Consensusability and Global Optimality of Discrete-Time Linear Multiagent Systems

The consensusability and global optimality problems are solved for the discrete-time linear multiagent system (MAS) with marginally stable and strictly unstable dynamics. A unified framework is proposed by capturing the maximal disc-guaranteed gain margin (GGM) of the discrete-time linear quadratic regulator (LQR). Sufficient and necessary conditions on consensusability are established. Two bounds of the consensus region are derived only in terms of the unstable eigenvalues of the agent' dynamics. For the single-input MAS, by proving that the radius of the consensus region exactly equals the reciprocal of the Mahler measure of the agent' dynamics, we incidentally reveal the relation between the maximal GGM and the intrinsic entropy rate of the system dynamics for single-input discrete-time linear systems. By employing the inverse optimal control approach, it is proved that the globally optimal consensus is achieved, if and only if the associated Laplacian matrix is a simple matrix and all its nonzero eigenvalues can be radially projected into a specific subset of the consensus region. Moreover, the limitation on the eigenvalues vanishes for the marginally stable MAS. As an application of the global optimality, the minimum-energy-distributed consensus control problem is solved for the marginally stable MAS. Finally, a numerical example is given to demonstrate the effectiveness of the obtained results.