Stabilisation of discrete-time bilinear systems by constant control inputs

In this paper, the stabilisation problem of discrete-time bilinear systems by using constant control inputs is considered. It is shown that the spectrum of a matrix derived from the system matrices plays a key role in the stabilisation design. Sufficient conditions for the systems to be stabilizable by constant control inputs are presented in the cases when the derived matrix has no complex eigenvalue as well as in the cases when it has complex eigenvalues. Particularly, it is proved that, if the derived matrix has only nonzero real eigenvalues, then the systems can always be stabilised by constant control inputs. Finally, simulations are given to demonstrate the obtained results.