--- title: Scalable Bilinear Transformation (SBT) url: https://www.emergentmind.com/topics/scalable-bilinear-transformation-sbt type: topic --- # Scalable Bilinear Transformation (SBT) 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 [2601.09549]. ## 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-$z$ transform: \[ s = \frac{1}{T} \frac{z-1}{\alpha z + (1-\alpha)} \] Introducing a time-scaling factor $T' = \beta T$, the final SBT mapping is: \[ s = \frac{1}{\beta T} \frac{z-1}{\alpha z + (1-\alpha)}, \quad \alpha \in [0.5, 1],\ \beta > 0 \] Here, $\alpha$ (shape factor) modulates the weighting between current and previous sample contributions, and $\beta$ (time factor) rescales the effective integration window. Special cases include backward Euler $(\alpha=1,\ \beta=1)$, forward Euler $(\alpha=0,\ \beta=1)$, Tustin $(\alpha=0.5,\ \beta=1)$, and pre-warped Tustin $(\alpha=0.5,\ \beta=\tan(\omega_nT/2)/(\omega_nT/2))$ [2601.09549]. ## 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 $s$-domain pole $s_0 = \sigma_0 + j\omega_0$ under SBT is: \[ \sigma_s = \frac{1}{\beta T} \frac{ \alpha (\gamma_z-1)^2 + (\gamma_z-1) + \alpha\zeta_z^2 }{ [\alpha\gamma_z+(1-\alpha)]^2 + [\alpha\zeta_z]^2 } \] \[ \omega_s = \frac{1}{\beta T} \frac{\zeta_z}{ [\alpha\gamma_z+(1-\alpha)]^2 + [\alpha\zeta_z]^2 } \] with $z = e^{\beta s_0 T}$, $\gamma_z = \Re\left\{ \frac{z-1}{\alpha z + (1-\alpha)} \right\}$, $\zeta_z = \Im\left\{ \frac{z-1}{\alpha z + (1-\alpha)} \right\}$. Increasing $\alpha \to 1$ yields greater damping of resonances, while $\alpha \to 0.5$ minimizes this effect. By suitable choice of $\beta$, frequency warping can be nullified exactly at a target frequency. This simultaneous, decoupled control over both artifacts is a distinctive property of SBT [2601.09549]. ## 3. Relationship to Existing Discretization Methods SBT subsumes and extends classical discretization frameworks. The table below catalogs key instantiations (all with $\beta=1$, unless otherwise stated): | Method | $\alpha$ | $\beta$ | $s$–$z$ Mapping | |---------------------|----------|-----------------------------|-----------------------------------------------------------| | Backward Euler | 1 | 1 | $s=(z-1)/(Tz)$ | | Forward Euler | 0 | 1 | $s=(z-1)/T$ | | Tustin (bilinear) | $1/2$ | 1 | $s=(2/T)(z-1)/(z+1)$ | | Prewarped Tustin | $1/2$ | $\tan(\omega_nT/2)/(\omega_nT/2)$ | $s=(1/\beta T)(z-1)/(\frac{1}{2}z+\frac{1}{2})$ | 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 [2601.09549]. ## 4. Parameter Selection and Optimization Strategy Parameter choice for $(\alpha,\ \beta)$ is task-dependent: - **Straightforward/prewarped design**: $\alpha=0.5$, $\beta=\tan(\omega_nT/2)/(\omega_nT/2)$. This guarantees zero frequency error at the target frequency $\omega_n$ and minimal resonance damping. - **Optimal tuning**: Minimize a user-defined loss, e.g., integrated squared error between discrete and continuous frequency response over $[\omega_{\min}, \omega_{\max}]$: \[ \min_{0.5 \leq \alpha \leq 1,\ \beta > 0} Q_\mathrm{loss}(\alpha,\beta),\quad Q_\mathrm{loss}=\int_{\omega_{\min}}^{\omega_{\max}} \big| G_\mathrm{d}(j\omega; \alpha,\beta) - G(j\omega) \big|^2 d\omega \] In most practical applications, the explicit prewarped choice is nearly optimal, but full loss-based optimization can offer marginal improvements if required [2601.09549]. ## 5. Application to Resonant and Quasi-Resonant Controllers SBT delivers explicit-form coefficient mappings for rational $z$-domain approximations of continuous controllers. For the Quasi-Resonant (QR) controller \[ G_{QR}(s) = \frac{2K_r \omega_c s}{s^2 + 2\omega_c s + \omega_n^2}, \] SBT substitution ($s\mapsto \frac{1}{\beta T} \frac{z-1}{\alpha z + (1-\alpha)}$), and cross-multiplication yield a second-order IIR difference equation with coefficients \[ \begin{align*} a_2 &= 2\alpha\beta K_r\omega_c T, \\ a_1 &= - (4\alpha - 2)\beta K_r\omega_c T, \\ a_0 &= - (2 - 2\alpha)\beta K_r\omega_c T, \\ b_2 &= 1 + 2\alpha\beta\omega_c T + (\alpha\beta\omega_n T)^2, \\ b_1 &= -2 - (4\alpha-2)\beta\omega_c T + 2\alpha(1-\alpha)(\beta\omega_n T)^2, \\ b_0 &= 1 - (2-2\alpha)\beta\omega_c T + [(1-\alpha)\beta\omega_n T]^2, \end{align*} \] as enumerated in Table III of [2601.09549]. 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 $|\Delta G| < 0.02$ dB with the prewarped choice. - PLECS simulation of magnitude RMSE across frequencies $0$–$10$ kHz reports $RMSE_\mathrm{SBT} = 0.31 \times RMSE_\mathrm{SOTA}$. - Experimental realization on a TMS320F28P65 board for grid-tied inverter regulation achieves $RMSE_\mathrm{SBT} = 0.75 \times RMSE_\mathrm{SOTA}$. - Total harmonic distortion improvement under injected disturbance, measured as THDi, yields $5.44\pm0.04\,\%$ for PI+QR SBT, compared to $38.73\pm0.06\,\%$ (PI), $37.92\pm0.06\,\%$ (PI+QR Euler), $10.78\pm0.05\,\%$ (PI+QR Tustin), and $5.54\pm0.04\,\%$ (PI+QR SOTA) [2601.09549]. 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 $s$-to-$z$ mapping techniques, introducing tunable control over frequency and amplitude distortions via global $(\alpha, \beta)$ 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 [2601.09549]. Source: https://www.emergentmind.com/topics/scalable-bilinear-transformation-sbt