An observations-weighted (OW) controller is presented for linear discrete-time stochastic single-input single-output systems. A dual criterion to be minimized includes the usual OW-type cost function, but has in addition the terms of the sensitivity and complementary sensitivity weightings. A stabilizing controller is obtained by using the polynomial equation approach. The dual criterion OW controller proposed here is shown to be a usual OW controller with perturbed weighting functions for the tracking error and control signals. Furthermore, it is asserted that the resulting controller is useful for improving the robustness of the usual OW controller and is very simple to implement. Thus, the controller may be suitable for use in robust self-tuning control systems.