Small-controllability of two-dimensional state-affine nonlinear systems

For most nonlinear systems, it is in general a difficult task to obtain algebraic controllability criteria. In this paper, we consider small-controllability of discrete-time state-affine nonlinear systems. We first focus on the systems in dimension two with single-input and improve a previous controllability criterion. That is, we derive a sufficient algebraic criterion for small-controllability of the systems, which is easier to apply than the previous one. We then show that if the state-affine nonlinear systems are bilinear, a necessary and sufficient algebraic criterion for small-controllability can be obtained using invariant sets. We also extend the derived controllability criteria to continuous-time state-affine nonlinear systems and to discrete-time multi-input state-affine nonlinear systems. Examples are given to illustrate the derived controllability criteria of this paper.