Unlike earlier work, this paper identifies parts of the state space where it is possible to invoke feedback linearization and uses the closed-loop paradigm of Kouvaritakis et al. (2000) to accommodate input constraints while utilizing an extremely efficient online receding horizon strategy. The key to this development is a partial invariance property that is derived through the use of linear difference inclusion made possible due to the bilinear nature of the models considered here. For all other parts of the state space the proposed algorithm switches to a bilinear controller which is designed to give invariance and feasibility over a low complexity polytope. Without increasing computational complexity, the use of bilinear controllers affords extra degrees of freedom with which to maximize the region of attraction of the algorithm. The improvements in terms of this and in terms of closed loop output performance are shown by means of a numerical example to be very significant.