Koopman-Based Optimal Tracking Control of Nonlinear Dynamical Systems

A data-driven procedure to approximate the Koopman operator is a leading technology in researching the properties of nonlinear dynamical systems. The Koopman operator is an infinite-dimensional linear operator that can be used to predict and control a nonlinear system in a linear manner. In this paper, the reformulation and control problems of complex nonlinear dynamical systems are explored. The extended dynamical mode decomposition method and the deep neural network method are applied, respectively, to obtain a finite approximation of the Koopman operator. Then, we present Koopman-based optimal tracking controller for nonlinear dynamical systems by using the linear quadratic tracking control strategy. Finally, simulation results demonstrate that the linear embedding of nonlinear systems by data-driven procedure can approximate the states of the nonlinear systems and the desired output of nonlinear systems can be optimally tracked based on the designed controller of the linear embedding system.