Abstract
The concept of controllability from control theory is applied to weighted and directed networks with heterogenous linear or linearized node dynamics subject to exogenous inputs, where the nodes are grouped into leaders and followers. Under this framework, the controllability of the controlled network can be decomposed into two independent problems: the controllability of the isolated leader subsystem and the controllability of the extended follower subsystem. Some necessary and/or sufficient conditions for the controllability of the leader-follower network are derived based on matrix theory and graph theory. In particular, it is shown that a single-leader network is controllable if it is a directed path or cycle, but it is uncontrollable for a complete digraph or a star digraph in general. Furthermore, some approaches to improving the controllability of a heterogenous network are presented. Some simulation examples are given for illustration and verification.
Highlights
Recent technological advances have stimulated broad interests in the notion of network controllability [1,2,3,4,5,6,7,8,9,10,11,12], which captures the ability to control aggregated dynamics of a networked system and guide it to a desired state by using limited external inputs [13, 14]
It is of both theoretical and practical importance to study the controllability of networked systems with nonidentical node dynamics, which can help develop a better understanding of the interplay between the complexity of the overall network topology and the collective dynamics of a networked system
The controllability problem for a leader-follower multiagent system was proposed by Tanner [1], who formulated it as the classical controllability of a single-input linear system and derived a necessary and sufficient algebraic condition in terms of the eigenvalues and eigenvectors of a submatrix of the graph’s Laplacian matrix
Summary
Recent technological advances have stimulated broad interests in the notion of network controllability [1,2,3,4,5,6,7,8,9,10,11,12], which captures the ability to control aggregated dynamics of a networked system and guide it to a desired state by using limited external inputs [13, 14]. The generators of a power network have different physical parameters and are certainly different from motors, which together form a heterogenous network with nonidentical node dynamics. The controllability problem for a leader-follower multiagent system was proposed by Tanner [1], who formulated it as the classical controllability of a single-input linear system and derived a necessary and sufficient algebraic condition in terms of the eigenvalues and eigenvectors of a submatrix of the graph’s Laplacian matrix. Some sufficient algebraic conditions were derived for a multileader system with time delays in the states, where both single and double integrator dynamics were considered [10]. The classic concept of controllability from control theory is extended to weighted and directed complex networks with nonidentical node dynamics in a systematic way. Necessary and/or sufficient conditions on node dynamics and network topology for controllability are given in either algebraic or graph-theoretic forms.
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