Robust control of uncertain multi-inventory systems via linear matrix inequality

We consider a continuous time linear multi-inventory system with unknown demands bounded within ellipsoids and controls bounded within ellipsoids or polytopes. We address the problem of ε-stabilising the inventory since this implies some reduction of the inventory costs. The main results are certain conditions under which ε-stabilisability is possible through a saturated linear state feedback control. All the results are based on a linear matrix inequalities approach and on some recent techniques for the modelling and analysis of polytopic systems with saturations. Numerical simulations are provided.     

Robust control strategies for multi-inventory systems with average flow constraints

In this paper, we consider multi-inventory systems in the presence of uncertain demand. We assume that (i) demand is unknown but bounded in an assigned compact set and (ii) the control inputs (controlled flows) are subject to assigned constraints. Given a long-term average demand, we select a nominal flow that feeds such a demand. In this context, we are interested in a control strategy that meets at each time all possible current demands and achieves the nominal flow in the average. We provide necessary and sufficient conditions for such a strategy to exist and we characterize the set of achievable flows. Such conditions are based on linear programming and thus they are constructive. In the special case of a static flow (i.e. a system with 0-capacity buffers) we show that the strategy must be affine. The dynamic problem can be solved by a linear-saturated control strategy (inspired by the previous one). We provide numerical analysis and illustrative examples.     

Non-linear protocols for optimal distributed consensus in networks of dynamic agents

We consider stationary consensus protocols for networks of dynamic agents with fixed topologies. At each time instant, each agent knows only its and its neighbors’ state, but must reach consensus on a group decision value that is function of all the agents’ initial state. We show that the agents can reach consensus if the value of such a function is time-invariant when computed over the agents’ state trajectories. We use this basic result to introduce a non-linear protocol design rule allowing consensus on a quite general set of values. Such a set includes, e.g., any generalized mean of order p of the agents’ initial states. As a second contribution we show that our protocol design is the solution of individual optimizations performed by the agents. This notion suggests a game theoretic interpretation of consensus problems as mechanism design problems. Under this perspective a supervisor entails the agents to reach a consensus by imposing individual objectives. We prove that such objectives can be chosen so that rational agents have a unique optimal protocol, and asymptotically reach consensus on a desired group decision value. We use a Lyapunov approach to prove that the asymptotical consensus can be reached when the communication links between nearby agents define a time-invariant undirected network. Finally we perform a simulation study concerning the vertical alignment maneuver of a team of unmanned air vehicles.     

Optimization of Long-Run Average-Flow Cost in Networks With Time-Varying Unknown Demand

We consider continuous-time robust network flows with capacity constraints and unknown but bounded time-varying demand. The problem of interest is to design a control strategy off-line with no knowledge of the demand realization. Such a control strategy regulates the flow on-line as a function of the realized demand.      

Consensus in Noncooperative Dynamic Games: A Multiretailer Inventory Application

We focus on Nash equilibria and Pareto optimal Nash equilibria for a finite horizon noncooperative dynamic game with a special structure of the stage cost. We study the existence of these solutions by proving that the game is a potential game. For the single-stage version of the game, we characterize the aforementioned solutions and derive a consensus protocol that makes the players converge to the unique Pareto optimal Nash equilibrium. Such an equilibrium guarantees the interests of the players and is also social optimal in the set of Nash equilibria. For the multistage version of the game, we present an algorithm that converges to Nash equilibria, unfortunately, not necessarily Pareto optimal. The algorithm returns a sequence of joint decisions, each one obtained from the previous one by an unilateral improvement on the part of a single player. We also specialize the game to a multiretailer inventory system.     

Boolean-controlled systems via receding horizon and linear programing

We consider dynamic systems controlled by boolean signals or decisions. We show that in a number of cases, the receding horizon formulation of the control problem can be solved via linear programing by relaxing the binary constraints on the control. The idea behind our approach is conceptually easy: a feasible control can be forced by imposing that the boolean signal is set to one at least one time over the horizon. We translate this idea into constraints on the controls and analyze the polyhedron of all feasible controls. We specialize the approach to the stabilizability of switched and impulsively controlled systems. This work was supported by MURST-PRIN 2007ZMZK5T “Decisional model for the design and the management of logistics networks characterized by high interoperability and information integration”.