[Paper Review] Invariant asymptotic observers

Silvère Bonnabel, Philippe Martin+1 more|arXiv (Cornell University)|Dec 7, 2006

Adaptive Control of Nonlinear Systems13 citations

TL;DR

This paper introduces invariant asymptotic observers for non-linear systems by leveraging physical symmetries, using the Cartan moving-frame method to construct symmetry-preserving correction terms. It proves exponential convergence with almost global stability for a non-holonomic car and semi-global stability for an inertial navigation system, validated through simulations with noise and bias.

ABSTRACT

This paper presents three non-linear asymptotic observers corresponding to three examples of engineering interest: a chemical reactor, a non-holonomic car, and an inertial navigation system. For each example, the design is based on physical symmetries. This motivates the theoretical development of invariant observers, i.e, symmetrypreserving observers. We consider an observer to consist in a copy of the system equation and a correction term, and we give a constructive method (based on the Cartan moving-frame method) to find all the symmetry-preserving correction terms. They rely on an invariant frame (a classical notion) and on an invariant output-error, a less standard notion precisely defined here. For each example, the convergence analysis relies also on symmetries consideration with a key use of invariant state-errors. For the non-holonomic car and the inertial navigation system, the invariant state-errors is shown to obey an autonomous differential equation independent of the system trajectory. This allows us to prove exponential convergence, with almost global stability for the non-holonomic car and with semi-global stability for the inertial navigation system. Simulations including noise and bias show the practical interest of such invariant asymptotic observers for the inertial navigation system.

Motivation & Objective

• To develop non-linear asymptotic observers that preserve the physical symmetries of engineering systems.
• To address the challenge of designing observers with guaranteed convergence for non-linear systems lacking linear structure.
• To establish a systematic method for constructing correction terms that maintain symmetry in observer dynamics.
• To prove stability and convergence using invariant state-errors, independent of system trajectories.

Proposed method

• The observer is structured as a copy of the system dynamics with a symmetry-preserving correction term.
• The Cartan moving-frame method is used to construct an invariant frame and an invariant output-error, a novel concept defined in the paper.
• Correction terms are derived from the invariant frame and invariant output-error to ensure symmetry preservation.
• The invariant state-error is defined and shown to satisfy an autonomous differential equation independent of the system trajectory.
• Convergence is analyzed by studying the dynamics of the invariant state-error, which simplifies stability analysis.
• The method is applied to three systems: a chemical reactor, a non-holonomic car, and an inertial navigation system.

Experimental results

Research questions

• RQ1 How can physical symmetries in non-linear systems be exploited to design observers with guaranteed convergence?
• RQ2 What is the role of invariant output-error in constructing symmetry-preserving correction terms?
• RQ3 Can the invariant state-error be shown to evolve according to an autonomous system independent of the trajectory?
• RQ4 What stability guarantees can be derived when the invariant state-error follows an autonomous equation?
• RQ5 How does the proposed method perform under noise and bias in practical applications like inertial navigation?

Key findings

• For the non-holonomic car, the invariant state-error evolves according to an autonomous differential equation, enabling proof of almost global exponential convergence.
• For the inertial navigation system, the invariant state-error also follows an autonomous equation, leading to semi-global exponential convergence.
• Simulations demonstrate robust performance under noise and bias, confirming practical utility in inertial navigation.
• The construction of correction terms is fully systematic and based on the Cartan moving-frame method, ensuring symmetry preservation.
• The invariant output-error is a key innovation, enabling the design of correction terms that respect system symmetries.
• The theoretical framework applies broadly to non-linear systems with symmetries, offering a unified observer design approach.