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Symmetry-based observers for ODE systems

Stefano Battilotti

Abstract

In this paper we introduce an observer design framework for ordinary differential equation (ODE) systems based on various types of existing or even novel one-parameter symmetries (exact, asymptotic and variational) ending up with a certain number of semi-global and global observers, with bounded or unbounded system's solutions and with infinite- or finite-time convergence. We compare some of these symmetry-based observers with existing observers, recovering for instance the same performances of high-gain semiglobal observers and the finite-time convergence capabilities of sliding mode observers, while obtaining novel global observers where existing techniques are not able to provide any.

Symmetry-based observers for ODE systems

Abstract

In this paper we introduce an observer design framework for ordinary differential equation (ODE) systems based on various types of existing or even novel one-parameter symmetries (exact, asymptotic and variational) ending up with a certain number of semi-global and global observers, with bounded or unbounded system's solutions and with infinite- or finite-time convergence. We compare some of these symmetry-based observers with existing observers, recovering for instance the same performances of high-gain semiglobal observers and the finite-time convergence capabilities of sliding mode observers, while obtaining novel global observers where existing techniques are not able to provide any.

Paper Structure

This paper contains 20 sections, 8 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Notation I
  3. Group of transformations and prolongations
  4. One-parameter group of transformations
  5. Time scales transformation
  6. Prolongation of groups and generators
  7. Symmetries of ODE systems
  8. Notation II
  9. Definitions, examples and practical aspects
  10. Contracting symmetries
  11. Symmetry-based observers
  12. Semiglobal symmetry-based observers
  13. Global symmetry-based observers
  14. SI-norm estimators
  15. Global symmetry-based observers with infinite-time convergence
  16. ...and 5 more sections

Key Result

Proposition III.1

The set ${\mathcal{V}} ({\mathfrak p})$ is open and the mapping $( t, \mathsf{z}) \in {\mathcal{V}} ({\mathfrak p}) \xmapsto{ \Psi_{\mathfrak p} } \Psi_{\mathfrak p} ( t, \mathsf{z}) \mathrel{\ensurestackMath{\stackon[1pt]{=}{\Delta}}} \Psi ({\mathfrak p} , t, \mathsf{z}) \in {\mathcal{V}} (-{

Theorems & Definitions (25)

  • Definition III.1
  • Proposition III.1
  • Definition III.2
  • Definition III.3
  • Proposition III.2
  • Proposition III.3
  • Definition IV.1
  • Example IV.1
  • Example IV.2
  • Example IV.3
  • ...and 15 more