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L₁ Discretization for Sampled-Data Controller Synthesis via piecewise linear approximation

IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2016年第5期61卷 1143-1157页

作者： Kim, Jung Hoon Hagiwara, Tomomichi Kyoto Univ Dept Elect Engn Kyoto 6158510 Japan Korea Inst Sci & Technol Ctr Robot Res Seoul 136791 South Korea

This paper develops a new discretization method with piecewise linear approximation for the L-1 optimal controller synthesis problem of sampled-data systems, which is the problem of minimizing the L infinity-induced norm of sampled-data systems. We apply fast-lifting on the top of the lifting technique, by which the sampling interval [0, h) is divided into M subintervals with an equal width. The signals on each subinterval are then approximated by linear functions by introducing two types of 'linearizing operators' for input and output, which leads to piecewise linear approximation of sampled-data systems. By using the arguments of preadjoint operators, we provide an important inequality that forms a theoretical basis for tackling the L-1 optimal controller synthesis problem of sampled-data systems more efficiently than the conventional method. More precisely, a mathematical basis for the piecewise linear approximation method associated with the convergence rate is shown through this inequality, and this suggests that the piecewise linear approximation method may drastically outperform the conventional method in the L-1 optimal controller synthesis problem of sampled-data systems. We then provide a discretization procedure of sampled-data systems by which the L-1 optimal controller synthesis problem is converted to the discrete-time l(1) optimal controller synthesis problem. Finally, effectiveness of the proposed method is demonstrated through a numerical example.

关键词： Fast-lifting L-1 optimal control piecewise linear approximation sampled-data systems

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piecewise-linear approximation of Self-Attention and Its Accuracy-Aware Training for Area-Efficient Vision Transformer Inference Accelerator

Piecewise-Linear Approximation of Self-Attention and Its Acc...

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IEEE International Symposium on Quality Electronic Design

作者： Teppei Kawamura Yutaka Masuda Tohru Ishihara Graduate School of Informatics Nagoya University Nagoya Japan

ISBN: (数字)9798331509422

ISBN: (纸本)9798331509439

Vision Transformer (ViT) is a neural network model specialized for image processing, introduced in 2021. The key feature of ViT is self-attention, which allows it to capture a wide range of correlations within an image effectively. However, its biggest drawback is its hardware-hungry characteristics. This paper proposes an areaefficient hardware implementation for self-attention. The proposed approach introduces piecewise linear approximation to both multiplication and Softmax functions. Additionally, we showcase a training method suitable for multiplication and Softmax functions, where the corresponding circuits are significantly simplified through the piecewise-linear approximation. Evaluation using CIFAR-10 demonstrated that our approximation to self-attention achieved comparable image classification accuracy to the model without the approximation. Moreover, with the proposed method, we experimentally confirmed that the area of the self-attention processing circuit could be reduced by 64.0% compared to the existing method applying 8-bit integer quantization to all the weights in self-attention.

关键词： Training Computer vision Accuracy Quantization (signal) piecewise linear approximation Circuits Neural networks Transformers Integrated circuit modeling Image classification

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On the number of segments needed in a piecewise linear approximation

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2010年第2期234卷 437-446页

作者： Frenzen, C. L. Sasao, Tsutomu Butler, Jon T. USN Postgrad Sch Dept Elect & Comp Engn Monterey CA 93943 USA USN Postgrad Sch Dept Appl Math Monterey CA 93943 USA Kyushu Inst Technol Dept Comp Sci & Elect Iizuka Fukuoka 8208502 Japan

The introduction of high-speed circuits to realize an arithmetic function f as a piecewise linear approximation has created a need to understand how the number of segments depends on the interval a <= x <= b and the desired approximation error E. For the case of optimum non-uniform segments, we show that the number of segments is given as s(epsilon) similar to c/root epsilon, (epsilon -> 0(+)), where c = 1/4 integral(b)(a)root vertical bar f ''(x)vertical bar dx. Experimental data shows that this approximation is close to the exact number of segments for a set of 14 benchmark functions. We also show that, if the segments have the same width (to reduce circuit complexity), then the number of segments is given by s(epsilon) similar to c/root epsilon (epsilon -> 0(+)), where c = (b-a)root vertical bar f ''vertical bar max/4. Published by Elsevier B.V.

关键词： piecewise linear approximation Numeric function generators

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Nearest piecewise linear approximation of fuzzy numbers

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FUZZY SETS AND SYSTEMS 2013年 233卷 26-51页

作者： Coroianu, Lucian Gagolewski, Marek Grzegorzewski, Przemyslaw Univ Oradea Dept Math & Informat Oradea 410087 Romania Polish Acad Sci Syst Res Inst PL-01447 Warsaw Poland Warsaw Univ Technol Fac Math & Informat Sci PL-00662 Warsaw Poland

The problem of the nearest approximation of fuzzy numbers by piecewise linear 1-knot fuzzy numbers is discussed. By using 1-knot fuzzy numbers one may obtain approximations which are simple enough and flexible to reconstruct the input fuzzy concepts under study. They might be also perceived as a generalization of the trapezoidal approximations. Moreover, these approximations possess some desirable properties. Apart from theoretical considerations approximation algorithms that can be applied in practice are also given. (C) 2013 Elsevier B.V. All rights reserved.

关键词： approximation of fuzzy numbers Fuzzy number piecewise linear approximation

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EPLA-DSTree: Extending piecewise linear approximation on a Dynamic Segmentation Tree Index in Sensor-Cloud Systems

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IEEE ACCESS 2021年 9卷 123742-123753页

作者： Xie, Bin Li, Qiuhong Wang, Yang Wang, Peng Nanjing Univ Sci & Technol Sch Comp Sci & Engn Nanjing 210094 Peoples R China Shanghai Chang Jiang Intelligent Data Technol Co Shanghai 201600 Peoples R China Fudan Univ Comp Coll Shanghai 200433 Peoples R China

In sensor-cloud applications, a huge amount of time series are generated. Efficient similarity search approaches are necessary for processing these sensor time series data. Concerning of time series search, whole matching similarity search and subsequence similarity search are two main research focuses. In this paper, we study the whole matching similarity search problem. We propose EPLA-DSTree, which extends piecewise linear approximation on a dynamic segmentation tree index for whole matching on time series. Compared with DSTree, EPLA-DSTree improves data locality of nodes by a better time series representation. EPLA-DSTree has a tighter lower bound for nodes which leads to a better query performance. Experiments show that it has a less index building time and a better query performance. To meet the requirements of sensor-cloud applications, we present an parallel EPLA-DSTree on MapReduce, which is a popular cloud programming model.

关键词： Time series analysis Programmable logic arrays Indexes piecewise linear approximation Cloud computing Upper bound Search methods Time series search sensor cloud index

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PARALLEL APPROACHES TO piecewise-linear approximation

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SIGNAL PROCESSING 1994年第3期37卷 415-423页

作者： PAPAKONSTANTINOU, G TSANAKAS, P MANIS, G National Technical University of Athens Department of Electrical and Computer Engineering Zografou Campus GR-157 73 Zografou Greece

Two parallel algorithms have been developed for the piecewise linear approximation (PLA) of digitised curves. The first one is a new general purpose PLA algorithm, based on certain improvements of a serial algorithm. The second one is a peak preserving PLA algorithm particularly suited for the ECG waveform approximation. Both algorithms have been fully implemented, tested and evaluated on a distributed memory parallel architecture, using the OCCAM language. The derived results for both algorithms are encouraging, since they lead to optimal curve approximations, and they are amenable to real-time PLA applications.

关键词： piecewise linear approximation PARALLEL PROCESSING ECG WAVE-FORMS

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A linear programming approach to difference-of-convex piecewise linear approximation

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2024年第2期312卷 493-511页

作者： Kazda, Kody Li, Xiang Queens Univ Dept Chem Engn 19 Div St Kingston ON K7L 3N6 Canada

We address the problem of finding continuous piecewise linear (CPWL) approximations of determinis-tic functions of any dimension that satisfy any predefined error-tolerance, while keeping the number of polytopes that partition the approximation domain low. Specifically, we focus on overcoming the major computational bottleneck of the CPWL approximation Algorithm (CPWL-AA) that has been proposed in the recent literature. CPWL-AA uses the difference-of-convex CPWL representation to search CPWL ap-proximations which can partition the approximation domain to have polytopes of any shape. A computa-tional bottleneck of the method is to solve a mixed-integer linear program (MILP) in which the number of binary variables is large for many problems of practical interest. In this paper, we overcome this by intro-ducing a method that obtains a high quality solution of the MILP by iteratively solving a linear program (LP). We further reduce the computational expense by developing a method that treats some constraints in the LP problem as lazy constraints. Through a computational study we demonstrate that the proposed methods substantially reduce the computation time of CPWL-AA, while maintaining high quality CPWL approximations. With this, we demonstrate that we can generate CPWL approximations that satisfy pre-defined error-tolerances on functions of up to five dimensions within reasonable solution times.& COPY;2023 Elsevier B.V. All rights reserved.

关键词： Integer programming piecewise linear approximation MILP Difference -of -convex

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OPTIMUM UNIFORM piecewise linear-approximation OF PLANAR CURVES

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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 1986年第1期8卷 67-75页

作者： DUNHAM, JG Department of Electrical Engineering Southern Methodist University Dallas TX 75275

Two-dimensional digital curves are often uniformly approximated by polygons or piecewise linear curves. Several algorithms have been proposed in the literature to find such curves. We present an algorithm that finds a piecewise linear curve with the minimal number of segments required to approximate a curve within a uniform error with fixed initial and final points. We compare our optimal algorithm to several suboptimal algorithms with respect to the number of linear segments required in the approximation and the execution time of the algorithm.

关键词： piecewise linear approximation approximation algorithms piecewise linear techniques Iterative algorithms Shape Pattern analysis Pattern classification Feature extraction Merging Strips Author Keywords scan-along Optimal approximation piecewise linear approximation planar curve segmentation polygonal approximation

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A flexible model based on piecewise linear approximation for the analysis of left truncated right censored data with covariates, and applications to Worcester Heart Attack Study data and Channing House data

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STATISTICS IN MEDICINE 2024年第2期43卷 233-255页

作者： Ganguly, Ayon Mitra, Debanjan Balakrishnan, Narayanaswamy Kundu, Debasis Indian Inst Technol Guwahati Dept Math Gauhati Assam India Indian Inst Management Udaipur Quantitat Methods Div Udaipur Rajasthan India McMaster Univ Dept Math & Stat Hamilton ON Canada Indian Inst Technol Kanpur Dept Math & Stat Kanpur Uttar Pradesh India

Left truncated right censored (LTRC) data arise quite commonly from survival studies. In this article, a model based on piecewise linear approximation is proposed for the analysis of LTRC data with covariates. Specifically, the model involves a piecewise linear approximation for the cumulative baseline hazard function of the proportional hazards model. The principal advantage of the proposed model is that it does not depend on restrictive parametric assumptions while being flexible and data-driven. Likelihood inference for the model is developed. Through detailed simulation studies, the robustness property of the model is studied by fitting it to LTRC data generated from different processes covering a wide range of lifetime distributions. A sensitivity analysis is also carried out by fitting the model to LTRC data generated from a process with a piecewise constant baseline hazard. It is observed that the performance of the model is quite satisfactory in all those cases. Analyses of two real LTRC datasets by using the model are provided as illustrative examples. Applications of the model in some practical prediction issues are discussed. In summary, the proposed model provides a comprehensive and flexible approach to model a general structure for LTRC lifetime data.

关键词： cumulative hazard function left truncation maximum likelihood estimators piecewise linear approximation prediction proportional hazards model right censoring

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piecewise linear approximation for identifying wind power ramp events

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AUTOMATIKA 2023年第4期64卷 1306-1314页

作者： Jayalakshmi, J. Linda, M. Mary Ponjesly Coll Engn Dept EEE Nagercoil Tamil Nadu India KIT Kalaignar karunanidhi Inst Technol Dept EEE Kannampaalayam Tamil Nadu India

WPREs (wind power ramp events) are one of the most critical factors affecting the security and protection of the electrical system. Accurate ramp event detection may help power systems better manage extreme events and reduce financial damage. In this study, We present an improved piecewise linear approximation for recognizing wind ramps in Kanyakumari district. In practise, wind power ramps can be decreased by properly managing and dispatching flexible reserve and associated services. This necessitates the use of proper ramp detection techniques as well as precise ramp forecasts. The method's plan to break down wind power signal into increasing with increasing ramps, making ramp identification easier and ensuring that all conceivable ramps of varying lengths are identified. Using observed wind power data, the ramp detection method is used to assess the performance of an energy wind farm. The results reveal that identifying wind power ramps using the segmentation method is equivalent to optical ramp identification.

关键词： Wind power ramp event improved optimal ramp detection algorithm piecewise linear approximation

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