Unified gain formula for control diagrams with matrix-based transfer functions

The conventional Mason's gain formula is limited to single-input single-output (SISO) systems and is inadequate for coupled multi-input multi-output (MIMO) systems as it does not consider matrix operations. To fill this gap, this paper introduces a novel unified gain formula for control diagrams with matrix-based transfer functions. The formula is developed using partitioned matrix theory and feature extraction methods. Applying the partitioned matrix concept, the transfer function between any input and output of three type control diagrams is derived. A unified gain formula for input u to output y of the control diagram is then proposed based on feature extraction. Subsequently, the unified gain formula for any input and output of the control system is proposed based on the deduced equivalent relationship between the original and moved disturbance input nodes. Lastly, case studies demonstrate the effectiveness of the proposed unified gain formula, verifying its potential for use in both SISO and MIMO systems.