  In this work we study a distributed optimal output consensus problem for
heterogeneous linear multi-agent systems where the agents aim to reach
consensus with the purpose of minimizing the sum of private convex costs. Based
on output feedback, a fully distributed control law is proposed by using the
proportional-integral (PI) control technique. For strongly convex cost
functions with Lipschitz gradients, the designed controller can achieve
convergence exponentially in an undirected and connected network. Furthermore,
to remove the requirement of continuous communications, the proposed control
law is then extended to periodic and event-triggered communication schemes,
which also achieve convergence exponentially. Two simulation examples are given
to verify the proposed control algorithms.
