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Singular Perturbation: When the Perturbation Parameter Becomes a State-Dependent Function

Tengfei Liu, Zhong-Ping Jiang

Abstract

This paper presents a new systematic framework for nonlinear singularly perturbed systems in which state-dependent perturbation functions are used instead of constant perturbation coefficients. Under this framework, general results are obtained for the global robust stability and input-to-state stability of nonlinear singularly perturbed systems. Interestingly, the proposed methodology provides innovative solutions beyond traditional singular perturbation theory for emerging control problems arising from nonlinear integral control, feedback optimization, and formation-based source seeking.

Singular Perturbation: When the Perturbation Parameter Becomes a State-Dependent Function

Abstract

This paper presents a new systematic framework for nonlinear singularly perturbed systems in which state-dependent perturbation functions are used instead of constant perturbation coefficients. Under this framework, general results are obtained for the global robust stability and input-to-state stability of nonlinear singularly perturbed systems. Interestingly, the proposed methodology provides innovative solutions beyond traditional singular perturbation theory for emerging control problems arising from nonlinear integral control, feedback optimization, and formation-based source seeking.
Paper Structure (19 sections, 11 theorems, 139 equations, 3 figures)

This paper contains 19 sections, 11 theorems, 139 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and Literature Review
  3. Main Contributions of This Paper
  4. Motivational Examples
  5. Nonlinear Integral Feedback
  6. Feedback Optimization
  7. Formation-Based Source Seeking
  8. Problem Formulation
  9. Main Results
  10. Perturbation Functions in Special Forms
  11. Tuning the Time Scale of the Reduced-Order Subsystem
  12. Tuning the Time Scale of the Boundary-Layer Subsystem
  13. Applications
  14. Nonlinear Integral Control
  15. Feedback Optimization
  16. ...and 4 more sections

Key Result

Theorem 1

Under Assumptions assumption.steadystate and assumption.stability, the system plant.slow--plant.fast is ISS with $(x,z-\varphi(0))$ as the state and $(d,w)$ as the input, if $\rho_s$ and $\rho_f$ satisfy the following conditions: Moreover, for certain $\sigma\in\mathcal{K}_{\infty}$ being continuously differentiable on $(0,\infty)$ and satisfying $\gamma_s(r)<\sigma(r)<\gamma_f^{-1}(r)$ for all $

Figures (3)

  • Figure 1: Configuration of a control system with nonlinear integral feedback.
  • Figure 2: Formation-Based Source Seeking.
  • Figure 3: Singularly perturbed system \ref{['plant.slow']}--\ref{['plant.fast']} as an interconnected system, where $\varphi$ represents the steady-state map of the $z$-subsystem.

Theorems & Definitions (22)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 2
  • Example 1
  • Corollary 1
  • Remark 6
  • ...and 12 more