Eigenvector-based intergroup connection of low rank for hierarchical multi-agent dynamical systems

Abstract

This paper proposes an eigenvector-based method for analysis and design of hierarchical networks for multi-agent systems. We first define the concept of eigen-connection by characterizing low rank information flow between layers based on the eigenvector of lower level interconnection structures. It is shown that the resulting intergroup interconnections affect only a few eigenvalues of interconnection structures in the lower layer, and we derive explicit expressions for shifted eigenvalues. Then a procedure for designing hierarchical networks that result in desirable eigenvalue distributions is proposed, where the eigen-connection is used for a key to move undesirable eigenvalues selectively. The effectiveness of the procedure is demonstrated by a numerical example.

An algebraic approach to hierarchical LQR synthesis for large-scale dynamical systems

“…Define the set A L ⊂ R n×n by (2). If A, A ⊤ , BR −1 B ⊤ , Q ∈ A L , then the stabilizing solution to the Riccati equation (6) belongs to A L . Proof: Note that A 1 = ⟨G A 1 ⟩ forms an algebra as mentioned in the previous subsection.…”

“…Hence, we can say that the concept of low rank interconnection studied in [4]- [6] matches up with the hierarchical LQ optimal control in comparison with the sparsity or the small gain property.…”

“…If ∆ 2 = b 2 1 ⊤ n2 , the average output of the subsystems in each group is shared among the groups and the upper layer interaction based on the average value affects the subsystems in each group in the same manner as the control input does. Such a structure is studied in [4]- [6] as a hierarchical network with low-rank interconnections. Although this is a specific example, the situation is quite common to systems with hierarchical structures.…”

“…The information about subsystems is transmitted through an aggregation process according to levels of hierarchy. Motivated by this fact, hierarchies are employed in the interconnection structure for multi-agent systems and analyzed based on the eigenvalue distributions [2]- [6]. In contrast to these, the objective of this paper is to design control laws for systems possessing hierarchical structures inherently.…”

Layering in networks: The case of biochemical systems

“…If a network is modelled as solving an optimisation problem, layering corresponds to the partition of the problem into sub-problems at lower layers, coordinated by a master problem at a higher layer. Such coordination is seen in the hierarchical layering of networks of communicating autonomous systems [28], [30], [32]. Here a large communication network is partitioned into sub-networks, each of which coordinate a subset of agents, and which also communicate with one another (through one leader, or some other low-rank information) via a higherlayer communication network.…”