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Approximation Filters for Nonlinear Systems
Published in Jitendra R. Raol, Girija Gopalratnam, Bhekisipho Twala, Nonlinear Filtering, 2017
Jitendra R. Raol, Girija Gopalratnam, Bhekisipho Twala
One needs to specify how a base tessellation may be computed, for example, for spheres, the first step is to generate the set of points xi. The equations for the half-space containing the manifold and being bordered by the tangent space at each point xi is computed, which comes down to finding a base for the space orthogonal to the tangent space at xi; for spheres, this is trivial. The intersection of these half-spaces is a polytope with a one-to-one correspondence between facets and generating points. Projecting the facets towards the origin will generate a tessellation, and for spheres this will be a Voronoi tessellation if the ‘natural’ embedding is used. Each region is given by the set of projected vertices of the corresponding polytope facet, and as a part of the tessellation task, the volume of each region should also be computed. For the 2-sphere this can be done exactly. In the general case we approximate the volumes on the manifold by the volume of the polytope facets. A facet can be reconstructed from the projected vertices by projecting back to the (embedded) tangent space at the generating point. For spheres the ideal total volume is known, and any difference between the sum of the volumes of the regions and the ideal total volume is compensated by scaling all volumes by a normalizing constant.
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.
Efficiency in Real-Time Motion Planning
Published in Changliu Liu, Te Tang, Hsien-Chung Lin, Masayoshi Tomizuka, Designing Robot Behavior in Human-Robot Interactions, 2019
Changliu Liu, Te Tang, Hsien-Chung Lin, Masayoshi Tomizuka
Figure 4.3a illustrates a convex feasible set in (4.11d)) for one q and i when X ⊂ ℝ2. The left hand side of (4.11d) represents the tangent plane of the distance function d∗ at the reference point xqr. Due to the cone structure of d∗, the tangent plane touches the boundary of the obstacle. The convex feasible set is the projection of the positive portion of the tangent plane onto ℝ2, which is a half space. The half space is maximal in the sense that the distance from the reference point to the boundary of the half space is maximized, which is equal to the distance from the reference point to the obstacle. With this observation, we can construct the convex feasible set using a purely geometric method without differentiation. For any reference point xr and any convex obstacle O, denote the closest point on O to xr as b∗. The convex feasible set for xr with respect to O is just the half space that goes through b∗ and whose normal direction is along b∗ − xr as shown in Fig. 4.3b.
Event-triggered feedback control for discrete-time piecewise-affine systems subject to input saturation and bounded disturbance
Published in International Journal of Control, 2022
Wei Wu, Yifei Ma, Daniel Görges, Xuyang Lou
The state space is partitioned into polyhedral regions and the specification of the dynamics valid within each region. Moreover, polyhedral regions satisfying and for , where denotes the closure of . The regions are assumed to be polyhedron , with and . In other words, the polyhedron is a convex set generated by the intersection of a finite number of halfspaces. The symbol ℘ is the index set of these polyhedral cells. In addition, the index set ℘ is divided into , where is the set of indices that contains the origin, that is , and is the set of indices that does not contain the origin, that is .
A representation of generalized convex polyhedra and applications
Published in Optimization, 2020
Nguyen Ngoc Luan, Nguyen Dong Yen
The intersection of a finite number of closed half-spaces of a finite-dimensional Euclidean space is called a polyhedral convex set (a convex polyhedron in brief). By convention, the intersection of an empty family of closed half-spaces is the whole space. Therefore, emptyset and the whole spaces are two special polyhedra. Due to [1, Theorem 19.1], for every given convex polyhedron one can find a finite number of points and a finite number of directions such that the polyhedron can be represented as the sum of the convex hull of those points and the convex cone generated by those directions. The converse is also true. This celebrated result is attributed [1, p. 427] primarily to Minkowski [2] and Weyl [3, 4]. By using the result, it is easy to derive fundamental solution existence theorems in linear programming. Note that the just cited representation formula for finite-dimensional polyhedral convex sets has many other applications in mathematics. As an example, one can refer to the elegant proofs of the necessary and sufficient second-oder conditions for a local solution and for a locally unique solution in quadratic programming, which were given by Contesse [5] in 1980; see [6, pp. 50–63] for details.
On mode reconstructability and reconstructability sets of piecewise linear systems
Published in International Journal of Control, 2021
Manuel Mera, Francisco Javier Bejarano
A hyperplane (affine hyperplane) divides into two half-spaces, denoted , called upper and lower half-spaces, respectively. We can write the closed half-spaces as the sets: An open half-space is the interior of a closed half-space , which we denote as . We characterise each one of the switching sets as the set resulting from the finite intersection of the closed lower half-spaces and the open upper half-spaces of the hyperplanes in a hyperplane arrangement. Thus, we consider the case when the partition used in system (1) is generated by an arrangement of affine hyperplanes, that is, each set is defined as where , for are such that , and . This partition is extensively used in piecewise systems modelling and analysis (e.g. Nakamura & Hamada, 1990; Rodrigues & How, 2003; Sang & Tao, 2011). Concerning the recontructibility, some previous results can be found in Chaib et al. (2005) and Boutat et al. (2004); however, there only sufficient conditions are given. In the present paper, we propose a way to determinate the mode reconstructability of the system (1) with partition (2); the conditions obtained here are necessary and sufficient. In the way of presenting the main result, we characterise some sets of the state space where the system is always mode reconstructible, and also we give some sets of the state space where the system is not mode reconstructible.