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Observability of Linear Time-Invariant Systems with Relative Measurements: A Geometric Approach

Ioannis Raptis

Abstract

This paper explores the observability and estimation capability of dynamical systems using predominantly relative measurements of the system's state-space variables, with minimal to no reliance on absolute measurements of these variables. We concentrate on linear time-invariant systems, in which the observation matrix serves as the algebraic representation of a graph object. This graph object encapsulates the availability of relative measurements. Utilizing algebraic graph theory and abstract linear algebra (geometric) tools, we establish a link between the structure of the graph of relative measurements and the system-theoretic observability subspace of linear systems. Special emphasis is given to multi-agent networked systems whose dynamics are governed by the linear consensus protocol. We demonstrate the importance of absolute information and its placement to the system's dynamics in achieving full-state estimation. Finally, the analysis shifts to the synthesis of a distributed observer with relative measurements for single integrator dynamics, exemplifying the relevance of the preceding analytical findings. We support our theoretical analysis with numerical simulations.

Observability of Linear Time-Invariant Systems with Relative Measurements: A Geometric Approach

Abstract

This paper explores the observability and estimation capability of dynamical systems using predominantly relative measurements of the system's state-space variables, with minimal to no reliance on absolute measurements of these variables. We concentrate on linear time-invariant systems, in which the observation matrix serves as the algebraic representation of a graph object. This graph object encapsulates the availability of relative measurements. Utilizing algebraic graph theory and abstract linear algebra (geometric) tools, we establish a link between the structure of the graph of relative measurements and the system-theoretic observability subspace of linear systems. Special emphasis is given to multi-agent networked systems whose dynamics are governed by the linear consensus protocol. We demonstrate the importance of absolute information and its placement to the system's dynamics in achieving full-state estimation. Finally, the analysis shifts to the synthesis of a distributed observer with relative measurements for single integrator dynamics, exemplifying the relevance of the preceding analytical findings. We support our theoretical analysis with numerical simulations.

Paper Structure

This paper contains 9 sections, 21 theorems, 37 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation, Terminology, and Mathematical Preliminaries
  3. Problem Statement
  4. Observability of Special RSS Observation Matrices
  5. Observability of General RSSs
  6. Observability Analysis of the Single Measurement Agreement Dynamics
  7. Design of a Distributed Estimator for the Single-Integrator Multi-Agent System
  8. Numerical Simulations
  9. Conclusions

Key Result

Theorem 1

The matrix pair $(A,C)$ is observable if and only if the rank of the $nm\times n$ observability matrix $\mathcal{O}$ in (eq:observability_matrix) is $n$.

Figures (4)

  • Figure 1: Examples of different graph topologies and observation symmetric anchor nodes: (a) only the anchor node $2$ is observation symmetric; (b) all the nodes of the graph are observation symmetric.
  • Figure 2: Information-sharing links and information-exchanged variables (local measurements and estimates) between node $i$ and its adjacent nodes for generating its estimate $\hat{x}_{i}$.
  • Figure 3: Gradual comparison between the actual and estimated states of a SIMA system at distinct time instances for $n=8960$. Each gray-scale pixel in the image grid represents the numerical value of a state-space variable, which maintains a constant value, e.g., $x_{i}(t)=c_{i}$, where $c_{i}\in\left[0\:255\right]$. The initial state estimates, $\hat{x}_{i}(0)$, are uniformly drawn from the gray-scale range interval $\left[0\:255\right]$.
  • Figure 4: Reference time-varying signal (dashed lines), actual states of the single-integrator (upper plot, solid lines), and estimated states (lower plot, solid lines), all plotted with respect to time.

Theorems & Definitions (45)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • ...and 35 more