(243b) Data-Driven Nonlinear Stabilization Using Operator-Theoretic Lyapunov-Based Model Predictive Control

Many chemical processes are characterized by complex nonlinear models that can describe their dynamics to near-perfect accuracy. However, analysis and control based on these nonlinear models is extremely challenging owing to great computational burden stemming from their high-dimensional nature and often lacking robustness and guarantees on the closed-loop stability. Fortunately, the rise of big data and advances in machine learning and sensor technologies have resulted in a resurgence of operator theoretic frameworks for the control of dynamical systems. The distinguishing feature of the operator theoretic methods is that they facilitate a linear representation (in the function space) for a nonlinear dynamical system (in the state space). Formulating nonlinear dynamics in terms of a linear system is appealing since it has the potential to enable prediction, estimation and control of nonlinear systems using linear systems theory.

Since its revival by Mezic [1], Koopman operator theory [2] 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 [5]. 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 [6]. 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.

Literature cited

[1] Mezic, I. Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dynamics, 41(1-3): 309-325, 2005.

[2] Koopman, B.O. Hamiltonian systems and transformation in Hilbert space. Proceedings of National Academy of Sciences USA, 17(5):315, 1931.

[3] Korda, M. and Mezic, I. Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control. Automatica, 93:149–160, 2018.

[4] Surana, A. and Banaszuk, A. Linear observer synthesis for nonlinear systems using Koopman operator framework. IFAC Papers On Line, 49(18):716-723, 2016.

[5] Mhaskar, P., El-Farra, N.H. and Christofides, P.D. Stabilization of nonlinear systems with state and control constraints using Lyapunov-based predictive control. Systems & Control Letters, 55(8):650– 659, 2006.

[6] Williams, M.O., Rowley, C.W. and Kevrekidis, I.G. A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition. Journal of Nonlinear Science, 25(6):1307–1346, 2015.