State-Space Averaging Revisited via Reconstruction Operators
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912779716263936 |
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| author | Yang, Yuxin Zhou, Hang Song, Hourong Hredzak, Branislav |
| author_facet | Yang, Yuxin Zhou, Hang Song, Hourong Hredzak, Branislav |
| contents | This paper presents an operator-theoretic reconstruction of an equivalent continuous-time LTI model from an exact sampled-data (Poincaré-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18340 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | State-Space Averaging Revisited via Reconstruction Operators Yang, Yuxin Zhou, Hang Song, Hourong Hredzak, Branislav Systems and Control This paper presents an operator-theoretic reconstruction of an equivalent continuous-time LTI model from an exact sampled-data (Poincaré-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. |
| title | State-Space Averaging Revisited via Reconstruction Operators |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2512.18340 |