Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonquadratic global asymptotic stability certificates for saturated linear feedbacks

Andrea Cristofaro, Luca Zaccarian

TL;DR

This work tackles the problem of achieving global asymptotic stability for linear plants driven by bounded saturated inputs, particularly when the open-loop matrix $A_0$ is not Hurwitz (ANCBI case). It introduces relaxed positivity conditions for sign-indefinite extended quadratic Lyapunov forms that incorporate the input deadzone, enabling GAS certificates through nonquadratic Lyapunov functions and an invariance-principle argument. The authors develop a unified single-input framework with explicit matching conditions and extend the approach to multi-input systems via LMI-based conditions, facilitating convex stability analysis for higher-order ANCBI plants. A downward cart-pendulum case study demonstrates the method’s practicality, and the results yield both analytical and computable certificates for GAS, including new conditions on saturated integral-oscillator configurations.

Abstract

We establish sufficient conditions for positive (semi-)definiteness, with or without radial unboundedness, for nonquadratic Lyapunov function constructed as sign-indefinite quadratic forms involving the state and the deadzone of a suitable input. We then use these conditions to build weak nonquadratic Lyapunov functions establishing global asymptotic stability of linear systems in feedback through a saturation, leveraging invariance principles. Our results are shown to be non-conservative (necessary and sufficient) for a family of well known prototypical examples of linear SISO feedbacks that are not globally exponentially stabilizable (the so-called ANCBI plants). Our multi-input extension leads to convex stability analysis tests, formulated as linear matrix inequalities that are applicable to ANCBI non-globally-exponentially-stabilizable plants.

Nonquadratic global asymptotic stability certificates for saturated linear feedbacks

TL;DR

This work tackles the problem of achieving global asymptotic stability for linear plants driven by bounded saturated inputs, particularly when the open-loop matrix is not Hurwitz (ANCBI case). It introduces relaxed positivity conditions for sign-indefinite extended quadratic Lyapunov forms that incorporate the input deadzone, enabling GAS certificates through nonquadratic Lyapunov functions and an invariance-principle argument. The authors develop a unified single-input framework with explicit matching conditions and extend the approach to multi-input systems via LMI-based conditions, facilitating convex stability analysis for higher-order ANCBI plants. A downward cart-pendulum case study demonstrates the method’s practicality, and the results yield both analytical and computable certificates for GAS, including new conditions on saturated integral-oscillator configurations.

Abstract

We establish sufficient conditions for positive (semi-)definiteness, with or without radial unboundedness, for nonquadratic Lyapunov function constructed as sign-indefinite quadratic forms involving the state and the deadzone of a suitable input. We then use these conditions to build weak nonquadratic Lyapunov functions establishing global asymptotic stability of linear systems in feedback through a saturation, leveraging invariance principles. Our results are shown to be non-conservative (necessary and sufficient) for a family of well known prototypical examples of linear SISO feedbacks that are not globally exponentially stabilizable (the so-called ANCBI plants). Our multi-input extension leads to convex stability analysis tests, formulated as linear matrix inequalities that are applicable to ANCBI non-globally-exponentially-stabilizable plants.
Paper Structure (12 sections, 12 theorems, 68 equations, 3 figures)

This paper contains 12 sections, 12 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Nonquadratic positivity of sign-indefinite extended forms
  3. Lyapunov construction for GAS but not GES
  4. Single-input results
  5. Lyapunov construction for single-input systems
  6. Saturated integrator
  7. Saturated double integrator
  8. Saturated oscillator
  9. Saturated integral oscillator
  10. General architecture of Lyapunov functions
  11. Case study: downward cart-pendulum
  12. Conclusions

Key Result

Proposition 1

Condition eq:LiuAkasaka holds if and only if $A_0$ has eigenvalues in the closed left-half plane with Jordan blocks of dimension at most 2 for the eigenvalues at the origin and simple Jordan blocks for the eigenvalues in the rest of the imaginary axis.

Figures (3)

  • Figure 1: Downward spring-cart-pendulum: model.
  • Figure 2: Downward cart-pendulum: displacement states $(p,\dot{p})$ (top) and angular states $(\vartheta,\dot\vartheta)$ (bottom).
  • Figure 3: Downward cart-pendulum: Lyapunov functions \ref{['eq:Vspecific']} (top) and saturated input (bottom)

Theorems & Definitions (28)

  • Proposition 1
  • Theorem 1
  • proof
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Theorem 2
  • proof
  • Lemma 1
  • ...and 18 more