Controllability and near-controllability of a class of discrete-time inhomogeneous bilinear systems

In this paper, a class of discrete-time inhomogeneous bilinear systems whose controllability cannot be checked by the classical results is considered. By using an algebraic technique, it is shown that the systems are never controllable. Near-controllability of the systems is thus studied and, by defining a novel transition operator, an algebraic necessary and sufficient criterion is established, where the control inputs to achieve state transition can also be computed. Furthermore, the established near-controllability result is applied to study nearly controllable subspaces of the discrete-time homogeneous bilinear systems. Although such systems were investigated in the previous work, a new and larger class of nearly controllable subspaces of the homogeneous systems is derived, which cannot be found by the previous work due to different algebraic structures. Finally, an algorithm for solving the control inputs to achieve state transition as well as examples are given to demonstrate the results of this paper.