Abstract
A smoothed representation (based on natural exponential and logarithmic functions) for the canonical piecewise-linear model, is presented. The result is a completely differentiable formulation that exhibits interesting properties, like preserving the parameters of the original piecewise-linear model in such a way that they can be directly inherited to the smooth model in order to determine their parameters, the capability of controlling not only the smoothness grade, but also the approximation accuracy at specific breakpoint locations, a lower or equal overshooting for high order derivatives in comparison with other approaches, and the additional advantage of being expressed in a reduced mathematical form with only two types of inverse functions (logarithmic and exponential). By numerical simulation examples, this proposal is verified and well-illustrated.
Highlights
Piecewise-linear models are widely used in diverse fields, such as circuit theory, image processing, system identification, economics and financial analysis, etc (Chua and Ying 1983; Chua and Deng 1985; Hasler and Schnetzler 1989; Yamamura and Ochiai 1992; Russo 2006; Feo and Storace 2004, 2007; Brooks 2008)
The factor that prevalently motivates the use of this type of models is the simplicity of their structure which let them be efficiently implemented in both algorithms and hardware
Jimenez‐Fernandez et al SpringerPlus (2016) 5:1612 there are many reported piecewise-linear models (Chua and Kang 1977; Kang and Chua 1978; Chua and Deng 1988; Kahlert and Chua 1990; Guzelis and Goknar 1991; Pospisil 1991; Kevenaar et al 1994; Leenaerts and Van-Bokhoven 1998; Julian et al 1999; Li et al 2001), due to its compact formulation, the most popular is the so-called canonical piecewise-linear representation (Chua and Kang 1977) which is given by the following theorem: Theorem 1 Any single-valued piecewise-linear function with at most σ breakpoints β1 < β2 < . . . < βσ, can be represented by the expression σ y(x) = a + bx + ci|x − βi|
Summary
Piecewise-linear models are widely used in diverse fields, such as circuit theory, image processing, system identification, economics and financial analysis, etc (Chua and Ying 1983; Chua and Deng 1985; Hasler and Schnetzler 1989; Yamamura and Ochiai 1992; Russo 2006; Feo and Storace 2004, 2007; Brooks 2008). It is important to mention that, in accordance with literature such lack of differentiability has been overcome by substituting the basis-function of the piecewise-linear model (in this case, the absolute-value) for its smooth approximation.
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