Koopman model predictive control of nonlinear dynamical systems

Abstract

This chapter presents a class of linear predictors for nonlinear controlled dynamical systems. The basic idea is to lift (or embed) the nonlinear dynamics into a higher dimensional space where its evolution is approximately linear. This is achieved by extending the Koopman operator framework to controlled dynamical systems and applying the extended dynamic mode decomposition (EDMD) with a particular choice of basis functions leading to a predictor in the form of a finite-dimensional linear controlled dynamical system. In numerical examples, the linear predictors obtained in this way exhibit a performance superior to existing linear predictors such as those based on local linearization or the so-called Carleman linearization. Importantly, the procedure to construct these linear predictors is completely data-driven and extremely simple—it boils down to a nonlinear transformation of the data (the lifting) and a linear least-squares problem in the lifted space that can be readily solved for large datasets. These linear predictors can be readily used to design controllers for the nonlinear dynamical system using linear controller design methodologies. We focus in particular on model predictive control (MPC) and show that MPC controllers designed in this way enjoy computational complexity of the underlying optimization problem comparable to that of MPC for a linear dynamical system with the same number of control inputs and the same dimension of the state space. Importantly, linear inequality constraints on the state and control inputs as well as nonlinear constraints on the state can be imposed in a linear fashion in the proposed MPC scheme. Similarly, cost functions nonlinear in the state variable can be handled in a linear fashion. We treat the full-state measurement case as well as the input–output case and demonstrate the approach with numerical examples.