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Distributed State Estimation for Linear Time-invariant Systems with Aperiodic Sampled Measurement

Shimin Wang, Ya-Jun Pan, Martin Guay

TL;DR

This work tackles distributed state estimation for linear time-invariant systems when individual agents have only partial measurements and communications occur at aperiodic, asynchronous times. It develops a distributed observer design with an emulation-based, hybrid stability analysis that guarantees exponential convergence of the estimation error under a computable upper bound on the sampling interval, $h_{\max}$, and a lower bound on the coupling gain, $\\gamma_{\max}$. The approach relies on Kalman observable decomposition to handle jointly observable but locally unobservable pairs $(A, C_i)$ and provides a concrete procedure to compute $h_{\max}$ via Algorithm, ensuring $\,\\hat{x}_i(t) \\to x(t)$ for all agents. The method is applied to a jointly observable tracking problem, yielding a distributed control law that achieves asymptotic tracking using only local sampled measurements and neighbor estimates, even when no agent individually observes the full state. This framework extends distributed estimation by relaxing local observability requirements and offering explicit sampling-interval guarantees, with practical implications for resource-efficient sensor networks and networked control systems.

Abstract

This paper deals with the state estimation of linear time-invariant systems using distributed observers with local sampled-data measurement and aperiodic communication. Each observer agent perceives partial information of the system to be observed but does not satisfy the observability condition. Consequently, distributed observers are designed to exponentially estimate the state of the system to be observed by time-varying sampling and asynchronous communication. Additionally, explicit upper bounds on allowable sampling periods for convergent estimation errors are given. Finally, a numerical example is provided to demonstrate the validity of the theoretical results

Distributed State Estimation for Linear Time-invariant Systems with Aperiodic Sampled Measurement

TL;DR

This work tackles distributed state estimation for linear time-invariant systems when individual agents have only partial measurements and communications occur at aperiodic, asynchronous times. It develops a distributed observer design with an emulation-based, hybrid stability analysis that guarantees exponential convergence of the estimation error under a computable upper bound on the sampling interval, , and a lower bound on the coupling gain, . The approach relies on Kalman observable decomposition to handle jointly observable but locally unobservable pairs and provides a concrete procedure to compute via Algorithm, ensuring for all agents. The method is applied to a jointly observable tracking problem, yielding a distributed control law that achieves asymptotic tracking using only local sampled measurements and neighbor estimates, even when no agent individually observes the full state. This framework extends distributed estimation by relaxing local observability requirements and offering explicit sampling-interval guarantees, with practical implications for resource-efficient sensor networks and networked control systems.

Abstract

This paper deals with the state estimation of linear time-invariant systems using distributed observers with local sampled-data measurement and aperiodic communication. Each observer agent perceives partial information of the system to be observed but does not satisfy the observability condition. Consequently, distributed observers are designed to exponentially estimate the state of the system to be observed by time-varying sampling and asynchronous communication. Additionally, explicit upper bounds on allowable sampling periods for convergent estimation errors are given. Finally, a numerical example is provided to demonstrate the validity of the theoretical results
Paper Structure (12 sections, 7 theorems, 62 equations, 5 figures, 1 algorithm)

This paper contains 12 sections, 7 theorems, 62 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Assumptions
  3. Agent's dynamics
  4. Graph theory basics
  5. Problem Formulation
  6. Assumptions and Lemmas
  7. Main Results
  8. Aperiodic Sampled-Data Distributed Observers Design
  9. Convergence Analysis
  10. Application to Jointly Observable Tracking Problem
  11. Numerical Example
  12. Conclusions

Key Result

Lemma 1

*zhang2015constructing Suppose that the communication network $\mathcal{G}=(\mathcal{V},\mathcal{E})$ is strongly connected. Let $\theta=\hbox{col}(\theta_1,\cdots,\theta_N)$ be the left eigenvector of the Laplacian matrix $\mathcal{L}$ associated with the eigenvalue $0$, i.e., $\mathcal{L}^T\theta=

Figures (5)

  • Figure 1: Schematic of Jointly Observable Tracking Problem
  • Figure 2: Communication topology $\bar{\mathcal{G}}$
  • Figure 3: Sampling intervals as a function of the sampling time.
  • Figure 4: Estimation errors of all agents, $i=1,\cdots,5$.
  • Figure 5: Tracking errors of all node dynamics, $i=1,\cdots,5$.

Theorems & Definitions (12)

  • Definition 1: Sampled-Data Distributed Observer
  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Remark 3
  • Lemma 3
  • Remark 4
  • Lemma 4
  • Corollary 1
  • ...and 2 more