Sparsity Promotion in State Feedback Controller Design

A globally convergent algorithm for synthesis of sparse optimal state feedback (SOSF) control of linear time-invariant (LTI) systems is proposed. This problem is known to be NP-hard due to its combinatorial nature. A structured H-2 norm controller design problem is intrinsically non-convex, even if a fixed structure is known in advance. The proposed algorithm minimizes the H-2 norm performance index, and simultaneously regularizes the sparsity of the control structure using the l(1) norm. It guarantees that the solution converges to a stationary point of the original problem. The algorithm is implemented using Linear Matrix Inequalities (LMIs) which are efficient, reliable and extendable to other control problems. The performance of the proposed algorithm is illustrated with application to several examples. The results are also compared with Alternating Direction Method of Multiplier (ADMM) for sparse LQR design (as a special case of the H-2 norm problem). In most instances, the appropriate selection of the control structure leads to a near optimal performance index with a substantially reduced number of communication links.