Global Linearization of Parameterized Nonlinear Systems with Stable Equilibrium Point Using the Koopman Operator
This work addresses the problem of analyzing and controlling nonlinear systems for researchers in dynamical systems and control theory, offering incremental theoretical advancements.
The study tackled the global linearization of parameterized nonlinear systems with stable equilibrium points using the Koopman operator, achieving a transformation into finite-dimensional linear systems that depend continuously on parameters, and derived conditions for parameter-independent global bilinearization in control-affine systems.
The Koopman operator framework enables global analysis of nonlinear systems through its inherent linearity. This study aims to clarify spectral properties of the Koopman operators for nonlinear systems with control inputs. To this end, we treat the inputs as parameters throughout this paper. We then introduce the Koopman operator for a parameterized dynamical system with a globally exponentially stable equilibrium point and analyze how eigenfunctions of the operator depend on the parameter. As a main result, we obtain a global linearization, which enables one to transform the nonlinear system into a finite-dimensional linear system, and we show that it depends continuously on the parameter. Subsequently, for a control-affine system, we investigate a condition under which the transformation providing a global bilinearization does not depend on the parameter. This provides the condition under which the global bilinearization for the control-affine system is independent of the parameter.