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Non-linear Symmetry-preserving Observer on Lie Groups

S. Bonnabel, P. Martin, P. Rouchon

arXiv:0707.2286v2math.OC

TL;DR

The paper addresses the design of observers for systems on finite-dimensional Lie groups with symmetries. It develops an explicit, intrinsic framework and, for left-invariant systems with right equivariant output, derives observers with autonomous error equations. The resulting observers converge locally around trajectories, with global behavior independent of the trajectory.

  • Problem

    The paper addresses observer design for systems on Lie groups with symmetries, including convergence issues in the general symmetry-preserving observer theory.

  • Method

    The paper develops an explicit and intrinsic Lie-group framework for symmetry-preserving observers and analyzes permanent trajectories and right-equivariant outputs.

  • Results

    The theory yields locally convergent observers around any trajectory, including observers with autonomous error equations whose global behavior is trajectory-independent.

  • Takeaways & Limitations

    The framework extends symmetry-preserving observer design to Lie-group systems while retaining explicit formulas and trajectory-independent global error behavior in the considered case.

Abstract

from arXiv · show

In this paper we give a geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries. The design and the error (between true and estimated state) equation are explicit and intrinsic. We consider also a particular case: left-invariant systems on Lie groups with right equivariant output. The theory yields a class of observers such that error equation is autonomous. The observers converge locally around any trajectory, and the global behavior is independent from the trajectory, which reminds of the linear stationary case.

I. INTRODUCTION

The paper develops an explicit, intrinsic framework for symmetry-preserving observers on Lie groups, extending prior work with convergence results. It identifies observers that converge locally around broad classes of trajectories and can have trajectory-independent error behavior.

  • The paper develops a proper theory of symmetry-preserving observers on Lie groups with explicit and intrinsic observer design and error equations.The formulas are globally defined and do not require the implicit function theorem.
  • The framework addresses convergence issues not treated in earlier general symmetry-preserving observer theory.It introduces trajectories around which convergent observers can be constructed.
  • Permanent trajectories extend equilibrium points for systems with symmetries, enabling locally convergent observers around any permanent trajectory.The construction is compared with making a Luenberger observer around an equilibrium point.
  • For left-invariant systems with right equivariant output, the theory derives observers with autonomous error equations and local convergence around any trajectory.This case includes engineering applications such as attitude estimation and inertial navigation.

II. SYMMETRY-PRESERVING OBSERVERS ON LIE GROUPS

This section formulates systems, group actions, and input-output transformations underlying symmetry-preserving observers. The observer construction is designed to respect left-invariance under the group action.

  • The framework considers systems with state x ∈ G, known input u ∈ U, and output y ∈ Y, where G is a Lie group.The input and output spaces are smooth manifolds, with Euclidean spaces as examples.
  • A left-invariant system with equivariant output remains unaffected by the specified transformations of state, input, and output.The state transformation uses left multiplication, while inputs and outputs transform through ψ_g and ρ_g.
  • The paper builds observers that respect the system's left-invariance by adapting a constructive symmetry-preserving method to Lie groups.The construction uses the group action and its induced tangent-space maps.

1) Invariant pre-observers:

Invariant pre-observers are characterized through functions of group-action invariants, providing a structured form for observer design. Convergence is a separate requirement rather than part of the pre-observer definition.

  • 1) Invariant pre-observers:: For variables (x,z) ∈ G×R^s, the invariant I(x,z)=φ_x^-1(z) provides functionally independent scalar invariants for constructing invariant functions.Any invariant real-valued function can be expressed as a function of the components of I(x,z).
  • 1) Invariant pre-observers:: An invariant pre-observer has dynamics d x̂/dt = F(x̂,u,y) satisfying the required invariance relation under group transformations.Its component functions are constrained to vanish when the estimate equals the true state in the transformed coordinates.
  • 1) Invariant pre-observers:: A pre-observer becomes an asymptotic observer only when the estimation error converges to the identity for every sufficiently close initial condition.The definition of a pre-observer itself does not establish convergence.

2) Invariant state-error dynamics:

The invariant error dynamics are explicit and depend on the trajectory only through the invariant term I(x,u). When ψ_g(u)≡u, they become autonomous and can be locally stabilized by linear gain tuning.

  • The invariant error η obeys a differential equation coupled to the system trajectory only through I(x,u)=ψ_x−1(u).
  • When ψ_g(u)≡u, the invariant error dynamics are independent of the state trajectory.This trajectory-independence underlies the autonomous-error property highlighted for the non-holonomic car example.
  • Near the identity, the error is parameterized as η=exp(εξ), with ξ in the Lie algebra and ε small.
  • The linearized invariant error equation is expressed in the fixed Lie algebra tangent space up to second-order terms in ε.
  • The linearization's gains can be tuned using linear techniques to achieve local convergence.

B. Local convergence around permanent trajectories

Permanent trajectories are those for which the invariant input is constant, extending equilibrium-point reasoning to trajectories on Lie groups. The construction yields invariant observers that converge locally around every permanent trajectory.

  • Permanent trajectories extend local convergence analysis from equilibrium points to arbitrary trajectories with constant invariant input.
  • A permanent trajectory is defined by a time-independent invariant input I(x(t),u(t))=Ī.
  • For a permanent trajectory, x(t)=x(0)exp(tw̄), corresponding up to left translation to a one-parameter subgroup.
  • Assuming the pair (A,C) is observable, pole placement for A+L̄C gives an invariant, locally convergent observer around any permanent trajectory associated with ū.
  • In the non-holonomic car example, permanent trajectories include constant-speed lines and circles.
  • In inertial navigation, permanent trajectories include helicoidal motion and degenerate uniformly accelerated lines or coordinated turns.

III. LEFT INVARIANT DYNAMICS AND RIGHT EQUIVARIANT OUTPUT

For left-invariant dynamics with right-equivariant outputs, the framework applies to systems modeled on Lie groups, including vehicles and rigid bodies with body-fixed measurements. It produces observers with autonomous error equations.

  • The framework treats left-invariant dynamics on Lie groups as motions of generalized rigid bodies with configuration space G.
  • Left multiplication preserves the dynamics, so transformed trajectories X(t)=gx(t) satisfy the same equations.
  • A right-equivariant output map models measurements of state quantities expressed in a body-fixed frame.
  • The theory yields nonlinear observers with autonomous error equations for cart-like vehicles and rigid bodies in space using body-fixed measurements.
  • If the output dimension is smaller than the state-space dimension, the system is necessarily not observable.

2) Observability:

Under left-invariant dynamics and a right-equivariant output, distinct transformed trajectories can produce identical outputs, establishing non-observability when the output dimension is smaller than the state dimension.

  • When dim y<dim g, distinct group elements x1 and x2 can satisfy h(x1)=h(x2).
  • Left invariance makes g1x(t) and g2x(t) trajectories whenever x(t) is a trajectory.
  • Right equivariance preserves equality of outputs for these transformed trajectories: h(g1x(t))=h(g2x(t)).
  • Because the transformed trajectories are distinct yet share the same output for all time, the system is not observable.

3) Applying the general theory of section II:

The paper applies its symmetry-based observer theory to left-invariant systems with right-equivariant outputs in two formulations. The resulting observer has an autonomous error equation independent of the state trajectory.

  • One formulation uses inputs (t, h(e)) and defines an output map H(x, u) = h(x) = ρ_x(h(e)).
  • A second formulation treats ω_s(t) as the input and defines ψ_g through the differential of the interior automorphism of G.
  • The theory is applied to left-invariant dynamics by exchanging the roles of left and right multiplication.
  • The observer class uses smooth scalar functions L_i satisfying L_i(h(e)) = 0 and preserves the stated group transformations.
  • The error equation is autonomous and independent of the trajectory t 7→x(t).

6) First order approximation:

The first-order design linearizes the invariant error near the identity and selects output-based correction functions through an adjoint operator. Under a definiteness condition, the observers converge locally exponentially around any trajectory, illustrated by magnetic-aided attitude estimation.

  • 6) First order approximation: Choosing L(y) = K(Dh(e))^T(y − h(e)) uses the adjoint of Dh(e) to construct the correction term.
  • 6) First order approximation: For K > 0, ∥ξ∥2 is a Lyapunov function for the first-order error dynamics.
  • B. A class of non-linear first-order convergent observers: A negative-definite symmetric-part condition on the selected functions L_i gives locally exponentially convergent nonlinear observers around any trajectory.
  • IV. BRIEF EXAMPLE: MAGNETIC-AIDED ATTITUDE ESTIMATION: In the attitude example, the magnetic-field output has dimension 2 while the SO(3) state space has dimension 3, so the system is not observable without an additional assumption.
  • IV. BRIEF EXAMPLE: MAGNETIC-AIDED ATTITUDE ESTIMATION: Under quasi-stationary flight, gravity and magnetic-field measurements form the output y = (R^-1G, R^-1B), to which the theory can be applied.

V. CONCLUSION

The conclusion presents a general intrinsic framework for symmetry-preserving observers on Lie groups and emphasizes trajectory-independent behavior. The constructed observers converge around any trajectory for a general class of systems.

  • The paper completes a general framework for symmetry-preserving observers with Lie-group state spaces.
  • The observers and their error equations are intrinsically and globally defined.
  • The observers converge around any trajectory, while global behavior is independent of the trajectory and time-varying inputs.
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