Towards a Data-Driven Bilinear Koopman Operator for Controlled Nonlinear Systems and Sensitivity Analysis

A Koopman operator is a linear operator that can describe the evolution of the dynamical states of any arbitrary uncontrolled dynamical system in a lifting space of infinite dimension. In practice, analysts consider a lifting space of finite dimension with a guarantee to gain accuracy on the state prediction as the order of the operator increases. For controlled systems, a bilinear description of the Koopman operator is necessary to account for the external input. Additionally, bilinear state-space model identification is of interest for two main reasons: some physical systems are inherently bilinear and bilinear models of high dimension can approximate a broad class of nonlinear systems. Nevertheless, no well-established technique for bilinear system identification is available yet, even less in the context of Koopman. This paper offers perspectives in identifying a bilinear Koopman operator from data only. Firstly, a bilinear Koopman operator is introduced using subspace identification methods for the accurate prediction of controlled nonlinear systems. Secondly, the method is employed for sensitivity analysis of nonlinear systems where it is desired to estimate the variation of a measured output given the deviation of a constitutive parameter of the system. The efficacy of the methods developed in this paper are demonstrated on two nonlinear systems of varying complexity.