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In particular, expertise has been developed in the design of systems obtained by interconnecting subsystems, for which the combination of the input-output approach with (convex [BTN01, BV04]) optimization tools seems to be particularly effective. Convincing results have already been obtained, ranging from upstream methodological contributions (e.g. [PKZS23,ACPKS23, LKD+17]) to their application to problems of strong practical interest (e.g. [PKS+21,KSCB16, GFS11]), and even to patent deposit (e.g. [PKZ+17,CGK10,CK13]).

Scientific background of the project The ongoing integration of information technologies into engineering systems is radically changing the possibilities in a wide range of applications (energy production and distribution, telecommunications, transportation of goods and people, Industry 4.0, medicine, intelligent buildings, etc.), but at the cost of a drastic increase in the complexity of the associated design problems. In addition to having to meet ever more stringent and even new requirements (performance, safety, security, energy efficiency, cost, etc.), it must explicitly take into account the complexity of the interconnection (large size, degraded communication, live-machine interface, etc.) of intrinsically complex and heterogeneous systems. To meet these challenges, traditional design methods based on simulation and a trial-and-error approach often appear to be limited, and it has become necessary to develop adapted methods that enable efficient design. System and Control theory and Signal Processing are natural candidates for the development of such design support methods. As representatives of cybernetics and information theory in engineering, these two disciplines provide the complementary System and Signal views needed to capture the full complexity of design problems. In addition to bridging the disciplines, their abstract formalism makes possible to represent systems and their interconnections while taking into account the practical constraints of specifications. Systems are represented as subsystems that interact with each other and with their environment through signals. Constraints are then imposed on these signals to characterize the subsystems and their interconnections, but also to translate the specifications.

In order to cope with the design complexity of modern systems, the use of computing power appears to be essential. Certain conflicting objectives then arise (computation time, optimality of the solution, design and implementation time of the algorithm used, etc.). For the engineering researcher, who has to develop and compare methods for different practical problems, a good compromise is given by the use of the class of convex optimization problems. In particular, this class is known to have good numerical solution properties, allowing efficient solution, with a computation time between a few seconds and a few minutes for a mediumsized problem with a few hundred optimization variables, and easy implementation, which has popularized its use in engineering sciences [BTN01,BV04]. The main difficulty in using convex optimization lies in the formulation of the problem in such a form, which often requires the development of reformulation or relaxation techniques.

Problem and Objective of the thesis This project proposes to address the issue of efficient analysis and synthesis of interconnected heterogeneous systems, particularly in the context of signal estimation through filtering (Figure 1). This problem is particularly relevant for large-scale systems (e.g., energy distribution networks, sensor networks, gene regulation networks, etc.).

A first strategy to tackle this problem is to consider the global system and use classical methods of analysis and synthesis. However, in the case of large systems, this type of approach will generally lead to very large optimization problems. A second strategy is to describe the overall system as a collection of subsystems, modeled by a characterization on the input and output signals of each subsystem. This type of approach has the advantage of greatly reducing the complexity of the optimization problems. In addition, for an application of estimation filter synthesis, this idea makes it possible to reduce the order of the filters obtained. This second strategy has recently led to significant results in the special case of homogeneous subsystems, i.e., those represented by the same model (see, for example, [ACPKS23, PKZS23, KSCB16]). However, in the more general case of heterogeneous subsystems, this approach tends to be conservative, i.e. it does not necessarily allow to find a solution even if one exists. The main suspected cause of this conservatism is the input-output characterization performed for each subsystem independently of the others, which implicitly assumes that the subsystems are independent and therefore that their models have no similarities.

The objective of this thesis is to overcome this problem by exploring an original idea : introducing dependency between subsystem characterizations to take into account similarities (e.g. algebraic or topological) in their modeling. The goal is to improve the trade-off between algorithmic complexity and conservatism by finding a balance between the two strategies described above. The interest and limitations of this idea will be illustrated in particular by a signal estimation filter synthesis application. The main challenge will be to formalize the type of dependency between subsystem characterizations that can be included in the analysis and synthesis (estimation filter) methods, while preserving the convex character of the optimization problems to be solved. An alternative path to consider will be the use of convex relaxations. As the expected contributions are mainly methodological, the results will be valorized mainly through presentations at international conferences and publications in leading journals in the field of Control and System theory., Interested candidates are warmly invited to send an e-mail containing a resume, transcript, and a short message of presentation and motivation to the advisors team (see e-mail addresses at the beginning of this document). The recruitment process consists of three stages :

1. Application deadline : April 30, 2026. Oral interview by the advisors team and selection of the candidate.

2. Oral interview by the EEA Doctoral School Board in late May/early June.

3. Final result : first half of June. Career prospects The future PhD will develop a set of skills that can be applied in a wide range of professional environments. In particular, the following careers are targeted : researcher, PhD engineer, R&D engineer, in the public or private sector.

Master Degree or equivalent, Candidate profile We are looking for candidates with a master's degree in System and Control or Signal Processing (MSc or MEng) and an excellent academic record. Those with a general degree and strong skills in applied mathematics are also encouraged to apply. Interest in developing optimization-based methods and experience with Matlab are also valued. Specific Requirements

The École Centrale de Lyon (ECL) is a public scientific, cultural and professional institution. Member of the Ecoles Centrales group and the Écoles Nationales d'Ingénieurs network,ECL trains high-level generalist engineers, specialized engineers, masters students and doctoral candidates. The school hosts 2,500 engineering students and trainees, 300 master students and more than 250 doctoral students. It is characterized by recognized research supported by 6 research laboratories. ECL's research activities are directed to and for the business world through numerous industrial contracts. The Ampère-lab is a joint research unit (CNRS, Ecole Centrale de Lyon, INSA Lyon, Université Lyon 1) of more than 150 researchers based in Lyon, France, working on the rational use of energy in systems in relation to their environment. The research carried out by the Automatic Control for System Engineering department includes the development of methods and tools for optimizing and controlling the dynamic behavior of systems in a wide range of application domains, in collaboration with other departments of the laboratory and other engineering laboratories. The combination of theoretical and applied dimensions of this research constitutes its great originality. Over the last few years, the advisors have been working on the possibilities offered by Systems, Control and Signal approaches for the development of methods for the design/understanding of systems from different disciplines (electronics, electrical engineering, mechanics, biology, etc.).