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Two Decades of Multidimensional Systems Research and Future Trends
Published in Krzysztof Gałkowski, Jeff David Wood, Multidimensional Signals, Circuits and Systems, 2001
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 such a space. If the coefficients vary about the nominal values, 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 intenal 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 ((Kogan 1995). for example), since Kharitonov's trend-setting publications.
Appendices
Published in Craig A. Tovey, Linear Optimization and Duality, 2020
Second, a polyhedron is the set sum of a polytope and a polyhedral cone. A polytope is a bounded polyhedron. It is equal to the set of convex combinations of its finitely many extreme points. The polyhedral cone can be characterized as the set of nonnegative linear combinations of a finite set of vectors, which are extreme directions of the polyhedron.
Enumerating extreme points of the polytopes of stochastic tensors: an optimization approach
Published in Optimization, 2020
A polytope is the convex hull of a finite set of points in (see, e.g. [3, p. 8] or [4, p. 4]). Equivalently, a polytope is a bounded intersection of finitely many closed halfspaces in the form ([4, p. 29]), where A is an real matrix for some positive integer m and . The dimension of the polytope is the minimum of dimensions of all affine spaces that contain . It turns out that the dimension of is equal to the dimension of the null space of A; that is, .