I am part of the Optimal Control at Large research project, issued by the European Research Council under the Horizon 2020 Advanced Grant OCAL.

# Data-driven control

### Data-driven optimal control with a relaxed linear program

The linear programming (LP) approach has a long history in the theory of approximate dynamic programming. When it comes to computation, however, the LP approach often suffers from poor scalability. In this work, we introduce a relaxed version of the Bellman operator for q-functions and prove that it is still a monotone contraction mapping with a unique fixed point. In the spirit of the LP approach, we exploit the new operator to build a relaxed linear program (RLP). Compared to the standard LP formulation, our RLP has only one family of constraints and half the decision variables, making it more scalable and computationally efficient. For deterministic systems, the RLP trivially returns the correct q-function. For stochastic linear systems in continuous spaces, the solution to the RLP preserves the minimizer of the optimal q-function, hence retrieves the optimal policy. Theoretical results are backed up in simulation where we solve sampled versions of the LPs with data collected by interacting with the environment. For general nonlinear systems, we observe that the RLP again tends to preserve the minimizers of the solution to the LP, though the relative performance is influenced by the specific geometry of the problem.

### On the synthesis of Bellman inequalities for data-driven optimal control

In the context of the linear programming (LP) approach to data-driven control, one assumes that the dynamical system is unknown but can be observed indirectly through data on its evolution. Both theoretical and empirical evidence suggest that a desired suboptimality gap is often only achieved with massive exploration of the state-space. In case of linear systems, we discuss how a relatively small but sufficiently rich dataset can be exploited to generate new constraints offline and without observing the corresponding transitions. Moreover, we show how to reconstruct the associated unknown stage-costs and, when the system is stochastic, we offer insights on the related problem of estimating the expected value in the Bellman operator without re-initializing the dynamics in the same state-input pairs.

# Networked systems

### Disturbance suppression in networks with clustering techniques

We address the problem of local disturbance suppression in networked systems. The aim is to detect a suitable cluster which is able to locally adsorb a disturbance by means of an appropriate redistribution of control load among its nodes, such that no external node is affected. Traditional clustering measures are not suitable for our purpose, since they do not explicitly take into account the structural conditions for disturbance containment. We propose a new measure based on the concept of degree of freedom for a cluster, and we introduce a heuristic procedure to quickly select a set of nodes according to this measure. Finally, we show an application of the method in the context of DC microgrids voltage control.

### Secondary control methods for power distribution models

Secondary control architectures for islanded direct-current microgrids are getting interest since they are necessary to manage the voltage references in order to properly distribute the time-varying load demand. To this aim, we propose three different optimization-based secondary control approaches considering the internal units constraints, losses minimization and the continuous satisfaction of the load demand. The described approaches are based on a centralized, distributed and cluster-based optimization strategy, respectively.

### Voltage stabilization in DC microgrids with energy-based control

We investigates the application of passivity-based nonlinear control to the problem of primary voltage stabilization in medium-voltage DC microgrids (MVDC mGs) given by the interconnection of nonlinear distributed generation units (DGUs) and power lines. To this aim, we propose nonlinear local regulators which steer the voltage at the output terminal of each DGU to a reference value. Each controller can be explicitly synthesized relying on DGU parameters, voltage reference values of the neighboring DGUs and resistance of the neighboring power lines. The control design enables plug-and-play (PnP) operations: a plug-in or -out of a DGU requires only the update of regulators of neighboring DGUs without spoiling the stability of overall mG.