(172h) Data-Driven Stabilization of Nonlinear Dynamical Systems: A Koopman Operator Approach

Nonlinear systems abound in nature. Yet, a universal feedback design for stabilizing nonlinear dynamics remains a daunting challenge unlike its linear counterpart. An alternative approach that has gained a lot of attention recently in this direction is the Koopman operator-theoretic description of dynamical systems [1]. Koopman operator facilitates a linear representation of an unforced nonlinear dynamical system, which is made possible by shifting focus from state space to the space of functions called observables (functions of states). This means that the spectral properties of the Koopman operator (i.e., eigenvalues and eigenfunctions) encode global information associated with geometric properties such as Lyapunov functions and measures, and therefore facilitates nonlinear stability analysis and controller synthesis [2]. The authors in [3] extended this idea to controlled systems and proposed a stabilizing feedback controller which relies on control Lyapunov functions (CLF) in the Koopman space. However, they did not provide any theoretical analysis on the stability of the original nonlinear system. CLFs were also employed in our previous work [4] where a feedback controller was designed in the Koopman space using Lyapunov constraints within a model predictive control framework. However, the limitation is that the CLF was derived for the original system which requires an explicit mathematical expression of the original nonlinear dynamics, which is often not available.

To address these issues, this work seeks to derive a stabilizing feedback controller based on the Koopman operator representation of the original nonlinear system. To do so, first, Koopman system identification is applied to derive a bilinear representation of the dynamics (i.e., Koopman operator theory, when extended to include actuation, gives rise to a bilinear term of state and input) [5]. Then, a predictive controller is formulated in the space of Koopman eigenfunctions by using an auxiliary CLF based bounded controller as a constraint which enables the characterization of stability of the Koopman bilinear system. Finally, a stability criterion is presented that guarantees stability of the original closed loop system in the Lyapunov sense, which comes from the stability of the Koopman bilinear system. Specifically, if there exists a continuously differentiable inverse mapping from the Koopman space to the original state space, then the difference between the predicted state and the original state can be shown to be bounded at all times. This, along with the guaranteed stability of the predicted state (owing to the stability of the bilinear system), will ensure the stability of the closed-loop system under the implementation of the proposed controller. Unlike [4], the feedback controller design proposed in this work is completely data-driven and does not require any a priori knowledge of the original system. Moreover, please note that the CLFs are derived for the bilinear system which is much more computationally affordable than the original nonlinear system. In fact, the search for CLFs can be focused to a class of quadratic functions, by formulating an optimization problem, which is known to effectively characterize the stability region of simpler systems like the (Koopman) bilinear systems. The application of the proposed method is illustrated on a couple of canonical examples in chemical engineering and mechanical engineering. Overall, the proposed method has the potential to control complex nonlinear systems using bilinear, simple surrogate models.

Literature cited:

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

[2] A. Mauroy and I. Mezic. Global stability analysis using the eigenfunctions of the Koopman operator, IEEE Transactions on Automatic Control, 61:3356–3369, 2016.

[3] B. Huang, X. Ma, and M. Vaidya, Feedback stabilization using koopman operator, in IEEE 57th Annual Conference on Decision and Control (CDC), Miami Beach, FL, Dec 17-19 2018, pp. 6434–6439.

[4] A. Narasingam and J. S. Kwon, Koopman Lyapunov-based model predictive control of nonlinear chemical process systems, AIChE J., 65:e16743, 2019.

[5] A. Surana and A. Banaszuk, “Linear observer synthesis for nonlinear systems using koopman operator framework,” IFAC-PapersOnLine, 49(18):716–723, 2016.