A Distributed Linear-Quadratic Discrete-Time Game Approach to Multi-Agent Consensus

In this paper we propose a distributed game theoretic approach to the multi-agent consensus problem, where we represent the consensus problem with multiple decision-makers as a Linear Quadratic Discrete-Time Game (LQDTG) over a connected communication graph. The players - taken here as double integrators - minimize individual finite-horizon cost functions (which depend on decisions by other players) and aim at reaching a Nash equilibrium. This involves a set of coupled Riccati difference equations, which cannot be solved in a distributed manner. Here we propose a distributed strategy for reaching a Nash equilibrium that is based on an associated multi-agent system evolving not on the vertices but on the edges of the underlying communication graph. This makes it possible to move the coupling between agents from the cost function to the agent dynamics, and to formulate an LQR problem with a decoupled cost that allows a distributed solution on the edges of the graph. Mapping this solution to local control inputs for the real agents (the nodes of the graph) requires however knowledge of the whole network. To arrive at a distributed control strategy, we propose an iterative, gradient-based construction of the individual control inputs that is based only on locally available information, but requires repeated exchange of information between neighboring agents within a single sampling interval. The resulting distributed strategy can be implemented in a receding horizon manner, and the optimal first-step state feedback gain matrices for the edge system can be calculated offline and stored by each agent. Consensus will be reached iff the edge system is stable under this state feedback law.