Geometric criteria for controllability and near-controllability of driftless discrete-time bilinear systems

Geometric approaches have been well developed for both linear and nonlinear systems, which are useful in the analysis and design of control systems. However, geometric approaches have been less considered for discrete-time bilinear systems. In this paper, controllability and near-controllability of driftless discrete-time bilinear systems are dealt with from a geometric point of view. More specifically, geometric characterizations are presented for controllability and near-controllability as well as for controllable subspaces and nearly controllable subspaces of the systems. Differently from the classical geometric approach for linear systems, it is shown that, for the bilinear systems, a controllable subspace has to be determined by two linear subspaces, while a nearly controllable subspace is determined by one linear subspace. As a result, geometric criteria for controllability, near-controllability, controllable subspaces, and nearly controllable subspaces of the systems are respectively derived, which are also applied to multi-input discrete-time bilinear systems and to linear time-invariant systems with a multiplicative perturbation. Examples are given to illustrate the derived geometric criteria.