Mixed integer predictive control and shortest path reformulation

Mixed integer predictive control deals with optimizing integer and real control variables over a receding horizon. The mixed integer nature of controls might be a cause of intractability for instances of larger dimensions. To tackle this little issue, we propose a decomposition method which turns the original $n$-dimensional problem into $n$ indipendent scalar problems of lot sizing form. Each scalar problem is then reformulated as a shortest path one and solved through linear programming over a receding horizon. This last reformulation step mirrors a standard procedure in mixed integer programming. The approximation introduced by the decomposition can be lowered if we operate in accordance with the predictive control technique: i) optimize controls over the horizon ii) apply the first control iii) provide measurement updates of other states and repeat the procedure

On robustness and dynamics in (un)balanced coalitional games

We build upon control theoretic concepts like robustness and dynamics to better accommodate all the situations where the coalitions’ values are uncertain and subject to changes over time. The proposed robust dynamic framework provides an alternative perspective on the study of sequences of coalitional games or interval valued games. For a sequence of coalitional games, either balanced or unbalanced, we analyze the key roles of instantaneous and average games. Instantaneous games are obtained by freezing the coalitions’ values at a given time and come into play when coalitions’ values are known. On the other hand, average games are derived from averaging the coalitions’ values up to a given time and are key part of our analysis when coalitions’ values are unknown. The main theoretical contribution of our paper is a design method of allocation rules that return solutions in the core and/or $\epsilon$-core of the instantaneous and average games. Theoretical results are then specialized to a simulated example to shed light on the impact of the design method and on the performance of the resulting allocation rules

Finite Alphabet Control of Logistic Networks with Discrete Uncertainty

We consider logistic networks in which the control and disturbance inputs take values in finite sets. We derive a necessary and sufficient condition for the existence of robustly control invariant (hyperbox) sets. We show that a stronger version of this condition is sufficient to guarantee robust global attractivity, and we construct a counterexample demonstrating that it is not necessary. Being constructive, our proofs of sufficiency allow us to extract the corresponding robust control laws and to establish the invariance of certain sets. Finally, we highlight parallels between our results and existing results in the literature, and we conclude our study with two simple illustrative examples

Robust Dynamic Cooperative Games

Classical cooperative game theory is no longer a suitable tool for those situations where the values of coalitions are not known with certainty. Recent works address situations where the values of coalitions are modelled by random variables. In this work we still consider the values of coalitions as uncertain, but model them as unknown but bounded disturbances. We do not focus on solving a specific game, but rather consider a family of games described by a polyhedron: each point in the polyhedron is a vector of coalitions’ values and corresponds to a specific game. We consider a dynamic context where while we know with certainty the average value of each coalition on the long run, at each time such a value is unknown and fluctuates within the bounded polyhedron. Then, it makes sense to define “robust” allocation rules, i.e., allocation rules that bound, within a pre- defined threshold, a so-called complaint vector while guaranteeing a certain average (over time) allocation vector. We also present as motivating example a joint replenishment application

Consensus for switched networks with unknown but bounded disturbances

We consider stationary consensus protocols for networks of dynamic agents with switching topologies. The measure of the neighbors' state is affected by Unknown But Bounded disturbances. Here the main contribution is the formulation and solution of what we call the $\epsilon$-consensus problem, where the states are required to converge in a tube of ray $\epsilon$ asymptotically or in finite time.Comment: 18 pages, 3 figures. The manuscript has been submitted for the Special issue on Control and optimization in Cooperative Networks. Submitted to SIAM SICO

Opinion Dynamics in Social Networks through Mean-Field Games

Emulation, mimicry, and herding behaviors are phenomena that are observed when multiple social groups interact. To study such phenomena, we consider in this paper a large population of homogeneous social networks. Each such network is characterized by a vector state, a vector-valued controlled input and a vector-valued exogenous disturbance. The controlled input of each network is to align its state to the mean distribution of other networks’ states in spite of the actions of the disturbance. One of the contributions of this paper is a detailed analysis of the resulting mean field game for the cases of both polytopic and L2 bounds on controls and disturbances. A second contribution is the establishment of a robust mean-field equilibrium, that is, a solution including the worst-case value function, the state feedback best-responses for the controlled inputs and worst-case disturbances, and a density evolution. This solution is characterized by the property that no player can benefit from a unilateral deviation even in the presence of the disturbance. As a third contribution, microscopic and macroscopic analyses are carried out to show convergence properties of the population distribution using stochastic stability theory

Dynamic demand and mean-field games

Within the realm of smart buildings and smart cities, dynamic response management is playing an ever-increasing role thus attracting the attention of scientists from different disciplines. Dynamic demand response management involves a set of operations aiming at decentralizing the control of loads in large and complex power networks. Each single appliance is fully responsive and readjusts its energy demand to the overall network load. A main issue is related to mains frequency oscillations resulting from an unbalance between supply and demand. In a nutshell, this paper contributes to the topic by equipping each signal consumer with strategic insight. In particular, we highlight three main contributions and a few other minor contributions. First, we design a mean-field game for a population of thermostatically controlled loads (TCLs), study the mean-field equilibrium for the deterministic mean-field game and investigate on asymptotic stability for the microscopic dynamics. Second, we extend the analysis and design to uncertain models which involve both stochastic or deterministic disturbances. This leads to robust mean-field equilibrium strategies guaranteeing stochastic and worst-case stability, respectively. Minor contributions involve the use of stochastic control strategies rather than deterministic, and some numerical studies illustrating the efficacy of the proposed strategies