The multivariable generalized predictive control law is derived in a stochastic setting which closely follows the usual polynomial solution of the LQG problem. A constraint is applied that both the control and future predicted control are calculated using the same time-invariant controller. The conditions which must apply for a time-invariant controller to exist are discussed. The assumptions which must be made to ensure a GPC type of control law emerges are also clear from the analysis, which provides insights into the relationship between LQG and GPC control. The form of the two degree-of-freedom control law obtained closely parallels the GPC solution but ensures closed-loop stability. The solution of the multi-step cost-function problem is shown to provide an LQG type of control action and a set of predicted future controls and outputs. This is of importance when input and output constraints are to be applied based on the future predicted signals. The control signal can be found using the vector-matrix form of the solution and it is very similar to the GPC form of controllers. However, the stability properties for the feedback loop are much improved, relative to the traditional GPC or LRPC algorithms.