Discrete Affine Research Articles

Analysis and synthesis of discrete nonlinear passive systems via affine T–S fuzzy models

The article considers the analysis and synthesis problem for the discrete nonlinear systems, which are represented by the discrete affine Takagi–Sugeno (T–S) fuzzy models. The state feedback fuzzy controller design methodology is developed to guarantee that the affine T–S fuzzy models achieve Lyapunov stability and strict input passivity. In order to find a suitable fuzzy controller, an Iterative Linear Matrix Inequality (ILMI) algorithm is employed in this article to solve the stability conditions for the closed-loop affine T–S fuzzy models. Finally, the application of the proposed fuzzy controller design methodology is manifested via a numerical example with computer simulations.

Classification of Three-Dimensional Integrable Scalar Discrete Equations

We classify all integrable three-dimensional scalar discrete affine linear equations Q 3 = 0 on an elementary cubic cell of the lattice $${\mathbb Z}^3$$ . An equation Q 3 = 0 is called integrable if it may be consistently imposed on all three-dimensional elementary faces of the lattice $${\mathbb Z}^4$$ . Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prove that the only non-trivial (non-linearizable) integrable equation from this class is the well-known dBKP-system.

Synthesis of nonlinear discrete control systems via time-delay affine Takagi-Sugeno fuzzy models

The affine Takagi-Sugeno (TS) fuzzy model played a more important role in nonlinear control because it can be used to approximate the nonlinear systems more than the homogeneous TS fuzzy models. Besides, it is known that the time delays exist in physical systems and the previous works did not consider the time delay effects in the analysis of affine TS fuzzy models. Hence a parallel distributed compensation based fuzzy controller design issue for discrete time-delay affine TS fuzzy models is considered in this paper. The time-delay effect is considered in the discrete affine TS fuzzy models and the stabilization issue is developed for the nonlinear time-delay systems. Finally, a numerical simulation for a time-delayed nonlinear truck-trailer system is given to show the applications of the present approach.

Discrete Symmetries of Systems of Isomonodromic Deformations of Second-Order Fuchsian Differential Equations

We compute the discrete affine group of Schlesinger transformations for isomonodromic deformations of a Fuchsian system of second-order differential equations. These transformations are treated as isomorphisms between the moduli spaces of logarithmic sl(2)-connections with given eigenvalues of the residues on ℙ1. The discrete structure is computed with the use of the modification technique for bundles with connections. The result generalizes the well-known classical computations of symmetries of the hypergeometric equation, the Heun equation, and the sixth Painleve equation.

Affine Defects and Gravitation

We argue that the structure general relativity (GR) as a theory of affine defects is deeper than the standard interpretation as a metric theory of gravitation. Einstein-Cartan theory (EC), with its inhomogenous affine symmetry, should be the standard-bearer for GR-like theories. A discrete affine interpretation of EC (and gauge theory) yields topological definitions of momentum and spin (and Yang Mills current), and their conservation laws become discrete topological identities. Considerations from quantum theory provide evidence that discrete affine defects are the physical foundation for gravitation.

Closed-form discrete fractional and affine Fourier transforms

The discrete fractional Fourier transform (DFRFT) is the generalization of discrete Fourier transform. Many types of DFRFT have been derived and are useful for signal processing applications. We introduce a new type of DFRFT, which are unitary, reversible, and flexible; in addition, the closed-form analytic expression can be obtained. It works in performance similar to the continuous fractional Fourier transform (FRFT) and can be efficiently calculated by the FFT. Since the continuous FRFT can be generalized into the continuous affine Fourier transform (AFT) (the so-called canonical transform), we also extend the DFRFT into the discrete affine Fourier transform (DAFT). We derive two types of the DFRFT and DAFT. Type 1 is similar to the continuous FRFT and AFT and can be used for computing the continuous FRFT and AFT. Type 2 is the improved form of type 1 and can be used for other applications of digital signal processing. Meanwhile, many important properties continuous FRFT and AFT are kept in the closed-form DFRFT and DAFT, and some applications, such as filter design and pattern recognition, are also discussed. The closed-form DFRFT we introduce has the lowest complexity among all current DFRFTs that is still similar to the continuous FRFT.

Minimal (max,+) Realization of Convex Sequences

We show that the minimal dimension of a linear realization over the (max,+) semiring of a convex sequence is equal to the minimal size of a decomposition of the sequence as a supremum of discrete affine maps. The minimal-dimensional realization of any convex realizable sequence can thus be found in linear time. The result is based on a bound in terms of minors of the Hankel matrix.

Affine Systems inL2([formula omitted]d): The Analysis of the Analysis Operator

Discrete affine systems are obtained by applying dilations to a given shift-invariant system. The complicated structure of the affine system is due, first and foremost, to the fact that it is not invariant under shifts. Affine frames carry the additional difficulty that they are “global” in nature: it is the entire interaction between the various dilation levels that determines whether the system is a frame, and not the behaviour of the system within one dilation level. We completely unravel the structure of the affine system with the aid of two new notions:the affine product, anda quasi-affine system. This leads to a characterization of affine frames; the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the Fourier domain. Several results, such as a general oversampling theorem follow from these characterizations. Most importantly, the affine product can be factored during a multiresolution analysis construction, and this leads to a complete characterization of all tight frames that can be constructed by such methods. Moreover, this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution. Of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions, and that the freedom offered by redundancy can be fully exploited in these constructions.

Transformation of digital images by discrete affine applications

In this paper I am interested in transformation of digitized images by affine applications; the discrete nature of a computer screen makes this operation rather difficult. I propose to discretize real affine applications in order to obtain quasi-affine transformations which are similar to the calculus of real applications in fixed point arithmetic. Until now, the computer graphists avoided this digitization, replacing it by other methods (inverse mapping, over-sampling) which require heavy calculation. I present an algorithm for the determination of reciprocal images of a point by a quasi-affine transformation; this algorithm will permit the transformation of digitized images by discrete affine applications. In general, a one-to-one mapping is not obtained, but it is possible to determine where the points are without antecedent or those with more than one antecedent. In addition if the affine application is sufficiently dilating, the associated discrete transformation is inversible. The algorithm I propose uses integer arithmetic with only tests and additions (no multiplication nor division) so that it can be quickly performed.

Affine Texture Mapping and Antialiasing Using Integer Arithmetic

Texture mapping techniques are very useful for generating more realistic images. However, texture compression, generally induced by geometric transformations, is at the origin of aliasing artifacts especially the well-known “moire” patterns. Two discrete affine texture mapping methods based exclusively on integer arithmetic are presented here. This original approach of discrete affine mapping is adequate for antialiasing in the case of compressed textures.

A combinatorial theory of Grünbaum's new regular polyhedra, Part II: Complete enumeration

The new regular polyhedra as defined by Branko Grunbaum in 1977 (cf. [5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. [3], [2] or [1]) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grunbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grunbaum—is suggested rather by the group theoretical investigations in [4], it turns out that the list of examples given in [5] is essentially complete except for one additional polyhedron. So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE 3 and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE 3, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself. In addition, a new classification scheme for regular polygons inE n is worked out in §9.