Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control
2016/11/30 by Milan Korda, Igor Mezić · 1,125 citations
Engineering · Mathematics · Physics and Astronomy · #Advanced Control Systems Optimization #Applied mathematics #Artificial intelligence #Computer science #Control (management) #Control theory (sociology) #Fluid Dynamics and Turbulent Flows #Linear model #Linear system #Mathematical analysis #Mathematics #Model Reduction and Neural Networks #Model predictive control #Nonlinear model #Nonlinear system #Operator (biology) #Physics #Statistics #math.OC
paper · pdf · doi:10.1016/j.automatica.2018.03.046
published in Automatica 93, 149-160 (Elsevier BV)
openalex created_date 2017/01/06 · arxiv created 2018/03/23 · arxiv updated 2018/03/26 · openalex publication_date 2018/03/28 · openalex updated_date 2026/08/05
Abstract
This paper presents a class of linear predictors for nonlinear controlled dynamical systems. The basic idea is to lift the nonlinear dynamics into a higher dimensional space where its evolution is approximately linear. In an uncontrolled setting, this procedure amounts to a numerical approximation of the Koopman operator associated to the nonlinear dynamics. In this work, we extend the Koopman operator to controlled dynamical systems and compute a finite-dimensional approximation of the operator in such a way that this approximation has the form a 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 data sets. 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 of the same size. 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. Numerical examples (including a high-dimensional nonlinear PDE control) demonstrate the approach with the source code available online.
Citations
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