{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T07:07:58Z","timestamp":1784185678892,"version":"3.55.0"},"reference-count":25,"publisher":"MDPI AG","issue":"14","license":[{"start":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T00:00:00Z","timestamp":1784073600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper addresses indirect sensing in resonant, oscillating, and periodically forced sensors, where the physical measurand is not directly available as a static output but is encoded in the dynamic response of the device. The sensor and its excitation are described as a single autonomous system, in which the excitation phase and the slowly varying measurand define a compact state representation after the decay of transients. Within this setting, delayed samples of the available output define an observation map that can be inverted, under suitable smoothness and observability conditions, to reconstruct the measurand in a deadbeat-like fashion. Compared with a preliminary conference study based on a simplified scalar-output RLC benchmark, the present work extends the formulation to vector-valued outputs, introduces a local conditioning indicator based on the Jacobian matrix, and focuses on a micromechanical sensing case with nonlinear electromechanical transduction. The inverse observation map is approximated by a feedforward neural network trained on synthetic data generated from the autonomous model. The methodology is applied to a vibratory MEMS gyroscope, where the signed angular rate is reconstructed from a delayed-output sequence combining the nonlinear capacitive current readout and the known AC drive reference. The augmented output is introduced to overcome the lack of observability affecting the raw current signal over signed angular-rate ranges. Numerical results show accurate reconstruction in ideal conditions and provide a preliminary robustness assessment under additive output noise.<\/jats:p>","DOI":"10.3390\/s26144504","type":"journal-article","created":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T03:34:23Z","timestamp":1784172863000},"page":"4504","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Delay-Embedded Neural Reconstruction for Indirect Sensing in Electrical and Micromechanical Oscillating Systems"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-7097-9639","authenticated-orcid":false,"given":"Francesco","family":"Grimaldi","sequence":"first","affiliation":[{"name":"Department of Electrical, Electronic and Information Engineering (DEI), Alma Mater Studiorum, University of Bologna (UniBo), 40136 Bologna, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christian","family":"Geminiani","sequence":"additional","affiliation":[{"name":"Department of Electrical, Electronic and Information Engineering (DEI), Alma Mater Studiorum, University of Bologna (UniBo), 40136 Bologna, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5079-5065","authenticated-orcid":false,"given":"Andrea","family":"Tilli","sequence":"additional","affiliation":[{"name":"Department of Electrical, Electronic and Information Engineering (DEI), Alma Mater Studiorum, University of Bologna (UniBo), 40136 Bologna, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,7,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1749","DOI":"10.1109\/JSEN.2011.2157679","article-title":"Atomic sensors\u2014A review","volume":"11","author":"Kitching","year":"2011","journal-title":"IEEE Sens. 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