THE JACOBI EIGENVALUE CRITERION: A DYNAMIC EXTENSION AND STABILITY THEOREM

Abstract

The Jacobi Eigenvalue criterion is a recently developed, steady state, scaling independent tool for the selection of variable pairing for multiloop control systems (Mijares et al., 1986). This criterion is based on the Jacobi Iteration method for solving sets of linear equations or obtaining the inverse of a matrix. A dynamic extension of the Jacobi Eigenvalue criterion which consists in evaluating the spectral radius of the Jacobi Iteration matrix in the frequency range of interest is presented. The Jacobi Eigenvalue criterion is inherently a diagonal dominance measure, and as such, is readily applied to develop an extension of Rosenbrock's Nyquist stability theorem. The extended Jacobi Eigenvalue criterion is then compared to different interaction and weak coupling measures used for design of decentralized control structures which are based on conditions for generalized diagonal dominance and H-matrices. Two examples are used to illustrate the use of the Jacobi Eigenvalue criterion and its relationship with the other weak coupling measures.