How to ensure sufficient data richness for the estimation of stochastic dynamical systems in finite time?

For most real-world dynamical systems, input–output models are developed for control, optimization, prediction, or diagnosis. However, the system dynamics are often unknown. Data-driven modeling, combining system identification and machine learning techniques, provides an effective strategy to determine a model from input–output data collected during excitation experiments. The user selects a structure that groups several candidate models. These models are ranked according to their ability to explain the data. The identified model is the one with the optimal score. In the current era of Big Data, many studies are now considering the use of large number of data in order to obtain the most accurate model. However, having a large dataset is not enough in practice! Indeed, an inappropriate choice of excitation may lead to ambiguity: several different models may achieve the same score, making the identification incorrect and potentially leading to dangerous scenarios if the model is later used for diagnosis or control.

The property that guarantees the uniqueness of the optimal model is called data informativity. It indicates whether the data contain sufficient information about the system dynamics. Introduced in the 1980s [5], this concept led to the establishment of necessary and sufficient conditions on excitation for the identification of linear time-invariant (LTI) systems [1–4]. However, these studies focused on the asymptotic case, assuming an infinite amount of data, which is unrealistic. More recently, informativity has been studied in the context of a finite number of data points [7,8], either in the noise-free case (using Willems’ lemma [9]) or with deterministic and bounded noise (set-membership method [8]). In practice, these assumptions are rarely satisfied since noise is often stochastic.

The central question of this thesis is the development of necessary and sufficient conditions on excitation to guarantee the informativity of a finite number of data affected by stochastic noise, in the framework of linear system identification. The approach proposed in a recent work at CRAN [6] will be used as a starting point. However, the simplifying assumptions in [6] limit its applicability. The objective is therefore to extend the analysis to more general scenarios, particularly closed-loop identification and linear parameter-varying systems, which are better suited for complex systems.

References
[1] Bazanella et al., “Necessary and sufficient conditions for uniqueness of the minimum in prediction error identification,” Automatica, 48(8):1621–1630, 2012.
[2] Colin et al., “Closed-loop identification of MIMO systems in the prediction error framework: Data informativity analysis,” Automatica, 121:109171, 2020.
[3] Colin et al., “Data informativity for the open-loop identification of MIMO systems in the prediction error framework,” Automatica, 117:109000, 2020.
[4] Gevers et al., “Informative data: How to get just sufficiently rich?,” Proceedings of the 47th IEEE Conference on Decision and Control, pp.1962–1967, 2008.
[5] Ljung, System Identification: Theory for the User, Prentice Hall, 1999.
[6] Sleiman et al., “Data informativity for prediction error identification of stochastic LTI systems with repeated finite-time experiments in open-loop,” 2025.
[7] van Waarde et al., “A behavioral approach to data-driven control with noisy input–output data,” IEEE Transactions on Automatic Control, 69(2):813–827, 2023.
[8] van Waarde et al., “Data informativity: A new perspective on data-driven analysis and control,” IEEE Transactions on Automatic Control, 65(11):4753–4768, 2020.
[9] Willems et al., “A note on persistency of excitation,” Systems & Control Letters, 54(4):325–329, 2005.

Created in 1980, the “Centre de Recherche en Automatique de Nancy (CRAN)” is a joint research unit (UMR 7039) shared by the University of Lorraine (UL) and the CNRS  “Institut des sciences de l’information et de leurs interactions (INS2I)” and “Institut des Sciences de l’Ingénierie et des Systèmes (INSIS)”. It also hosts researchers from the “Lorraine Cancer Institute” (ICL, Centre de lutte contre le cancer), the University Hospital., the Regional Hospital Center of Metz-Thionville and the LIST in Luxembourg-City.As of January 1, 2022, the laboratory has 106 professor-researchers, 4 emeritus, 9 CNRS researchers, 9 other researchers from UL, ICL and CHU or external organizations, 13 post-doctoral fellows, 90 doctoral students and 34 (including 29 permanent and 5 fixed-term contracts) engineers, technicians or administrators. It is part of the Charles Hermite Automatique, Informatique, Mathématiques de Lorraine Research Federation and the Automatique, Mathématiques, Informatique et leurs Interactions (AM2I) scientific pole of the University of Lorraine.

Based on digital sciences, the laboratory is internationally recognized for its activities in the fields of signal and image processing, control and computer engineering, as well as for its work in health in connection with biology and neuroscience. Today, its fundamental and applied research enables it to accompany the changes in society and goes beyond the traditional industrial issues: energy production, management of the intelligent city or transport. In health, it is opening up to diagnosis and care in cancerology and neurology.
They are crossing sociology, listening to social behaviors and opinion dynamics, and investing in the field of sustainable development, in the service of the circular economy and ecological systems.

This Ph.D. offer will take place within the department Control Identification Diagnostics.

We are looking for a candidate who has graduated or is in the final year of a Master’s program or an engineering school degree with skills in control engineering, system identification, data analysis, machine learning or applied mathematics. A good level of English (min B2) is required and proficiency in French is not mandatory.