On the Near Controllability of Single-Input Rank-One Bilinear Systems

In this exposition, we study the controllability of discrete-time bilinear systems. It is well known that in the case of rank-one bilinear systems, bilinear controllability requires the controllability and observability of its associated linear system. A third criterion, known as the greatest common divisor (GCD) condition, is also necessary to ensure controllability. In our recent work we showed that this condition can be relaxed, resulting in a nearly controllable bilinear system. Here we present a new constructive proof, capturing the interplay between the pseudo-inputs (inputs of the associated linear system) and the original inputs driving the bilinear system. This constructive proof provides insights to derive an algorithm to design an input sequence for the original bilinear system, based on the sequence of pseudo-inputs of the associated linear system.