Structural Controllability of Networked Relative Coupling Systems

Abstract

This paper studies controllability of networked systems in which subsystems are of general high-order linear dynamics and coupled through relative state variables, from a structure perspective. The purpose is to search conditions for subsystem dynamics and subsystem interaction topologies, under which there exists a set of weights for the interaction links such that the associated networked system can be controllable (i.e., structural controllability). Three types of subsystem interaction fashions are considered, which are 1) each subsystem is singleinput-single-output (SISO), 2) each subsystem is multiple-inputmultiple-output (MIMO), and the interaction weights for different channels between two subsystems can be different, and 3) each subsystem is MIMO but the interaction weights between two subsystems are the same. Necessary and/or sufficient conditions for structural controllability are given. These conditions indicate that, under certain conditions on the subsystem dynamics, the whole system is structurally controllable, if and only if the network topology is globally input-reachable. Finally, these results are extended to the case where subsystem dynamics are fixed but the interaction topologies are switching. A promising point of the structure analysis taken in this paper is that, it can handle certain subsystem heterogeneities, which are illustrated by some practical systems, including the liquid-level systems, the power networks and the mechanical systems.

Structural Controllability of Undirected Diffusive Networks With Vector-Weighted Edges

“…To proceed with our proof, we need the following lemma. Lemma 2 (Lemma 8 of [16]): Given four matrices H, P, G and Λ, suppose the following conditions hold: 1) H, P and G are of the dimensions k × n, k × m and n × k respectively; 2) Whenever there exists one l ∈ {1, ..., k} such that G il = 0 and H li = 0 (resp. G il = 0 and P li = 0), it implies that [GH] ij = 0 (resp.…”

“…Specially, when r = 1, (1)-( 2) becomes a networked system with SISO subsystems. Readers are referred to [14,16] for more examples for networked systems with vector-weighted edges.…”

“…For example, in some networks consisting of both physical coupling and cyber coupling, the physical channels and the cyber ones between two subsystems can have different weights. See more examples in [14][15][16]. If we use a graph to denote the subsystem interaction topology (i.e., network topology), then each edge of the graph may have a vectorvalued or matrix-valued weight as introduced in [14,15].…”

On the Generic Low-Rank Matrix Completion

“…This system is said to be controllable, if rank[λI n − A, B] = n for all λ ∈ C. A complex value λ violating the aforementioned condition is called an uncontrollable mode. In the case where only the zero-nonzero structure of [A, B] is available, this system is called structurally controllable if for almost all realizations of [A, B], the corresponding numerical system is controllable [17,19]. The proposed GLRMC framework could answer, given a structurally controllable system (A, B), whether there generically exists a numerical perturbation [δA, δB] with a prescribed zero-nonzero structure, such that the perturbed system (A + δA, B + δB) has an uncontrollable zero mode.…”

“…In other words, a property is generic, if for almost all 1 parameters in the corresponding space, this property holds; the parameters violating the property have zero measure. Genericity has attracted special attention in control theory [17][18][19], rigidity theory [20], etc., but little in the LRMC.…”