Optimal Network Topology Design in Composite Systems for Structural Controllability

This article deals with structural controllability of a linear time invariant (LTI) <italic>composite</italic> system consisting of several circuits/subsystems. We consider subsystems with structurally similar state matrices, i.e., the zero/nonzero pattern of the state matrices of the subsystems are the same, but dynamics can be different due to different numerical values. Structurally similar subsystems arise in large circuits implemented using many similar smaller circuits and agent-based networks consisting of homogeneous agents. The subsystems may not be structurally controllable individually; however, structural controllability of the composite system can be achieved by sharing state information. Sharing of information among subsystems incurs cost and our aim is to design a structurally controllable composite system with minimum information sharing. Minimizing information sharing is the same as minimizing the number of interaction links between subsystems, referred to as <italic>interconnections</italic>. An <italic>optimal network topology</italic> is one with a minimum number of interconnections. This article presents a closed-form expression for the minimum number of interconnections for structural controllability and derives a polynomial time algorithm to find an optimal network topology. The algorithm is based on a minimum weight perfect matching algorithm and a so-called edge reconstruction process. The minimum number of interconnections required is formulated in terms of two indices we define in the article: <italic>maximum commonality index</italic> (<inline-formula><tex-math notation="LaTeX">$\alpha _{{{\mathscr N}}}$</tex-math></inline-formula>) and <italic>dilation index</italic> (<inline-formula><tex-math notation="LaTeX">$\beta _{{\mathcal {I}}}$</tex-math></inline-formula>). Loosely speaking, more connectedness of subsystems leads to a lower total value of <inline-formula><tex-math notation="LaTeX">$\alpha _{{{\mathscr N}}}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\beta _{{\mathcal {I}}}$</tex-math></inline-formula>. Further, <inline-formula><tex-math notation="LaTeX">$\alpha _{{{\mathscr N}}}$</tex-math></inline-formula> decreases if the subsystems have fewer numbers of connected components. We apply and verify our general result to special cases that arise in control, where the minimum number can be more directly obtained. The special cases considered are structurally cyclic subsystems, irreducible subsystems, and subsystems in controller canonical form.