Any block-chain system of LTI and polynomial static blocks has an exact finite-dimensional Koopman embedding, which becomes bilinear when the linear blocks have no feedthrough.
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Exact Finite Koopman Embedding of Block-Oriented Polynomial Systems
Any block-chain system of LTI and polynomial static blocks has an exact finite-dimensional Koopman embedding, which becomes bilinear when the linear blocks have no feedthrough.