(243b) Data-Driven Nonlinear Stabilization Using Operator-Theoretic Lyapunov-Based Model Predictive Control
Since its revival by Mezic , Koopman operator theory  has been the focus of many research efforts as a popular tool for data-driven analysis and control of nonlinear dynamical systems [3,4]. It is particularly attractive because of its ability to provide global linearizations in a larger domain (in some cases the entire basin of attraction) compared to local linearizations. However, the tightness of the obtained linear predictors is not well established and guarantees on the closed-loop stability are not well studied. In this work, we design a stabilizing feedback control strategy for nonlinear systems which relies on control Lyapunov function (CLF). Specifically, we propose to integrate Koopman based linear predictors with Lyapunov based model predictive control (LMPC) scheme which is known for its explicit characterization of stability properties and guaranteed closed-loop stabilization in the presence of state and input constraints . By obtaining linear representations of the system in the observable space, we design LMPC based controllers in the same observable space that stabilize the linear system. We then show that the stability properties of the higher dimensional linear system (in the observable space) are inherited by the original nonlinear system under certain assumptions. Importantly, we show that embedding linear predictors in the controller formulation yields a completely standard convex (quadratic) optimization problem (provided the state and input constraints are linear) within the LMPC framework. Therefore, the large library of efficient solvers available for linear MPC can be readily used to solve the proposed Koopman based LMPC scheme for the control of nonlinear systems.
To demonstrate the proposed method we consider closed-loop stabilization of a well-mixed, nonisothermal continuously stirred tank reactor (CSTR) example with an exothermic second order reaction. In the first step we identify a linear time invariant state-space model (in the observable space) using a numerical algorithm for approximating the Koopman operator . We show that the Koopman based linear models display superior prediction performance compared to two standard approaches: local linearization and a subspace identification method. We also show that starting from the initial condition, the closed-loop trajectory of the CSTR under the proposed Koopman LMPC scheme is successfully stabilized to the target.
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