Multivariable controllers are often avoided in the chemical process industries in favor of simpler diagonal or block-diagonal controllers. Such "decentralized" controllers are desirable because they result in control systems with fewer tuning parameters and greater failure tolerance. However, the ensuing simplicity in controller design must be weighted against the interactions which result from ignoring the off-diagonal system blocks. These can lead to performance deterioration and even instability. The purpose of an Interaction Measure (IM) is to indicate under what conditions the stability of the diagonal loops/blocks will guarantee that of the complete system.
One such measure, the Relative Gain Array (RGA), has found widespread acceptance both in industry and academia despite its empirical basis. This measure, in fact, has sound theoretical justifications. Rigorous relationships are derived in this study linking the RGA to closed-loop stability and robustness with respect to model uncertainty.
Using the notion of Structured Singular Value, a new dynamic IM is also defined for multi-variable systems under feedback with diagonal or block-diagonal controllers. This measure, the µ IM, can be used to select the "best" variable pairings for the controller as well as predict the stability of the decentralized control system. Its steady-state value also provides a sufficient condition for achieving offset-free performance with the closed-loop system. The relationship of this new IM with Rijnsdorp' IM and Rosenbrock's Direct Nyquist Array is clarified.
Finally, it is shown how the µ IM, in conjunction with the RGA, can form the basis of a novel and useful methodology for the design of decentralized controllers.
