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This article considers the problem of minimal control inputs to affect the system states such that the resulting system is structurally controllable. This problem and the dual problem of minimal observability are claimed to have no polynomial-order exact solution and, therefore, are NP-hard. Here, adopting a graph-theoretic approach, this problem is solved for general nonlinear (and also structure-invariant) systems, and a P-order solution is proposed. In this direction, the dynamical system is modeled as a directed graph, called system digraph, and two types of graph components are introduced, which are tightly related to structural controllability. Two types of nodes, which are required to be affected (or driven) by an input, called driver nodes, are defined, and minimal number of these driver nodes are obtained. Polynomial-order complexity of the given algorithms to solve the problem ensures applicability of the solution for analysis of large-scale dynamical systems. The structural results in this article are significant as compared with the existing literature, which offer approximate and computationally less-efficient, e.g. Gramian-based, solutions for the problem, while this article provides an exact solution with the lower computational complexity and applicable for controllability analysis of nonlinear systems.

Minimal Driver Nodes for Structural Controllability of Large-Scale Dynamical Systems: Node Classification