  This paper studies a class of distributed online convex optimization problems
for heterogeneous linear multi-agent systems. Agents in a network, knowing only
their own outputs, need to minimize the time-varying costs through neighboring
interaction subject to time-varying coupled inequality constraints. Based on
the saddle-point technique, we design a continuous-time distributed controller
which is shown to achieve constant regret bound and sublinear fit bound,
matching those of the standard centralized online method. We further extend the
control law to the event-triggered communication mechanism and show that the
constant regret bound and sublinear fit bound are still achieved while reducing
the communication frequency. Additionally, we study the situation of
communication noise, i.e., the agent's measurement of the relative states of
its neighbors is disturbed by a noise. It is shown that, if the noise is not
excessive, the regret and fit bounds are unaffected, which indicates the
controller's noise-tolerance capability to some extent. Finally, a numerical
simulation is provided to support the theoretical conclusions.
