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As a new type of output feedback control for a stochastic discrete-time state-space system, a finite memory control (FMC) is proposed and its properties are investigated. Instead of using an internal state involved with old data, the FMC is required to utilise inputs and outputs on the recent finite time together with the future reference signals, and minimise a receding horizon quadratic performance criterion. As the linear quadratic Gaussian (LQG) controls with infinite memory can be separated into the linear quadratic (LQ) controls and the Kalman filters, the proposed FMC consists of a receding horizon control (RHC) for state feedback control and a minimum variance finite impulse response (MVFIR) filter for state estimation. It is shown that the closed-loop poles are made up of those involved with the RHC and the rest of zeros. The stability is also shown to be guaranteed under some condition. A numerical example using a two-dimensional free-body system is given to illustrate the performance of the proposed control.