In this chapter, we will discuss approximate and noisy realization problems of affine dynamical systems, which realize any input response map, equivalently, as an input/output map with causality. Affine dynamical systems were proposed and the realization problem of the systems were solved in the reference [Matsuo & Hasegawa, 2003]. We characterized the finite-dimensionality of affine dynamical systems. We obtained the same results as ones established in linear system theory.
A criterion for canonical finite-dimensional affine dynamical systems was given. There uniquely exists a quasi-reachable standard system in the isomorphic class of finite-dimensional canonical affine dynamical systems. We obtained a criterion for the behavior of finite-dimensional affine dynamical systems. We also gave a procedure on how to obtain the quasi-reachable standard system from an input response map.
For our discussion of approximate and noisy realization problems of affine dynamical systems, we will need a partial realization algorithm of the systems which was not stated in the reference. We will obtain a partial realization algorithm in section 7.3. The proofs about the results of the algorithm will be stated in the appendix of this chapter.