Observer Design for a Class of ODE -- Continuum-PDE Cascade Systems Inspired by a Control-Theoretic Model of Large-Scale Arterial Networks of Blood Flow

TL;DR

This paper develops a backstepping observer for large-scale ODE-continuum PDE cascade systems, inspired by blood flow models in arterial networks, demonstrating stability, optimal approximation, and numerical implementation.

Contribution

It introduces a novel backstepping-based observer design for infinite-dimensional cascade systems, with applications to blood flow estimation in arterial networks.

Findings

Proves exponential stability of the observer error system.

Establishes a connection between backstepping kernels and Sylvester equations.

Demonstrates the approach with numerical simulations using realistic parameters.

Abstract

We develop a backstepping-based observer design for a class of ODE - continuum-PDE cascade systems, which can be viewed as the limit, of a finite collection of ODE - $2 \times 2$ hyperbolic systems, as the number of individual PDE system components tends to infinity. The large-scale collection of ODE - $2 \times 2$ hyperbolic systems is motivated by a dynamic model that we present, of a network of peripheral arteries, to which central (aortic) blood flow/pressure enters. We address a case in which average (boundary) measurements, over the ensemble dimension, are available, which is motivated by the availability of non-invasive, peripheral flow/pressure measurements. Exponential stability of the estimation error system is shown by proving well-posedness of the kernel equations and constructing a Lyapunov functional. We also establish that part of the backstepping kernels derived coincide with the solution of a Sylvester equation. We then apply the continuum-based observer for state estimation of the large-scale counterpart and, in particular, of the blood flow system, introducing an approach for optimal construction of continuum approximations. We also introduce an implementation method, adopting a spectral-based approach for computing the observer dynamics, which we illustrate in an academic, numerical simulation example. Furthermore, we illustrate the design in the problem of central flow/pressure estimation using realistic parameters and flow/pressure waveforms.