Direct transformation to the hyperbolic observer canonical form for the considered PDE–ODE class

Establish a direct coordinate transformation that maps linear single‑input single‑output systems consisting of two coupled first‑order transport partial differential equations on z ∈ [0,1] attached to a finite‑dimensional boundary ordinary differential equation, with boundary measurement at the opposite end, into the hyperbolic observer canonical form (HOCF).

Background

The paper treats linear SISO hyperbolic PDE–ODE systems composed of two counter‑propagating first‑order transport PDEs with spatially varying coefficients, bidirectionally coupled at one boundary to a finite‑dimensional observable ODE, and with the output measured at the opposite boundary. The authors construct the hyperbolic observer canonical form (HOCF) via a two‑step procedure: first mapping the system to observability coordinates defined by a neutral functional differential equation, and then mapping to the HOCF.

In this context, the authors explicitly note that, for this class of systems, they are not aware of a direct transformation of the governing equations to the HOCF. This identifies an unresolved question of whether such a direct transformation exists and how to obtain it, beyond the intermediate observability‑coordinate approach developed in the paper.