Approximating Constrained Minimum Cost Input-Output Selection for Generic Arbitrary Pole Placement in Structured Systems

This paper is about minimum cost constrained selection of inputs and outputs in structured systems for generic arbitrary pole placement. The input-output set is constrained in the sense that the set of states that each input can influence and the set of states that each output can sense is pre-specified. Our goal is to optimally select an input-output set that the system has no structurally fixed modes. Polynomial time algorithms do not exist for solving this problem unless P = NP. To this end, we propose an approximation algorithm by splitting the problem in to three sub-problems: a) minimum cost accessibility problem, b) minimum cost sensability problem and c) minimum cost disjoint cycle problem. We prove that problems a) and b) are equivalent to the weighted set cover problem. We also show that problem c) can be solved using a minimum cost perfect matching algorithm. Using these, we give an approximation algorithm which solves the minimum cost generic arbitrary pole placement problem. The proposed algorithm incorporates an approximation algorithm to solve the weighted set cover problem to solve a) and b) and a minimum cost perfect matching algorithm to solve c). Further, we show that the algorithm has polynomial complexity and gives an order optimal O(logn) approximate solution to the minimum cost input-output selection for generic arbitrary pole placement problem, where n denotes the number of states in the system.