Figure 3
On the far left, a vertical rectangular column labeled “x subscript k” contains three circular nodes arranged vertically inside a narrow red-outlined box. From this column, multiple diagonal lines fan out to the right into a larger rectangular block labeled “Encoder”, which is outlined in green. Inside the encoder, eight circular nodes are arranged vertically, with dense diagonal connections forming a layered neural network pattern. From the right side of the encoder, connections lead into a narrow vertical column of five circular nodes labeled “z subscript k”, enclosed in a light yellow box. From this column, a horizontal arrow extends to the right into a square block labeled “A of (theta subscript A)”, which contains a grid pattern of small squares. A line exits the right side of this block and curves downward toward a circular summation node marked with a plus symbol. Below the “z subscript k” column, a second narrow vertical column labeled “u subscript k” contains four circular nodes inside a blue-outlined box. A horizontal arrow extends from this column into another square block labeled “B of (theta subscript B)”, which also contains a grid pattern. A line exits this block and curves upward into the same circular summation node. From the summation node, a horizontal arrow extends to the right into another narrow vertical column labeled “z subscript k plus 1”, again shown as stacked five circular nodes inside a light yellow box. From this column, multiple diagonal connections spread into a second large green-outlined rectangular block labeled “Decoder”, which mirrors the encoder structure with eight circular nodes and dense interconnections. From the right side of the decoder, connections converge into a final narrow vertical column labeled “x subscript k plus 1”, containing three circular nodes inside a red-outlined box.Architecture of the proposed DK-MPC framework. The encoder maps the quadrotor state xk into a latent space representation zk. The dynamics in this latent space are modeled linearly using matrices A(θA) and B(θB), which evolve the system according to the control input uk. The decoder then reconstructs the next state xk+1 from the predicted latent state zk+1. This integration of neural networks with Koopman-based linear dynamics enables accurate forward prediction and serves as the foundation for the DK-MPC controller