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A New Type of Nonlinear Disturbance Rejection

Simon Kuang, Xinfan Lin

TL;DR

The paper tackles the lack of a nonlinear high-pass analogue for disturbance rejection in continuous-time nonlinear systems and introduces a dissipativity-based framework to obtain quantitative guarantees. It defines nonlinear high-pass behavior through $NHP(1)$ and $ANHP(1)$, linking the input derivative $\dot{u}$ to output stability and providing nonasymptotic bounds via an $Lp$-Barbalat perspective. Through a suite of SISO examples (linear, nonlinear with $i\propto \sinh v$, slow nonlinear, and sector-bounded resistors), the authors illustrate how the framework yields provable performance and robust-control results, including nonlinear PI controllers with finite-time-domain guarantees. The work also discusses compositionality of nonlinear high-pass operators and proposes a nonlinear Padé-like NP(1) approximation to analyze asymptotics, indicating potential for formal verification and extension to higher-order behavior.

Abstract

Asymptotic disturbance rejection (equivalently tracking) for nonlinear systems has been studied only in qualitative terms (the state is asymptotically stable under bounded disturbances). We show how to prove quantitative performance guarantees for the nonlinear servomechanism problem. Our technique originates by applying a gain inequalities point of view to an ad fontes reexamination of the linear problem: what is the nonlinear equivalent of a sensitivity transfer function with a zero at the origin? We answer: a nonlinear input-output system is high-pass if its output is stable with respect to the \emph{derivative} of the input. We first show that definition generalizes high-pass resistor-capacitor circuit analysis to accommodate nonlinear resistors. We then show that this definition generalizes the steady-state disturbance rejection property of integral feedback controllers for linear systems. The theoretical payoff is that low-frequency disturbance rejection is captured by a quantitative, non-asymptotic output cost bound. Finally, we raise theoretical questions about compositionality of nonlinear operators.

A New Type of Nonlinear Disturbance Rejection

TL;DR

The paper tackles the lack of a nonlinear high-pass analogue for disturbance rejection in continuous-time nonlinear systems and introduces a dissipativity-based framework to obtain quantitative guarantees. It defines nonlinear high-pass behavior through and , linking the input derivative to output stability and providing nonasymptotic bounds via an -Barbalat perspective. Through a suite of SISO examples (linear, nonlinear with , slow nonlinear, and sector-bounded resistors), the authors illustrate how the framework yields provable performance and robust-control results, including nonlinear PI controllers with finite-time-domain guarantees. The work also discusses compositionality of nonlinear high-pass operators and proposes a nonlinear Padé-like NP(1) approximation to analyze asymptotics, indicating potential for formal verification and extension to higher-order behavior.

Abstract

Asymptotic disturbance rejection (equivalently tracking) for nonlinear systems has been studied only in qualitative terms (the state is asymptotically stable under bounded disturbances). We show how to prove quantitative performance guarantees for the nonlinear servomechanism problem. Our technique originates by applying a gain inequalities point of view to an ad fontes reexamination of the linear problem: what is the nonlinear equivalent of a sensitivity transfer function with a zero at the origin? We answer: a nonlinear input-output system is high-pass if its output is stable with respect to the \emph{derivative} of the input. We first show that definition generalizes high-pass resistor-capacitor circuit analysis to accommodate nonlinear resistors. We then show that this definition generalizes the steady-state disturbance rejection property of integral feedback controllers for linear systems. The theoretical payoff is that low-frequency disturbance rejection is captured by a quantitative, non-asymptotic output cost bound. Finally, we raise theoretical questions about compositionality of nonlinear operators.
Paper Structure (19 sections, 12 theorems, 42 equations)

This paper contains 19 sections, 12 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Systems theory
  4. Robust control
  5. Contributions
  6. Systems theory
  7. Robust control
  8. Preliminaries
  9. Overview of dissipativity theory
  10. Definition of a nonlinear high pass filter
  11. Examples: SISO filters
  12. A linear high-pass filter
  13. A nonlinear high-pass filter
  14. A slow nonlinear high-pass filter
  15. A sector-bounded nonlinear resistor
  16. ...and 4 more sections

Key Result

Theorem 1

Suppose that the system eq:state_space is ANHP(1). Then if $\int_0^T \beta(|\dot u(t)|)\dif t < \infty$ and $\int_0^T \beta_1(|\dot u(t)|)\dif t < \infty$, then $y(t) \to 0$.

Theorems & Definitions (26)

  • Definition 1
  • Definition 2: Weakly finite-gain stable
  • Definition 3: NHP(1)
  • Definition 4: ANHP(1)
  • Theorem 1
  • proof
  • Corollary 1
  • Remark 1
  • Proposition 1
  • proof
  • ...and 16 more