Observability-Preserving Discretizations

Overview

Many vibration-control problems for beams and other distributed systems are modeled by partial differential equations, while sensors and actuators are available only at the boundary, such as at the tip of a beam. A boundary controller uses measurements like displacement, velocity, or rotation to suppress vibrations throughout the structure. Its success depends on whether the sensor captures enough information about the full system. A system is exactly observable when boundary measurements collected over a finite time are sufficient to determine the energy of the initial state.

One might expect this observability property to be retained when the continuous system is approximated using standard numerical methods such as finite differences or finite elements. However, these discretizations can introduce spurious high-frequency modes that are poorly detected by the boundary sensor, causing the discrete model to lose uniform observability as the mesh is refined.

In this project, we developed and compared two remedies for a certain class of beam models:

• Direct Fourier filtering, which removes the problematic portion of the discrete spectrum.
• Order-reduced finite differences (ORFD), which build the desired observability behavior into the discretization itself.

Models and results covered

The project brings together several related studies:

• hinged multilayer Mead–Marcus beam models with an arbitrary number of layers, (Aydin et al., 2023; Özer & Aydin, 2022)
• cantilevered Mead–Marcus sandwich beams (Özer & Aydin, 2023),
• fully clamped Euler–Bernoulli beam equations (Aydin et al., 2024), and
• extensions toward strongly coupled piezoelectric systems (Aydin et al., 2026).

Across these settings, standard discretizations create artificial modes that are invisible to the boundary sensor, while filtering or structure-preserving discretization restores a numerically reliable model.

Spectral behavior and filtering

The eigenvalues of the corresponding control problem showcase the spurious modes. In the unfiltered standard finite-difference model, the high-frequency modes drift toward the imaginary axis. A direct approach is to eliminate these modes from the spectrum altogether. Another approach is to redesign the discretization. The two approaches serve related but slightly different purposes.

Direct Fourier filtering is especially useful when the problematic spectral branch can be identified explicitly. It trims away the artificial part of the discrete spectrum and recovers uniform observability on the retained subspace. This gives a practical fix for standard discretizations and leads to robust sensor and actuator design in the multilayer beam setting. (Özer & Aydin, 2022; Aydin et al., 2023)

ORFD takes a more structural approach. Instead of correcting the spectrum after discretization, it modifies the discretization itself so that the numerical model respects the same observability mechanism as the continuous PDE. This is particularly valuable in settings where spectral filtering is difficult or inconvenient, such as cantilevered beam models and fully clamped Euler–Bernoulli problems. (Özer & Aydin, 2023; Aydin et al., 2024)

Unfiltered: the standard discretization retains spurious high-frequency activity and decays slowly.

Filtered: direct Fourier filtering removes the unstable portion of the numerical spectrum and restores effective damping.

ORFD: the order-reduced discretization preserves the desired observability behavior directly, without needing an added filter.

In conclusion, a reduced model for control should preserve boundary observability, not just state convergence. For these beam problems:

• the unfiltered standard discretization can misrepresent the high-frequency behavior that matters for sensing and feedback
• direct Fourier filtering repairs the model by removing the spurious modes and
• order-reduced finite differences give a cleaner long-term solution by preserving the correct structure at the discretization stage.

Selected Presentations on the Project

• 62nd IEEE Conference on Decision and Control, Singapore, December 2023 — A Novel Finite Difference-Based Model Reduction and a Sensor Design for a Multilayer Smart Beam with Arbitrary Number of Layers
• Joint Mathematics Meetings (JMM), Washington, DC, January 2026 — A New Semi-Discretization of the Fully Clamped Euler–Bernoulli Beam Preserving Boundary Observability Uniformly
• SIAM Pacific Northwest Sectional Meeting, Vancouver, Washington, May 2022 — Avoiding Sensor Data Filtering by a Novel Model Reduction for the PDE System of a Multi-Layer Sandwich Beam