Graphical Interpretations of Steady-State Interaction Measures

Abstract

The relative gain, the relative disturbance gain, and the singular value decomposition are commonly used methods to analyze multivariable linear processes. In this paper, graphical interpretations of these interaction measures are described, and additional graphical techniques for visualizing steady-state process interactions are proposed. For systems with two inputs and two outputs, these interaction measures are directly related to the angle of intersection and degree of rotation of constant output contours, or, in the case of singular value decomposition, constant sums of squares contours. Extensions to larger systems and the relative disturbance gain are identified in terms of projections between one-dimensional subspaces. The relationship between these interaction measures and measures of association in multivariate statistical analysis is established. The impact of process nonlinearities on interaction measures is identified graphically as deformations of the constant output norm contours. The use of graphical methods is demonstrated for mixing and high-purity distillation examples.