On necessary and sufficient conditions of the consensusability for second-order discrete multi-agent systems

In this paper, necessary and sufficient conditions of the consensusability are proposed for second-order discrete multi-agent systems without assuming that all the eigenvalues of each agent´s dynamical equation lie on or outside the unit circle. With these conditions, all the possible control gains realizing the consensus are obtained. The obtained results show that, if each agent´s dynamical equation has a stable eigenvalue, the condition Π_j|λ_j^u(A)| <; (1 + r)/(1-r) in [1] is not necessary anymore, where λ_j^u(A) denotes each unstable eigenvalue of the coefficient matrix A of every agent and r the eigenratio of the graph. For some numerical examples not satisfying the above condition, consensus protocols are designed and some simulations show the correctness of the obtained results.