Recent Impacts of Multidimensional Systems Research

Abstract

A polynomial or a rational function (matrix), characterizing a single-input single-output (multi-input multi-output) system, has the coefficients for parameters. The number of such free parameters defines the dimension of the space and a system with fixed parameters may be represented by a point in parameter space. If at least one coefficient varies about its nominal value, a region in parameter space is generated. This region characterizes a family of systems instead of one fixed system. When the coefficients vary independently of each other within specified compact intervals, an interval system is generated. A well explored case when the coefficients do not vary independently occurs when the region in parameter space is a bounded polyhedral set. A polyhedral set is formed from the intersection of a finite number of closed half-spaces and could be unbounded. A bounded polyhedral set is a convex polytope and vice versa. For an interval system, the polytope degenerates into a boxed domain or a hyper-rectangle. Extensive documentation of research results concerned with the extraction of information about the complete polytope from a very small subset of the polytope with respect to the property of stability for both continuous-time and discrete-time systems is available in several recent texts ([237], [238], [239], for example), since Kharitonov’s trend-setting publications [240] followed by the edge theorem [241]. The goal is to obtain tests for invariance of useful properties of sets of distinguished classes of functions from tests on a small subset of such functions. Properties of concern in control