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Passivity theorems for input-to-state stability of forced Lur'e inclusions and equations, and consequent entrainment-type properties

Chris Guiver

Abstract

A suite of input-to-state stability results are presented for a class of forced differential inclusions, so-called Lur'e inclusions. As a consequence, semi-global incremental input-to-state stability results for systems of forced Lur'e differential equations are derived. The results are in the spirit of the passivity theorem from control theory as both the linear and nonlinear components of the Lur'e inclusion (or equation) are assumed to satisfy passivity-type conditions. These results provide a basis for the analysis of forced Lur'e differential equations subject to (almost) periodic forcing terms and, roughly speaking, ensure the existence and attractivity of (almost) periodic state- and output-responses, comprising another focus of the present work. One ultimate aim of the study is to provide a robust and rigorous theoretical foundation for a well-defined and tractable ``frequency response'' of forced Lur'e systems.

Passivity theorems for input-to-state stability of forced Lur'e inclusions and equations, and consequent entrainment-type properties

Abstract

A suite of input-to-state stability results are presented for a class of forced differential inclusions, so-called Lur'e inclusions. As a consequence, semi-global incremental input-to-state stability results for systems of forced Lur'e differential equations are derived. The results are in the spirit of the passivity theorem from control theory as both the linear and nonlinear components of the Lur'e inclusion (or equation) are assumed to satisfy passivity-type conditions. These results provide a basis for the analysis of forced Lur'e differential equations subject to (almost) periodic forcing terms and, roughly speaking, ensure the existence and attractivity of (almost) periodic state- and output-responses, comprising another focus of the present work. One ultimate aim of the study is to provide a robust and rigorous theoretical foundation for a well-defined and tractable ``frequency response'' of forced Lur'e systems.
Paper Structure (10 sections, 14 theorems, 193 equations, 6 figures, 1 table)

This paper contains 10 sections, 14 theorems, 193 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Lur'e differential inclusions --- preliminaries
  3. A passivity theorem for Lur'e differential inclusions
  4. A passivity theorem for semi-global incremental stability of forced Lur'e differential equations
  5. Consequences of incremental stability --- response to almost periodic forcing terms
  6. Examples
  7. Summary
  8. Further properties of correspondence $F$
  9. Proof of Lemma \ref{['lem:inclusion_F0_prep']}
  10. Proof of Lemma \ref{['lem:sg_technical_preparation']}

Key Result

Lemma 2.1

Consider the Lur'e inclusion eq:lure_inclusion and introduce the statement: Assume that there exist continuously differentiable $V : \mathbb R^n \to \mathbb R_+$ and $\alpha_i \in {\mathcal{K}}_\infty$, $i= 1,2$, such that The following statements hold.

Figures (6)

  • Figure 1.1: Forced Lur'e system
  • Figure 6.1: Coupled mass-spring system
  • Figure 6.2: Numerical results from Example \ref{['ex:two_mass']}. (a) Graph of the functions $v_p$ (black) and $v_{\rm s}$ (grey). (b) Graphs of $z_i(t; x_0, v_{\rm p})$ against $t$. (c) Graphs of $\dot z_i(t; x_0, v_{\rm p})$. See main text for full description.
  • Figure 6.3: (a) Graphs of $z_i(t; 0, v)$ against $t$. (b) Graphs of $\dot z_i(t; 0, v)$ against $t$. See legend or main text in Example \ref{['ex:two_mass']} for full description.
  • Figure 6.4: Graphs of $\dot z_i(t; z_i(0), v_{\rm s})$ against $t$. See legend or main text in Example \ref{['ex:two_mass']} for full description.
  • ...and 1 more figures

Theorems & Definitions (32)

  • Lemma 2.1
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • proof : Proof of Theorem \ref{['thm:iss_inclusion']}
  • Theorem 3.5
  • ...and 22 more