On the Controllability of Matrix-Weighted Networks

Abstract

This letter examines the controllability of consensus dynamics on matrix-weighed networks from a graph-theoretic perspective. Unlike the scalar-weighted networks, the rank of weight matrix introduces additional intricacies into characterizing the dimension of controllable subspace for such networks. Specifically, we investigate how the definiteness of weight matrices influences the dimension of the controllable subspace. In this direction, graph-theoretic characterizations of the lower and upper bounds on the dimension of the controllable subspace are provided by employing, respectively, distance partition and almost equitable partition of matrix-weighted networks. Furthermore, the structure of an uncontrollable input for such networks is examined. Examples are then provided to demonstrate the theoretical results.

Controllability and observability of linear multi-agent systems over matrix-weighted signed networks

“…Remark 3. Compared with the case that all the matrix weights are positive definite in Pan et al (2020), Definition 1 applies in the case that the matrix weights are negative definite or semi negative definite. Therefore, this extension is necessary and meaningful.…”

“…Lemma 1 shows the relationship between the characteristic matrix and the Laplacian matrix. For the first-order system, the upper bound of the controllable subspace can be obtained by the expression of the controllable subspace in the geometric theory of linear systems Pan et al (2020). However, how to deal with the unknown coefficient matrix is a challenge for general linear dynamics and worth further discussion.…”

“…From Theorem 1 and Corollary 1, matrices 1 and need to be found to make = 1 and = , (0 ≤ ≤ , 0 ≤ ≤ ) hold. There is no such limitation for the similar results in Pan et al (2020), so it is necessary to further discuss this condition.…”

“…Then the conditions of Theorem 1 and Corollary 1 both hold in this case. It should be noted that the model considered in this case is the same to the one in Pan et al (2020).…”

“…For instance, additional complexity is introduced because the rank of weight matrix is not fixed when describing the dimension of controllable subspace. Pan, Shao, Mesbahi, Xi and Li (2020) analyzed the controllability of matrix-weighted network by using the method of almost equitable partition and distance partition and gave the upper and lower bounds of controllable subspace. Notably, the weight matrix is assumed to be positive definite or positive semi-definite in their work.…”