## Section: New Results

### Control design and networked systems

#### Control design for hydro-electric power-plants

Participants : C. Canudas de Wit [Contact person] , S. Gerwig [Feb 2014–Mar 2016] , F. Garin, B. Sari [Alstom] .

This is the study of collaborative and resilient control of hydro-electric power-plants, in collaboration with Alstom. The goal is to improve performance of a hydro-electric power-plant outside its design operation conditions, by cancellation of oscillations that occur in such an operation range. Indeed, current operation of power-plants requires to operate on a variety of conditions, often different from the ones initially considered when designing the plant. At off-design operation pressure, the hydraulic turbine exhibits a vortex rope below the runner. This vortex generates pressure fluctuations after the turbine and can excite the hydraulic pipes. Indeed the water is compressible and the pipe walls elastic, so the system can oscillate. The goal is to damp these pressure oscillations as they create vibrations in the system and can lead to damages. Our first contribution [23] has been to model the effect of the vortex rope on the hydraulic system as an external perturbation source acting on pipes. The pipes themselves are described with equations taking into account water compressibility and pipe-wall elasticity. The resulting model is nonlinear with hyperbolic functions in the equations (analogous to high-frequency transmission lines), from which we obtain a suitably linearized model. This model can then be used for control design.

#### Collaborative source seeking

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

The problem of source localization consists in finding the point or the spatial region from which a quantity of interest is being emitted. We consider collaborative source seeking, where various moving devices, each equipped with a sensor, share information to coordinate their motion towards the source. We focus on the case where information can only be shared locally (with neighbor agents) and where the the agents have no global position information, and only limited relative information (bearing angle of neighbor agents). This setup is relevant when GPS navigation is not available, as in underwater navigation or in cave exploration, and when relative position of neighbors is vision-based, making it easier to measure angles than distances. In [16] we propose and analize a control law, which is able to bring and keep the agents on a circular equispaced formation, and to steer the circular formation towards the source via a gradient-ascent technique; the circular equispaced formation is beneficial to a good approximation of the gradient from local pointwise measurements. This algorithm is different from the ones present in the literature, because it can cope with our above-described restrictive assumptions on the available position information.

#### Distributed control and game theory: self-optimizing systems

Participants : F. Garin [Contact person] , B. Gaujal [POLARIS] , S. Durand.

The design of distributed algorithms for a networked control system composed of multiple interacting agents, in order to drive the global system towards a desired optimal functioning, can benefit from tools and algorithms from game theory. This is the motivation of the Ph.D. thesis of Stéphane Durand, a collaboration between POLARIS and NECS teams. The first results of this thesis concern the complexity of a classical algorithm in game theory, the Best Response Algorithm, an iterative algorithm to find a Nash Equilibrium. For potential games, Best Response Algorithm converges in finite time to a pure Nash Equilibrium. The worse-case convergence time is known to be exponential in the number of players, but surprisingly it turns out that on average (over the possible values of the potentials) the complexity is much smaller, only linearly growing, see [27], [26], [22].