{"paper":{"title":"Learning a Contracting KKL-observer with Local Optimal Guarantees","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Neural networks learn KKL observers that stay globally contracting yet locally match the minimum-energy estimator.","cross_cats":["cs.SY"],"primary_cat":"eess.SY","authors_text":"Clara Luc\\'ia Galimberti, Daniele Astolfi, Johan Peralez, Madiha Nadri, Vincent Andrieu","submitted_at":"2026-05-13T12:48:47Z","abstract_excerpt":"The Kazantzis-Kravaris-Luenberger (KKL) observer provides a general framework for nonlinear state estimation by immersing the system dynamics into a stable linear or nonlinear latent dynamics. However, the performance of KKL observers relies heavily on the specific choice of these latent dynamics, which is often heuristic. This paper proposes a methodology to learn a KKL observer that combines global stability guarantees with local optimality. We derive a condition on the latent dynamics such that the observer locally mimics the behavior of a Minimum Energy Estimator (Mortensen observer). We t"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We derive a condition on the latent dynamics such that the observer locally mimics the behavior of a Minimum Energy Estimator (Mortensen observer). We then employ Deep Learning to approximate the KKL transformation and the latent dynamics, using neural network architectures that structurally enforce the contraction property.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That a suitable latent dynamics satisfying the derived local-optimality condition exists for the target systems and that neural networks with contraction-enforcing architectures can accurately approximate the required KKL transformation and latent dynamics.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Learns contracting KKL observers via deep learning that locally match minimum energy estimators with global stability guarantees.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Neural networks learn KKL observers that stay globally contracting yet locally match the minimum-energy estimator.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1f3c4b7ae657ff737579e8ad562e796e442352b54e06f33d512746edc2782e1a"},"source":{"id":"2605.13453","kind":"arxiv","version":1},"verdict":{"id":"2ac00b95-0cde-404f-8bf4-0b92e50045fa","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:57:21.558890Z","strongest_claim":"We derive a condition on the latent dynamics such that the observer locally mimics the behavior of a Minimum Energy Estimator (Mortensen observer). We then employ Deep Learning to approximate the KKL transformation and the latent dynamics, using neural network architectures that structurally enforce the contraction property.","one_line_summary":"Learns contracting KKL observers via deep learning that locally match minimum energy estimators with global stability guarantees.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That a suitable latent dynamics satisfying the derived local-optimality condition exists for the target systems and that neural networks with contraction-enforcing architectures can accurately approximate the required KKL transformation and latent dynamics.","pith_extraction_headline":"Neural networks learn KKL observers that stay globally contracting yet locally match the minimum-energy estimator."},"references":{"count":18,"sample":[{"doi":"","year":2006,"title":"Andrieu, V. and Praly, L. (2006). On the existence of a Kazantzis–Kravaris/Luenberger observer.SIAM Journal on Control and Optimization, 45(2), 432–456. Beik Mohammadi, H., Hauberg, S., Arvanitidis, G","work_id":"9da62dd5-8490-48b8-af47-02fbedbc18c1","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Figueroa, N., Neumann, G., and Rozo, L. (2024). Neural contractive dynamical systems. InICLR, 49097–49120","work_id":"06bd2b80-9f2a-4e8b-bba3-a8129622d87a","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"Bernard, P., Andrieu, V., and Astolfi, D. (2022). Observer design for continuous-time dynamical systems.Annual Reviews in Control, 53, 224–248","work_id":"a205f1c4-3f19-4c18-abaf-2788c57d4ed5","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"Brivadis, L., Andrieu, V., Bernard, P., and Serres, U. (2023). Further remarks on KKL observers.Systems & Control Letters, 172, 105429","work_id":"7e82b998-4c28-4d21-b6ca-a2e59c1b82c5","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"Buisson-Fenet, M., Bahr, L., Morgenthaler, V., and Di Meglio, F. (2023). Towards gain tuning for numerical KKL observers.IFAC-PapersOnLine, 56(2), 4061–4067","work_id":"c4bb2816-0389-4183-9a4e-536a8c9f8c39","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":18,"snapshot_sha256":"1496578442054900a429fbd28bd1fcd1d93866b5bcc5f403df5c1ae0462f862b","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}