A New Fuzzy Lyapunov Approach to Non-Quadratic Stabilization of Takagi-Sugeno Fuzzy Models

A New Fuzzy Lyapunov Approach to Non-Quadratic Stabilization of Takagi-Sugeno Fuzzy ModelsIn this paper, new non-quadratic stability conditions are derived based on the parallel distributed compensation scheme to stabilize Takagi-Sugeno (T-S) fuzzy systems. We use a non-quadratic Lyapunov function as a fuzzy mixture of multiple quadratic Lyapunov functions. The quadratic Lyapunov functions share the same membership functions with the T-S fuzzy model. The stability conditions we propose are less conservative and stabilize also fuzzy systems which do not admit a quadratic stabilization. The proposed approach is based on two assumptions. The first one relates to a proportional relation between multiple Lyapunov functions and the second one considers an upper bound to the time derivative of the premise membership functions. To illustrate the advantages of our proposal, four examples are given.

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Synthesis of an LMI-based fuzzy control system with guaranteed cost performance: a piecewise Lyapunov approach

Sba Controle & Automação Sociedade Brasileira de Automatica ◽

10.1590/s0103-17592006000200009 ◽

2006 ◽

Vol 17 (2) ◽

pp. 213-225 ◽

Cited By ~ 6

Author(s):

Natache S. D. Arrifano ◽

Vilma A. Oliveira ◽

Lúcia V. Cossi

Keyword(s):

Lyapunov Functions ◽

Fuzzy Systems ◽

State Feedback ◽

Linear Matrix ◽

Cost Performance ◽

Lyapunov Approach ◽

Analysis And Design ◽

Quadratic Lyapunov Functions ◽

Guaranteed Cost ◽

Takagi Sugeno

A new stability analysis and design of a fuzzy switching control based on uncertain Takagi-Sugeno fuzzy systems are proposed. The fuzzy system adopted is composed by a family of local linear uncertain systems with aggregation. The control design proposed uses local state feedback gains obtained from an optimization problem with guaranteed cost performance formulated in the context of linear matrix inequalities and a fuzzy switching scheme built from local Lyapunov functions. The global stability is guaranteed by considering a class of piecewise quadratic Lyapunov functions. Examples are given to illustrate the applicability of the proposed approach.

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Stability Analysis of Interconnected Fuzzy Systems Using the Fuzzy Lyapunov Method

Mathematical Problems in Engineering ◽

10.1155/2010/734340 ◽

2010 ◽

Vol 2010 ◽

pp. 1-10 ◽

Cited By ~ 17

Author(s):

Ken Yeh ◽

Cheng-Wu Chen

Keyword(s):

Stability Analysis ◽

Lyapunov Functions ◽

Fuzzy Systems ◽

Lyapunov Method ◽

Lyapunov Stability Theory ◽

Linear Matrix ◽

Stability Conditions ◽

Fuzzy Lyapunov Function ◽

Quadratic Lyapunov Functions ◽

The Stability

The fuzzy Lyapunov method is investigated for use with a class of interconnected fuzzy systems. The interconnected fuzzy systems consist ofJinterconnected fuzzy subsystems, and the stability analysis is based on Lyapunov functions. Based on traditional Lyapunov stability theory, we further propose a fuzzy Lyapunov method for the stability analysis of interconnected fuzzy systems. The fuzzy Lyapunov function is defined in fuzzy blending quadratic Lyapunov functions. Some stability conditions are derived through the use of fuzzy Lyapunov functions to ensure that the interconnected fuzzy systems are asymptotically stable. Common solutions can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. Finally, simulations are performed in order to verify the effectiveness of the proposed stability conditions in this paper.

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Stability analysis of Takagi-Sugeno fuzzy systems via LMI: methodologies based on a new fuzzy Lyapunov function

Sba Controle & Automação Sociedade Brasileira de Automatica ◽

10.1590/s0103-17592011000600011 ◽

2011 ◽

Vol 22 (6) ◽

pp. 664-676 ◽

Cited By ~ 5

Author(s):

Leonardo Amaral Mozelli ◽

Reinaldo Martinez Palhares

Keyword(s):

Stability Analysis ◽

Lyapunov Function ◽

Fuzzy Systems ◽

Time Derivative ◽

Time Varying ◽

Membership Functions ◽

Fuzzy Lyapunov Function ◽

Numerical Tools ◽

Takagi Sugeno ◽

First Time

Stability analysis of TS fuzzy systems can be much improved by resorting to fuzzy Lyapunov functions, since they are parameterized by membership functions and can better characterize the time-varying feature of these systems by means of the information regarding the first time-derivative of the membership functions. In this paper an enhanced fuzzy Lyapunov function is used to develop stability conditions that evaluate also the second time-derivative of membership functions, improving the time-varying characterization of TS systems. By using diferent strategies to consider the information regarding such derivatives and employing some numerical tools that decouple system from Lyapunov function matrices new LMI tests are developed. Numerical examples illustrate the effectiveness of those methodologies.

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Generalized non-monotonic Lyapunov functions for analysis and synthesis of Takagi-Sugeno fuzzy systems

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-200262 ◽

2020 ◽

Vol 39 (3) ◽

pp. 4147-4158

Author(s):

Pedro H.S. Coutinho ◽

Márcia L.C. Peixoto ◽

Márcio J. Lacerda ◽

Miguel Bernal ◽

Reinaldo M. Palhares

Keyword(s):

Lyapunov Functions ◽

Fuzzy Systems ◽

State Of The Art ◽

Disturbance Attenuation ◽

Linear Matrix ◽

Matrix Inequalities ◽

Current State ◽

Quadratic Lyapunov Functions ◽

Analysis And Synthesis ◽

Takagi Sugeno

This paper presents new stability and stabilisation conditions in the form of linear matrix inequalities for discrete-time Takagi-Sugeno fuzzy systems; they are derived considering a class of non-quadratic Lyapunov functions with multi-parametric non-monotonic terms, which significantly enhances the feasibility set of current state-of-the-art results. In addition, extensions to cope with the disturbance attenuation control problem are included. Benchmark numerical examples are provided to illustrate the effectiveness of the proposed approach.

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A Simplified Output Regulator for a Class of Takagi-Sugeno Fuzzy Models

Mathematical Problems in Engineering ◽

10.1155/2015/148173 ◽

2015 ◽

Vol 2015 ◽

pp. 1-18 ◽

Cited By ~ 8

Author(s):

Tonatiuh Hernández-Cortés ◽

Jesús A. Meda Campaña ◽

Luis A. Páramo Carranza ◽

Julio C. Gómez Mancilla

Keyword(s):

Fuzzy Systems ◽

Membership Functions ◽

Numerical Examples ◽

Fuzzy Models ◽

Mathematical Expressions ◽

Takagi Sugeno

This paper is devoted to solve the regulation problem on the basis of local regulators, which are combined using “new” membership functions. As a result, the exact tracking of references is achieved. The design of linear local regulators is suggested in this paper, but now adequate membership functions are computed in order to ensure the proper combination of the local regulators in the interpolation regions. These membership functions, which are given as mathematical expressions, solve the fuzzy regulation problem in a relative simple way. The form of the new membership functions is systematically derived for a class of Takagi-Sugeno (T-S) fuzzy systems. Some numerical examples are used to illustrate the viability of the proposed approach.

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Continuous Stability TS Fuzzy Systems Novel Frame Controlled by a Discrete Approach and Based on SOS Methodology

Mathematics ◽

10.3390/math9233129 ◽

2021 ◽

Vol 9 (23) ◽

pp. 3129

Author(s):

Ameni Ellouze ◽

Omar Kahouli ◽

Mohamed Ksantini ◽

Ali Rebhi ◽

Nidhal Hnaien ◽

...

Keyword(s):

Fuzzy Systems ◽

Discrete Systems ◽

Continuous Systems ◽

Fuzzy Models ◽

Discrete Approach ◽

Quadratic Lyapunov Functions ◽

The Stability ◽

Takagi Sugeno ◽

Stability Results ◽

Discrete Controller

Generally, the continuous and discrete TS fuzzy systems’ control is studied independently. Unlike the discrete systems, stability results for the continuous systems suffer from conservatism because it is still quite difficult to apply non-quadratic Lyapunov functions, something which is much easier for the discrete systems. In this paper and in order to obtain new results for the continuous case, we proposed to connect the continuous with the discrete cases and then check the stability of the continuous TS fuzzy systems by means of the discrete design approach. To this end, a novel frame was proposed using the sum of square approach (SOS) to check the stability of the continuous Takagi Sugeno (TS) fuzzy models based on the discrete controller. Indeed, the control of the continuous TS fuzzy models is ensured by the discrete gains obtained from the Euler discrete form and based on the non-quadratic Lyapunov function. The simulation examples applied for various models, by modifying the order of the Euler discrete fuzzy system, are presented to show the effectiveness of the proposed methodology.

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