Products of Generalized Stochastic Matrices With Applications to Consensus Analysis in Networks of Multiagents With Delays

Product theory of stochastic matrices provides a powerful tool in the consensus analysis of discrete-time multiagent systems. However, the classic theory cannot deal with networks with general coupling coefficients involving negative ones, which have been discussed only in very few papers due to the technicalities involved. Motivated by these works, here we developed some new results for the products of matrices which generalize that of the classical stochastic matrices by admitting negative entries. Particularly, we obtained a generalized version of the classic Hajnal inequality on this generalized matrix class. Based on these results, we proved some convergence results for a class of discrete-time consensus algorithms with time-varying delays and general coupling coefficients. At last, these results were applied to the analysis of a class of continuous-time consensus algorithms with discrete-time controller updates in the existence of communication/actuation delays.