Chaohou Liu | Search arXiv

arXiv2026

A Novel $αβ$-Approximation Method Based on Numerical Integration for Discretizing Continuous Systems

In this article, we propose a novel discretization method based on numerical integration for discretizing continuous systems, termed the $αβ$-approximation or Scalable Bilinear Transformation (SBT). In contrast to existing methods, the proposed method consists of two factors, i.e., shape factor ($α$) and time factor ($β$). Depending on the discretization technique applied, we identify two primary distortion modes in discrete resonant controllers: frequency warping and resonance damping. We further provide a theoretical explanation for these distortion modes, and demonstrate that the performance of the method is superior to all typical methods. The proposed method is implemented to discretize a quasi-resonant (QR) controller on a control board, achieving 25\% reduction in the root-mean-square error (RMSE) compared to the SOTA method. Finally, the approach is extended to discretizing a resonant controller of a grid-tied inverter. The efficacy of the proposed method is conclusively validated through favorable comparisons among the theory, simulation, and experiments.

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arXiv2025

Optimized Design of the Generalized Bilinear Transformation for Discretizing Analog Systems

A common approach to digital system design involves transforming a continuous-time (s-domain) transfer function into the discrete-time (z-domain) using methods such as Euler or Tustin. These transformations are shown to be specific cases of the Generalized Bilinear Transformation (GBT), characterized by a design parameter, $α$, whose physical interpretation and optimal selection remain inadequately explored. In this paper, we propose an alternative derivation of the GBT derived by employing a new hexagonal shape to approximate the enclosed area of the error function, and we define the parameter $α$ as a shape factor. We reveal, for the first time, the physical meaning of $α$ as the backward rectangular ratio of the proposed hexagonal shape. Through domain mapping, the stable range of is rigorously established to be [0.5, 1]. Depending on the operating frequency and the chosen $α$, we observe two distinct distortion modes, i.e., the magnitude and phase distortion. We further develop an optimal design method for $α$ by minimizing a normalized magnitude or phase error objective function. The effectiveness of the proposed method is validated through the design and testing of a low-pass filter (LPF), demonstrating strong agreement between theoretical predictions and experimental results.

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