Coupling Structures in Model-Based Systems: An Analytical Framework Based on Axiomatic Design

The engineering design of complex systems increasingly relies on model-based methods that generate large-scale models with a vast multitude of couplings between the functional requirements and design parameters. The so-called Axiomatic Design Approach tries to characterize and evaluate the couplings between functional requirements (FRs) and design parameters (DPs) by imposing a square matrix form for the mapping. In full generality however, the physical modeling of complex systems produces rectangular matrices due to an a priori unequal number of FRs and DPs, as dictated by physical laws and already shown with mathematical rigour in previous works. By using the full rectangular form of the coupling matrix, this paper presents an analytical framework that uses classical sensitivity-based analysis to identify, quantify, and visualize coupling structures represented by rectangular design matrices. Normalized visualizations of the Jacobian matrix in the form of so-called heatmaps then help to reveal intrinsic subsystem structures and highlight areas of strong internal couplings and minimal inter-subsystem interactions. The heatmap thus helps to detect weakly coupled subsystem blocks, supports a more systematic decomposition and allows a targeted optimization in a loose analogy to the original idea of the so-called Independence Axiom in axiomatic design. This methodology permits the sequential or even independent optimization of subsystems or parts thereof, facilitating a reliable and efficient system-level design. The proposed approach is demonstrated through two applications: a Formula Student race car and a small satellite design. The case studies illustrate how rectangular matrix analysis can provide a practical basis for guiding system decomposition and design decisions in model-based engineering systems.