MIMO and Multi-Group Comparison Problem in Process Control: A Multivariate Statistical Framework for Systems Represented with Frequency Response Functions

Characterizing complex Multi-Input Multi-Output (MIMO) systems presents two issues: feedback controllers mask fault variance, rendering single-variable monitoring ineffective, and repeated testing across multiple conditions inflates false positive rates due to the multiple comparisons problem. This study proposes a multivariate statistical framework to resolve these limitations. We extend a statistical library that identifies system dynamics via $\mathit{\text{Frequency}}\mathit{\text{Response}}\mathit{\text{Functions}}$ (FRFs) by transforming them into time-domain $\mathit{\text{Pseudo-Impulse}}\mathit{\text{Responses}}$ (PIRs). This functional representation captures the complete dynamic signature of the system, offering a richer diagnostic profile than traditional static scalar metrics. The framework, originally developed for SISO systems, is extended to the MIMO case by introducing supervectors that aggregate multiple PIRs, evaluated using $\mathit{\text{Permutational}}\mathit{\text{Multivariate}}\mathit{\text{Analysis}}\mathit{\text{of}}\mathit{\text{Variance}}$ (PERMANOVA). This non-parametric approach handles the high dimensionality and complex correlation structures inherent in MIMO functional data. We demonstrate that the PIR Supervector is more powerful than single-variable analysis, and the MIMO PERMANOVA approach outperforms traditional SISO anomaly detection. By distinguishing between normal operation, external thermal disturbance effectively compensated within the investigated operating range, and severe parametric faults that standard univariate methods miss, the framework provides a single, rigorous metric for specific fault diagnosis ($p<0.001$).