Observability of complex systems via conserved quantities

TL;DR

This paper explores how conserved quantities in nonlinear dynamical systems can be used to improve the observability of system states, especially in biological models, by linking conserved quantities to measurement strategies.

Contribution

It establishes conditions under which conserved quantities enhance system observability and demonstrates their application in biological models for better state estimation.

Findings

Conserved quantities can increase the set of measurable variables for observability.

Differential embeddings reveal the role of conserved quantities in system analysis.

Application to biological models shows practical benefits in state measurement.

Abstract

Many systems in biology, physics, and engineering are modeled by nonlinear dynamical systems where the states are usually unknown and only a subset of the state variables can be physically measured. Can we understand the full system from what we measure? In the mathematics literature, this question is framed as the observability problem. It has to do with recovering information about the state variables from the observed states (the measurements). In this paper, we relate the observability problem to another structural feature of many models relevant in the physical and biological sciences: the conserved quantity. For models based on systems of differential equations, conserved quantities offer desirable properties such as dimension reduction which simplifies model analysis. Here, we use differential embeddings to show that conserved quantities involving a set of special variables provide more flexibility in what can be measured to address the observability problem for systems of interest in biology. Specifically, we provide conditions under which a collection of conserved quantities make the system observable. We apply our methods to provide alternate measurable variables in models where conserved quantities have been used for model analysis historically in biological contexts.