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Journal articles on the topic 'Piecewise affine functions'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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1

Gorokhovik, V. V., O. I. Zorko, and G. Birkhoff. "Piecewise affine functions and polyhedral sets∗." Optimization 31, no. 3 (1994): 209–21. http://dx.doi.org/10.1080/02331939408844018.

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2

Eghbal, Najmeh, Naser Pariz, and Ali Karimpour. "Discontinuous piecewise quadratic Lyapunov functions for planar piecewise affine systems." Journal of Mathematical Analysis and Applications 399, no. 2 (2013): 586–93. http://dx.doi.org/10.1016/j.jmaa.2012.09.054.

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3

Veeser, Andreas. "Positivity Preserving Gradient Approximation with Linear Finite Elements." Computational Methods in Applied Mathematics 19, no. 2 (2019): 295–310. http://dx.doi.org/10.1515/cmam-2018-0017.

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AbstractPreserving positivity precludes that linear operators onto continuous piecewise affine functions provide near best approximations of gradients. Linear interpolation thus does not capture the approximation properties of positive continuous piecewise affine functions. To remedy, we assign nodal values in a nonlinear fashion such that their global best error is equivalent to a suitable sum of local best errors with positive affine functions. As one of the applications of this equivalence, we consider the linear finite element solution to the elliptic obstacle problem and derive that its e

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4

Cheraghi-Shami, Farideh, Ali-Akbar Gharaveisi, Malihe M. Farsangi, and Mohsen Mohammadian. "Discontinuous Lyapunov functions for a class of piecewise affine systems." Transactions of the Institute of Measurement and Control 41, no. 3 (2018): 729–36. http://dx.doi.org/10.1177/0142331218771138.

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In this paper, a Lyapunov-based method is provided to study the local asymptotic stability of planar piecewise affine systems with continuous vector fields. For such systems, the state space is supposed to be partitioned into several bounded convex polytopes. A piecewise affine function, not necessarily continuous on the boundaries of the polytopic partitions, is proposed as a candidate Lyapunov function. Then, sufficient conditions for the local asymptotic stability of the system, including a monotonicity condition at switching instants, are formulated as a linear programming problem. In addi

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5

Kripfganz, Anita, and R. Schulze. "Piecewise affine functions as a difference of two convex functions." Optimization 18, no. 1 (1987): 23–29. http://dx.doi.org/10.1080/02331938708843210.

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6

Kari, Jarkko. "Piecewise Affine Functions, Sturmian Sequences and Wang Tiles." Fundamenta Informaticae 145, no. 3 (2016): 257–77. http://dx.doi.org/10.3233/fi-2016-1360.

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7

Donovan, G. C., J. S. Geronimo, and D. P. Hardin. "Compactly Supported, Piecewise Affine Scaling Functions on Triangulations." Constructive Approximation 16, no. 2 (2000): 201–19. http://dx.doi.org/10.1007/s003659910009.

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8

Xu, Jun, Ton J. J. van den Boom, Bart De Schutter, and Shuning Wang. "Irredundant lattice representations of continuous piecewise affine functions." Automatica 70 (August 2016): 109–20. http://dx.doi.org/10.1016/j.automatica.2016.03.018.

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9

Nguyen, Ngoc Anh, and Sorin Olaru. "A family of piecewise affine control Lyapunov functions." Automatica 90 (April 2018): 212–19. http://dx.doi.org/10.1016/j.automatica.2017.12.052.

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10

Adeeb, S., and V. G. Troitsky. "Locally piecewise affine functions and their order structure." Positivity 21, no. 1 (2016): 213–21. http://dx.doi.org/10.1007/s11117-016-0411-7.

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11

Kristensen, Jan, and Filip Rindler. "Piecewise affine approximations for functions of bounded variation." Numerische Mathematik 132, no. 2 (2015): 329–46. http://dx.doi.org/10.1007/s00211-015-0721-x.

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12

Xu, Jia, Ton van den Boom, and Bart De Schutter. "Optimistic optimization for continuous nonconvex piecewise affine functions." Automatica 125 (March 2021): 109476. http://dx.doi.org/10.1016/j.automatica.2020.109476.

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13

MORINAGA, Eiji, and Kenji HIRATA. "L2-Gain Analysis of Piecewise Affine Systems via Piecewise Quadratic Storage Functions." Transactions of the Society of Instrument and Control Engineers 40, no. 4 (2004): 405–14. http://dx.doi.org/10.9746/sicetr1965.40.405.

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14

Giesl, Peter, and Sigurdur Hafstein. "Existence of piecewise affine Lyapunov functions in two dimensions." Journal of Mathematical Analysis and Applications 371, no. 1 (2010): 233–48. http://dx.doi.org/10.1016/j.jmaa.2010.05.009.

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15

Xu, Jun, Ton J. J. van den Boom, Bart De Schutter, and Xiong-Lin Luo. "Minimal Conjunctive Normal Expression of Continuous Piecewise Affine Functions." IEEE Transactions on Automatic Control 61, no. 5 (2016): 1340–45. http://dx.doi.org/10.1109/tac.2015.2465212.

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16

Gaubert, Stéphane, Zheng Qu, and Srinivas Sridharan. "Maximizing concave piecewise affine functions on the unitary group." Optimization Letters 10, no. 4 (2015): 655–65. http://dx.doi.org/10.1007/s11590-015-0951-y.

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17

Waitman, Sérgio, Paolo Massioni, Laurent Bako, and Gérard Scorletti. "Incremental L2-gain stability of piecewise-affine systems with piecewise-polynomial storage functions." Automatica 107 (September 2019): 224–30. http://dx.doi.org/10.1016/j.automatica.2019.05.050.

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18

KURGANSKYY, OLEKSIY, IGOR POTAPOV, and FERNANDO SANCHO-CAPARRINI. "REACHABILITY PROBLEMS IN LOW-DIMENSIONAL ITERATIVE MAPS." International Journal of Foundations of Computer Science 19, no. 04 (2008): 935–51. http://dx.doi.org/10.1142/s0129054108006054.

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In this paper we analyze the dynamics of one-dimensional piecewise maps. We show that one-dimensional piecewise affine maps are equivalent to pseudo-billiard or so called “strange billiard” systems. We also show that use of more general classes of functions lead to undecidability of reachability problem for one-dimensional piecewise maps.

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19

Chen, Yunsai, Yongjie Pang, Zhao Yang, and Liang Ma. "A New Robust Nonfragile Controller Design Scheme for a Class of Hybrid Systems through Piecewise Affine Models." Mathematical Problems in Engineering 2018 (October 21, 2018): 1–13. http://dx.doi.org/10.1155/2018/1875610.

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This paper investigates the robust H∞ nonfragile control problem for a class of discrete-time hybrid systems based on piecewise affine models. The objective is to develop an admissible piecewise affine nonfragile controller such that the resulting closed-loop system is asymptotically stable with robust H∞ performance γ. By employing a state-control augmentation methodology, some new sufficient conditions for the controller synthesis are formulated based on piecewise Lyapunov functions (PLFs). The controller gains can be obtained via solving a set of linear matrix inequalities. Simulation examp

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20

F. Hafstein, Sigurdur, Christopher M. Kellett, and Huijuan Li. "Computing continuous and piecewise affine lyapunov functions for nonlinear systems." Journal of Computational Dynamics 2, no. 2 (2015): 227–46. http://dx.doi.org/10.3934/jcd.2015004.

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21

Szűcs, Alexander, Michal Kvasnica, and Miroslav Fikar. "Optimal Piecewise Affine Approximations of Nonlinear Functions Obtained from Measurements." IFAC Proceedings Volumes 45, no. 9 (2012): 160–65. http://dx.doi.org/10.3182/20120606-3-nl-3011.00061.

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22

Gaudioso, Manlio, Giovanni Giallombardo, Giovanna Miglionico, and Adil M. Bagirov. "Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations." Journal of Global Optimization 71, no. 1 (2017): 37–55. http://dx.doi.org/10.1007/s10898-017-0568-z.

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23

Gang, Ting-Ting, Jun Yang, Qing Gao, Yu Zhao, and Jianbin Qiu. "A Fuzzy Approach to Robust Control of Stochastic Nonaffine Nonlinear Systems." Mathematical Problems in Engineering 2012 (2012): 1–17. http://dx.doi.org/10.1155/2012/439805.

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This paper investigates the stabilization problem for a class of discrete-time stochastic non-affine nonlinear systems based on T-S fuzzy models. Based on the function approximation capability of a class of stochastic T-S fuzzy models, it is shown that the stabilization problem of a stochastic non-affine nonlinear system can be solved as a robust stabilization problem of the stochastic T-S fuzzy system with the approximation errors as the uncertainty term. By using a class of piecewise dynamic feedback fuzzy controllers and piecewise quadratic Lyapunov functions, robust semiglobal stabilizatio

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24

Ng, K. F., and X. Y. Zheng. "Error Bounds of Constrained Quadratic Functions and Piecewise Affine Inequality Systems." Journal of Optimization Theory and Applications 118, no. 3 (2003): 601–18. http://dx.doi.org/10.1023/b:jota.0000004873.30548.ca.

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25

Baier, Robert, Philipp Braun, Lars Grüne, and Christopher M. Kellett. "Numerical Calculation of Nonsmooth Control Lyapunov Functions via Piecewise Affine Approximation." IFAC-PapersOnLine 52, no. 16 (2019): 370–75. http://dx.doi.org/10.1016/j.ifacol.2019.11.808.

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26

Conti, Sergio, Matteo Focardi, and Flaviana Iurlano. "Approximation of fracture energies with p-growth via piecewise affine finite elements." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 34. http://dx.doi.org/10.1051/cocv/2018021.

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The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation GSBDp(Ω), p ∈ (1, ∞), their treatment is however hindered by the very low regularity of those functions and by the lack of appropriate density results. We construct here an approximation of GSBDp functions, for p ∈ (1, ∞), with functions which are Lipschitz continuous away from a jump set which is a finite union of closed subsets of C1 hypersurfaces. The strains of the approximating functions converge strongly in Lp to the strain of the target, an

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27

Mousavi, Seyedahmad, та Jinglai Shen. "Solution uniqueness of convex piecewise affine functions based optimization with applications to constrained ℓ1 minimization". ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 56. http://dx.doi.org/10.1051/cocv/2018061.

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In this paper, we study the solution uniqueness of an individual feasible vector of a class of convex optimization problems involving convex piecewise affine functions and subject to general polyhedral constraints. This class of problems incorporates many important polyhedral constrained ℓ1 recovery problems arising from sparse optimization, such as basis pursuit, LASSO, and basis pursuit denoising, as well as polyhedral gauge recovery. By leveraging the max-formulation of convex piecewise affine functions and convex analysis tools, we develop dual variables based necessary and sufficient uniq

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28

Opschoor, Joost A. A., Philipp C. Petersen, and Christoph Schwab. "Deep ReLU networks and high-order finite element methods." Analysis and Applications 18, no. 05 (2020): 715–70. http://dx.doi.org/10.1142/s0219530519410136.

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Approximation rate bounds for emulations of real-valued functions on intervals by deep neural networks (DNNs) are established. The approximation results are given for DNNs based on ReLU activation functions. The approximation error is measured with respect to Sobolev norms. It is shown that ReLU DNNs allow for essentially the same approximation rates as nonlinear, variable-order, free-knot (or so-called “[Formula: see text]-adaptive”) spline approximations and spectral approximations, for a wide range of Sobolev and Besov spaces. In particular, exponential convergence rates in terms of the DNN

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29

Ben-Amram, Amir M. "Mortality of iterated piecewise affine functions over the integers: Decidability and complexity." Computability 4, no. 1 (2015): 19–56. http://dx.doi.org/10.3233/com-150032.

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30

Comaschi, F., B. A. G. Genuit, A. Oliveri, W. P. M. H. Heemels, and M. Storace. "FPGA Implementations of Piecewise Affine Functions Based on Multi-Resolution Hyperrectangular Partitions." IEEE Transactions on Circuits and Systems I: Regular Papers 59, no. 12 (2012): 2920–33. http://dx.doi.org/10.1109/tcsi.2012.2206490.

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31

Milani, Bası́lio E. A. "Piecewise-affine Lyapunov functions for discrete-time linear systems with saturating controls." Automatica 38, no. 12 (2002): 2177–84. http://dx.doi.org/10.1016/s0005-1098(02)00193-0.

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32

Oliveri, Alberto, Martin Reimers, and Marco Storace. "Automatic Domain Partitioning of Piecewise-Affine Simplicial Functions Implementing Model Predictive Controllers." IEEE Transactions on Circuits and Systems II: Express Briefs 62, no. 9 (2015): 886–90. http://dx.doi.org/10.1109/tcsii.2015.2435971.

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33

Li, Huijuan, Sigurður Hafstein, and Christopher M. Kellett. "Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems." Journal of Difference Equations and Applications 21, no. 6 (2015): 486–511. http://dx.doi.org/10.1080/10236198.2015.1025069.

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34

Sun, Weiquan, Min Li, and Kexin Wang. "Approximate explicit model predictive control using high-level canonical piecewise-affine functions." International Journal of Automation and Control 6, no. 1 (2012): 66. http://dx.doi.org/10.1504/ijaac.2012.045441.

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35

Fu, Shasha, Jianbin Qiu, and Wenqiang Ji. "Non-fragile control of fuzzy affine dynamic systems via piecewise Lyapunov functions." Frontiers of Computer Science 11, no. 6 (2016): 937–47. http://dx.doi.org/10.1007/s11704-016-6138-6.

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36

Shen, Ruobing, Bo Tang, Leo Liberti, Claudia D’Ambrosio, and Stéphane Canu. "Learning discontinuous piecewise affine fitting functions using mixed integer programming over lattice." Journal of Global Optimization 81, no. 1 (2021): 85–108. http://dx.doi.org/10.1007/s10898-021-01034-x.

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37

Wang, Ning, Ying Wang, and Maolong Lv. "Fuzzy Adaptive DSC Design for an Extended Class of MIMO Pure-Feedback Non-Affine Nonlinear Systems in the Presence of Input Constraints." Mathematical Problems in Engineering 2019 (January 2, 2019): 1–14. http://dx.doi.org/10.1155/2019/4360643.

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A novel adaptive fuzzy dynamic surface control (DSC) scheme is for the first time constructed for a larger class of (multi-input multi-output) MIMO non-affine pure-feedback systems in the presence of input saturation nonlinearity. First of all, the restrictive differentiability assumption on non-affine functions has been canceled after using the piecewise functions to reconstruct the model for non-affine nonlinear functions. Then, a novel auxiliary system with bounded compensation term is firstly introduced to deal with input saturation, and the dynamic system employed in this work designs a b

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38

Griewank, Andreas, and Andrea Walther. "Polyhedral DC Decomposition and DCA Optimization of Piecewise Linear Functions." Algorithms 13, no. 7 (2020): 166. http://dx.doi.org/10.3390/a13070166.

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For piecewise linear functions f : R n ↦ R we show how their abs-linear representation can be extended to yield simultaneously their decomposition into a convex f ˇ and a concave part f ^ , including a pair of generalized gradients g ˇ ∈ R n ∋ g ^ . The latter satisfy strict chain rules and can be computed in the reverse mode of algorithmic differentiation, at a small multiple of the cost of evaluating f itself. It is shown how f ˇ and f ^ can be expressed as a single maximum and a single minimum of affine functions, respectively. The two subgradients g ˇ and − g ^ are then used to drive DCA a

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39

Lorent, Andrew. "The two-well problem with surface energy." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 136, no. 4 (2006): 795–805. http://dx.doi.org/10.1017/s030821050000473x.

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Let Ω be a bounded Lipschitz domain in R2, let H be a 2 × 2 diagonal matrix with det(H) = 1. Let ε > 0 and consider the functional over AF ∩ W2,1(Ω), where AF is the class of functions from Ω satisfying affine boundary condition F. It can be shown by convex integration that there exists F ∉ SO(2) ∪ SO(2)H and u ∈ AF with I0(u) = 0. Let 0 < ζ1 < 1 < ζ2 < ∞, .In this paper we begin the study of the asymptotics of mε ≔ infBF∩W2,1Iε for such F. This is one of the simplest minimization problems involving surface energy for which we can hope to see the effects of convex integration so

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40

Steentjes, Tom R. V., Alina I. Doban, and Mircea Lazar. "Construction of continuous and piecewise affine Lyapunov functions via a finite-time converse." IFAC-PapersOnLine 49, no. 18 (2016): 13–18. http://dx.doi.org/10.1016/j.ifacol.2016.10.132.

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41

Brox, Piedad, Macarena C. Martínez-Rodríguez, Erica Tena-Sánchez, Iluminada Baturone, and Antonio J. Acosta. "Application specific integrated circuit solution for multi-input multi-output piecewise-affine functions." International Journal of Circuit Theory and Applications 44, no. 1 (2015): 4–20. http://dx.doi.org/10.1002/cta.2058.

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42

Hu, Jun, and Mira Schedensack. "Two low-order nonconforming finite element methods for the Stokes flow in three dimensions." IMA Journal of Numerical Analysis 39, no. 3 (2018): 1447–70. http://dx.doi.org/10.1093/imanum/dry021.

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Abstract In this paper, we propose two low-order nonconforming finite element methods (FEMs) for the three-dimensional Stokes flow that generalize the nonconforming FEM of Kouhia & Stenberg (1995, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Eng, 124, 195–212). The finite element spaces proposed in this paper consist of two globally continuous components (one piecewise affine and one enriched component) and one component that is continuous at the midpoints of interior faces. We prove that the discrete Korn in

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43

Brox, Piedad, Javier Castro-Ramirez, Macarena C. Martinez-Rodriguez, et al. "A Programmable and Configurable ASIC to Generate Piecewise-Affine Functions Defined Over General Partitions." IEEE Transactions on Circuits and Systems I: Regular Papers 60, no. 12 (2013): 3182–94. http://dx.doi.org/10.1109/tcsi.2013.2265962.

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44

Poggi, T., F. Comaschi, and M. Storace. "Digital Circuit Realization of Piecewise-Affine Functions With Nonuniform Resolution: Theory and FPGA Implementation." IEEE Transactions on Circuits and Systems II: Express Briefs 57, no. 2 (2010): 131–35. http://dx.doi.org/10.1109/tcsii.2010.2040316.

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45

Chen, Yu, Yue Sun, Chun-Sen Tang, Yu-Gang Su, and Aiguo Patrick Hu. "Characterizing regions of attraction for piecewise affine systems by continuity of discrete transition functions." Nonlinear Dynamics 90, no. 3 (2017): 2093–110. http://dx.doi.org/10.1007/s11071-017-3786-5.

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46

Sakamoto, Masaru, Duo Dong, Takashi Hamaguchi, Yutaka Ota, Toshiaki Itoh, and Yoshihiro Hashimoto. "Nonlinear Systems Approximation Using a Piecewise Affine Model Based on a Radial Basis Functions Network." Journal of Chemical Engineering of Japan 39, no. 10 (2006): 1078–84. http://dx.doi.org/10.1252/jcej.39.1078.

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47

Guo, Haifeng, Jianbin Qiu, Hui Tian, and Huijun Gao. "Fault detection of discrete-time T–S fuzzy affine systems based on piecewise Lyapunov functions." Journal of the Franklin Institute 351, no. 7 (2014): 3633–50. http://dx.doi.org/10.1016/j.jfranklin.2013.03.012.

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48

Conti, Sergio, Matteo Focardi, and Flaviana Iurlano. "Which special functions of bounded deformation have bounded variation?" Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 1 (2017): 33–50. http://dx.doi.org/10.1017/s030821051700004x.

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Functions of bounded deformation (BD) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation (BV), but are less well understood. We discuss here the relation to BV under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that BD functions that are piecewise affine on a Caccioppoli partition are in GSBV, and we prove that SBDp functions are approximately co

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49

Liu, Zhilin, Lutao Liu, Jun Zhang, and Xin Yuan. "Model Predictive Control of Piecewise Affine System with Constrained Input and Time Delay." Abstract and Applied Analysis 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/937274.

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A model predictive control (MPC) is proposed for the piecewise affine (PWA) systems with constrained input and time delay. The corresponding operating region of the considered systems in state space is described as ellipsoid which can be characterized by a set of vector inequalities. And the constrained control input of the considered systems is solved in terms of linear matrix inequalities (LMIs). An MPC controller is designed that will move the PWA system with time delay from the current operating point to the desired one. Multiple objective functions are used to relax the monotonically decr

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50

Zhu, Yanzheng, and Wei Xing Zheng. "Multiple Lyapunov Functions Analysis Approach for Discrete-Time-Switched Piecewise-Affine Systems Under Dwell-Time Constraints." IEEE Transactions on Automatic Control 65, no. 5 (2020): 2177–84. http://dx.doi.org/10.1109/tac.2019.2938302.

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