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Steady State Distributed Kalman Filter

Francisco Rego

Abstract

One of the main challenges in set-based state estimation is the trade-off between accuracy and computational complexity, which becomes particularly critical for systems with time-varying dynamics. Accurate set representations such as polytopes, even when encoded as Constrained Zonotopes (CZs) or Constrained Convex Generators (CCGs), typically lead to a progressive growth of the set description, requiring order reduction procedures that increase the online computational burden. In this paper, we propose a fixed structure and computationally efficient approach for guaranteed state estimation of discrete-time Linear Time-Varying (LTV) systems using CCG formulations. The proposed method expresses the state enclosure explicitly in terms of a fixed number of past inputs and measurements, resulting in a constant-size set description and avoiding the need for online order reduction. Numerical results illustrate the effectiveness and computational advantages of the proposed method.

Steady State Distributed Kalman Filter

Abstract

One of the main challenges in set-based state estimation is the trade-off between accuracy and computational complexity, which becomes particularly critical for systems with time-varying dynamics. Accurate set representations such as polytopes, even when encoded as Constrained Zonotopes (CZs) or Constrained Convex Generators (CCGs), typically lead to a progressive growth of the set description, requiring order reduction procedures that increase the online computational burden. In this paper, we propose a fixed structure and computationally efficient approach for guaranteed state estimation of discrete-time Linear Time-Varying (LTV) systems using CCG formulations. The proposed method expresses the state enclosure explicitly in terms of a fixed number of past inputs and measurements, resulting in a constant-size set description and avoiding the need for online order reduction. Numerical results illustrate the effectiveness and computational advantages of the proposed method.
Paper Structure (16 sections, 1 theorem, 26 equations, 1 figure, 2 algorithms)

This paper contains 16 sections, 1 theorem, 26 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Mathematical background
  4. Best Linear Unbiased Estimation
  5. Kalman filtering and steady-state covariances
  6. Problem Definition
  7. Networked system
  8. Steady-state distributed estimation objective
  9. Distributed observer synthesis based on BLUE
  10. Time-varying distributed BLUE formulation
  11. Fixed-gain steady-state distributed observer
  12. Offline gain computation
  13. Online distributed observer
  14. Numerical results
  15. Conclusion
  16. ...and 1 more sections

Key Result

Theorem 1

Consider the matrices $\eta_i\in\mathbb{R}^{\left|\mathcal{N}^i\right|n\times Nn}$, defined by $\eta_i:=\operatorname{row}\left(\boldsymbol{e}_j,j\in\mathcal{N}^i\right)\otimes I_n$, where vector $\boldsymbol{e}_i$ is a column vector with all entries equal to $0$ except for entry $i$ which is $1$, $ whereas global covariance matrix is described by where $T_t$ is defined as $T_t:=\left[T^{ij}_t\ri

Figures (1)

  • Figure 1: Average norm of the estimation errors for different estimation strategies.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Remark : Convergence considerations
  • proof