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A Minimal Cardinality Solution to Fitting Sawtooth piecewise-linear functions

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2022年第3期192卷 930-959页

作者： Allen, Cody de Oliveira, Mauricio Univ Calif San Diego San Diego CA 92103 USA

In this paper, we explore a method to parameterize a linear function with jump discontinuities, which we refer to as a "sawtooth" function, and then develop theory and algorithms for estimating the function parameters from noisy data in a least-squares framework. Because there will always exist a sawtooth function that exactly fits a given data set, one is led to bounding the maximum number of jumps the sawtooth function can have in order to obtain reasonable practical estimates. The main contribution of the paper is a proof that cardinality of the optimal solutions to a relaxation of the associated least-squares problem in which a constraint on the cardinality of the solutions is replaced by a 1-norm constraint on the vector of jumps is a monotonic function of the parameter of the relaxation. This property allows one to avoid dealing with the combinatorial cardinality constraint and quickly explore solutions using the proposed convex relaxation. A fitting algorithm based on the proposed results is developed and illustrated with a simple numerical example.

关键词： Least squares Cardinality constraint piecewise-linear functions

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Digital architectures realizing piecewise-linear multivariate functions: Two FPGA implementations

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INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS 2011年第1期39卷 1-15页

作者： Storace, Marco Poggi, Tomaso Univ Genoa Biophys & Elect Engn Dept Via Opera Pia 11A I-16145 Genoa Italy

Digital architectures for the circuit realization of multivariate piecewise-linear (PWL) functions are reviewed and compared. The output of the circuits is a digital word representing the value of the PWL function at the n-dimensional input. In particular, we propose two architectures with different levels of parallelism/complexity. PWL functions with n=3 inputs are implemented on an FPGA and experimental results are shown. The accuracy in the representation of PWL functions is tested through three benchmark examples, two concerning three-variate static functions and one concerning a dynamical control system defined by a bi-variate PWL function. Copyright (C) 2009 John Wiley & Sons, Ltd.

关键词： piecewise-linear functions digital architecture FPGA nonlinear circuit

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Research-oriented junior/senior design projects: An analog circuit design example

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IEEE TRANSACTIONS ON EDUCATION 2004年第1期47卷 93-99页

作者： Halalu, WM Elwakil, AS Univ Sharjah Dept Elect & Elect Engn Sharjah U Arab Emirates

Briefly, the advantages of offering research-oriented design projects for undergraduate students of particular interest and talent are discussed. A successful research-based analog circuit design project is described as an example. The project is concerned with developing bipolar transistor circuit cells capable of realizing arbitrary nonlinear current transfer characteristics in piecewise-linear form using the Static Translinear Principle (STLP). The obtained results are shown.

关键词： analog electronics nonlinear circuits piecewise-linear functions translinear circuits

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Design of a nonlinear controller based on a piecewise-linear Hammerstein model

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SYSTEMS & CONTROL LETTERS 2008年第4期57卷 332-339页

作者： Dolanc, Gregor Strmcnik, Stanko Jozef Stefan Inst Dept Syst & Control Ljubljana 1000 Slovenia

This paper presents a nonlinear controller, which can be used to control single-input single-output nonlinear systems that can be approximated in terms of Hammerstein models. The proposed nonlinear controller is based on the principles of a classic linear pole placement controller and the piecewise-linear Hammerstein model. The controller can be used to control processes with highly nonlinear or even discontinuous static functions. while keeping simple controller structure and a very low computational burden. (C) 2007 Elsevier B.V. All rights reserved.

关键词： nonlinear control nonlinear systems discontinuous systems piecewise-linear Hammerstein model piecewise-linear functions pole placement controller design

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Identification of nonlinear systems using a piecewise-linear Hammerstein model

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SYSTEMS & CONTROL LETTERS 2005年第2期54卷 145-158页

作者： Dolanc, G Strmcnik, S Jozef Stefan Inst Dept Syst & Control Ljubljana 1000 Slovenia

In the paper a method for nonlinear system identification is proposed. It is based on a piecewise-linear Hammerstein model, which is linear in the parameters. The model and the identification algorithm are adapted to allow the parameter identification in the presence of a special form of the excitation signal. The identification method is derived from a recursive least-squares algorithm, which is properly adapted to take into account the proposed model structure and the properties of the identification signal. The applicability of the approach is illustrated by an example in which a discontinuous nonlinear static function is connected to a dynamic block. (C) 2004 Elsevier B.V. All rights reserved.

关键词： nonlinear systems Hammerstein systems piecewise-linear functions system identification parameter estimation

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Model-predictive dynamic control allocation scheme for reentry vehicles

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JOURNAL OF GUIDANCE CONTROL AND DYNAMICS 2007年第1期30卷 100-113页

作者： Luo, Yu Serrani, Andrea Yurkovich, Stephen Oppenheimer, Michael W. Doman, David B. Ohio State Univ Dept Elect & Comp Engn Columbus OH 43210 USA USAF Res Lab Control Design & Anal Branch Wright Patterson AFB OH 45433 USA

Allocation of control authority among redundant control effectors, under hard constraints, is an important component of the inner loop of a reentry vehicle guidance and control system. Whereas existing control allocation schemes generally neglect actuator dynamics, thereby assuming A static relationship between control surface deflections and moments about a three-body axis, in this work a dynamic control allocation scheme is developed that implements a form, of model-predictive control. In the approach proposed here, control allocation is posed as a sequential quadratic programming problem with constraints, which can also be cast into a linear complementarity problem and therefore solved in a finite number of iterations. Accounting directly for nonnegligible dynamics of the actuators with hard constraints, the scheme extends existing algorithms by providing asymptotic tracking of time-varying input commands for this class of applications. To illustrate the effectiveness of the proposed scheme, a high-fidelity simulation for an experimental reusable launch vehicle is used, in which, results are compared with those of static control allocation schemes in situations of actuator failures.

关键词： CONSTRAINED CONTROL ALLOCATION NONlinear CONTROL ALLOCATION piecewise-linear functions PROGRAMMING APPROACH STABILITY EFFICIENT X-33

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Simplicial computation: A methodology to compute vector-vector multiplications with reduced complexity

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INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS 2021年第11期49卷 3766-3788页

作者： Julian, P. Andreou, A. G. Villemur, M. Rodriguez, N. Silicon Austria Labs Embedded Syst Div Linz Austria UNS CONICET IIIE DIEC Bahia Blanca Buenos Aires Argentina Johns Hopkins Univ ECE Dept Baltimore MD USA

In this paper, we propose the use of the simplicial algorithm, originally proposed to implement piecewise-linear functions, to compute a digital vector-vector multiplication (VVM) without multiplications. The main contributions of the proposed methodology are (a) an improved error propagation with respect to parameter quantization;(b) a more efficient digital implementation with respect to area, energy, and speed in the case of large number of inputs;and (c) an intrinsic capability to produce multivariable nonlinear processing. We show that when quantization of inputs and parameters are considered, the simplicial method achieves the same accuracy with fewer representation bits for the parameters, assuming the same quantization for the inputs. Actually, in the particular case of a large number of inputs, the simplicial method needs half the number of parameter bits of a linear combination plus one.

关键词： linear combination piecewise-linear functions vector-vector multiplication

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Nonparametric Random Variate Generation Using a piecewise-linear Cumulative Distribution Function

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COMMUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION 2012年第4期41卷 449-468页

作者： Kaczynski, W. Leemis, L. Loehr, N. McQueston, J. US Mil Acad Dept Math Sci West Point NY 10996 USA Coll William & Mary Dept Math Williamsburg VA 23185 USA Virginia Polytech Inst & State Univ Dept Math Blacksburg VA 24061 USA

The standard approach to solving the interpolation problem for a trace-driven simulation involving a continuous random variable is to construct a piecewise-linear cdf that fills in the gaps between the data values. Some probabilistic properties of this estimator are derived, and three extensions to the standard approach (matching moments, weighted values, and right-censored data) are presented, along with associated random variate generation algorithms. The algorithm is a nonparametric blackbox variate generator requiring only observed data from the user.

关键词： Lifetime distributions Modeling piecewise-linear functions Simulation

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Mesh Topological Optimization for Improving piecewise-linear Image Registration

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JOURNAL OF MATHEMATICAL IMAGING AND VISION 2010年第2期37卷 166-182页

作者： Gonzalez, Javier Arevalo, Vicente Univ Malaga Dept Syst Engn & Automat E-29071 Malaga Spain

This paper presents a mutual-information based optimization algorithm for improving piecewise-linear (PWL) image registration. PWL-registration techniques, which are well-suited for registering images of the same scene with relative local distortions, divide the images in conjugate triangular patches that are individually mapped through affine transformations. For this process to be accurate, each pair of corresponding image triangles must be the projections of a planar surface in space;otherwise, the registration incurs in errors that appear in the resultant registered image as local distortions (distorted shapes, broken lines, etc.). Given an initial triangular mesh onto the images, we propose an optimization algorithm that, by swapping edges, modifies the mesh topology looking for an improvement in the registration. For detecting the edges to be swapped we employ a cost function based on the mutual information (MI), a metric for registration consistency more robust to image radiometric differences than other well-known metrics such as normalized cross correlation (NCC). The proposed method has been successfully tested with different sets of test images, both synthetic and real, acquired from different angles and lighting conditions.

关键词： Image registration piecewise-linear functions Mesh optimization Greedy search Mutual information

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Bivariate Nonparametric Random Variate Generation Using a piecewise-linear Cumulative Distribution Function

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COMMUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION 2012年第4期41卷 469-496页

An extension of the univariate case of nonparametric random variate generation using a piecewise-linear cumulative distribution function is developed. The method is a blackbox variate generation technique requiring only data pairs from the modeler. The technique is a novel nonparametric approach to density estimation, and generating variates for simulation is accomplished without explicitly computing the estimated joint density, thereby speeding up random point generation. The method presented effectively captures marginal distributions with multiple modes. The algorithm presented uses the convex hull of the observed data as a preliminary support, then generates the first element of the two-dimensional random vector via inversion of the marginal piecewise-linear cdf, and the second element from a conditional weighted piecewise-linear cdf created from selected values of the second variable.

关键词： Density estimation Marginal distributions Modeling piecewise-linear functions

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