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Harnessing Membership Function Dynamics for Stability Analysis of T-S Fuzzy Systems

Donghwan Lee, Do-Wan Kim

TL;DR

This work addresses the challenge of stability analysis for continuous-time Takagi-Sugeno fuzzy systems, where traditional methods incur conservatism due to MF derivative bounds. It introduces a novel augmented Lyapunov function and an augmented state that explicitly includes MF dynamics, yielding a derivative-free LMI condition: $P \succ 0$ and $P\Phi(\alpha,\beta) + \Phi(\alpha,\beta)^T P \prec 0$ for all $ (\alpha,\beta) \in \Lambda_r \times \Lambda_p$. By modeling MF Jacobians and leveraging the augmented system, the approach reduces conservatism and avoids MF-derivative bounds, with finite-dimensional LMIs obtained via standard relaxations and structure-preserving slack variables. Comparative analyses suggest the method is less conservative than quadratic stability and does not require MF derivative bounds, at the cost of increased computational load. The framework offers a practical path for robust stability analysis of T-S fuzzy systems and points toward future extensions using sum-of-squares and further relaxations to support broader applicability.

Abstract

The main goal of this paper is to develop a new linear matrix inequality (LMI) condition for the asymptotic stability of continuous-time Takagi-Sugeno (T-S) fuzzy systems. A key advantage of this new condition is its independence from the bounds on the time-derivatives of the membership functions, a requirement present in the existing approaches. This is achieved by introducing a novel fuzzy Lyapunov function that incorporates an augmented state vector. Notably, this augmented state vector encompasses the membership functions, allowing the dynamics of these functions to be integrated into the proposed condition. This inclusion of additional information about the membership function serves to reduce the conservativeness of the suggested stability condition. To demonstrate the effectiveness of the proposed method, examples are provided.

Harnessing Membership Function Dynamics for Stability Analysis of T-S Fuzzy Systems

TL;DR

This work addresses the challenge of stability analysis for continuous-time Takagi-Sugeno fuzzy systems, where traditional methods incur conservatism due to MF derivative bounds. It introduces a novel augmented Lyapunov function and an augmented state that explicitly includes MF dynamics, yielding a derivative-free LMI condition: and for all . By modeling MF Jacobians and leveraging the augmented system, the approach reduces conservatism and avoids MF-derivative bounds, with finite-dimensional LMIs obtained via standard relaxations and structure-preserving slack variables. Comparative analyses suggest the method is less conservative than quadratic stability and does not require MF derivative bounds, at the cost of increased computational load. The framework offers a practical path for robust stability analysis of T-S fuzzy systems and points toward future extensions using sum-of-squares and further relaxations to support broader applicability.

Abstract

The main goal of this paper is to develop a new linear matrix inequality (LMI) condition for the asymptotic stability of continuous-time Takagi-Sugeno (T-S) fuzzy systems. A key advantage of this new condition is its independence from the bounds on the time-derivatives of the membership functions, a requirement present in the existing approaches. This is achieved by introducing a novel fuzzy Lyapunov function that incorporates an augmented state vector. Notably, this augmented state vector encompasses the membership functions, allowing the dynamics of these functions to be integrated into the proposed condition. This inclusion of additional information about the membership function serves to reduce the conservativeness of the suggested stability condition. To demonstrate the effectiveness of the proposed method, examples are provided.
Paper Structure (11 sections, 9 theorems, 45 equations, 3 figures, 2 tables)

This paper contains 11 sections, 9 theorems, 45 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Takagi--Sugeno (T--S) fuzzy system
  5. Quadratic stability
  6. Fuzzy Lyapunov stability
  7. Augmented system
  8. Fuzzy model of Jacobian
  9. Augmented system model
  10. Main results
  11. Comparative analysis

Key Result

Lemma 1

Let us consider any symmetric matrices $\Upsilon _{i} \in {\mathbb R}^{n\times n},i \in {\cal I}_r$. The following PMI holds: if $\Upsilon _{i} \prec 0$ holds for all $i \in {\cal I}_r$.

Figures (3)

  • Figure 1: Three main directions to reduce the conservatism
  • Figure 2: Interconnected system viewpoint of T-S fuzzy system
  • Figure 3: Estimated computational cost $\ln (N_d^2{N_l})$ of several approaches for different cases $n=2,r=3$, $n=3,r=3$, and $n=4,r=3$

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Lemma 1: wang1996approach
  • Lemma 2: tanaka1998fuzzy
  • Lemma 3: tanaka2004fuzzy
  • proof
  • Lemma 4: tanaka2003multiple
  • Lemma 5: mozelli2009reducing
  • Proposition 1
  • proof
  • ...and 6 more