Minimal controller structure for generic pole placement

In this paper we address the generic pole placement problem for a system represented by differential algebraic equations. The genericity aspect is relevant when dealing with large dynamical systems where the plant equations are sparse. We capture the sparsity structure of the plant equations into an edge weighted and undirected bipartite graph. We propose an algorithm that furnishes a `minimal´ controller structure for achieving generic arbitrary pole placement: minimality in the sense of the sparsity within controller equations. More precisely, we introduce a procedure to come up with a set of controller equations such that, in addition to generically achieving arbitrary pole placement, the bipartite graph constructed for this controller has the minimum number of edges amongst all controllers that generically achieve arbitrary pole placement. The algorithm we propose involves finding a minimum number of paths that cover a given set of vertices corresponding to plant equations. We introduce an integer that captures the extent of MIMO features inside the plant equations, since this turns out to crucially decide the minimum number of required edges. This paper´s minimal controller structure problem and the proposed solution turn out to also solve the problem of generically completing a given rectangular polynomial matrix into a unimodular matrix using the minimum number of nonzero entries.