On verification and design of input matrix for robust linear systems: Complexity and polynomially solvable cases

This article deals with the robustness of large-scale structured systems in terms of controllability when subject to failure of links from the inputs to the state variables (i.e., input-links). Firstly, we consider a deletion problem of determining the minimum number of input-links, if removed, lead to a structurally uncontrollable system. This problem is known to be NP-hard. We prove that it remains NP-hard even for strongly connected systems. We develop efficient polynomial time methods to solve this problem optimally/suboptimally under suitable assumptions imposed on the generic rank of the state matrix. The assumptions imposed are often satisfied by a large class of systems. These methods mainly use the notion of Dulmage–Mendelsohn decomposition of bipartite graphs and minimum vertex cover problem for undirected graphs. Secondly, we consider an addition problem whose goal is to identify a set of input-links of minimum cardinality to be added between the existing inputs and the state variables in order to preserve structural controllability with respect to failure of an arbitrary input-link. We establish that this particular problem is NP-hard and even inapproximable to a multiplicative factor of $logp$, where $p$ is the number of critical input-links in the system. Additionally, we identify several practically relevant tractable cases associated with this problem. Finally, an example illustrating the usefulness of the methods developed is given in this article.