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

TL;DR

This paper introduces a graph-theoretic method to efficiently identify the minimal set of driver nodes needed for the structural controllability of large-scale nonlinear dynamical systems, providing exact solutions with polynomial complexity.

Contribution

It presents a novel polynomial-order algorithm for determining minimal driver nodes in nonlinear systems, improving upon existing approximate and less efficient methods.

Findings

Provides a polynomial-order solution for large-scale systems

Identifies two types of driver nodes related to controllability

Offers exact solutions for nonlinear and structure-invariant systems

Abstract

This paper 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 \textit{system digraph}, and two types of graph components are introduced which are tightly related with structural controllability. Two types of nodes which are required to be affected (or driven) by an input, called \textit{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 paper are significant as compared to the existing literature which offer approximate and computationally less-efficient, e.g. Gramian-based, solutions for the problem, while this paper provides exact solution with lower computational complexity and applicable for controllability analysis of nonlinear systems.