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Transforming the canonical piecewise-linear model into a smooth-piecewise representation
Victor M Jimenez-Fernandez1, Maribel Jimenez-Fernandez2, Hector Vazquez-Leal1
1Faculty of Electronic Instrumentation, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Zona Universitaria, 91000 Xalapa, Veracruz Mexico.
Springerplus
|September 22, 2016
Summary
A novel smoothed representation of the piecewise-linear model is introduced using exponential and logarithmic functions. This differentiable formulation enhances parameter inheritance, controls smoothness, and improves approximation accuracy.
Area of Science:
- Mathematics
- Numerical Analysis
- Computational Modeling
Background:
- Piecewise-linear models are widely used but often lack differentiability.
- Existing smoothing techniques may not preserve model parameters or offer precise control over approximation.
Purpose of the Study:
- To present a novel, completely differentiable smoothed representation for the canonical piecewise-linear model.
- To introduce a method that allows direct parameter inheritance from the original model.
- To enable control over smoothness grade and approximation accuracy at breakpoints.
Main Methods:
- Development of a smoothed representation using natural exponential and logarithmic functions.
- Formulation of a completely differentiable mathematical model.
- Verification through numerical simulation examples.
Main Results:
- The proposed smoothed model directly inherits parameters from the original piecewise-linear model.
- The method allows adjustable smoothness and approximation accuracy at specific locations.
- It demonstrates lower or equal overshooting for high-order derivatives compared to other methods.
- The formulation is mathematically concise, utilizing only logarithmic and exponential functions.
Conclusions:
- The presented smoothed representation offers a versatile and accurate alternative to traditional piecewise-linear models.
- This approach facilitates direct parameter transfer and fine-tuned control over model behavior.
- Numerical simulations confirm the efficacy and advantages of the proposed method.
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