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Sensor Placement on a Cantilever Beam Using Observability Gramians

Natalie L. Brace, Nicholas B. Andrews, Jeremy Upsal, Kristi A. Morgansen

TL;DR

The paper addresses optimal sensor placement for continuum systems by deriving both analytical and empirical observability Gramians for a PDE model of a freely vibrating Euler-Bernoulli cantilever beam. It develops a finite-mode truncation to create a linear ODE representation and constructs observability measures to guide sensor placement, while also formulating a convex-relaxed optimization problem for selecting sensor locations. The authors demonstrate that analytical and empirical Gramians align for this linear continuum system and show that optimally placed sensors improve UKF estimation performance, achieving substantial reductions in the state covariance compared to random placements. This work provides a framework for extending observability-based sensor placement to more complex, potentially nonlinear continuum systems and highlights practical implications for structural health monitoring and related sensing networks.

Abstract

Working from an observability characterization based on output energy sensitivity to changes in initial conditions, we derive both analytical and empirical observability Gramian tools for a class of continuum material systems. Using these results, optimal sensor placement is calculated for an Euler-Bernoulli cantilever beam for the following cases: analytical observability for the continuum system and analytical observability for a finite number of modes. Error covariance of an Unscented Kalman Filter is determined for both cases and compared to randomly placed sensors to demonstrate effectiveness of the techniques.

Sensor Placement on a Cantilever Beam Using Observability Gramians

TL;DR

The paper addresses optimal sensor placement for continuum systems by deriving both analytical and empirical observability Gramians for a PDE model of a freely vibrating Euler-Bernoulli cantilever beam. It develops a finite-mode truncation to create a linear ODE representation and constructs observability measures to guide sensor placement, while also formulating a convex-relaxed optimization problem for selecting sensor locations. The authors demonstrate that analytical and empirical Gramians align for this linear continuum system and show that optimally placed sensors improve UKF estimation performance, achieving substantial reductions in the state covariance compared to random placements. This work provides a framework for extending observability-based sensor placement to more complex, potentially nonlinear continuum systems and highlights practical implications for structural health monitoring and related sensing networks.

Abstract

Working from an observability characterization based on output energy sensitivity to changes in initial conditions, we derive both analytical and empirical observability Gramian tools for a class of continuum material systems. Using these results, optimal sensor placement is calculated for an Euler-Bernoulli cantilever beam for the following cases: analytical observability for the continuum system and analytical observability for a finite number of modes. Error covariance of an Unscented Kalman Filter is determined for both cases and compared to randomly placed sensors to demonstrate effectiveness of the techniques.
Paper Structure (16 sections, 3 theorems, 39 equations, 7 figures)

This paper contains 16 sections, 3 theorems, 39 equations, 7 figures.

Table of Contents

  1. INTRODUCTION
  2. System Models
  3. Observability Tools
  4. Linear Observability for Finite-Dimensional Systems
  5. Linear Observability for Infinite-Dimensional Systems
  6. Empirical Observability Gramian
  7. Measures of Observability
  8. Local unobservability index
  9. Local estimation condition number
  10. Optimal Sensor Placement
  11. Analytical Results
  12. Continuum Analytical Observability Gramian
  13. Continuum Empirical Observability Gramian
  14. Single Sensor Observability
  15. Numerical Results
  16. ...and 1 more sections

Key Result

Theorem 1

The observability Gramian, $W_\infty$, for system $\Sigma_\infty$ is given by where

Figures (7)

  • Figure 1: Model of a cantilever beam.
  • Figure 2: First ten mode shapes for an Euler-Bernoulli beam fixed at the left end.
  • Figure 3: The objective function of the observability Gramian, $J(W_o) = \kappa + w \nu$, plotted along the length of the beam for two to ten modes.
  • Figure 4: Second derivative of mode shapes with respect to $x$ for modes one through ten.
  • Figure 5: Optimal sensor locations for $n_\phi=8$ modes for the continuum and truncated systems ($\Sigma_\infty$ and $\Sigma_n$, respectively).
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Theorem 1: Continuum analytical observability Gramian
  • proof
  • Theorem 2: Continuum empirical observability Gramian
  • proof
  • Theorem 3: Single sensor analytical observability
  • proof