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

TL;DR

This paper introduces the αβ-approximation, termed the Scalable Bilinear Transformation (SBT), as a two-parameter framework for discretizing continuous systems via numerical integration. By adding a time factor $β$ to the bilinear form, SBT enables more accurate placement of resonant poles and reduces discretization-induced distortions such as frequency warping and resonance damping, outperforming Euler, Tustin, and state-of-the-art methods. The authors derive SBT from SHA-based numerical integration, establish stability conditions (notably $α \ge 0.5$ with a stability circle in the $z$-plane), and show that SBT encompasses classic discretizations as special cases. They apply SBT to discretize a quasi-resonant controller and extend the approach to a grid-tied inverter, with comprehensive theory, simulations, and hardware experiments validating significant performance gains, including a 25% RMSE reduction over SOTA and a THDi improvement to 5.44%. Overall, SBT offers a practical, computationally efficient, and highly accurate discretization strategy for digital resonant controllers in power electronics and related continuous systems.

Abstract

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.

Figures (16)

• Figure 1: Bode plot of quasi-resonant controller under different discretization methods. (resonant frequency = 950 Hz, sampling frequency =20 kHz)
• Figure 2: Different approximation approaches based on the numerical integration. (a) Rectangular approximation. (b) Trapezoidal approximation. (c) Hexagonal approximation. (d) Scalable Hexagonal Approximation (SHA).
• Figure 3: Mapping of s-plane to z-plane
• Figure 4: Frequency response of the QR controllers
• Figure 5: Magnitude error under different discretization methods