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A linear dissipativity approach to incremental input-to-state stability for a class of positive Lur'e systems

Violaine Piengeon, Chris Guiver

TL;DR

The paper addresses incremental stability for a class of forced, positive Lur'e systems by leveraging linear dissipativity theory. It establishes an incremental stability framework under two key hypotheses that encode a linear dissipation structure and an incremental bound on the nonlinearity, yielding two main trajectory-difference bounds; these yield incremental ISS and input/output stability with linear gains. Consequences include convergence to equilibria under convergent forcing and controlled responses to almost periodic inputs, with explicit results for equilibrium existence and equi-convergence. The findings extend existing absolute stability and ISS results by exploiting positivity to obtain tractable dissipation inequalities and by treating multi-input multi-output (MIMO) cases, including almost periodic forcing behavior; this advances robust understanding of nonlinear positive Lur'e networks and informs potential extensions to observer design and synchronization problems.

Abstract

Incremental stability properties are considered for certain systems of forced, nonlinear differential equations with a particular positivity structure. An incremental stability estimate is derived for pairs of input/state/output trajectories of the Lur'e systems under consideration, from which a number of consequences are obtained, including the incremental exponential input-to-state stability property and certain input-output stability concepts with linear gain. Incremental stability estimates provide a basis for an investigation into the response to convergent and (almost) periodic forcing terms, and is treated presently. Our results show that an incremental version of the real Aizerman conjecture is true for positive Lur'e systems when an incremental gain condition is imposed on the nonlinear term, as we describe. Our argumentation is underpinned by linear dissipativity theory -- a property of positive linear control systems.

A linear dissipativity approach to incremental input-to-state stability for a class of positive Lur'e systems

TL;DR

The paper addresses incremental stability for a class of forced, positive Lur'e systems by leveraging linear dissipativity theory. It establishes an incremental stability framework under two key hypotheses that encode a linear dissipation structure and an incremental bound on the nonlinearity, yielding two main trajectory-difference bounds; these yield incremental ISS and input/output stability with linear gains. Consequences include convergence to equilibria under convergent forcing and controlled responses to almost periodic inputs, with explicit results for equilibrium existence and equi-convergence. The findings extend existing absolute stability and ISS results by exploiting positivity to obtain tractable dissipation inequalities and by treating multi-input multi-output (MIMO) cases, including almost periodic forcing behavior; this advances robust understanding of nonlinear positive Lur'e networks and informs potential extensions to observer design and synchronization problems.

Abstract

Incremental stability properties are considered for certain systems of forced, nonlinear differential equations with a particular positivity structure. An incremental stability estimate is derived for pairs of input/state/output trajectories of the Lur'e systems under consideration, from which a number of consequences are obtained, including the incremental exponential input-to-state stability property and certain input-output stability concepts with linear gain. Incremental stability estimates provide a basis for an investigation into the response to convergent and (almost) periodic forcing terms, and is treated presently. Our results show that an incremental version of the real Aizerman conjecture is true for positive Lur'e systems when an incremental gain condition is imposed on the nonlinear term, as we describe. Our argumentation is underpinned by linear dissipativity theory -- a property of positive linear control systems.
Paper Structure (14 sections, 9 theorems, 102 equations, 3 figures, 1 table)

This paper contains 14 sections, 9 theorems, 102 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Matrices, vectors and norms
  5. Function spaces
  6. A positive linear dissipation inequality
  7. Lur'e systems
  8. Incremental stability
  9. Consequences of incremental stability
  10. Convergence properties
  11. Response to almost periodic forcing
  12. Examples
  13. Summary
  14. Appendix

Key Result

Lemma 2.1

Consider the positive linear control system eq:lti, let $\xi \geq 0$ be given and assume that $A + \xi I$ is Hurwitz. Let $(r,q) \in \mathbb R^{\texttt{m}}_+ \times \mathbb R^{\texttt{p}}_+$ be given. The following statements are equivalent. If either statement above holds and, additionally, then $p \gg 0$.

Figures (3)

  • Figure 1.1: Block diagram of forced four-block Lur'e system
  • Figure 6.1: Simulation results from Example \ref{['ex:1']}. See main text for a description.
  • Figure 6.2: Numerical simulations from Example \ref{['ex:2']}. See main text for a description.

Theorems & Definitions (21)

  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lem:dissipation']}
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 4.1
  • proof : Proof of Theorem \ref{['thm:incremental']}
  • Corollary 4.2
  • proof : Proof of Corollary \ref{['cor:incremental_stability']}
  • Remark 4.3
  • Lemma 5.1
  • ...and 11 more