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Predictability of Observables of Dynamical Systems

Xinyu Liu, Dongbin Xiu

Abstract

We study the evolution of observables of dynamical systems. For linear systems, we show that observables satisfy a closed differential equation whose minimal order is determined by the dynamical system and observation operator. This yields a minimal order closure and an equivalent discrete delay representation of the observable dynamics. For nonlinear systems we introduce the notion of diminishing ambiguity, which provides a framework under which the instantaneous observable dynamics can be approximately determined from sufficiently long output history, resulting in delay differential equation representation. These results clarify when observable dynamics can be inferred from past history without knowledge of the dynamical system and its full state.

Predictability of Observables of Dynamical Systems

Abstract

We study the evolution of observables of dynamical systems. For linear systems, we show that observables satisfy a closed differential equation whose minimal order is determined by the dynamical system and observation operator. This yields a minimal order closure and an equivalent discrete delay representation of the observable dynamics. For nonlinear systems we introduce the notion of diminishing ambiguity, which provides a framework under which the instantaneous observable dynamics can be approximately determined from sufficiently long output history, resulting in delay differential equation representation. These results clarify when observable dynamics can be inferred from past history without knowledge of the dynamical system and its full state.
Paper Structure (14 sections, 9 theorems, 61 equations)

This paper contains 14 sections, 9 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contribution of this work
  4. Preliminaries
  5. Linear Systems with Linear Observations
  6. Krylov Subspace and Minimal Order Closure
  7. Delay Representations
  8. Scalar Observable
  9. Vector Observables
  10. Nonlinear Systems and Diminishing Ambiguity
  11. $T$-history Determinism
  12. Diminishing Ambiguity
  13. Conclusion
  14. Proof for Theorem \ref{['thm:linear']}

Key Result

Lemma 3.1

(Stabilization index) There exists a stabilization index $r\le n-1$ such that $\dim \mathcal{V}_{r-1}<\dim \mathcal{V}_r=\dim \mathcal{V}_{r+1}\leq n$, where $\dim \mathcal{V}_{-1}=0.$

Theorems & Definitions (23)

  • Definition 2.1: Admissible histories
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3: Relation to Cayley--Hamilton
  • Remark 3.4
  • Corollary 3.5: Invariance under differentiation
  • proof
  • Theorem 3.6: Discrete delay representation
  • proof
  • ...and 13 more