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Synchronous Models and Fundamental Systems in Observer Design

Pieter van Goor, Robert Mahony

TL;DR

The paper defines a synchronous model as an error-based extension of an internal model, and shows a system admits a synchronous lift if and only if its accessibility Lie algebra is finite-dimensional and complete. It then identifies fundamental systems as those induced by transitive group actions on homogeneous spaces, establishing a one-to-one correspondence between fundamental systems and systems with synchronous lifts. For fundamental systems, a constructive synchronous observer architecture is developed, enabling discretisation and composition of correction terms while guaranteeing Lyapunov-based convergence. Three illustrative examples demonstrate how synchrony, symmetry construction, and observer design operate in practice, with implications for robust, almost-global estimation on manifolds and Lie groups in robotics and related fields.

Abstract

This paper introduces the concept of a synchronous model as an extension of the internal model concept used in observer design for dynamical systems. A system is said to contain a synchronous model of another if there is a suitable error function between the two systems that remains stationary for all of the trajectories of the two systems. A system is said to admit a synchronous lift if a second system containing a synchronous model exists. We provide necessary and sufficient conditions that a system admits a synchronous lift and provide a method to construct a (there may be many) lifted system should one exist. We characterise the class of all systems that admit a synchronous lift by showing that they consist of fundamental vector fields induced by a Lie group action, a class of system we term fundamental systems. For fundamental systems we propose a simple synchronous observer design methodology, for which we show how correction terms can be discretised and combined easily, facilitating global characterisation of convergence and performance. Finally, we provide three examples to demonstrate the key concepts of synchrony, symmetry construction, and observer design for a fundamental system.

Synchronous Models and Fundamental Systems in Observer Design

TL;DR

The paper defines a synchronous model as an error-based extension of an internal model, and shows a system admits a synchronous lift if and only if its accessibility Lie algebra is finite-dimensional and complete. It then identifies fundamental systems as those induced by transitive group actions on homogeneous spaces, establishing a one-to-one correspondence between fundamental systems and systems with synchronous lifts. For fundamental systems, a constructive synchronous observer architecture is developed, enabling discretisation and composition of correction terms while guaranteeing Lyapunov-based convergence. Three illustrative examples demonstrate how synchrony, symmetry construction, and observer design operate in practice, with implications for robust, almost-global estimation on manifolds and Lie groups in robotics and related fields.

Abstract

This paper introduces the concept of a synchronous model as an extension of the internal model concept used in observer design for dynamical systems. A system is said to contain a synchronous model of another if there is a suitable error function between the two systems that remains stationary for all of the trajectories of the two systems. A system is said to admit a synchronous lift if a second system containing a synchronous model exists. We provide necessary and sufficient conditions that a system admits a synchronous lift and provide a method to construct a (there may be many) lifted system should one exist. We characterise the class of all systems that admit a synchronous lift by showing that they consist of fundamental vector fields induced by a Lie group action, a class of system we term fundamental systems. For fundamental systems we propose a simple synchronous observer design methodology, for which we show how correction terms can be discretised and combined easily, facilitating global characterisation of convergence and performance. Finally, we provide three examples to demonstrate the key concepts of synchrony, symmetry construction, and observer design for a fundamental system.

Paper Structure

This paper contains 16 sections, 7 theorems, 90 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Smooth Manifolds
  4. Systems
  5. Lie groups and algebras
  6. Synchronous Models
  7. Existence of a Synchronous Lift
  8. Fundamental Systems
  9. Observer Architecture for Fundamental Systems
  10. Synchronous Observer Design for Fundamental Systems
  11. Constructive Observers through Lyapunov Design
  12. Examples
  13. Rotating Bearings
  14. Deriving the Symmetry of a Unicycle
  15. Velocity-Aided Attitude
  16. ...and 1 more sections

Key Result

Lemma 4.3

Consider two systems $f^\dag : \mathbb{L} \to \mathfrak{X}(\mathcal{G})$ and $f : \mathbb{L} \to \mathfrak{X}(\mathcal{M})$ and suppose that $f^\dag$ is a synchronous lift of $f$. Then, for every vector field $g \in \mathrm{Lie}(\mathop{\mathrm{im}}\nolimits(f))$, there exists a synchronous lift $g^

Figures (3)

  • Figure 1: A Synchronous Observer Architecture. Inputs $v$ and outputs $y^\ell$, $\ell = 1,..., n$, are used by the synchronous lift $f^\dag$ and correction functions $\Gamma^\ell$, respectively. The error between the observer state $\hat{X}$ and the system state $\xi$ is given by the error function $e(\hat{X}, \xi)$. The state estimate is obtained through the reconstruction function $\varphi(\hat{X})$.
  • Figure 2: Estimation performance of the velocity-aided attitude observer combining discretised update terms for magnetometer and GNSS velocity measurements. The times at which GNSS and magnetometer updates are applied are highlighted in red and orange, respectively.
  • Figure 3: Estimation error and Lyapunov value evolution for the velocity-aided attitude observer combining discretised update terms for magnetometer and GNSS velocity measurements. The times at which GNSS and magnetometer updates are applied are highlighted in red and orange, respectively.

Theorems & Definitions (21)

  • Definition 3.1: Error Function
  • Definition 3.2: Synchronous Model
  • Definition 4.1
  • Definition 4.2
  • Lemma 4.3
  • proof
  • Theorem 4.4
  • proof
  • Lemma 4.5
  • proof
  • ...and 11 more