On the LQG problem with correlated noise and its relation to minimum variance control

The linear quadratic Gaussian (LQG) stochastic control problem with correlated dynamic and observation noise and no information delay is studied. An explicit feedback solution is given for finite as well as infinite time problems. These results are then applied to minimum variance control of single-input single-output ARMAX systems. The LQG controller and the minimum variance controller obtained using input-output methods are shown to be identical for any system delay, filling an apparent gap in linear stochastic control theory.