PI-FAS-CLO: physics-informed fully actuated system-coordinate learning observer

This paper develops a physics-informed fully actuated system-coordinate learning observer (PI-FAS-CLO) for sensor-sparse nonlinear output-feedback systems. PI-FAS-CLO reconstructs the fully actuated system (FAS)-coordinate stack comprising the measured output and its unmeasured derivatives. Learned drift and input-matrix functions retain an input-affine high-order channel, while input-matrix factorisation enforces a positive lower singular-value bound. Integrated output moments provide measurement-derived affine initialisation and weak channel supervision without pointwise highest-order derivative targets. Training combines multistep output and coordinate losses with weak channel residuals. A discrete-time linear matrix inequality enables synthesis of a constant observer gain. Under uniform bounds and discrete contraction, sampled-data analysis establishes a practical estimation-error bound. Across 12 conditions on a third-order benchmark, PI-FAS-CLO reduces mean coordinate error relative to a validation-selected pointwise counterpart by 13.4% and 17.2% on in-distribution and shifted tests. At higher measurement-noise levels, PI-FAS-CLO achieves lower mean coordinate error than the learning-based high-gain observer. On a separately trained two-link mechanical benchmark, PI-FAS-CLO achieves the lowest mean velocity error among five observers under high noise across nominal, payload, and higher-frequency tests. With a common sampled controller, the PI-FAS-CLO-based closed loop achieves the lowest mean tracking error.