Assembling lattices from discrete building blocks enables the composition of large, heterogeneous, and easily reconfigurable objects with desirable mass-to-stiffness ratios. This type of building system may also be referred to as a digital material, as it is constituted from discrete, error-correcting components. Researchers have demonstrated various active structures and even robotic systems that take advantage of the reconfigurable, mass-efficient properties of discrete lattice structures. However, the existing literature has predominantly used open-loop control strategies, limiting the performance of the presented systems. In this paper, we present a novel approach to feedback control of digital lattice structures, leveraging real-time measurements of the system dynamics. We introduce an actuated voxel which constitutes a novel means for actuation of lattice structures. Our control method is based on the Extended Dynamical Mode Decomposition algorithm in conjunction with the Linear Quadratic Regulator and the Koopman Model Predictive Control. The key advantage of our approach lies in its purely data-driven nature, without the need for any prior knowledge of a system's structure. We illustrate the developed method via real experiments with custom-built flexible lattice beam, showing its ability to accomplish various tasks even with minimal sensing and actuation resources. In particular, we address two problems: stabilization together with disturbance attenuation, and reference tracking.