In the absence of sparsity constraints, i.e., at , the optimal controller
Localization can be enforced by truncating the optimal centralized controller. This, however, may introduce performance degradation and even instability of the closed-loop system. (Additional information is provided in the Network with 100 unstable nodes example.) Instead of using simple truncation, the alternating direction method of multipliers enforces sparsity in the -minimization step and improves the quadratic performance in the -minimization step. This mechanism of alternating between promoting sparsity and optimizing the quadratic performance plays an important role in striking a balance between these two competing objectives.
For illustration, we use cardinality function, weighted norm, and sum-of-logs as the sparsity-promoting penalty functions. For all three cases, the optimal position and velocity feedback gains become diagonal matrices for large values of . It is noteworthy that the optimal sparse feedback gain, with only of nonzero elements relative to the optimal centralized controller , introduces performance loss of only (compared to ).