Scalable Bilinear Transformation (SBT)

• SBT is a parameterized discretization framework that maps continuous-time systems to digital equivalents using tunable shape (α) and time (β) factors.
• It generalizes classical methods like Tustin and Euler, providing explicit control over frequency warping and resonance damping in resonant systems.
• Empirical evaluations demonstrate that SBT achieves lower RMS error and improved harmonic performance compared to state-of-the-art discretization techniques.

The Scalable Bilinear Transformation (SBT) constitutes a principled, flexible discretization framework for mapping continuous-time systems to discrete-time realizations in digital control and signal processing. SBT introduces two user-defined degrees of freedom—a shape factor ($\alpha$) and a time factor ($\beta$)—providing a unified formalism that encompasses and generalizes classical discretization rules, such as Tustin’s method (bilinear), forward and backward Euler, and pre-warped bilinear transforms. SBT directly addresses primary distortions endemic to discrete approximations of resonant systems, namely frequency warping and resonance damping, permitting targeted compensation via explicit manipulation of $\alpha$ and $\beta$. Empirical evaluations, including digital quasi-resonant controller implementation and grid-tied inverter control, demonstrate that SBT achieves substantial reductions in RMS error compared to state-of-the-art alternatives while imposing negligible additional computational burden (Chen et al., 14 Jan 2026).

1. Mathematical Formulation and Derivation

SBT is derived via numerical quadrature applied to the integral form of the continuous-time integrator $u(t) = \int e(t)\,dt$, evaluated over a sampling interval $[(n-1)T, nT]$. Rather than adopting fixed-area approximation rules, SBT parameterizes the interval division using $\alpha$:

$$u(n) - u(n-1) = (1-\alpha)T\,e(n-1) + \alpha T\,e(n)$$

The sampled system is then mapped to the $z$-domain, yielding a general $s$-to-$\beta$0 transform:

$\beta$1

Introducing a time-scaling factor $\beta$2, the final SBT mapping is:

$\beta$3

Here, $\beta$4 (shape factor) modulates the weighting between current and previous sample contributions, and $\beta$5 (time factor) rescales the effective integration window. Special cases include backward Euler $\beta$6, forward Euler $\beta$7, Tustin $\beta$8, and pre-warped Tustin $\beta$9 (Chen et al., 14 Jan 2026).

2. Analysis of Frequency Warping and Resonance Damping

SBT provides a theoretical framework for quantifying two critical sources of distortion in discrete resonant controllers:

• Frequency warping: Discretization causes nonlinear mapping of frequency, particularly problematic for resonant poles.
• Resonance damping: The real part of mapped poles can deviate from their continuous counterparts, introducing artificial damping.

The transformation of an $\alpha$0-domain pole $\alpha$1 under SBT is:

$\alpha$2

$\alpha$3

with $\alpha$4, $\alpha$5, $\alpha$6.

Increasing $\alpha$7 yields greater damping of resonances, while $\alpha$8 minimizes this effect. By suitable choice of $\alpha$9, frequency warping can be nullified exactly at a target frequency. This simultaneous, decoupled control over both artifacts is a distinctive property of SBT (Chen et al., 14 Jan 2026).

3. Relationship to Existing Discretization Methods

SBT subsumes and extends classical discretization frameworks. The table below catalogs key instantiations (all with $\beta$0, unless otherwise stated):

Method	$\beta$1	$\beta$2	$\beta$3–$\beta$4 Mapping
Backward Euler	1	1	$\beta$5
Forward Euler	0	1	$\beta$6
Tustin (bilinear)	$\beta$7	1	$\beta$8
Prewarped Tustin	$\beta$9	$u(t) = \int e(t)\,dt$0	$u(t) = \int e(t)\,dt$1

A direct consequence is that SBT can reproduce the error-minimization properties of pre-warped Tustin, and extend these further by minimizing amplitude and phase distortion simultaneously over parameterized intervals (Chen et al., 14 Jan 2026).

4. Parameter Selection and Optimization Strategy

Parameter choice for $u(t) = \int e(t)\,dt$2 is task-dependent:

• Straightforward/prewarped design: $u(t) = \int e(t)\,dt$3, $u(t) = \int e(t)\,dt$4. This guarantees zero frequency error at the target frequency $u(t) = \int e(t)\,dt$5 and minimal resonance damping.
• Optimal tuning: Minimize a user-defined loss, e.g., integrated squared error between discrete and continuous frequency response over $u(t) = \int e(t)\,dt$6:

$u(t) = \int e(t)\,dt$7

In most practical applications, the explicit prewarped choice is nearly optimal, but full loss-based optimization can offer marginal improvements if required (Chen et al., 14 Jan 2026).

5. Application to Resonant and Quasi-Resonant Controllers

SBT delivers explicit-form coefficient mappings for rational $u(t) = \int e(t)\,dt$8-domain approximations of continuous controllers. For the Quasi-Resonant (QR) controller

$u(t) = \int e(t)\,dt$9

SBT substitution ($[(n-1)T, nT]$0), and cross-multiplication yield a second-order IIR difference equation with coefficients

$[(n-1)T, nT]$1

as enumerated in Table III of (Chen et al., 14 Jan 2026). SBT applies analogously to multi-stage controllers such as PI+QR for grid-tied inverters, generating stable and high-fidelity digital implementations without heuristic tuning.

6. Empirical Evaluation and Performance

Simulation and hardware-in-the-loop experiments validate SBT's efficacy across multiple performance metrics:

• Frequency-response modeling on a QR controller demonstrates SBT achieves a magnitude error at the target frequency $[(n-1)T, nT]$2 dB with the prewarped choice.
• PLECS simulation of magnitude RMSE across frequencies $[(n-1)T, nT]$3–$[(n-1)T, nT]$4 kHz reports $[(n-1)T, nT]$5.
• Experimental realization on a TMS320F28P65 board for grid-tied inverter regulation achieves $[(n-1)T, nT]$6.
• Total harmonic distortion improvement under injected disturbance, measured as THDi, yields $[(n-1)T, nT]$7 for PI+QR SBT, compared to $[(n-1)T, nT]$8 (PI), $[(n-1)T, nT]$9 (PI+QR Euler), $\alpha$0 (PI+QR Tustin), and $\alpha$1 (PI+QR SOTA) (Chen et al., 14 Jan 2026).

A plausible implication is that SBT can be broadly adopted for precision-critical digital control applications, offering superior accuracy without increased implementation complexity.

7. Summary and Significance

The Scalable Bilinear Transformation formalizes a parameterized extension to established $\alpha$2-to-$\alpha$3 mapping techniques, introducing tunable control over frequency and amplitude distortions via global $\alpha$4 design. It unifies prior indirect discretization methods, supports analytic optimization, and has demonstrated superiority in both simulated and real-world digital control tasks with respect to RMS error and harmonic suppression. SBT is thus positioned as a general-purpose tool for the principled discretization of continuous-time systems, particularly where high-fidelity resonance preservation is required (Chen et al., 14 Jan 2026).