  Rate splitting (RS) systems can better deal with imperfect channel state
information at the transmitter (CSIT) than conventional approaches. However,
this requires an appropriate power allocation that often has a high
computational complexity, which might be inadequate for practical and large
systems. To this end, adaptive power allocation techniques can provide good
performance with low computational cost. This work presents novel robust and
adaptive power allocation technique for RS-based multiuser multiple-input
multiple-output (MU-MIMO) systems. In particular, we develop a robust adaptive
power allocation based on stochastic gradient learning and the minimization of
the mean-square error between the transmitted symbols of the RS system and the
received signal. The proposed robust power allocation strategy incorporates
knowledge of the variance of the channel errors to deal with imperfect CSIT and
adjust power levels in the presence of uncertainty. An analysis of the
convexity and stability of the proposed power allocation algorithms is
provided, together with a study of their computational complexity and
theoretical bounds relating the power allocation strategies. Numerical results
show that the sum-rate of an RS system with adaptive power allocation
outperforms RS and conventional MU-MIMO systems under imperfect CSIT.
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