This article presents an input-output simulation approach to controlling multi-affine systems for linear temporal logic (LTL) specifications, which consists of the following steps. First, the state space is partitioned into rectangles, each of which satisfies atomic LTL propositions. Then, we study the control of multi-affine systems on rectangles, including the control based on the exit sub-region to drive all trajectories starting from a rectangle to exit through a facet and the control to stabilise the multi-affine system towards a desired point. With the proposed controllers, a finitely abstracted transition system is constructed which is shown to be input-output simulated by the rectangular transition system of the multi-affine system. Since the input-output simulation preserves LTL properties, the controller synthesis of the multi-affine system for LTL specifications is achieved by designing a nonblocking supervisor for the abstracted transition system and by implementing the resulting supervisor to the original multi-affine system.