Completeness and other properties of input-output based finite approximations

We consider discrete-time plants that interact with their controllers via fixed discrete alphabets in the absence of exogenous inputs. For this class of systems, we revisit a general procedure for constructing a sequence of finite state approximate models starting from finite length sequences of input and output signal pairs. After providing intuition for the proposed construction, we show that it satisfies the desired properties of ρ/μ approximations. We then propose a readily verifiable sufficient condition for ensuring finiteness of the approximation error gain. Finally, we show that the proposed construct satisfies an interesting 'completeness' property, making it attractive as a medium for studying fundamental limitations of finite state approximation and state observation under coarse sensing.