Abstract
In this paper the discrete-time system is treated. And it is discussed the realization problem of almost-linear systems to be the special sub-class of pseudo-linear systems. Almost-linear systems are the pseudo-linear systems whose state transition equation are represented as the sum of free motion and linear action by input, where the one of pseudo-linear systems are represented as sum of free-motion and non-linear action by input.
The established realization theorem of pseudo-linear systems being based on, the realization problem of almost-linear systems is discussed.
Generally speaking, the realization problem is stated as follows. Let I/O be the set of input/output map to be a black-box and CD be the category of dynamical systems which have the same behavior as the given black-box, it is intended to obtain the following realization theorem. Realization theorem: ⌈For any given input/output map α∈I/O, there exist at least one dynamical systems σ∈CD which have the behavior α. Let any σ1, σ2∈CD have the same behavior, then σ1 is isomorphic to σ2 in sense of category CD.⌋
In this paper, let I/O be the set of any input/output map with causality, time-invarince and affinity (this map is called a time-invariant, affine input-response map) and let CD be the category of canonical almost-linear systems, then the realization theorem of almost-linear systems is obtained.
Modified impulse response being introduced, it is shown that any time-invariant, affine input-response map can be characterized by it. “So-called” linear systems can be given as specific examples of almost-linear systems. Where “so-called” linear systems mean linear systems with a non-zero initial state.
