Verification of consensus in networks of heterogeneous LTI agents

A framework for robust stability of large scale systems consisting of linear time-invariant systems interconnected over a network will be surveyed and applied to solve heterogeneous consensus problems. The purpose of distributed consensus algorithms is to reach an agreement regarding a certain quantity of interest that depends on the state of all systems. In most of consensus literature, dynamics of all agents in the network are assumed to be the same and of low dimensions. This simplifies the analysis and, for the most elementary networks, stability and the rate of convergence can be determined from the eigenvalues of the interconnection matrix. Here we discuss analogous results for the case where the individual dynamics are heterogeneous and possibly of infinite dimensions. Our criterion for consensus resembles the classical Nyquist criterion. An interesting aspect of this criterion is that in some instances a three-dimensional plot is required in order to make analysis accurate.