Relaxed Stability Conditions Research Articles

Relaxed Stability Conditions for Discrete-Time T–S Fuzzy Systems via Double Homogeneous Polynomial Approach

This paper deals with the problem of stability analysis for discrete-time Takagi–Sugeno (T–S) fuzzy systems. The double homogeneous polynomially parameter-dependent (DHPPD) Lyapunov function is proposed, which is expressed in the double homogeneous polynomial form not only of the membership functions but also of the state variables. Based on the DHPPD Lyapunov function, a relaxed stability condition is derived. Additionally, the complete square matricial representation of homogeneous polynomials is considered to further reduce the conservatism. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed approach.

Novel Summation Inequalities and Their Applications to Stability Analysis for Systems With Time-Varying Delay

The inequality technique plays an important role in stability analysis for time-delay systems. This technical note presents a new sequence of novel summation inequalities by introducing some free matrices, which includes the newly-developed Wirtinger-based and free-matrix-based summation inequalities as special cases. Moreover, the idea can be easily extended to the multiple-summation-inequality case. Based on the proposed inequalities, relaxed stability conditions are obtained for systems with time-varying delay. Numerical examples are given to demonstrate the effectiveness of the proposed approach.

Relaxed stability conditions for polynomial‐fuzzy‐model‐based control system with membership function information

This study presents a new approach for stability analysis of polynomial-fuzzy-model-based (PFMB) control system using membership function information. For the purpose of extracting the regional information of the membership functions, the operating domain is partitioned into several sub-domains. In each sub-domain, the boundaries of every single membership function overlap term and the numerical relation among all the membership function overlap terms are represented as a group of inequalities. Through the S-procedure, the regional membership function information is taken into account in the stability analysis to relax the stability conditions. The operating domain partition scheme naturally arises the motivation of constructing the PFMB control system with the sub-domain fuzzy controllers. Each polynomial fuzzy controller works in its corresponding sub-domain such that the compensation capability of controller is enhanced. The sum-of-squares (SOS) approach is proposed to obtain the stability conditions of the PFMB control system using the Lyapunov stability theory. The PFMB control system studied in this study has the feature that the number of fuzzy rules and the membership function shapes of the polynomial fuzzy controller can be designed independently from the polynomial fuzzy model. To verify the stability analysis result, a numerical example is given to demonstrate the validity of the proposed method.

불완전한 전반부 정합 하에서의 이산 T-S 퍼지 모델에 대한 완화된 안정화 조건

불완전한 전반부 정합 하에서의 이산 T-S 퍼지 모델에 대한 완화된 안정화 조건

Relaxed stability and stabilisation conditions for continuous‐time Takagi–Sugeno fuzzy systems using multiple‐parameterised approach

This study deals with the problem of stability analysis and control design for continuous-time Takagi–Sugeno fuzzy systems. Based on generalised fuzzy Lyapunov function (FLF) and fuzzy controller, relaxed stability and stabilisation conditions in terms of linear matrix inequalities are obtained. The generalised FLF is homogeneous polynomially parameter dependent on MFs with arbitrary degree. As the degree of the Lyapunov function increases, the conservatism of the obtained stability and stabilisation conditions is gradually reduced. In addition, it can be proved that the conditions generated in this study can include the existing ones found in the literature as special cases. Compared with the existing methods, no additional slack variables are introduced in stability analysis, and hence the same or less conservative results can be obtained with a lower computational cost. Numerical examples illustrate the effectiveness of the proposed method.

Stabilization of Interval Type-2 Polynomial-Fuzzy-Model-Based Control Systems

In this paper, the stability of polynomial-fuzzy-model-based (PFMB) systems equipped with mismatched interval type-2 (IT2) membership functions is investigated. Unlike the membership-function-independent methods, the information and properties of IT2 membership functions are considered in the stability analysis and contained in the stability conditions in terms of sum-of-squares (SOS) based on the Lyapunov stability theory. Three methods, demonstrating their own advantages, are proposed to conduct the stability analysis for the IT2 PFMB control systems. In the first one, we divide the operating domain into subdomains and then conduct the stability analysis incorporating the information and properties of the IT2 membership functions in subdomains. Through this approach, the stability conditions can be further relaxed compared with the membership-function-independent analysis. Polynomial functions are adopted in the second method to approximate the IT2 membership functions. The advantage of this method compared with the first one is that richer information of IT2 membership functions is considered without increasing the number of SOS conditions. In the third one, we combine the advantages of both the first and the second method offering a new approach which utilizes the information and properties of the lower and upper IT2 membership functions in subdomains through simpler polynomial approximation functions. It can be shown that more relaxed stability conditions can be obtained compared with the first two methods. Numerical examples and simulations are presented to verify the effectiveness of the proposed methods.

More Relaxed Non‐Quadratic Stabilization Conditions Using Ts Open Loop System and Control Law Properties

AbstractThis paper proposes more relaxed stabilization conditions based on a non‐quadratic Lyapunov function (NQLF) and parallel distributed compensator (PDC) controller. The conditions are derived in terms of linear matrix inequalities (LMIs) by introducing three slack matrices based on the properties of TS membership functions, an open loop system and a PDC controller. These slack matrices are utilized to decouple the LMI variables from the TS system and the input matrices. Therefore, the proposed approach greatly reduces the number of LMI conditions and improves feasibility by providing more degrees of freedom compared to recently published studies. Moreover, local stability and stabilization conditions are considered to handle the time derivatives of membership functions appearing in the stabilization synthesis of the TS closed‐loop system with PDC controller. Finally, several examples are presented to demonstrate the advantages of the proposed approaches.

Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams

We discuss numerical strategies to deal with PDE systems describing traffic flows, taking into account a density threshold, which restricts the vehicle density in the situation of congestion. These models are obtained through asymptotic arguments. Hence, we are interested in the simulation of approached models that contain stiff terms and large speeds of propagation. We design schemes intended to apply with relaxed stability conditions.

Relaxed conditions for stability of time-varying delay systems

In this paper, the problem of delay-dependent stability analysis of time-varying delay systems is investigated. Firstly, a new inequality which is the modified version of free-matrix-based integral inequality is derived, and then by aid of this new inequality, two novel lemmas which are relaxed conditions for some matrices in a Lyapunov function are proposed. Based on the lemmas, improved delay-dependent stability criteria which guarantee the asymptotic stability of the system are presented in the form of linear matrix inequality (LMI). Two numerical examples are given to describe the less conservatism of the proposed methods.

Relaxed stability conditions for the Takagi–Sugeno fuzzy system using a polynomial non‐quadratic Lyapunov function

In this study, sum-of-squares (SOS)-based relaxed stability conditions for the continuous-time Takagi–Sugeno (T–S) fuzzy system are derived using a polynomial non-quadratic Lyapunov function (PNQLF). The proposed methods consist of two approaches. Like the previous studies, the first uses the upper bounds on a time derivative of the fuzzy weighting function (TDFWF), and the other directly uses the TDFWF in order to obtain relaxed results and enlarge the domain of attraction. Moreover, unlike the conventional non-quadratic Lyapunov function, the newly proposed PNQLF uses different fuzzy weighting functions from those of the system in order to solve the existing problems. Finally, the effectiveness of the proposed methods is demonstrated by some numerical examples.

Relaxed stability conditions for continuous‐time Takagi–Sugeno fuzzy systems based on a new upper bound inequality

The important issue of reducing the conservatism of feasible stability criteria for continuous‐time Takagi–Sugeno fuzzy systems is studied in this article. In order to obtain more advanced result than previous ones, a new upper bound inequality is proposed and thus the properties of the normalized fuzzy weighting functions' time derivatives can be better used than the previous ones. In particular, the so‐called “redundant terms” considered in previous literature can be converted to “useful terms” which play a positive role in the underlying analysis process. Moreover, some useless additional variables and their derived inequalities are removed for enhancing the efficiency. Finally, an illustrative example is given to show the effectiveness of the proposed method. © 2016 Wiley Periodicals, Inc. Complexity 21: 289–295, 2016

Relaxed nonquadratic stabilization conditions for Markovian jump fuzzy systems with incomplete transition descriptions

This paper focuses on deriving relaxed mean-square stability and H∞ performance conditions for discrete-time MJFSs under a general description of transition probabilities. In the derivation, some available constraints are exploited on the basis of the property of transition probabilities, and then a two-stage relaxation scheme is proposed to reduce the conservatism that can arise from incomplete knowledge of probabilities as well as from fuzzy basis functions. Finally, nonquadratic stabilization conditions are derived in terms of LMIs, whose effectiveness is shown by two illustrative examples.

Robust fuzzy control for a hybrid magnetic bearings: the relaxed stabilization condition approach

In this paper, we propose robust fuzzy control for a hybrid magnetic bearings. The control objective of HMBs enables the rotor to rotate without any physical contact in spite of the nonlinearity and uncertainty of the concerned plants. To achieve the robust stability, we address the uncertainties of the given system based on the Takagi–Sugeno fuzzy model. Also, in order to maintain the relaxed stabilization condition, nonparallel distributed compensation control law, as analyzed by the parameter-dependent Lyapunov function, is applied to the HMBs with parametric uncertainties. The conditions for the robust controller are obtained in terms of solutions to linear matrix inequalities. Finally, simulation results for HMBs are used to demonstrate the feasibility of the proposed method.

Nonlinear MNL‐FDTD scheme for short‐gap electrostatic discharge simulation

A nonlinear and partially implicit finite-difference time-domain scheme with a relaxed stability condition is presented for fast full-wave simulation of electrostatic discharges occurred at a short gap. The nonlinear spark resistance model of Rompe and Weizel is directly incorporated in this scheme. Its stability, accuracy and computational efficiency are assessed in the application to a metallic structure system.

Modeling of pure acoustic wave in tilted transversely isotropic media using optimized pseudo-differential operators

Conventional modeling and imaging for tilted transversely isotropic (TTI) media may suffer from numerical instabilities and shear artifacts due to the coupling of the P- and SV-wave modes in the coupled equations. On the contrary, the decoupled equations for TTI media provide a more stable solution due to the separated P- and S-wave modes. Because the decoupled equations involve complicated pseudo-differential operators in space, it is more convenient to apply the pseudo-spectral method to these equations. However, the second-order time-stepping scheme of the pseudo-spectral method may suffer from time-stepping errors and instabilities for a large time step. We have developed an optimized pseudo-differential operator (OPO) that incorporates not only the spectral evaluation of the pseudo-differential operator but also a temporal correction term that would effectively compensate the time-stepping errors associated with the time-wavenumber domain extrapolation of the wave equation. The OPO was constructed through multiplying the symbol of the OPO by the normalized pseudo-Laplacian operator, which contains a variable compensation velocity. It was efficiently solved through the low-rank decomposition. We have applied OPOs to solve the TTI decoupled equation to simulate the pure acoustic wave. Our 2D and 3D synthetic results demonstrate that the proposed method has high accuracy in time and space with relaxed stability conditions compared with the conventional pseudo-spectral method. The low rank of symbols of OPOs makes the proposed method more efficient than the dispersion relation-based low-rank wave extrapolation and pseudo-analytical methods.

Improvement on stability conditions for continuous-time T–S fuzzy systems

This paper addresses the problem of stability analysis for continuous-time Takagi–Sugeno (T–S) fuzzy systems. A new series of line-integral Lyapunov functions (LILFs) is developed which can be seen as an extension of the existing LILF. Without the difficulty of dealing with the time derivatives of the membership functions, relaxed stability conditions are derived by combining the matrix Polya׳s theorem in terms of linear matrix inequalities. To further illustrate the effectiveness of the extended LILFs, the T–S fuzzy systems with time delay are also studied and corresponding relaxed stability conditions are obtained as well. Finally, several numerical examples are provided to illustrate the improvement of the proposed results.

Stabilization conditions for fuzzy control of uncertain fractional order non‐linear systems with random disturbances

A novel fractional order Takagi–Sugeno (T–S) fuzzy control method for a class of fractional order non-linear systems is proposed in this study. On the basis of the fractional order interval theory, the generalised T–S fuzzy model for a class of uncertain fractional order non-linear systems is presented. By using fractional order Lyapunov stability theorem, the more relaxed and practical sufficient stability conditions which are presented as a new set of linear matrix inequalities are put forward, which are guaranteed by rigorous mathematical derivation. The proposed control scheme is developed for fractional order non-linear systems stabilisation in the presence of random disturbances. Typical instances including the fractional order three-dimensional (3D) non-linear system and the fractional order 4D non-linear Lorenz hyperchaos are implemented. Simulation results indicate the designed fractional order T–S fuzzy control scheme works well compared with the existing method.

On the stability analysis of phasor and classic extremum seeking control

In this work we present a semi-global practical asymptotic stability analysis for phasor extremum seeking control with a general non-linear dynamic system. With the same technique applied to the classic band pass filter algorithm, we present a more relaxed (less constrained) semi-global practical asymptotic stability condition compared to earlier work. The results are based on a non approximated averaging for both control techniques.

Relaxed Stabilization Conditions for the T–S Fuzzy System with Input Constraints

The stabilization conditions of the discrete Takagi–Sugeno (T–S) fuzzy system are reduced by considering possible switching subregions. In addition, the stabilization conditions for the T–S fuzzy system are relaxed by representing the interactions among the fuzzy subsystems in a single matrix. However, these two concepts have not been applied together to the discrete T–S fuzzy system with constraints on the control input. The aim of this paper is to relax the stabilization conditions for the discrete T–S fuzzy system with constraints on the control input. The possible switching subregions fired by two successive states of the system are analyzed and utilized to reduce the stabilization conditions. The interactions of fuzzy subsystems within two subregions are integrated into a single matrix to relax the stabilization conditions. The relaxation and effectiveness of the proposed stabilization conditions are demonstrated by a numerical example and a mass-spring-damper system.

Robust H∞ stabilization conditions for a class of uncertain T–S fuzzy neutral systems with disturbance

In this paper, the H∞ stabilization problems for a class of uncertain Takagi–Sugeno (T–S) fuzzy neutral systems are investigated. Based on homogeneous polynomials and Pólya׳s theorem, a novel stabilization condition for uncertain T–S fuzzy neutral systems is proposed in terms of a linear matrix inequality (LMI) to guarantee the asymptotic stabilization of fuzzy neutral systems. In addition, by using a novel polynomial slack matrix, a more relaxed stabilization condition is developed. Lastly, a truck–trailer example and a numerical example are given to demonstrate that the proposed stabilization condition gives longer allowable delay times and lower desired level of disturbance attenuation γ than those obtained using some existing methods, showing the effectiveness and applicability of the proposed method.