Successive design of backstepping observers for parabolic PDE-ODE systems and its duality to state feedback stabilization
Abstract
The paper introduces a successive backstepping observer design for strictly feedforward parabolic PDE-ODE systems, in which the coupling structure determines the order of error stabilization and the corresponding transformations. First, a transformation based on a virtual measurement stabilizes the ODE observer error subsystem, which is most distal from the measurement, while decoupling it from the PDE error state. Second, a Volterra integral transformation is employed to stabilize the PDE error subsystem and to map the overall error dynamics into a cascade of exponentially stable ODE and PDE subsystems. The design is shown to be dual to a recently proposed multi-step state feedback design for parabolic PDE-ODE systems in strict feedback form, thus explaining the structure of the presented observer design.
Related
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BibTeX
@misc{gehring2026successive,
title = {{Successive design of backstepping observers for parabolic PDE-ODE systems and its duality to state feedback stabilization}},
author = {Nicole Gehring},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29693},
eprint = {2608.29693},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29693},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Successive design of backstepping observers for parabolic PDE-ODE systems and its duality to state feedback stabilization
AU - Nicole Gehring
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29693
UR - https://arxiv.org/abs/2608.29693
AB - The paper introduces a successive backstepping observer design for strictly feedforward parabolic PDE-ODE systems, in which the coupling structure determines the order of error stabilization and the corresponding transformations. First, a transformation based on a virtual measurement stabilizes the ODE observer error subsystem, which is most distal from the measurement, while decoupling it from the PDE error state. Second, a Volterra integral transformation is employed to stabilize the PDE error subsystem and to map the overall error dynamics into a cascade of exponentially stable ODE and PDE subsystems. The design is shown to be dual to a recently proposed multi-step state feedback design for parabolic PDE-ODE systems in strict feedback form, thus explaining the structure of the presented observer design.
ER -