Structurally Controllable and Observable Optimal Disjoint Decomposition of Complex Systems: An Optimization Framework

Decomposition of a dynamic complex system into smaller subsystems can significantly reduce the complexity and simplify the analysis, particularly in networked control design. This paper proposes a structurally controllable and observable optimal disjoint <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula>-decomposition for complex systems. The methodology for this decomposition is conducted through the perspective of graph theory in conjunction with an optimization framework. To achieve this kind of decomposition, an optimization problem is first presented for the disjoint <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula>-decomposition. Then, the suggested optimization problem is extended to address optimal disjoint <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula>-decomposition. Next, the resultant decomposition is developed to fulfill the structural controllability and observability of the subsystems. Afterward, the recommended optimization problem undergoes modifications to account for structural perturbations. Finally, the performance and effectiveness of the decomposition approach are evaluated. The simulation results demonstrate that it successfully decomposes a complex system under structural perturbation into controllable and observable weakly coupled subsystems. Additionally, this method outperforms existing methods in terms of both methodology and specifications.