Robots with flexible spines based on tensegrity structures have potential advantages over traditional designs with rigid torsos. However, these robots can be difficult to control due to their high-dimensional nonlinear dynamics. To overcome these issues, this work presents two controllers for tensegrity spine robots, using model-predictive control (MPC), and demonstrates the first closed-loop control of such structures. The first of the two controllers is formulated using only state tracking with smoothing constraints. The second controller, newly introduced in this work, tracks both state and input reference trajectories without smoothing. The reference input trajectory is calculated using a rigid-body reformulation of the inverse kinematics of tensegrity structures, and introduces the first feasible solutions to the problem for certain tensegrity topologies. This second controller significantly reduces the number of parameters involved in designing the control system, making the task much easier. The controllers are simulated with 2D and 3D models of a particular tensegrity spine, designed for use as the backbone of a quadruped robot. These simulations illustrate the different benefits of the higher performance of the smoothing controller versus the lower tuning complexity of the more general input-tracking formulation. Both controllers show noise insensitivity and low tracking error, and can be used for different control goals. The reference input tracking controller is also simulated against an additional model of a similar robot, thereby demonstrating its generality.