| Three mathematical tools for determining the robust stability of multivariable feedback systems are introduced. First, the numerical range technique extracts the phase and gain information as well as the condition of a multivariable transfer function matrix. A modified version of the numerical range can quantify the behavior of uncertainties associated with the transfer function which can lead to a precise assessment of system stability. Second, a family of eigenvalue inclusion region (EIR) techniques bound the movements of the eigenvalues of linear operators which are subjected to perturbation. In particular, the refined eigenvalue inclusion regions (REIR) approach, developed to combine advantages from the singular value decomposition and the numerical range concepts, generates tighter uncertainty bands around the nominal eigenloci than conventional norm-bounded methods. The uncertainty bands, when incorporated into the Characteristic Loci framework, can be effectively used to determine the robust stability of linear feedback systems.;To illustrate the applications of the approach, several examples are presented including the use of sector technique to evaluate the validity of order reduction for a distillation column.;Finally, the modified multivariable sector technique bounds the magnitudes of both linear and static nonlinear operators though the use of inside and outside sector conditions. The optimized bounds, combined with the Small-Gain theorem, constitute robust stability criteria of systems with static nonlinear elements. In order to extend the method to accommodate dynamic nonlinear systems, a polynomial transformation is introduced to create viable approximations that can be exploited within the sector bound setting. The magnitude of the discrepancies due to approximations are measured and taken into account when the stability is assessed. For the purpose of convenience, an index is defined such that the closed-loop stability is guaranteed when the index is less than unity. Using this index, regions of permissible operation can be mapped on state variable planes. |
