Realization of discrete-time internally bilinear systems

From the external descriptions algorithms are obtained for the construction of discrete-time, internally bilinear state-space models with unknown initial states. Both the homogeneous and the inhomogeneous bilinear systems are treated. In both cases explicit existence criteria for minimal realization as well as the state space isomorphism theorems are given. One unusual result is that for a given input/output map there exist minimal realizations which are not equivalent under basis transformations. In the special case of zero initial state the algorithm can be reduced to the one previously obtained by Isidori, and if the input-output map is linear, the algorithm reduces to that given by Ho and Kalman for linear systems. Finally, the abstract realization theory is given. We use the affine tensor product as a basic tool to describe both homogeneous and inhomogeneous bilinear systems. The state space is constructed via Nerode equivalence. This forms a theoretical basis for the realization algorithms obtained in this paper.