  In this paper, a distributed optimal steady-state regulation problem is
formulated and investigated for heterogeneous linear multi-agent systems
subject to external disturbances. We aim to steer this high-order multi-agent
network to a prescribed steady-state determined as the optimal solution of a
resource allocation problem in a distributed way. To solve this problem, we
employ an embedded control design and convert the formulated problem to two
simpler subproblems. Then, both state-feedback and output feedback controls are
presented under mild assumptions to solve this problem with disturbance
rejection. Moreover, we extend these results to the case with only real-time
gradient information by high-gain control techniques. Finally, numerical
simulations verify their effectiveness.
