Structural controllability of multiinput linear systems

Abstract

To understand the controllability of complex networks is a forefront problem relevant to different fields of science and engineering. Despite recent advances in network controllability theories, an outstanding issue is to understand the effect of network topology and nodal interactions on the controllability at the most fundamental level. Here we develop a universal framework based on local information only to unearth the most fundamental building blocks that determine the controllability. In particular, we introduce a network dissection process to fully unveil the origin of the role of individual nodes and links in control, giving rise to a criterion for the much needed strong structural controllability. We theoretically uncover various phase-transition phenomena associated with the role of nodes and links and strong structural controllability. Applying our theory to a large number of empirical networks demonstrates that technological networks are more strongly structurally controllable (SSC) than many social and biological networks, and real world networks are generally much more SSC than their random counterparts with intrinsic resilience and adaptability as a result of human design and natural evolution. * Electronic address: wenxuwang@bnu.edu.cn

Control principles of complex systems

“…The power of structural controllability comes from the fact that if a system is structurally controllable then it is controllable for almost all possible parameter realizations (Davison, 1977;Dion et al, 2003;Glover and Silverman, 1976;Hosoe and Matsumoto, 1979;Lin, 1974;Linnemann, 1986;Mayeda, 1981;Reinschke, 1988;Shields and Pearson, 1976). To see this, denote with S the set of all possible LTI systems that share the same zero-nonzero connectivity pattern as a structurally controllable system (A, B).…”

“…1974; Shields and Pearson, 1976). This is rooted in the fact that if a system (A 0 , B 0 ) ∈ S is uncontrollable, then for every > 0 there exists a controllable system (A, B) with ||A − A 0 || < and ||B − B 0 || < where || · || denotes matrix norm (Lee and Markus, 1968;Lin, 1974).…”

“…These two conditions can be accurately checked by inspecting the topology of the digraph G(A, B) without dealing with floatingpoint operations. Hence, this bypasses the numerical issues involved in evaluating Kalman's controllability rank 2 The structural controllability theorem also has a pure algebraic meaning (Shields and Pearson, 1976), which plays an important role in the characterization of strong structural controllability (Mayeda and Yamada, 1979).…”

Exact controllability of linear dynamical systems: A~geometrical approach

“…Controllability of a dynamical system has been largely studied by several authors and under many different points of view, see [1], [2], [3], [5], [6], [4], [9] for example. Among different aspects in which we can study the controllability we have the notion of structural controllability that has been proposed by Lin [7] as a framework for studying the controllability properties of directed complex networks where the dynamics of the system is governed by a linear system:ẋ(t) = Ax(t) + Bu(t); usually the matrix A of the system is linked to the adjacency matrix of the network, x(t) is a time dependent vector of the state variables of the nodes, u(t) is the vector of input signals, and B defines how the input signals are connected to the nodes of the network and is called the input matrix.…”

Identification of optimal measurement sets for complete flux elucidation in metabolic flux analysis experiments

“…In this article, we decide to bypass this difficulty by first enforcing a relaxed identifiability condition to generate a candidate measurement set and subsequently tightening the relaxation if necessary. This relaxed identifiability condition is constructed by using incidence structure analysis inspired from linear systems structural controllability analysis (Lin, 1974;Shields and Pearson, 1976). We demonstrate the applicability of the analysis in choosing flux and isotopic measurements, and present the framework used for experimental designs resulting in the maximum identifiable system at the minimum relative cost.…”