Learning Linear Surrogate Models of Nonlinear Systems

In general, most dynamical systems exhibit some sort ofnonlinear behavior. However, most control and identificationapplications rely on LTI models, which are only validlocally. In recent years, the Koopman framework hasgained popularity within the control and identification communities,proposing a global linear representation of nonlinearsystems. This is achieved through the embedding of thenonlinear state-space into a possibly infinite-dimensionallifted space where the dynamics are linear and governedby the Koopman operator. In practice, only a finite numberof lifting functions is used and, while the choice of thedictionary heavily impacts the representation quality of theresulted linear model, there are little to no systematic methodsfor the selection. We address this by combining a LeastSquares Support Vector Machine (LS-SVM) regression,to estimate the nonlinear state transition map, with the linearitycondition of the Koopman form