A Finite-Time Dual Method for Negotiation between Dynamical Systems

This work presents a distributed algorithm for online negotiations of an optimal control policy between dynamical systems. We consider a network of self-interested agents that must agree upon a common state within a specified finite time. The proposed algorithm exploits the distributed structure of the corresponding dual problem and uses a "shrinking horizon" property to enforce the finite-time constraint. It is shown that this algorithm evolves like a time-varying and linear dynamical system, parameterized by a scalar variable analogous to the step-size rule in subgradient methods. The convergence and performance properties of the system are studied in the context of error systems between the algorithm trajectories and a sequence of centralized optimal control trajectories. This analysis provides a simple linear matrix inequality condition for choosing a proper step-size rule and also gives conditions for when no step-size rule can guarantee uniform convergence of the error systems. These conditions are shown to be functions of communication graph Laplacian eigenvalues and the state and control weights of each agent. We also provide a lower bound on the horizon time that guarantees that the terminal state generated by the algorithm is $\delta$-close to an agreement state. The results are then demonstrated via a few numerical simulations. [PUBLICATION ABSTRACT]