Showing papers on "Piecewise published in 1996"

Computation of Piecewise Quadratic Lyapunov Functions for Hybrid Systems

Abstract: This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach.

Brunn–Minkowski inequality for multiplicities

Abstract: This is a convex subset of the real vector space P⊗ZR. In fact, it is a convex polytope; see [Br], where this polytope is discussed from an algebraic point of view. It is known that the total mass of m is a polynomial in m for su ciently large m (denote by k its degree) and m −k m(m · ) weak −→ ( )d ; where d is the Lebesgue measure supported on and the density ( ) is a piecewise-polynomial function; we will not use this piecewise polynomiality in this paper. Recall that a real function f de ned on a convex subset U of a vector space V is called concave, if

Transfer of Near-Infrared Multivariate Calibrations without Standards.

Abstract: A novel approach to the transfer of multivariate calibration is proposed. This method is based on the finite impulse response (FIR) filtering of a set of spectra to be transferred, using a spectrum on the target instrument to direct the filtering process. Often, the target spectrum is the mean of a calibration set. The method is compared against direct transfer and piecewise direct transfer on near-infrared reflectance spectra in two representative data sets. Results from these studies suggest that FIR transfer compares favorably with piecewise direct transfer in terms of accuracy and precision of the match of transferred spectra to the predictive calibration models developed on the target instrument. Unlike piecewise direct transfer, FIR transfer requires no measurement of standard samples on both the source and target spectrometers. Details and current limitations of the FIR transfer method are presented.

Time dependent spectral analysis of nonstationary time series

Abstract: Modeling of nonstationary stochastic time series has found wide applications in speech processing, biomedical signal processing, seismology, and failure detection. Data from these fields have often been modeled as piecewise stationary processes with abrupt changes, and their time-varying spectral features have been studied with the help of spectrograms. A general class of piecewise locally stationary processes is introduced here that allows both abrupt and smooth changes in the spectral characteristics of the nonstationary time series. It is shown that this class of processes behave as approximately piecewise stationary processes and can be used to model various naturally occuring phenomena. An adaptive segmentation method of estimating the time-dependent spectrum is proposed for this class of processes. The segmentation procedure uses binary trees and windowed spectra to nonparametrically and adaptively partition the data into approximately stationary intervals. Results of simulation studies dem...

Residual-Free Bubbles for the Helmholtz Equation

Abstract: The Galerkin method enriched with residual-free bubbles is considered for approximating the solution of the Helmholtz equation. Two-dimensional tests demonstrate the improvement over the standard Galerkin method and the Galerkin-least-squares method using piecewise bilinear interpolations.

Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point

Abstract: Consider a second-order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution. Consider also its numerical approximation by standard conforming finite element methods with, for example, fixed degree piecewise polynomials on a quasi-uniform mesh-family (the “h-method”). It will be shown that, if the finite element function spaces are locally symmetric about a point $x_0 $ with respect to the antipodal map $x \to x_0 - (x - x_0 )$, then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of $x_0 $. For piecewise polynomials of even degree, superconvergence occurs in function values; for piecewise polynomials of odd degree, it occurs in derivatives.

Quadratic stability analysis and design of continuous-time fuzzy control systems

Abstract: This paper presents a design method for fuzzy control systems. The method is based on a fuzzy state-space model A suitable piecewise smooth quadratic (PSQ) Lyapunov function is used to establish asymptotic stability of the closed-loop system. With the PSQ Lyapunov function a kind of global controller design method can be developed, and thus the disadvantage of using fixed P in the Lyapunov function can be overcome. Furthermore, a constructive algorithm is developed to obtain the stabilizing feedback control law. The controller design algorithm involves solving a set of certain algebraic Riccati equations. An example is given to illustrate the application of the method

Coding for a binary independent piecewise-identically-distributed source

Abstract: Two weighting procedures are presented for compaction of output sequences generated by binary independent sources whose unknown parameter may occasionally change. The resulting codes need no knowledge of the sequence length T, i.e., they are strongly sequential, and also the number of parameter changes is unrestricted. The additional-transition redundancy of the first method was shown to achieve the Merhav lower bound, i.e., log T bits per transition. For the second method we could prove that additional-transition redundancy is not more than 3/2 log T bits per transition, which is more than the Merhav bound; however, the storage and computational complexity of this method are also more interesting than those of the first method. Simulations show that the difference in redundancy performance between the two methods is negligible.

An improved, chattering free, VSC scheme for uncertain dynamical systems

Abstract: A control strategy proposed by the authors in previous works to reduce the effect of chattering led to the solution of a differential inequality involving the first and second derivatives of the control signal and a parameter for a first-order observer. The analysis was carried out assuming this parameter to be constant. In the present paper, this assumption is removed, thus allowing the solution of the differential inequality to be a function of a piecewise continuous first derivative of the control signal, without need for assuming a priori knowledge of bounds to the plant trajectories.

Computer-Aided Design With Spatial Rational B-Spline Motions

Abstract: Using rational motions it is possible to apply many fundamental B-spline techniques to the design of motions. The present paper summarizes the basic theory of rational motions and introduces a linear control structure for piecewise rational motions suitable for geometry processing. Moreover it provides algorithms for the calculation of the surface which is swept out by a moving polyhedron and examines interpolation techniques. The methods presented in this paper can be applied to various problems in computer animation as well as in robotics.

Piecewise smoothness, local invertibility, and parametric analysis of normal maps

Abstract: This paper is concerned with properties of the Euclidean projection map onto a convex set defined by finitely many smooth, convex inequalities and affine equalities. Under a constant rank constraint qualification, we show that the projection map is piecewise smooth PC1 hence Bouligand-differentiable, or directionally differentiable; and a relatively simple formula is given for the B-derivative. These properties of the projection map are used to obtain inverse and implicit function theorems for associated normal maps, using a new characterization of invertibility of a PC1 function in terms of its B-derivative. An extension of the implicit function theorem which does not require local uniqueness is also presented. Degree theory plays a major role in the analysis of both the locally unique case and its extension.

Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems

Abstract: Multigrid methods are very efficient iterative solvers for system of algebraic equations arising from finite element and finite difference discretization of elliptic boundary value problems. The main principle of multigrid methods is to complement the local exchange of information in point-wise iterative methods by a global one utilizing several related systems, called coarse levels, with a smaller number of variables. The coarse levels are often obtained as a hierarchy of discretizations with different characteristic meshsizes, but this requires that the discretization is controlled by the iterative method. To solve linear systems produced by existing finite element software, one needs to create an artificial hierarchy of coarse problems. The principal issue is then to obtain computational complexity and approximation properties similar to those for nested meshes, using only information in the matrix of the system and as little extra information as possible. Such algebraic multigrid method that uses the system matrix only was developed by Ruge. The prolongations were based on the matrix of the system by partial solution from given values at selected coarse points. The coarse grid points were selected so that each point would be interpolated to via so-called strong connections. Our approach is based on smoothed aggregation introduced recently by Vanek. First the set of nodes is decomposed into small mutually disjoint subsets. A tentative piecewise constant interpolation (in the discrete sense) is then defined on those subsets as piecewise constant for second order problems, and piecewise linear for fourth order problems. The prolongation operator is then obtained by smoothing the output of the tentative prolongation and coarse level operators are defined variationally.

Analysis and application of subdivision surfaces

Abstract: Subdivision surfaces are a convenient representation for modeling objects of arbitrary topological type In this dissertation, we investigate the analysis of a piecewise smooth subdivision scheme, and we apply the scheme to reconstruct objects from non-uniformly sampled data points Defined as the limit of repeated refinement of a mesh of 3D control points, subdivision surfaces require analysis to establish convergence to a well-defined, tangent plane smooth $(G\sp1)$ surface Recent research has focused on analyzing smooth surface schemes in which the rules are symmetrical about each vertex and edge However, a scheme for creating surfaces with sharp features has rules that do not exhibit this symmetry In this dissertation, we extend the use of eigenanalysis and characteristic maps to analyze a piecewise smooth subdivision scheme that generalizes quartic triangular B-spline surfaces Subdivision surfaces are suitable for optimized surface fitting and have been used in the reconstruction of objects from 3D data Previous methods have created accurate representations of objects from dense and uniform data samples As a practical low cost alternative, we present an algorithm for creating a subdivision surface from data sampled uniformly along closed curves and non-uniformly within the regions they enclose

Error estimates for finite element methods for scalar conservation laws

Abstract: In this paper, new a posteriors error estimates for the shock-capturing streamline diffusion (SCSD) method and the shock-capturing discontinuous galerkin (SCDG) method for scalar conservation laws are obtained. These estimates are then used to prove that the SCSD method and the SCDG method converge to the entropy solution with a rate of at least $h^{{1 / 8}} $ and $h^{{1 / 8}} $, respectively, in the $L^\infty (L^1 )$-norm. The triangulations are made of general acute simplices and the approximate solution is taken to be piecewise a polynomial of degree k. The result is independent of the dimension of the space.

Ericksen's bar revisited : Energy wiggles

Abstract: This paper addresses the non-uniqueness pointed out by Ericksen in his classical analysis of the equilibrium of a one-dimensional elastic bar with non-convex energy. According to Ericksen, for the bar in a hard device, the piecewise constant functions delivering the global minimum of the energy can have an arbitrary numberN of discontinuities in strain (phase-boundaries). Following some previous work in this area, we regularize the problem in order to resolve this degeneracy. We add two non-local terms to the energy density: one depends on the high (second) derivatives of the displacement, the other contains low (zero) derivatives. The low-derivative term (scaled with a constant β) introduces a strong non-locality, and simulates a three-dimensional interaction with the loading device, forcing the formation of layered microstructures in the process of energy minimization. The high-derivative (strain-gradient) term (scaled with a different constant α), represents a surface energy contribution which penalizes the formation of phase interfaces and prevents the infinite refinement of microstructures. In our description we consider the positions of interfaces as variables. This singles out in a natural way an infinite number of finite-dimensional subspaces, where all the essential nonlinearity is concentrated. In this way we can calculate explicitly the local minimizers (metastable states) and their energy, which turns out to be a multi-valued function of the interface positions and the imposed overall straind. Our approach thus gives an explicit framework for the study of the rich variety of finite-scale equilibrium microstructures for the bar and their stability properties. This allows for the study of a number of properties of phase transitions in solids; in particular their hysteretic behavior. Among our goals is the investigation of the phase diagram of the system, described by the functionN(d, α, β) giving the number of phase-boundaries in the absolute minimizer. We observe the somewhat counterintuitive effect that the energy at the global minimum, as a function of the overall strain, generically develops non-smooth oscillations (wiggles).

Brief Contributions Sigmoid Generators for Neural Computing Using Piecewise Approximations

Abstract: A piecewise second order approximation scheme is proposed for computing the sigmoid function. The scheme provides high performance with low implementation cost; thus, it is suitable for hardwired cost effective neural emulators. It is shown that an implementation of the sigmoid generator outperforms, in both precision and speed, existing schemes using a bit serial pipelined implementation. The proposed generator requires one multiplication, no look-up table and no addition. It has been estimated that the sigmoid output is generated with a maximum computation delay of 21 bit serial machine cycles representing a speedup of 1.57 to 2.23 over other proposals.

H∞ control of nonlinear continuous-time systems based on dynamical fuzzy models

Abstract: This paper presents a solution of the H∞ control problem for a class of continuous-time nonlinear systems. The method is based on a fuzzy dynamical model of the nonlinear system. A suitable piecewise differentiate quadratic (PDQ) Lyapunov function is used to establish asymptotic stability of the closed-loop system. Furthermore, a constructive algorithm is developed to obtain the stabilizing feedback control law. The controller design algorithm involves solving a set of suitable algebraic Riccati equations. An example is given to illustrate the application of the method

Nonlinearity estimation in Hammerstein systems based on ordered observations

Abstract: The nonlinear subsystem of a Hammerstein system is identified, i.e., its characteristic is recovered from input output ohservations of the whole system. The input and disturbance are white stochastic processes. The identified characteristic satisfies a piecewise Lipschitz condition only. Algorithms presented in the paper are calculated from ordered input-output observations, i.e., from pairs of observations arranged in a sequence in which input measurements increase in value. The mean integrated square error converges to zero as the number of observations tends to infinity. Convergence rates are insensitive to the shape of the probability density of the input signal. Results of numerical simulation are also shown.

Sigmoid generators for neural computing using piecewise approximations

Abstract: A piecewise second order approximation scheme is proposed for computing the sigmoid function. The scheme provides high performance with low implementation cost; thus, it is suitable for hardwired cost effective neural emulators. It is shown that an implementation of the sigmoid generator outperforms, in both precision and speed, existing schemes using a bit serial pipelined implementation. The proposed generator requires one multiplication, no look-up table and no addition. It has been estimated that the sigmoid output is generated with a maximum computation delay of 21 bit serial machine cycles representing a speedup of 1.57 to 2.23 over other proposals.

On the Gibbs phenomenon III: recovering exponential accuracy in a sub-interval from a spectral partial sum of a piecewise analytic function

Abstract: We continue the investigation of overcoming the Gibbs phenomenon, i.e., obtaining exponential accuracy at all points, including at the discontinuities themselves, from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. We show that if we are given the first N expansion coefficients of an $L^2 $ function $f(x)$ in terms of either the trigonometric polynomials or the Legendre polynomials, we can construct an exponentially convergent approximation to the point values of $f(x)$ in any sub-interval in which it is analytic.

Classification of Generic Singularities for the Planar Time-Optimal Synthesis

Abstract: This paper is concerned with control systems on the plane with control appearing linearly. It is known that under generic conditions the problem of reaching points from the origin in minimum time admits a regular synthesis. The minimum time function is piecewise smooth, possibly nondifferentiable on a set that is a finite union of embedded submanifolds of dimension 1 or 0, called singularities. The purpose of the present paper is to provide a classification of all types of singularities arising under generic conditions.