MEAN FIELD (NCE) FORMULATION OF ESTIMATION BASED LEADER–FOLLOWER COLLECTIVE DYNAMICS

Abstract

AbstractWe consider a leader-follower dynamic games model for large population systems where the agentshave linear stochastic dynamics and are coupled via their quadratic cost functions. The cost of eachleader is based on a tradeoff between moving toward a certain reference trajectory signal and stayingnear a weighted average of the members’ states. Followers react by tracking the weighted average ofthe leaders’ states. We approach this large population game problem by use of the so-called Mean Field(MF), or Nash Certainty Equivalence (NCE), methodology. First, as for the basic MF (NCE) framework,we show that the set of MF (NCE) control laws for leaders-followers possesses an almost sure  N -Nashequilibrium property for a population of size Nwhere  N goes to zero as Ngoes to infinity. Second,we consider the case where the leaders track a convex combination of their overall average and acertain reference trajectory signal which is unknown to the followers. The followers use a maximumlikelihood estimator on a sample of the leaders’ trajectories to identify the member of a given finiteclass of models which is generating the reference trajectory of the leaders. It is shown that subject toreasonable conditions the true reference trajectory model is identified in finite time with probability oneas the leaders’ population goes to infinity. Simulations for different cases are provided to demonstratethe effectiveness of the model.Keywords: Mean field stochastic control theory, leader-follower collective motion, maximum like-lihood ratio.