[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-124910-en":3,"doc-seo-124910-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},124910,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","Stable Linear Subspace Identification - A Machine Learning Approach","Machine Learning (ML) and linear system identification (SI) are traditionally developed separately. This paper introduces SIMBa, a family of discrete linear multi-step-ahead state-space SI methods implemented via backpropagation and automatic differentiation. SIMBa uses a novel linear-matrix-inequality-based free parametrization of Schur matrices to guarantee stability. Experiments show higher fitting performance than standard linear SI methods, with marked gains (often above 25%) over other stability-guaranteed SI approaches, across simulated and real input-output systems. The work is extended with potential for structured nonlinear model identification and provides open-source code.","Stable Linear Subspace Identification: A Machine Learning Approach  \nLoris Di Natale,† Muhammad Zakwan,† Bratislav Svetozarevic,  \nPhilipp Heer, Giancarlo Ferrari-Trecate, Colin N. Jones  \narXiv :2311 .03197v4 [ ee ss . SY] 26 Mar 2024  \nAbstract—Machine Learning (ML) and linear System Identification (SI) have been historically developed independently. In this paper, we leverage well-established ML tools — especially the automatic differentiation framework—to introduce SIMBa, a family of discrete linear multi-step-ahead state-space SI methods using backpropagation. SIMBa relies on a novel Linear-Matrix-Inequality-based free parametrization of Schur matrices to ensure the stability of the identified model.  \nWe show how SIMBa generally outperforms traditional linear state-space SI methods, and sometimes significantly, although at the price of a higher computational burden. This performance gap is particularly remarkable compared to other SI methods with stability guarantees, where the gain is frequently above 25% in our investigations, hinting at SIMBa’s ability to simultaneously achieve state-of-the-art fitting performance and enforce stability. Interestingly, these observations hold for a wide variety of input-output systems and on both simulated and real-world data, showcasing the flexibility of the proposed approach. We postulate that this new SI paradigm presents a great extension potential to identify structured nonlinear models from data, and we hence opensource SIMBa on [https://github.com/Cemempamoi/simba](https://github.com/Cemempamoi/simba).  \nI. INTRODUCTION  \nWhile linear System Identification (SI) matured decades ago [1], Machine Learning (ML) only rose to prominence in recent years, especially following the explosion of data collection and thanks to unprecedented computational power. In particular, large Neural Networks (NNs) have shown impressive performance on a wide variety of tasks [2], [3], leading to the recent boom of Deep Learning (DL) applications [4] . A key factor behind these successes has been the availability of efficient open-source libraries greatly accelerating the deployment of NNs, such as PyTorch and TensorFlow in Python. In particular, Automatic Differentiation (AD), atthe core of the backpropagation algorithm [5], the backbone of NN training, nowadays benefits from extremely efficient implementations.  \nGiven their effectiveness at grasping complex nonlinear patterns from data, NNs have recently been used for nonlinear system identification, where traditional SI methods struggle to compete [6]–[8] . NNs can be leveraged to create deep state-space models [9], deep subspace encoders [10], or  \nThis research was supported by the Swiss National Science Foundation under NCCR Automation, grant agreement 51NF40   180545.  \nL. Di Natale, B. Svetozarevic, and P. Heer are with the Urban Energy Systems Laboratory, Swiss Federal Laboratories for Materials Science and Technology (Empa), 8600 D¨ubendorf, Switzerland. L. Di Natale, M. Zakwan, G. Ferrari-Trecate, and C.N. Jones are with the Laboratoired’Automatique, Swiss Federal Institute of Technology Lausanne (EPFL), 1015 Lausanne, Switzerland.  \n† L. Di Natale and M. Zakwan contributed equally to this work. Corresponding author: L. Di Natale, [loris.dinatale@alumni.epfl.ch](loris.dinatale@alumni.epfl.ch).  \ndeep autoencoders [11], for example. While applying NNsto identify nonlinear systems can achieve good performance, it can underperform for linear systems, where methods assuming model linearity might achieve better accuracy [9] .  \nAlthough nonlinear SI has attracted a lot of attention in the last years, the identification of Linear Time Invariant (LTI) models is, however, still of paramount importance to many applications. Indeed, linear models come with extensive theoretical properties [12] and lead to convex optimization problems when combined with convex cost functions ina Model Predictive Controller (MPC) [13], for example. Moreover, to date, nu","cbCaieAB0sY34Qob","https://ap.wps.com/l/cbCaieAB0sY34Qob","pdf",369791,1,6,"English","en",105,"# Introduction\n## Subspace identification for linear systems\n## Enforcing stability","[{\"question\":\"What is SIMBa in this research?\",\"answer\":\"SIMBa is a family of discrete linear multi-step-ahead state-space system identification methods using backpropagation and automatic differentiation.\"},{\"question\":\"How does SIMBa guarantee stability of the identified model?\",\"answer\":\"It relies on a linear-matrix-inequality-based free parametrization of Schur matrices to ensure stability.\"},{\"question\":\"What results does the paper report compared with traditional methods?\",\"answer\":\"SIMBa generally outperforms traditional linear state-space SI methods, with sometimes large improvements, especially versus stability-guaranteed SI methods.\"}]","Stable Linear Subspace Identification - A Machine Learning Approach | PDF",1785895348,15,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"stable-linear-subspace-identification-a-machine-learning-approach","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/stable-linear-subspace-identification-a-machine-learning-approach/124910/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-05",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is SIMBa in this research?","Question",{"text":75,"@type":76},"SIMBa is a family of discrete linear multi-step-ahead state-space system identification methods using backpropagation and automatic differentiation.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does SIMBa guarantee stability of the identified model?",{"text":80,"@type":76},"It relies on a linear-matrix-inequality-based free parametrization of Schur matrices to ensure stability.",{"name":82,"@type":73,"acceptedAnswer":83},"What results does the paper report compared with traditional methods?",{"text":84,"@type":76},"SIMBa generally outperforms traditional linear state-space SI methods, with sometimes large improvements, especially versus stability-guaranteed SI methods.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]