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Output-Feedback System Level Synthesis via Dynamic Programming

System Level Synthesis (SLS) allows us to construct internally stabilizing controllers for large-scale systems. However, solving large-scale SLS problems is computationally expensive and the state-of-the-art methods consider only state feedback; output feedback poses additional challenges because the constraints are no longer uniquely row or column separable. We exploit the structure of the output-feedback SLS problem by vectorizing the multi-sided matrix multiplications in the SLS optimization constraints, which allows us to reformulate it as a discrete-time control problem and solve using two stages of dynamic programming (DP). Additionally, we derive an approximation algorithm that offers a faster runtime by partially enforcing the constraints, and show that this algorithm offers the same results. DP solves SLS up to $7$ times faster, with an additional 42% to 68% improvement using the approximation algorithm, than a convex program solver, and scales with large state dimensions and finite impulse response horizon.