In this paper, we consider the matrix-weighted consensus problems with
disturbances. To this end, we firstly propose a new disturbance observer
design for systems with unknown matched or mismatched disturbances
representable as a linear combination of time-varying basis functions.
Under some assumptions on the boundedness and persistent excitation of
the regression matrix, the disturbances can be precisely estimated at an
exponential rate and thus, can be compensated by a suitable compensation
input. Next, disturbance-observer based consensus algorithms are
proposed for matrix-weighted networks of single- and double-integrators
with matched or mismatched disturbances. We show that both matched and
mismatched disturbances can be estimated and actively compensated, and
the system globally asymptotically converges to a fixed point in the
kernel of the matrix-weighted Laplacian. Depending on the network
connectivity, the system can asymptotically achieve a consensus or a
cluster configuration. The disturbance-observer based consensus design
is further extended for higher-order integrators with disturbances.
Finally, simulation results are provided to support the mathematical
analysis.