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Transversally exponentially stable Euclidean space extension technique for discrete time systems

Soham Shanbhag, Dong Eui Chang

TL;DR

This work addresses extending Euclidean-space control techniques to discrete-time systems evolving on manifolds by enforcing a local attractor via a transversally stable extension. It presents a general theorem that augments the nominal update with a gradient term $-\alpha \nabla V(f(\cdot))$ and proves exponential convergence to $V^{-1}(0)$ under explicit bounds on curvature and step size. The authors demonstrate the method on a rigid-body rotation model on $S^3$ using quaternions and show substantial improvement in observer accuracy with a Luenberger-like design. The work enables leveraging linear discrete-time tools in manifold-aware settings and has broad practical impact for control and estimation on embedded manifolds.

Abstract

We propose a modification technique for discrete time systems for exponentially fast convergence to compact sets. The extension technique allows us to use tools defined on Euclidean spaces to systems evolving on manifolds by modifying the dynamics of the system such that the manifold is an attractor set. We show the stability properties of this technique using the simulation of the rigid body rotation system on the unit sphere $S^3$. We also show the improvement afforded due to this technique on a Luenberger like observer designed for the rigid body rotation system on $S^3$.

Transversally exponentially stable Euclidean space extension technique for discrete time systems

TL;DR

This work addresses extending Euclidean-space control techniques to discrete-time systems evolving on manifolds by enforcing a local attractor via a transversally stable extension. It presents a general theorem that augments the nominal update with a gradient term and proves exponential convergence to under explicit bounds on curvature and step size. The authors demonstrate the method on a rigid-body rotation model on using quaternions and show substantial improvement in observer accuracy with a Luenberger-like design. The work enables leveraging linear discrete-time tools in manifold-aware settings and has broad practical impact for control and estimation on embedded manifolds.

Abstract

We propose a modification technique for discrete time systems for exponentially fast convergence to compact sets. The extension technique allows us to use tools defined on Euclidean spaces to systems evolving on manifolds by modifying the dynamics of the system such that the manifold is an attractor set. We show the stability properties of this technique using the simulation of the rigid body rotation system on the unit sphere . We also show the improvement afforded due to this technique on a Luenberger like observer designed for the rigid body rotation system on .
Paper Structure (6 sections, 3 theorems, 23 equations, 2 figures, 1 table)

This paper contains 6 sections, 3 theorems, 23 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main Results
  3. Simulation
  4. Simulation on the unit sphere in quaternion space
  5. Simulation of observer on $S^3$
  6. Conclusion

Key Result

Theorem 1

Consider a discrete time dynamical system on the subset $U \subset \mathbb{R}^n$ where $f$ is a $C^1$ map from $U \times W$ to $U$, where $W \subset \mathbb{R}^m$. We make the following assumptions: Then, the modified dynamical system on $V^{-1}([0, \epsilon])$, converges to the set $V^{-1}(0)$ exponentially fast if $\alpha$ is chosen such that $0 < \alpha < \min\left(\frac{2}{d}, \frac{\delta}{

Figures (2)

  • Figure 1: Convergence of the state of the quaternion system to the manifold $S^3$
  • Figure 2: Convergence of the state of the quaternion system to the manifold $S^3$

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof