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Estimation of Regions of Attraction for Nonlinear Systems via Coordinate-Transformed TS Models

Artun Sel, Mehmet Koruturk, Erdi Sayar

Abstract

This paper presents a novel method for estimating larger Region of Attractions (ROAs) for continuous-time nonlinear systems modeled via the Takagi-Sugeno (TS) framework. While classical approaches rely on a single TS representation derived from the original nonlinear system to compute an ROA using Lyapunov-based analysis, the proposed method enhances this process through a systematic coordinate transformation strategy. Specifically, we construct multiple TS models, each obtained from the original nonlinear system under a distinct linear coordinate transformation. Each transformed system yields a local ROA estimate, and the overall ROA is taken as the union of these individual estimates. This strategy leverages the variability introduced by the transformations to reduce conservatism and expand the certified stable region. Numerical examples demonstrate that this approach consistently provides larger ROAs compared to conventional single-model TS-based techniques, highlighting its effectiveness and potential for improved nonlinear stability analysis.

Estimation of Regions of Attraction for Nonlinear Systems via Coordinate-Transformed TS Models

Abstract

This paper presents a novel method for estimating larger Region of Attractions (ROAs) for continuous-time nonlinear systems modeled via the Takagi-Sugeno (TS) framework. While classical approaches rely on a single TS representation derived from the original nonlinear system to compute an ROA using Lyapunov-based analysis, the proposed method enhances this process through a systematic coordinate transformation strategy. Specifically, we construct multiple TS models, each obtained from the original nonlinear system under a distinct linear coordinate transformation. Each transformed system yields a local ROA estimate, and the overall ROA is taken as the union of these individual estimates. This strategy leverages the variability introduced by the transformations to reduce conservatism and expand the certified stable region. Numerical examples demonstrate that this approach consistently provides larger ROAs compared to conventional single-model TS-based techniques, highlighting its effectiveness and potential for improved nonlinear stability analysis.

Paper Structure

This paper contains 17 sections, 2 theorems, 46 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions for Mathematical Optimization Related Concepts
  4. Linear Matrix Inequality
  5. Semi Definite Programming
  6. Relaxation in Mathematical Optimization
  7. Relaxation
  8. Definitions for Applications of the Methods
  9. Dynamical System
  10. Stability Definitions
  11. Analysis Problem in LTI systems
  12. Synthesis Problem in LTI systems
  13. Lyapunov Function and Stability of Nonlinear Dynamical Systems
  14. Convex Optimization Based Nonlinear Dynamical System Analysis
  15. Computation Using Method
  16. ...and 2 more sections

Key Result

Theorem 1

For eq:autonomous_NL_DYN_SYS, where the origin is the equilibrium point, If $\exists V(x): \mathcal{D} \rightarrow \mathbb{R}$continuously differentiable such that then $\mathbf{x} = 0$ is asymptotically stable; if, in addition, $\mathcal{D} = \mathbb{R}^n$ and $V(x)$ is radially unbounded, then $\mathbf{x} = 0$ is globally asymptotically stable. If $V(x)$ satisfies the constraints in $\Omega \su

Figures (2)

  • Figure 1: for the system computed with the method
  • Figure 2: (a) computed for the system whose state variable is $\bar{x}\in\bar{\mathcal{D}}$ (b) computed for the system whose state variable is $x\in\mathcal{D}$ by using the -method illustrated by red-shaded region and the boundary of the that is computed in $\bar{\mathcal{D}}$ domain and mapped to $\mathcal{D}$

Theorems & Definitions (17)

  • Remark 1
  • Definition 1: Stability in the Sense of Lyapunov
  • Definition 2: Asymptotic Stability
  • Definition 3: Exponential Stability
  • Definition 4: Global Stability
  • Definition 5: Local Stability
  • Remark 2
  • Theorem 1
  • Definition 6
  • Definition 7
  • ...and 7 more