Exact Finite Koopman Embedding of Block-Oriented Polynomial Systems

The challenge of finding exact and finite-dimensional Koopman embeddings of nonlinear systems has been largely circumvented by employing data-driven techniques to learn models of different complexities (e.g., linear, bilinear, input affine). Although these models may provide good accuracy, selecting the model structure and dimension is still ad-hoc and it is difficult to quantify the error that is introduced. In contrast to the general trend of data-driven learning, in this paper, we develop a systematic technique for nonlinear systems that produces a finite-dimensional and exact embedding. If the nonlinear system is represented as a network of series and parallel linear and nonlinear (polynomial) blocks, one can derive an associated Koopman model that has constant state and output matrices and the input influence is polynomial. Furthermore, if the linear blocks do not have feedthrough, the Koopman representation simplifies to a bilinear model.