A Linear Programming Approach to the Minimum Cost Sparsest Input Selection for Structured Systems

In this paper, we consider three related cost-sparsity induced optimal input selection problems for structural controllability using a unifying linear programming (LP) framework. More precisely, given an autonomous system and a constrained input configuration where whether an input can directly actuate a state variable, as well as the corresponding (possibly different) cost, is prescribed, the problems are, respectively, selecting the minimum number of input links, selecting the minimum cost of input links, and selecting the input links with the cost as small as possible while their cardinality is not exceeding a prescribed number, all to ensure structural controllability of the resulting systems. Current studies show that in the dedicated input case (i.e., each input can actuate only a state variable), the first and second problems are polynomially solvable by some graphtheoretic algorithms, while the general nontrivial constrained case is largely unexploited. In this paper, we formulate these problems as equivalent integer linear programming (ILP) problems. Under a weaker constraint on the prescribed input configurations than most of the currently known ones with which the first two problems are reportedly polynomially solvable, we show these ILPs can be solved by simply removing the integer constraints and solving the corresponding LP relaxations, thus providing a unifying algebraic method, rather than graph-theoretic, for these problems with polynomial time complexity. The key to our approach is the observation that the respective constraint matrices of the ILPs are totally unimodular.