Minimum Cost Feedback Selection for Arbitrary Pole Placement in Structured Systems

This paper addresses optimal feedback selection for arbitrary pole placement of structured systems when each feedback edge is associated with a cost. Given a structured system and a feedback cost matrix, our aim is to find a feasible feedback matrix of minimum cost that guarantees arbitrary pole placement of the closed-loop structured system. The same problem but with uniform costs has been considered recently, but the claim and the NP-hardness proof there has subtle flaws: we mention this briefly in our paper and then prove the NP-hardness of the feedback selection problem using a reduction from the weighted set cover problem. We next prove the polynomial time constant factor in approximability of the problem by showing that a constant factor approximation for the problem does not exist, unless the weighted set cover problem can also be approximated within a constant factor. We study a subclass of systems whose directed acyclic graph constructed using the strongly connected components of the state digraph is a line graph and the state bipartite graph has a perfect matching. We propose a polynomial time algorithm based on dynamic programming principles for optimal feedback selection on this class of systems. Further, over the same class of systems we relax the perfect matching assumption, and provide a polynomial time 2-optimal solution using a minimum cost perfect matching algorithm.