Marcello Colombino

Decentralized Convergence to Nash Equilibria in Constrained Deterministic Mean Field Control

This paper considers decentralized control and optimization methodologies for large populations of systems, consisting of several agents with different individual behaviors, constraints and interests, and influenced by the aggregate behavior of the overall population. For such large-scale systems, the theory of aggregative and mean field games has been established and successfully applied in various scientific disciplines. While the existing literature addresses the case of unconstrained agents, we formulate deterministic mean field control problems in the presence of heterogeneous convex constraints for the individual agents, for instance arising from agents with linear dynamics subject to convex state and control constraints. We propose several model-free feedback iterations to compute in a decentralized fashion a mean field Nash equilibrium in the limit of infinite population size. We apply our methods to the constrained linear quadratic deterministic mean field control problem.

Mean field constrained charging policy for large populations of Plug-in Electric Vehicles

Constrained charging control of large populations of Plug-in Electric Vehicles (PEVs) is addressed using mean field game theory. We consider PEVs as heterogeneous agents, with different charging constraints (plug-in times and deadlines). The agents minimize their own charging cost, but are weakly coupled by the common electricity price. We propose an iterative algorithm that, in the case of an infinite population, converges to the Nash equilibrium associated with a related decentralized optimization problem. In this way we approximate the centralized optimal solution, which in the unconstrained case fills the overnight power demand valley, via a decentralized procedure. The benefits of the proposed formulation in terms of convergence behavior and overall charging cost are illustrated through numerical simulations.

A Convex Characterization of Robust Stability for Positive and Positively Dominated Linear Systems

We provide convex necessary and sufficient conditions for the robust stability of linear positively dominated systems. In particular, we show that the structured singular value is always equal to its convex upper bound for nonnegative matrices and we use this result to derive necessary and sufficient Linear Matrix Inequality (LMI) conditions for robust stability that involve only the systems static gain. We show how this approach can be applied to test the robust stability of the Foschini-Miljanic algorithm for power control in wireless networks in presence of uncertain interference.

Convex characterization of robust stability analysis and control synthesis for positive linear systems

We present necessary and sufficient conditions for robust stability of positive systems. In particular we show that for such systems the structured singular value is equal to its convex upper bound and thus it can be computed efficiently. Using this property, we show that the problem of finding a structured static state feedback controller achieving internal stability, contractiveness, and internal positivity in closed loop remains convex and tractable even in the presence of uncertainty.

On the convexity of a class of structured optimal control problems for positive systems

We study a class of structured optimal control problems for positive systems in which the design variable modifies the main diagonal of the dynamic matrix. For this class of systems, we establish convexity of both the H2 and H∞ optimal control formulations. In contrast to previous approaches, our formulation allows for arbitrary convex constraints and regularization of the design parameter. We provide expressions for the gradient and subgradient of the H2 and norms and establish graph-theoretic conditions under which the H∞ norm is continuously differentiable. Finally, we develop a customized proximal algorithm for computing the solution to the regularized optimal control problems and apply our results for HIV combination drug therapy design.

On constrained mean field control for large populations of heterogeneous agents: Decentralized convergence to Nash equilibria

We consider mean field games in a large population of heterogeneous agents subject to convex constraints and coupled by a quadratic cost, which depends on the average population behavior. The problem of steering such population to a Nash equilibrium is usually addressed in the (mean field control) literature by formulating an iterative game between the agents and a central coordinator, that broadcasts at every step the average population behavior. Here we generalize this approach by allowing the central operator to filter such signal using a feedback mapping. We propose different classes of feedback mappings and we derive sufficient conditions guaranteeing convergence to a ε-Nash equilibrium, even for cases when the standard approach fails. We show that the deviation ε of each agent from its optimal cost, decreases at least linearly to zero with the increase of the population size. Contrary to the state of the art, the proposed approach guarantees convergence in the presence of heterogeneous convex constraints for the agents. Finally, we show how these results can be applied to regulate in a decentralized fashion the charging process of a large population of plug-in electric vehicles. Our findings give theoretical support and extend previous literature results.

Robust stability of a class of interconnected nonlinear positive systems

We present conditions for robust stability of a class of linear systems interconnected by uncertain nonlinear, norm-bounded functions. We show that such conditions can be reformulated as classical small gain like conditions for a related linear system. Under further assumptions that render the related linear system positive, we show that we can achieve sharp tractable conditions for robust stability of the original nonlinear system.

Quadratic two-team games

We consider stochastic quadratic two-player games where each player represents a team of agents subject to information constraints. We present conditions that guarantee the existence and uniqueness of a Nash equilibrium in the space of linear decentralized policies and we provide an iterative algorithm to compute such an equilibrium. The results are illustrated on a numerical example inspired from power systems security.

AEOLUS, the ETH Autonomous Model Sailboat

Path planning and control are particularly challenging tasks for a sailboat. In contrast to land vehicles or motorboats, the movement of a sailboat is heavily restricted by the wind direction. This paper focuses on the low-level control acting on the rudder and the sails. Specifically, a standard proportional controller and a non-linear controller have been implemented to track a reference heading. Further, special control algorithms that are activated during a tack or a jibe perform fast and smooth maneuvers. The path planner is based on the minimization of the weighted sum of different cost functions and allows for multi-objective optimization of the boat trajectory such as obstacle avoidance, time-to-target minimization and tactical behaviors.

Mutually Quadratically Invariant Information Structures in Two-Team Stochastic Dynamic Games

We formulate a two-team linear quadratic stochastic dynamic game featuring two opposing teams each with decentralized information structures. We introduce the concept of mutual quadratic invariance (MQI), which, analogously to quadratic invariance in (single team) decentralized control, defines a class of interacting information structures for the two teams under which optimal feedback control strategies are linear and easy to compute. We show that for zero-sum two-team dynamic games with MQI information structure, structured state feedback saddle-point equilibrium strategies can be computed from equivalent structured disturbance feedforward saddle point equilibrium strategies. We also show that there is a saddle-point equilibrium in linear strategies even when the teams are allowed to use nonlinear strategies. However, for nonzero-sum games we show via a counterexample that a similar equivalence fails to hold. The results are illustrated with a simple yet rich numerical example that illustrates the importance of the information structure for dynamic games.