Structural controllability of networked systems with general heterogeneous subsystems

Abstract

In this note, we investigate structural controllability of networked systems in which subsystems have high-order and heterogeneous dynamics and are coupled through relative state variables. It is aimed to search conditions for subsystem dynamics, under which structural controllability of the whole network essentially depends only on network topology as well as subsystem interconnection links. Necessary and/or sufficient conditions for structural controllability are given by simplifying subsystem dynamics. As an application, these criteria are utilized to design network topology that minimizes the number of subsystem interconnection links under the requirement of structural controllability without concerning subsystem dynamics, which is also much more efficient computationally than a direct design for the global system. The effectiveness of the criteria is demonstrated through several examples.

Strong structural controllability of colored structured systems

“…Another opportunity for future research is to extend our results to a wider range of applications. For example, based on the results of weak structural controllability of networks, numerous works have been reported from rather diverse perspectives on such topics as topology design [10], minimal input selection problem [19], and so on.…”

Further results for synchronization of two coupled Boolean networks

“…, then, an algebraic form for (8a) and (8b) is obtained, v(t + 1) = Fv(t), w(t + 1) = Gv(t)w(t), where F = 𝛿 8 [4,7,5,8,3,6,4,7], G = 𝛿 8 [3,3,3,3,3,4,3,4, … ] ∈  8×64 , the detailed information for matrix G is omitted here. Let 𝜉(t) = v(t)w(t) ∈ Δ 64 , then we get…”

“…Then, an algebraic form for (10a) and (10b) is obtained, v(t + 1) = Fv(t), w(t + 1) = Gv(t)w(t), where F = 𝛿 8 [3,7,8,8,1,5,6,6], G = 𝛿 8 [5, 1, 6, 2, 7, 3, 8, 4, … ] ∈  8×64 , the detailed information for matrix G is omitted here. Then, we get 64 and 𝛿 62 64 of (11) before and after {1, 2}-perturbation 𝜉(t + 1) = L𝜉(t), (11) where L = 𝛿 64 [21, 17, 22, 18, 23, 19, 24, 20,…”

“…To overcome this difficulty, Cheng et al derived the equivalent algebraic form of BNs by using the semi-tensor product (STP) of matrices in 2010 [4]. Then, the researches on BNs are further developed, includ-ing controllability, observability, optimal control, synchronization, output tracking, and state estimation [5][6][7][8][9][10][11][12][13][14][15][16].…”

Effect of the interdependence between subnets on the structural controllability of complex networks

“…For these problems, scientists have obtained a lot of results. In the works of literatures [9][10][11][12][13][14][15][16][17][18][19], the network topology, the dynamics and the heterogeneity of nodesystems, external control inputs, and the inter-coupling between two nodes are discussed altogether or, respectively, to determine whether they affect the controllability of complex networks, which have different topological structure and dynamic structure. Zhou [20] investigated the minimal number of inputs/outputs required to guarantee the controllability/observability of a system and proved that this minimal number is equal to the maximum geometric multiplicity of the system state transition matrix.…”