The Best Piecewise Linearization of Nonlinear Functions
ABSTRACT
In this paper, we propose a method for finding the best piecewise linearization of nonlinear functions. For this aim, we try to obtain the best approximation of a nonlinear function as a piecewise linear function. Our method is based on an optimization problem. The optimal solution of this optimization problem is the best piecewise linear approximation of nonlinear function. Finally, we examine our method to some examples.
In this paper, we propose a method for finding the best piecewise linearization of nonlinear functions. For this aim, we try to obtain the best approximation of a nonlinear function as a piecewise linear function. Our method is based on an optimization problem. The optimal solution of this optimization problem is the best piecewise linear approximation of nonlinear function. Finally, we examine our method to some examples.
Cite this paper
Mazarei, M. , Behroozpoor, A. and Kamyad, A. (2014) The Best Piecewise Linearization of Nonlinear Functions. Applied Mathematics, 5, 3270-3276. doi: 10.4236/am.2014.520305.
Mazarei, M. , Behroozpoor, A. and Kamyad, A. (2014) The Best Piecewise Linearization of Nonlinear Functions. Applied Mathematics, 5, 3270-3276. doi: 10.4236/am.2014.520305.
References
[1] Aranda-Bricaire, E., Kotta, U. and Moog, C. (1996) Linearization of Discrete-Time Systems. SIAM Journal on Control and Optimization, 34, 1999-2023.
http://dx.doi.org/10.1137/S0363012994267315 [2] Jouan, P. (2003) Immersion of Nonlinear Systems into Linear Systems Modulo Output Injection. SIAM Journal on Control and Optimization, 41, 1756-1778.
http://dx.doi.org/10.1137/S0363012901391706 [3] Sladecek, L. (2003) Exact Linearization of Stochastic Dynamical Systems by State Space Coordinate Transformation and Feedback Ig-Linearization. Applied Mathematics E-Notes, 3, 99-106. [4] Vahidian Kamyad, A., Hashemi Mehne, H. and Hashemi Borzabadi, A. (2005) The Best Linear Approximation for Nonlinear Systems. Applied Mathematics and Computation, 167, 1041-1061.
http://dx.doi.org/10.1016/j.amc.2004.08.002 [5] Herdem, S. and Koksal, M. (2002) A Fast Algorithm to Compute Steady-State Solution of Nonlinear Circuits by Piecewise Linearization. Computers and Electrical Engineering, 28, 91-101. [6] Gunnerud, V., Foss, B.A., Mckinnon, K.I.M. and Nygreen, B. (2012) Oil Production Optimization Solved by Piecewise Linearization in a Branch and Price Framework. Computers and Operations Research, 39, 2469-2477.
http://dx.doi.org/10.1016/j.cor.2011.12.013 [7] Ramos, J.I. and Garcia-Lopez, C.M. (1997) Nonstandard Finite Difference Equations for ODEs and 1-D PDEs Based on Piecewise Linearization. Applied Mathematics and Computations, 86, 11-36.
[1] Aranda-Bricaire, E., Kotta, U. and Moog, C. (1996) Linearization of Discrete-Time Systems. SIAM Journal on Control and Optimization, 34, 1999-2023.
http://dx.doi.org/10.1137/S0363012994267315 [2] Jouan, P. (2003) Immersion of Nonlinear Systems into Linear Systems Modulo Output Injection. SIAM Journal on Control and Optimization, 41, 1756-1778.
http://dx.doi.org/10.1137/S0363012901391706 [3] Sladecek, L. (2003) Exact Linearization of Stochastic Dynamical Systems by State Space Coordinate Transformation and Feedback Ig-Linearization. Applied Mathematics E-Notes, 3, 99-106. [4] Vahidian Kamyad, A., Hashemi Mehne, H. and Hashemi Borzabadi, A. (2005) The Best Linear Approximation for Nonlinear Systems. Applied Mathematics and Computation, 167, 1041-1061.
http://dx.doi.org/10.1016/j.amc.2004.08.002 [5] Herdem, S. and Koksal, M. (2002) A Fast Algorithm to Compute Steady-State Solution of Nonlinear Circuits by Piecewise Linearization. Computers and Electrical Engineering, 28, 91-101. [6] Gunnerud, V., Foss, B.A., Mckinnon, K.I.M. and Nygreen, B. (2012) Oil Production Optimization Solved by Piecewise Linearization in a Branch and Price Framework. Computers and Operations Research, 39, 2469-2477.
http://dx.doi.org/10.1016/j.cor.2011.12.013 [7] Ramos, J.I. and Garcia-Lopez, C.M. (1997) Nonstandard Finite Difference Equations for ODEs and 1-D PDEs Based on Piecewise Linearization. Applied Mathematics and Computations, 86, 11-36.