Figure 1
On the far left, a rectangular block labeled “Encoder” is shown. A horizontal arrow labeled “x subscript r” enters this block from the left. Inside the encoder, a neural network pattern appears with multiple small circular nodes arranged in vertical layers and connected by diagonal lines, with colored vertical side panels. From this encoder, a horizontal arrow labeled “z subscript r” exits to the right and enters a large rounded rectangular block labeled “D L-Koopman M P C”. Within the “D L-Koopman M P C” block, a rectangular block labeled “Model Predictive Control (M P C)” is positioned near the bottom center. Above and slightly to the right, a rounded rectangular box labeled “Linear Lifted Model” is placed. Inside this lifted model, a circular summation node with a plus symbol is located toward the upper-left area. Two rectangular blocks labeled “A of (theta)” positioned to the right and “B of (theta)” positioned below are connected with solid arrows into the summation node. A horizontal arrow labeled “z subscript k” enters the lifted model from the right. From the summation node, a dashed arrow labeled “z subscript k plus 1” extends downward into the M P C block. Another dashed arrow labeled “u bar subscript k” extends upward from the M P C block into the “B of (theta)” block. From the bottom of the M P C block, a vertical arrow labeled “u asterisk” exits downward from the “D L-Koopman M P C” block to a rectangular image labeled “Asc Tec Pelican Quadrotor”, showing a central body with four extended arms arranged in a cross pattern. To the right of the quadrotor, a second rectangular block labeled “Encoder” is shown with a similar neural network pattern of layered circular nodes and connecting lines. A horizontal arrow labeled “x subscript k” enters this encoder from the left. From this encoder, a vertical line extends upward along the right side, curves across the top, and connects back into the lifted model at the input labeled “z subscript k”. Another downward arrow from the encoder loops back to the left and upward to the M P C block.Schematic of the proposed DK-MPC framework for controlling the AscTec Pelcian quadrotor. The deep Koopman operator projects both the reference state xref and the current state x into a high-dimensional latent space where the system exhibits linear behavior. Using these latent representations and the corresponding linear dynamics, the MPC controller computes optimal control inputs u* to guide the quadrotor along the desired trajectory