Journées de l'optimisation 2009 Optimizations days

Energy-efficient and time-efficient reconfiguration strategies for formation flying of autonomous agents are presented. It is assumed that a finite set of possible formations is given, and that the probability of each formation in this set is known a priori. The idea is to move the agents to floating stations in the idle time, i.e. the time between the accomplishment of the last reconfiguration task and the issuance of the next reconfiguration command, to minimize the expected value of the energy consumption or reconfiguration time. In the energy-efficient strategy, the position of each floating station is derived as a function of the agent’s current position, and the weighted center of gravity of the set of possible positions for that agent. In the time-efficient strategy, on the other hand, it turns out that the problem of finding the position of the corresponding floating stations is non-convex. To address this issue, a method is provided to reduce the global minimum search space to a convex compact set. Consequently, a numerical procedure for finding an arbitrarily precise global minimum is proposed in this case. The effectiveness of the proposed strategies is illustrated via simulation.

We consider linear-quadratic-Gaussian (LQG) games with a major player and a large number of minor players. The minor players individually have negligible influence, but they collectively contribute mean field coupling terms in the individual dynamics and costs. We examine the major player's impact upon the mean field effect, and apply the Nash certainty equivalence approach for the design of decentralized asymptotic Nash strategies for all players. This is accomplished by using state space augmentation and identifying a certain interaction consistency relationship.

In this paper we develop an attitude synchronization algorithm for spacecraft formation flying. We are particularly interested to develop a control algorithm which enables (re)joining spacecraft to the formation (evolving network). In addition, it is required that the control algorithm considers the input saturation constraints, which could be present due to failures of attitude actuators. Unlike classical leader-follower approach, the proposed control algorithm enables existence and cooperation of more than one leader in the formation, which can communicate with the ground station and the remaining spacecraft in the formation. Stability of the spacecraft formation is analyzed by using Lyapunov's method and LaSalle's Invariance Principle. Simulation results are reported to demonstrate stability and performance of the proposed control algorithm.

We study the stochastic consensus problem in a large population system by using the so-called Nash Certainty Equivalence (NCE) Principle. The NCE methodology has been studied for dynamic games in a large population of stochastic agents. It specifies a family of consistent individual decentralized control laws which depend upon each agent’s state and the aggregate effect of other agents. In this paper, we formalize the large population stochastic consensus problem as a dynamic game problem in which the agents have similar dynamics and are coupled via their individual costs. Then for large but finite populations, we derive the standard consensus algorithm dynamics from mean field stochastic control NCE equations. While the NCE formulation requires a priori information on the mean values of the initial states, the standard consensus algorithms need no priori information, but instead use local state feedback on the network’s graph. Subject respectively to cost weighing and connectivity assumptions, both schemes converge to the same limits. Furthermore, in the NCE-Consensus formulation, each agent’s behavior is optimal with respect to other agents in a game theoretic Nash sense.

The computational intractability of the dynamic programming (DP) equations associated with optimal admission and routing in stochastic loss networks of any non-trivial size (Ma et al, 2006, 2008) leads one to consider suboptimal distributed game theoretic formulations of the problem. The special class of radial networks with a central core of infinite capacity is considered, and it is shown (under adequate assumptions) that an associated distributed admission control problem formulated as a game becomes tractable asymptotically, as the size of radial network grows to infinity. This is achieved by following a methodology first explored by M. Huang et. al. (2003, 2006-2008) in the context of large scale dynamic games for sets of weakly coupled linear stochastic control systems. At the established Nash equilibrium, each agent network reacts optimally with respect to a pair of Poisson processes (incoming and outgoing) processes, asymptotically capturing the influence of the mass on that agent. The rates of these mass Poisson process are approximated by deterministic infinite population limits (associated with the mean field or ensemble statistics of the random agents) which are solutions of a particular fixed point problem. An algorithm is presented for computing a (possibly non unique) Nash equilibrium and corresponding numerical experiments are reported.