Residual-Driven Adaptive Multi-Rate Quadratic Programming Framework for Nonlinear Analog Audio Circuit Emulation
This work extends our previously proposed Quadratic Programming (QP) approach for the emulation of nonlinear analog audio circuits by formalizing its main numerical ingredients and introducing a residual-driven adaptive multi-rate scheme. Starting from a state-space Differential Algebraic System of Equations (DAE) formulation, the nonlinear algebraic circuit device relations are replaced inside the QP by a first-order surrogate linear constraint, and the post-step nonlinear residual is shown to act as a valid defect indicator for adaptive step-size control. This yields a single-step simulation procedure that avoids the usual combination of nonlinear iterative solves and separate integration updates. The method is evaluated on a diode clipper, a BJT common-emitter amplifier, and a Colpitts oscillator, using SPICE as a baseline reference. The results show that adaptive step sizing considerably improves agreement with the reference solution, that the pseudo-inverse implementation is essentially equivalent to the full equality-constrained QP in the tested cases, and that the proposed formulation remains effective beyond the baseline clipper example, including for a self-oscillating circuit. These results position the proposed method as a promising bridge between SPICE-like interpretability and the efficiency demands of virtual analog (VA) audio applications.