On embedding the Koopmanization into controlled nonlinear systems, its comparison with the Carleman linearisation and concerning results: beyond the feedback linearisation

Abstract

The feedback linearisation has found its striking applications in many practical situations for control problems that exploit the ideas from the linear system theory. Towards linearisation techniques for controlled nonlinear systems, operator-theoretic frameworks from functional analysis have received attentions as well. The Koopman operator is a linear, but an infinite dimensional operator. Most notably, the novelties of the paper are the following. (i) Sketching a rigorous and comprehensive proof of the Koopmanization of the controlled nonlinear system using the notion of the operator-invariant subspace and unifying the minimiser inequality formula with the Koopmanized system. The proof unfolds that the Koopmanization meets the bilinearisation in control context with the augmented state space. (ii) A formal construction of an observer encompassing the generalised Riccati equation in the Koopman setting of the original controlled nonlinear system. The Carleman version of the generalised Riccati equation is rephrased as well. (iii) Unfolding the usefulness of the Koopman observer. To achieve extensive illustrations of the theory of the paper, we construct two Koopman observers for two practical control problems, i.e. a controlled Duffing system with a cubic nonlinearity and a controlled system with a square nonlinearity. Then, we compare the Koopman observers with the Carleman observers.