[Paper Review] Characterising Linear Spatio-Temporal Dynamical Systems in the Frequency Domain

Hua‐Liang Wei, Billings, Stephen A.|arXiv (Cornell University)|Oct 29, 2021

Control Systems and Identification32 references3 citations

TL;DR

This paper introduces the spatio-temporal transfer function (STTF), a frequency-domain framework for characterizing linear time-invariant (LTI) spatio-temporal dynamical systems. By extending classical transfer functions to spatial and temporal dimensions, the STTF enables analysis of system dynamics, stability, and response across space and frequency, offering a unified tool for modeling and control of distributed systems such as sensor networks or environmental processes.

ABSTRACT

A new concept, called the spatio-temporal transfer function (STTF), is introduced to characterise a class of linear time-invariant (LTI) spatio-temporal dynamical systems. The spatio-temporal transfer function is a natural extension of the ordinary transfer function for classical linear time-invariant control systems. As in the case of the classical transfer function, the spatio-temporal transfer function can be used to characterise, in the frequency domain, the inherent dynamics of linear time-invariant spatio-temporal systems. The introduction of the spatio-temporal transfer function should also facilitate the analysis of the dynamical stability of discrete-time spatio-temporal systems.

Motivation & Objective

• To develop a frequency-domain characterization tool for linear spatio-temporal dynamical systems.
• To extend classical transfer function concepts to systems with spatial and temporal dimensions.
• To enable stability analysis of discrete-time spatio-temporal systems through a unified framework.
• To provide a systematic method for modeling and analyzing distributed systems with spatially distributed inputs and outputs.
• To support the design and control of complex systems such as sensor networks, environmental systems, and large-scale infrastructure.

Proposed method

• Proposes the spatio-temporal transfer function (STTF) as a multidimensional generalization of the classical transfer function.
• Derives the STTF using the 2D Laplace or Z-transform to represent system dynamics in both space and time domains.
• Models the system as a linear time-invariant (LTI) operator acting on spatially and temporally varying inputs.
• Applies the STTF to analyze system responses, including gain and phase characteristics across spatial frequencies and temporal frequencies.
• Uses the STTF to assess dynamical stability by examining poles in the complex spatial-temporal frequency plane.
• Demonstrates the framework on theoretical and illustrative examples to validate its analytical power.

Experimental results

Research questions

• RQ1 How can classical frequency-domain tools be extended to systems with both spatial and temporal dynamics?
• RQ2 What is the mathematical form and structural properties of a transfer function for spatio-temporal LTI systems?
• RQ3 How does the STTF enable stability analysis of discrete-time spatio-temporal systems?
• RQ4 In what ways does the STTF improve the characterization of system behavior compared to time-domain or spatial-only methods?
• RQ5 Can the STTF be used to identify dominant modes and resonant frequencies in distributed systems?

Key findings

• The spatio-temporal transfer function (STTF) provides a natural extension of classical transfer functions to systems with spatial and temporal dimensions.
• The STTF enables full frequency-domain characterization of linear spatio-temporal systems, including gain, phase, and resonance behavior across space and time.
• The framework facilitates stability analysis by allowing pole-zero analysis in the 2D complex frequency plane (spatial and temporal frequencies).
• The STTF can be derived using 2D Laplace or Z-transforms, providing a rigorous mathematical foundation for system modeling.
• The method is applicable to discrete-time systems, enabling analysis of sampled-data spatio-temporal processes such as sensor arrays or grid-based simulations.
• The STTF offers a systematic approach to modeling and control of distributed systems, with potential applications in environmental monitoring, smart infrastructure, and networked systems.