State-Space Averaging Revisited via Reconstruction Operators

Abstract

This paper presents an operator-theoretic reconstruction of an equivalent continuous-time LTI model from an exact sampled-data (Poincar\'e-map) baseline of a piecewise-linear switching system. The rebuilding is explicitly expressed via matrix logarithms. By expanding the logarithm of a product of matrix exponentials using the Baker--Campbell--Hausdorff (BCH) formula, we show that the classical state-space averaging (SSA) model can be interpreted as the leading-order truncation of this exact reconstruction when the switching period is small and the ripple is small. The same view explains why SSA critically relies on low-frequency and small-ripple assumptions, and why the method becomes fragile for converters with more than two subintervals per cycle. Finally, we provide a complexity-reduced, SSA-flavoured implementation strategy for obtaining the required spectral quantities and a real-valued logarithm without explicitly calling eigen-decomposition or complex matrix logarithms, by exploiting $2\times 2$ invariants and a minimal real-lift construction.