Tensegrity structures represent a special class of tendon space structures, whose members may simultaneously perform the functions of strength, sensing, actuating and feedback control. Thus, these structures ideally match the definition of smart structures. This paper introduces the concept of controllable tensegrity as a new class of smart structures capable of large displacement. The kinematics and nonlinear dynamics of one element of this class is derived and analyzed. Pre-stressability conditions are given and a particular equilibrium identified. The equations of motion are then linearized about this equilibrium and linear parametric models generated. These are next used for controller design. For control system design some of the tendons are chosen as actuators and some as sensors and a family of dynamic controllers designed such that the control energy is minimized while requiring output variance constraints to be satisfied. Another family of controllers is designed such that the output variance is minimized while requiring input variance constraints to be satisfied. The performances of these controllers are evaluated.