## Section: New Results

### Distributed methods for control

#### Distributed control

Participants : A. Seuret [Contact person] , G. Rodrigues de Campos, L. Brinon-Arranz, D.V. Dimarogonas [KTH] , K.H. Johansson [KTH] .

Another particular effort has been provided to the design of distributed control laws for multi-agents systems. Three main contributions have been produced and can be summarized as follows.

In [44] , a new consensus algorithms for heterogeneous multi-agent systems is provided. A control strategy based on a consensus algorithm which is decoupled from the original systems is proposed. Consequently, its major advantage remains in the separation of the stability analysis of each subsystem and the distributed control algorithm. It is shown that our method allows using classical distributed consensus algorithms such as simple integrator consensus (with or without delay) and distributed consensus filter algorithms.

For many multi-robot applications it is interesting to impose a particular configuration for the robotic agents. This paper discusses the design and analysis of a distributed algorithm for the compact deployment of agents, where the behavior of each vehicle is only dependent on local information. The objective of the paper [72] is to achieve the most compact formation possible. To solve this problem we propose, in a first step, two uncorrelated controllers: one designed for dispersion with connectivity maintenance and a second designed to minimize inter-agent angles. An improved controller including variable gains, particularly designed to avoid singular configurations, is also provided. Lastly, we propose a sequential strategy composed of the two previously mentioned controllers and a stability analysis based on hybrid systems theory. Finally, some simulation results for different configurations supporting our theoretical results are presented.

#### Collaborative source seeking control

Participants : C. Canudas [Contact person] , R. Fabbiano, F. Garin.

The problem of source localization consists in finding the point or the spatial region from which a quantity of interest is being emitted; this goal can be pursued by one or several agents possibly cooperating each other. Source-seeking agents can be fixed sensors, that collect and exchange some information about the signal field and try to identify the position of the source (or the smallest region in which it is included), or moving devices equipped with one or more sensors, that physically reach the source in an individual or cooperative way.

Within the FeedNetBack European project, we have addressed the problem of collaborative source seeking with a fleet of autonomous underwater vehicles (UAVs). This topic was explored in the PhD thesis of Lara Brinon [61] , where a solution was proposed, based on circular formations with the center of the formation following a 2-dimensional movement in the direction of the gradient of the source. The gradient computation was achieved through an approximation using the point-wise measurements from the various vehicles.

In a more recent work [29] , we leave temporarily aside all issues of coordination and communication failures well-addressed in [61] , and we focus on the gradient computation formula. Under some assumptions on the source emission (isotropic diffusive source in steady-state, whose solution satisfies the Laplace equation), we show that there is an exact integral formula (based on the Poisson integral of harmonic functions) for the computation of the gradient at the center of a circle, using pointwise measurements along the circumference. This approach has two main advantages: it can be generalized in three (or more) dimensions, and it allows to compute also higher-order derivatives, which allow to find higher-order control laws, useful e.g. for non-holonomic vehicles. A relevant property is that such an integral formula exploits mathematical properties of the source density distribution (the fact that it is harmonic), but does not require the knowledge of an explicit expression for the density function. This makes our approach different from the main source-seeking techniques present in the literature, which either are based on a specific knowledge of the solution of the diffusion process, or make use of an extremum-seeking approach, exciting the system with a periodic signal so as to explore the field and collect enough information to reconstruct the gradient of the quantity of interest.

The latter work is part of the research of Ruggero Fabbiano during his Ph.D. studies.

#### Distributed real-time Simulation of numerical models

Participants : D. Simon [Contact person] , A. Ben Khaled [IFPEN] , M. Ben Gaid [IFPEN] .

The need of quick innovation in the automotive domain made simulation necessary at early stages of the development cycle. Vehicles and powertrains are complex systems where different domains are involved. Representative phenomenological models of powertrains have been developed and have been used in the design phase under domain dedicated tools. However, their use for controls validation using Model-In-the-Loop (MIL) and Hardware-In-the-Loop (HIL) was prevented due to performance limitation of widely used single-solver/single-core simulation approaches.

Multicore simulation for complex systems has been studied with a focus on simulation duration speedup. The methodology of parallelization across the model has been selected for such problem where strong interactions between the model components are observed. The current study showed that decoupling the model parts by relaxing their data dependencies is promising in term of simulation speed (by increasing the parallelism) and results accuracy. Besides, tests results on engine model showed that, with the model partitioning, it is possible to use efficiently variable-step solvers thanks to the decrease of the number of discontinuities, so the number of integration interrupts, in each subsystem [26] .

Further work will investigate in the combination of the use of variable-step solvers in split model with the use of multicore architecture for parallel computing, in order to improve the simulation speedup while keeping results accuracy under control.