculate a maximum bound for the upper bounds of the time derivatives of the membership functions. Moreover, some slack matrices are introduced that help to reduce conservativeness. The number of inequalities is few compared to the existing results in literature, which helps the feasibility in the case of large number of fuzzy rules. Simulation examples and comparison results demonstrate the merits of this method.

A Partially Implicit FDTD Method for the Wideband Analysis of Spoof Localized Surface Plasmons

A partially implicit finite-difference time- domain (FDTD) method with relaxed stability condition is formulated on the basis of combining the implicit Newmark-Beta and the explicit leapfrog time integrations. Its stability is investigated by the Fourier method. The 2-D open and closed structures supporting spoof localized surface plasmons are analyzed with the presented method. The computational time is reduced in comparison with the traditional explicit FDTD while maintaining the same level of accuracy.

A Novel HIE-FDTD Method with Large Time-Step Size [Programmer's Notebook

A novel hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) method that is applicable to analyze structures with fine-scale discretization in one or two directions is presented. The more relaxed stability condition and dispersion relation are analytically derived. Simulation results are compared with those obtained by the existing HIE-FDTD method.

Control Synthesis of Discrete-Time T-S Fuzzy Systems via a Multi-Instant Homogenous Polynomial Approach.

This paper deals with the problem of control synthesis of discrete-time Takagi-Sugeno fuzzy systems by employing a novel multiinstant homogenous polynomial approach. A new multiinstant fuzzy control scheme and a new class of fuzzy Lyapunov functions, which are homogenous polynomially parameter-dependent on both the current-time normalized fuzzy weighting functions and the past-time normalized fuzzy weighting functions, are proposed for implementing the object of relaxed control synthesis. Then, relaxed stabilization conditions are derived with less conservatism than existing ones. Furthermore, the relaxation quality of obtained stabilization conditions is further ameliorated by developing an efficient slack variable approach, which presents a multipolynomial dependence on the normalized fuzzy weighting functions at the current and past instants of time. Two simulation examples are given to demonstrate the effectiveness and benefits of the results developed in this paper.

A new interval type-2 fuzzy controller for stabilization of interval type-2 T–S fuzzy systems

This study proposes a new interval type-2 (IT2) fuzzy controller for stabilization of nonlinear systems subject to parameter uncertainties. An IT2 T–S fuzzy model is employed to represent the nonlinear systems. New membership-dependent stability conditions in terms of linear matrix inequalities (LMIs) are derived by employing fuzzy weighting-dependent Lyapunov functions. The information about the time derivative of upper and lower membership functions is considered to achieve more relaxed stability conditions. Compared to existing techniques in literature, simulation examples illustrate that the conservativeness of the obtained results is reduced.

LMI condition for sampled‐data fuzzy control of nonlinear systems

A novel linear matrix inequality (LMI) condition for the stability of the sampled-data fuzzy control system based on the Takagi-Sugeno fuzzy model is presented. Using the novel Lyapunov functional, the relaxed stability condition is presented for the sampled-data fuzzy control and represented in the LMI format. A simulation example is provided to verify the effectiveness of the proposed technique.

Affine parallel distributed compensator design for affine fuzzy systems via fuzzy Lyapunov function

This paper develops a novel stability analysis and robust controller design method for affine fuzzy systems. The emphasis of the paper is to present more relaxed stability conditions based on nonquadratic fuzzy Lyapunov function and affine parallel distributed compensation. At first, diffeomorphic transformations are used to treat more general class of nonlinear systems in a unified manner. Then, by introducing slack matrices, the Lyapunov matrices are decoupled from the feedback gain matrices and controller affine terms which lead to eliminate the structural constraints of Lyapunov matrices and consequently reduces the conservativeness of the proposed approach. Because of the bias terms, the stabilization conditions are obtained in terms of bilinear matrix inequalities. A nonsingular state transformation together with using the S-procedure and also slack variables lead to derive the stabilization conditions in the formulation of linear matrix inequalities which can be solved by convex optimization techniques. Moreover, H∞ controller is used to reject the disturbances. Finally, the merit and applicability of the proposed approach are demonstrated via comparative numerical and industrial case studies.

Relaxed Stability and Stabilization Conditions of Networked Fuzzy Control Systems Subject to Asynchronous Grades of Membership

This paper presents relaxed stability and stabilization conditions for Takagi-Sugeno (T-S) fuzzy systems under network environments subject to asynchronous grades of membership. Because of the introduction of a communication network, the favorable property in the point-to-point connection, that is, sharing the identical premises in the fuzzy plant and the fuzzy controllers cannot be arbitrarily employed. To widen the applicability of the fuzzy control method under network environments, a novel method is provided to reconstruct the synchronous time scale grades of membership at the controller as those in the plant. As a result, the aforementioned favorable property can be conditionally used in the derivation of the stability and stabilization criteria for the system under consideration controlled over a communication network. Numerical examples have been given to illustrate the effectiveness of the proposed approach.

A Dynamic Decoupling Approach to Robust T–S Fuzzy Model-Based Control

In this paper, a dynamic decoupling approach is proposed to improve the robust Takagi-Sugeno (T-S) fuzzy model-based control to cope with system uncertainty, input actuator nonlinearity, and input time delay. First, the basic dynamic decoupling concept is introduced by involving virtual input dynamics, such that the system uncertainty and control input are decoupled in each fuzzy rule. This leads to simplified linear matrix inequality (LMI) conditions. Next, the dynamic decoupling approach is extended to controlling uncertain systems with input actuator nonlinearity (e.g., saturation, quantization, dead-zone, etc.) or time-varying input delay. Due to the decoupling between uncertainty, actuator nonlinearity, and input delay, more relaxed stability conditions are obtained for the asymptotic stability and H ∞ control performance. Furthermore, the limit on the initial condition is removed when considering input saturation. Larger and faster time-varying state and input delays are allowed under fewer LMIs. Finally, to show the advantages of the developed control method, numerical simulations are carried out on an inverted pendulum (subject to either the saturation, quantization, or delay input), a delay mass-spring-damper system, and a delay truck-trailer system.

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Abstract: In this paper, we present and analyze a LQ (Linear Quadratic) inverse optimal state-consensus protocol for continuous-time multi-agent systems with undirected graph topology. By Lyapunov analysis of the state-consensus error dynamics, we show the sufficient conditions on the algebraic connectivity of the graph to guarantee LQ inverse optimality and closed-loop stability. A more relaxed stability condition is also provided in terms of the algebraic connectivity. Finally, a formation control protocol for multiple mobile robots is proposed based on the target LQ inverse optimal consensus protocol, and the simulation results are provided to verify the performance of the proposed LQ inverse formation control method.Keywords: inverse optimal consensus, multi-agent systems, formation control Copyright© ICROS 2014 I. 서론 (consensus control)  ! #$%&'()*+,-, ./01234 56*+[1-3]. 7 #(multi-agent system) 8, 9:; ?7@ABC??D(E?-, 8(F?GHIJ=(state variable) K%LMJ=(output variable) <J =1 NO?DPQ(RS? T(U[1,2]. VWX6Y/ Z[\# ]/ ./01^_`ab*+,-[3-14],  c/ 9:;de(formation control) f gbh[15-17]./i, jk./Z./01^_bh[8-12]. lmnop:6(graph topology) c/jk q(LQ: Linear Quadratic) rZ(suboptimal)  1[8]#g?s,-, .=tuvf@w(algebraic Riccati equation) [8-11], HI[x [10,11], 9Zyz {[12] <( c,:?|}f VWX6~r4gb*+. ]/Z|}f 1aa€%

Relaxed stability conditions for interval type-2 fuzzy-model-based control systems

Relaxed stability conditions for interval type-2 fuzzy-model-based control systems

On stability analysis for neural networks with interval time-varying delays via some new augmented Lyapunov–Krasovskii functional

This paper is concerned with the problem of stability analysis of neural networks with interval time-varying delays. It is assumed that the lower bound of time-varying delays is not restricted to be zero. By constructing a newly augmented Lyapunov–Krasovskii functional which has not been proposed yet and utilizing some integral information on activation function as elements of augmented vectors, an improved stability criterion with the framework of linear matrix inequalities (LMIs) is introduced in Theorem 1. Based on the result of Theorem 1 and utilizing the property of the positiveness of Lyapunov–Krasovskii functional, a further relaxed stability condition will be proposed in Theorem 2. The effectiveness and less conservatism of the proposed theorems will be illustrated via three numerical examples.

Stability Analysis of Polynomial-Fuzzy-Model-Based Control Systems Using Switching Polynomial Lyapunov Function

This paper investigates the stability problem of polynomial-fuzzy-model-based control system, which is formed by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop. A switching polynomial Lyapunov function consisting of a number of local polynomial Lyapunov functions is proposed to investigate the system stability. It demonstrates a nice property in favor of the stability analysis that each local polynomial Lyapunov function transits continuously to each other. As different local polynomial Lyapunov functions are employed to investigate the system stability according to the operating domain, relaxed stability conditions compared with the stability analysis result with a common Lyapunov function can be developed. In order to allow a greater design flexibility for the polynomial fuzzy controller, the proposed polynomial-fuzzy-model-based control scheme does not require that both the polynomial fuzzy model and polynomial fuzzy controller share the same premise membership functions. Stability conditions in terms of sum of squares are obtained to guarantee system stability and facilitate control synthesis. Simulation examples are given to verify the stability analysis results and demonstrate the effectiveness of the proposed polynomial fuzzy control scheme.