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Global Observer Design for a Class of Linear Observed Systems on Groups

Changwu Liu, Yuan Shen

Abstract

Linear observed systems on groups encode the geometry of a variety of practical state estimation problems. In this paper, we propose an observer framework for a class of linear observed systems by restricting a bi-invariant system on a Lie group to its normal subgroup. This structural property enables a system embedding of the original system into a linear time-varying system. An observer is constructed by first designing a Kalman-like observer for the embedded system and then reconstructing the group-valued state via optimization. Under an extrinsic observability rank condition, global exponential stability (GES) is achieved provided that one global optimum of the reconstruction optimization is found, reflecting the topological difficulties inherent to the non-Euclidean state space. Semi-global stability is guaranteed when input biases are jointly estimated. The theory is applied to the GES observer design for two-frame systems, capable of modeling a family of navigation problems. Simulations are provided to illustrate the implementation details.

Global Observer Design for a Class of Linear Observed Systems on Groups

Abstract

Linear observed systems on groups encode the geometry of a variety of practical state estimation problems. In this paper, we propose an observer framework for a class of linear observed systems by restricting a bi-invariant system on a Lie group to its normal subgroup. This structural property enables a system embedding of the original system into a linear time-varying system. An observer is constructed by first designing a Kalman-like observer for the embedded system and then reconstructing the group-valued state via optimization. Under an extrinsic observability rank condition, global exponential stability (GES) is achieved provided that one global optimum of the reconstruction optimization is found, reflecting the topological difficulties inherent to the non-Euclidean state space. Semi-global stability is guaranteed when input biases are jointly estimated. The theory is applied to the GES observer design for two-frame systems, capable of modeling a family of navigation problems. Simulations are provided to illustrate the implementation details.

Paper Structure

This paper contains 39 sections, 11 theorems, 92 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Group Theory Basics
  5. Error-State Extended Kalman Observer
  6. Observability and Matrix Riccati/Lyapunov Equations
  7. Introductory Examples
  8. Embedding-based Observer for LTI Systems
  9. Embedding-based Attitude Observer
  10. Embedding-based IMU-Landmark Pose Observer
  11. LTV Embedding of a Class of Linear Observed Systems on Groups
  12. A Class of Linear Observed Systems
  13. LTV Embedding of the Class of System
  14. Global Observer Design Using LTV embedding
  15. Observer Structure from the Embedding
  16. ...and 24 more sections

Key Result

Theorem 1

A Type-1 or Type-2 system defined by eq::def_type_1_sys or eq::def_type_2_sys can be embedded into an LTV system. The embedding map $\pi$ and the LTV system are given below. For simplicity, we consider only one measurement and omit the superscript $i$ in $y^{(i)}$ and $d^{(i)}$.

Figures (2)

  • Figure 1: Comparison of the proposed observer with the InEKF for attitude estimation. 50 runs are conducted from the same initial condition with large initial rotation error and identical noise configurations.
  • Figure 2: Comparison of the proposed method with the InEKF for IMU-landmark navigation. 50 runs are conducted from the same initial condition.

Theorems & Definitions (30)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • proof
  • Remark 5
  • Remark 6
  • ...and 20 more