Lattice piecewise affine representations on convex projection regions

In this paper, continuous piecewise affine (PWA) functions are realized using the lattice PWA representations on convex projection regions. Detailed proof for the realization is given. Compared with previous methods that develop lattice PWA representations based on base regions, the computational burden can be largely decreased for the current representation. Based on the result, continuous PWA functions can be more efficiently studied and used in modelling as well as approximation of nonlinear systems.

[1]  Piedad Brox Jiménez,et al.  Digital VLSI Implementation of Piecewise-Affine Controllers Based on Lattice Approach , 2015, IEEE Transactions on Control Systems Technology.

[2]  John Lygeros,et al.  Every continuous piecewise affine function can be obtained by solving a parametric linear program , 2013, 2013 European Control Conference (ECC).

[3]  Sven G. Bartels,et al.  Continuous selections of linear functions and nonsmooth critical point theory , 1995 .

[4]  Marco Storace,et al.  Piecewise linear implementation of nonlinear dynamical systems: from theory to practice , 2009 .

[5]  Leon O. Chua,et al.  Canonical piecewise-linear modeling , 1986 .

[6]  Leon O. Chua,et al.  Canonical piecewise-linear representation , 1988 .

[7]  Bart De Schutter,et al.  Equivalence of hybrid dynamical models , 2001, Autom..

[8]  Geoffrey E. Hinton,et al.  Rectified Linear Units Improve Restricted Boltzmann Machines , 2010, ICML.

[9]  Alberto Bemporad,et al.  Min-max control of constrained uncertain discrete-time linear systems , 2003, IEEE Trans. Autom. Control..

[10]  Ran Jin,et al.  Reconfigured piecewise linear regression tree for multistage manufacturing process control , 2012 .

[11]  Leo Breiman,et al.  Hinging hyperplanes for regression, classification, and function approximation , 1993, IEEE Trans. Inf. Theory.

[12]  Robert H. Wilkinson,et al.  A Method of Generating Functions of Several Variables Using Analog Diode Logic , 1963, IEEE Trans. Electron. Comput..

[13]  Alberto Bemporad,et al.  The explicit linear quadratic regulator for constrained systems , 2003, Autom..

[14]  Nadir Farhi.  Piecewise linear car-following modeling , 2011 .

[15]  J. M. Tarela,et al.  Region configurations for realizability of lattice Piecewise-Linear models , 1999 .

[16]  Shuning Wang,et al.  Generalization of hinging hyperplanes , 2005, IEEE Transactions on Information Theory.

[17]  Sergei Ovchinnikov,et al.  Discrete piecewise linear functions , 2008, Eur. J. Comb..

[18]  Manfred Morari,et al.  Multi-Parametric Toolbox 3.0 , 2013, 2013 European Control Conference (ECC).

[19]  Yoshua Bengio,et al.  Maxout Networks , 2013, ICML.

[20]  Bart De Schutter,et al.  Irredundant lattice representations of continuous piecewise affine functions , 2016, Autom..

[21]  Berin Martini,et al.  Large-Scale FPGA-based Convolutional Networks , 2011 .