Small-controllability of state-affine nonlinear systems

This paper is concerned with the small-controllability problem of state-affine nonlinear systems, where the control inputs can be sufficiently small. By proposing the characteristic equation, a necessary condition for small-controllability of discrete-time state-affine nonlinear systems is first given, which covers the classical results for bilinear systems. The proposed characteristic equation is also generalized to continuous-time state-affine nonlinear systems. Then, two classes of sufficient conditions for small-controllability of discrete-time state-affine nonlinear systems are formulated by algebraizing the local rank condition. In this way, not only the classical sufficient results applied for the degenerate systems are improved, but also the scope is extended to more general state-affine nonlinear systems. Finally, the sufficient conditions are generalized to the multi-input case and the continuous-time case. Examples are given to illustrate the small-controllability results of this paper.