Quantification of interaction in multiloop control systems using directed spectral decomposition

Interactions among control loops are a critical and challenging issue in the control of multivariable systems. The focus of this work is on the analysis and quantification of interactions in multiloop or decentralized control systems. Existing interaction measures suffer from one or more of the following limitations: (i) the lack of a direct connection to a performance metric, (ii) assumption of the availability of process models and (iii) the approximate and/or a heuristic nature of the approach to their development, resulting only in approximate indicators of interaction. This work presents an exact quantifier of interaction that arises out of a directional decomposition of the loop variance using methods of causality (directional) analysis in frequency-domain. The main result is that the spectrum of the filtered output can be decomposed into (i) an interaction-and-feedback invariant term and (ii) an interaction-dependent term. The associated filter can be derived from the closed-loop data and is related to the diagonal element of the multiloop sensitivity function. The invariant term for each output is the spectral density of that output when the corresponding loop is under open-loop conditions. It is further shown to be solely a function of the control pairing. Variance measures corresponding to the invariant and interaction terms are introduced. The utility of the measure is that it can be computed from closed loop data as well as from the process model. Applications to simulated systems and a real time distillation process are presented to demonstrate the theoretical ideas.