Derivation of integrated state equation for combined outputs-inputs vector of discrete-time linear time-invariant system and its application to reinforcement learning

For a discrete-time linear time-invariant partially observable system that satisfies the well-known algebraic condition of observability, we derive a fully observable system equation using an augmented state vector combining output and input sequences. The derived representation enables application of reinforcement learning methods for fully observable linear quadratic regulation problems to partially observable ones avoiding the problems invoked in existing methods.