## Research

Our research focuses on several aspects of distributed control and optimization:### Structured Robust Linear Control of Positive Systems

In classical linear control design decentralized and distributed control design is often equivalent to the synthesis of structured controllers. By imposing a zero-pattern to the controller one can decide which actuators have access to information form each sensor. Such stuctured control problems are hard in general. We focus on structured control design for positive systems in the presence of uncertainty and we show that for such systems the structured control problem can be solved efficiently.

A dynamical system is said to be positive if, for every nonnegative initial condition and nonnegative input, the output trajectory remains nonnegative for all time. Such systems arise naturally in several application areas including chemical reactions, population dynamics, job balancing in computer networks and consensus problems over graphs. Positive systems received increasing attention over the last decade not only for their practical relevance but also for their system theoretic properties. Recent results show that several classical hard control problems, especially those related to distributed and decentralized control, can be solved efficiently for positive systems. In [1] we show that robust stability for positive systems can be characterized as a convex feasibility problem and we use this result to design optimal distributed - decentralized controllers in the presence of uncertainty.

### Cooperative Distributed Model Predictive Control

For large-scale networks of systems, which are often subject to communication constraints, feedback controllers have to be operated in a distributed way. In particular, each system in the network has to take local control decisions based on local measurements and communication with neighboring systems. A possible approach, which is suited in particular for constrained networks of systems with a common objective function, is cooperative distributed model predictive control (MPC). This control scheme has many possible applications such as power grids, traffic networks or wind farms [2]. In cooperative distributed MPC, similarly as in centralized MPC, a globally defined finite horizon optimal control problem is solved in every time step. As opposed to centralized MPC however, this problem is solved by a distributed optimization method which works under the given communication constraints, such as the alternating direction method of multipliers or dual decomposition based on fast gradient updates. Consequently, research on cooperative distributed MPC is mainly concerned with the impact of distributed optimization on system properties such as stability, feasibility, etc. Such issues arise for instance in the formulation of stable cooperative distributed MPC controllers [3], [4], [5] or in the convergence properties of distributed optimization methods when used to solve MPC problems [6].

### Decentralized Mean Field Control of Large Populations of Systems

We investigate the control of large populations of dynamical systems. In this framework, each system (or agent) seeks a dynamical evolution that minimizes a cost-function depending not only on its own behavior, but also on the average behavior of the overall population. A centralized control scheme cannot be used to achieve this goal, mainly because in general the number of systems under consideration is too high for centralized computations and decisions; We use the theory of Mean Field Control (MFC) to devise decentralized control schemes that steer the population towards Nash Equilibrium. In [7] we use this scheme to control the charging of a large population of Plug-in Electric vehicles. For more information on the theorethical tools and the applications of mean field control please visit the page of the Demand Response Methods for Energy systems group .

### Synchronization in Complex Oscillator Networks

The emergence of synchronization in a network of coupled oscillators is a fascinating in various scientific disciplines ranging from neuroscience, physics, and biology to social networks and engineering. A coupled oscillator network is characterized by a population of heterogeneous oscillators and a graph describing the interaction among the oscillators. These two ingredients give rise to a rich dynamic behavior that keeps on fascinating the scientific community. An extensive survey on the theory of complex oscillator networks and their applications in power grids can be found in [8] for a tutorial. On the theory side of work, we focus on quantifying the trade-off between coupling and heterogeneity that leads to synchronization [9], [10]. On the application side, we are interested in how to use coupled oscillator theory for transient stability assessment in bulk power systems [11], distributed control and optimization in microgrids [12], [13], as well as novel computational approaches to power flow [14].